Spectral Analysis of Nonlinear Elastic Shapes 3030594939, 9783030594930

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James F. Doyle

Spectral Analysis of Nonlinear Elastic Shapes

Spectral Analysis of Nonlinear Elastic Shapes

James F. Doyle

Spectral Analysis of Nonlinear Elastic Shapes

James F. Doyle School of Aeronautics & Astronautics Purdue University West Lafayette, IN, USA

ISBN 978-3-030-59493-0 ISBN 978-3-030-59494-7 (eBook) https://doi.org/10.1007/978-3-030-59494-7 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

For Linda thank you again and again. I could not have done this without you.

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 5

1

Overview of Shapes and Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Deformed Shapes of Simple Slender Members. . . . . . . . . . . . . . . . . . . . . . . . 1.2 Modeling Continuous Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Structural Stiffness and Its Spectral Properties . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Spectral Shapes of Slender Members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 21 46 59 77

2

Shapes with Coupled Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Curved Beams and Arches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Deflections of Thin Curved Plates and Shells . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Long Structures with Open Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Spectral Analysis of Coupled Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 80 104 136 159 182

3

Nonlinear Elastic Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Stiffness of Nonlinearly Deformed Structures . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Large Deflections of Thin-Walled Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Structures with Initially Coupled Deformations . . . . . . . . . . . . . . . . . . . . . . . 3.4 Monitoring Changes of Shape and Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

185 185 202 229 242 264

4

Buckling Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Buckling Shapes of Straight Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Buckling of Arches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Plate and Shell Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Load Coupling of Deformation Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

267 267 292 309 335 354

vii

viii

5

Contents

Studies of Postbuckled Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Postbuckling of Beams and Arches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Plates and Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Mode Interactions with Softening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

355 355 371 393 406

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

Notation

Roman Letters a Radius, plate width A Cross-sectional area b, bi Thickness, depth, plate length, body force D, D¯ Plate flexural rigidity, Eh3 /12(1 − ν 2 ), D¯ = Gh3 /12(1 − ν 2 ) Unit vectors eˆi E, E ∗ Young’s modulus, E ∗ = E/(1 − ν 2 ) EI beam flexural stiffness Eij Lagrangian large strain tensor Member axial force, element nodal force F, Fˆ , F¯o F Generalized nodal force gi (x) Element shape function, Ritz function G Shear modulus h Beam or rod √ height, plate thickness i Complex −1, counter I Second moment of area, I = bh3 /12 for rectangle o J , J, Je Jacobian k, k1 , k2 Wavenumber K, [ k ], [ K ] Stiffness, stiffness matrices L Length M, Mx Moment M, [ m ], [ M ] Mass, mass matrices P (t), Pˆ , {P } Applied force history P Generalized applied load q, qu , qv , qw Distributed load r, R Radial coordinate, radius s Hoop coordinates [ R ] Rotation matrix t, ti Time, traction vector [ T ] Transformation matrix ix

x

Notation

Cartesian displacements Strain energy Member shear force, volume Potential of conservative load Beam width Work, external work Original rectilinear coordinates Deformed rectilinear coordinates

u, v, w U V V W W, We x o , y o , zo x, y, z Greek Letters α δ δij  , ij θ κ λ ν

ρo, ρ σ, σij φx , φy , φz [  ] ψ ω

Subtended angle of arch and curved plate Small quantity, variation, virtual Kronecker delta Determinant, increment Small quantity, strain Angular coordinate Plate curvature Eigenvalue Poisson’s ratio Total potential energy Mass density Stress Rotation Modal matrix Stress function Angular frequency

Special Symbols ∇2 [ ] { }   ¯ ˆ ˙

2

2

∂ ∂ Differential operator, ∂x 2 + ∂y 2 Square matrix, rectangular array Vector Diagonal matrix (bar) Local coordinates (hat) Vector (dot) Time derivative

Subscripts E, G, T i, j, k I, J, K ,

Elastic, geometric, tangent (total) stiffness matrix Continuum tensor components DoF enumerator (comma) Partial differentiation

Superscripts o 

Original configuration (prime) Derivative with respect to argument

Notation

xi

Abbreviations BC, pBC DoF, SDoF DKT EoM EVP FE FB MRT ODE PoVW

Boundary condition, periodic BC Degree of freedom, single DoF Discrete Kirchhoff triangular FE element Equation of motion Eigenvalue problem Finite element Free body Membrane with rotation triangular FE element Ordinary differential equation Principle of virtual work

Introduction

The order of historical events clearly shows the true position of the variational principle: It stands at the end of a long chain of reasoning as a satisfactory and beautiful condensation of the results. Max Born [1]

The most familiar characteristic of structures is their initial shape, the variety of these is to be seen everywhere. Less commonly perceived is the change of shape under loading: the young tree bending in the wind or laden with snow, the clenching of a fist or pointing of a finger, the opening of an umbrella. A change of shape is called a deformation. Most purpose-built engineering structures are designed to be stiff, that is, resist change of shape under load, and stiffness is a measure of this resistance to change of shape. When the loads can no longer be supported, the structure collapses and changes its shape. Sometimes this is complete collapse with catastrophic consequences as observed with some buildings after an earthquake, but sometimes it is only temporary as with the umbrella. The former is collapse due to loss of strength, the latter is collapse due to loss of stiffness and is called a buckling instability. Because there is recovery of shape on unloading the latter is also called elastic buckling; this book is exclusively concerned with elastic buckling. The first systematic study of the changes of elastic shapes was given in 1744 when Euler published an Addendum to his “A Method of Finding Curved Lines Enjoying Properties of Maximum . . . ” called “Elastic Curves” [4]. In it, he explored the natural shapes assumed by a loaded ribbon. Figure I.1a shows some shapes similar to those appearing in the Addendum. In referring to a shape like Fig. I.1b, Euler wrote that the problem of finding the shape of an elastic ribbon must be expressed as: That among all curves of the same length, which not only pass through the points A and B, but also are tangential to given straight lines at these points, that curve be determined in  ds which the value of is a minimum. R2

© Springer Nature Switzerland AG 2020 J. F. Doyle, Spectral Analysis of Nonlinear Elastic Shapes, https://doi.org/10.1007/978-3-030-59494-7_1

1

2

Introduction

Fig. I.1 A sampling of elastic shapes. (a) Selected frames from an FE generated movie of the axial compression of a long flexible ribbon under axial compression. (b) Euler’s statement of shape as a variational (extremization) problem

In this, R is the local radius of curvature and ds is the differential arc length given by 1 q = , R [1 + p2 ]3/2

 ds = dx 1 + p2 ,

p=

dy , dx

q=

d 2y dx 2

The shapes are therefore not arbitrary but possess this property that makes them quite special. Euler proceeds to solve this problem in the calculus of variations sense. While the problem is stated in terms of positions and tangents, Euler later states of his solution “this equation is very convenient for the most common method of bending ribbons when they are held either by forces or two fingers (which press in opposite directions) while at the same time the ribbon is stretched.” The two fingers refer to an applied end moment. He certainly understood that shapes are load dependent, what we here call nonlinear elastic shapes. To Euler, it appears that the shapes were not just the solution to a particular boundary value problem but embodied something much deeper. He wrote that because “there is an infinite variety of these elastic curves, it will be worth while to enumerate all the different kinds included in this class of curves. For in this way not only will the character of these curves be more profoundly perceived, but also, in any case whatsoever offered, it will be possible to decide from the mere figure into what class the curved formed ought to be put.” Euler has made the connection between shape and some intrinsic property of deformed systems. Understandably, this connection could not be made explicit at the time. Euler’s elastic curves have fascinated many researchers over the centuries and have led to many new branches of research such as elliptical integrals, capillaries, splines, and stability. For example, Max Born (Nobel prize winner in physics, 1954, for contributions to quantum mechanics), whose 1906 Ph.D. thesis title is “Stability of Elastic Curves in the Plane and Space,” said in his autobiography [1] that it was the stability aspects that fascinated him because stability problems give the clue to understanding structures be they natural, man-made, or cosmic and the

Introduction

3

elastic curve posed the problem in tractable form. The curve in Fig. I.1b is on the point of an instability, while the top curves in Fig. I.1a have already gone through two instabilities, Born’s thesis shows experimental photographs of similar shapes. References [3, 5] and the citations therein give a taste of current stability research and the terminology emanating from elastic curves. Another remarkable aspect of Euler’s Addendum is that he included the smallamplitude vibrations problem of the ribbon. Mathematically, this is different than the shape problem: one is reduced to quadratures, the other to solution of differential equations. It must nonetheless be so that Euler considered them related beyond just another example of shapes. For our purpose here, we like to imagine that Euler knew or sensed that small vibration responses give information about the state of the elastic shape of the structure; and, furthermore, the “state” being the elastic stiffness of the structure. We now know that stability is related to stiffness. We codify this information in the form of the spectral analysis of the elastic shapes. More specifically, we look at the spectral properties of the stiffness matrix; in this context, the word spectral is used to lay out in a hierarchical order certain properties of the system. Figure I.2a gives a sense of the general scheme of things: a change of shape causes a new stiffness, this new stiffness is computed through the strain energy. A spectral analysis of the stiffness informs insights and knowledge about the change of shape. The driver of the changes is the applied load and the information is the hierarchy of shapes/stiffnesses. Three spectral monitors are developed: Amplitude Spectrums are the decomposition of a deformed shape in terms of a bases set of shapes, Spectral Plots show the hierarchical change of stiffness with load, and Mode Amplitude Plots identify the coupling between modes. This is expounded in Fig. I.2b for a particular structure showing how the stiffness changes with load. We point out an interesting connection with some of the present work (especially, the amplitude spectrums) and what is happening in the graphics simulation world.

Fig. I.2 Outline of analysis plan. (a) Broad connection between nonlinear elastic shape, Stiffness, and strain energy. The symbol  means “a change of.” (b) Spectral analysis of nonlinear shapes and its change with load

4

Introduction

In an effort to make simulated motions more realistic animators are using spectral shapes as their bases; Refs. [2, 6] and their citations give a sense of how active this endeavor is. This is a confirmation that shapes, and the changes of shapes, continue to hold a fascination for researchers beyond just their utilitarian value.

Outline of the Book Chapter 1 is an overview of the mechanics of deformed shapes and the methods used for their analyses. It introduces virtual work (and its variational forms) as the fundamental principle on which the analytical methods are based. The key connection between stiffness and spectral analysis is developed here but its full elaboration and explanation does not occur until the nonlinear analyses of Chap. 3. Changes of shape are fundamentally nonlinear, and the structures most susceptible to exhibiting this nonlinear behavior are the thin-walled plate and shell structures. Chapter 2 introduces their linear analysis and Chap. 3 extends this to when there are large deflections and large rotations. What shells and nonlinear elastic shapes have in common is that they both exhibit coupled deformations. This aspect is explored in both chapters. Instabilities are associated with drastic changes of shape of a structure. The primary contributor to instabilities in structures is the nonlinear load/shape interaction effects on the stiffness. Chapter 4 develops the primary analytical tools used to identify buckling instabilities and Chap. 5 navigates through buckling instabilities to determine the new postbuckled equilibrium state. Spectral analysis is used to clarify the meaning of mode and explores the consequences resulting from the coupling of different modes. Throughout the chapters, we deal with models of different types. Our primary model for “solving the problem” is finite element (FE) based and represents the structure in terms of a large number of discrete unknowns. With the use of FE methods almost any structural problem can be solved in the sense that given the geometry, material properties, loads, and so on, responses can be generated. This is where a different level of model enters, one that helps explain the computed numbers; these are of the reduced model (or simplified model) type specifically meaning they involve a limited number of degrees of freedom (DoF) but not implying that they are crude or unsophisticated in any way. That is, when trying to understand a complex system, it is quite useful (and arguably necessary) to have available these simpler models—not as solutions per se but as organizational principles for seeing through the numbers produced by the FE codes. The spectral monitors are powerful tools aiding the construction of these models. To amplify on this last point, we take as a given that modern analyses of structures make use of FE methods; the challenge in the analyses is to understand the results, to understand the underlying mechanisms driving the results. To this end, there are a good number of example problems distributed throughout the chapters where computer programs are used to produce results, do computer experiments

References

5

as it were, but the analysis is directed toward explaining the results. Remember that, unlike the physical experiment, the FE solution can provide almost unlimited information about the solution presented in almost unlimited different forms. Therefore, postprocessing of the data is a very important stage and the spectral monitors are a new part of this stage.

References 1. Born, M.: My Life: Recollections of a Nobel Laureate. Scribner, New York (1968) 2. Haung, Q.-X., Wicke, M., Adams, B., Guibas, L.: Shape decomposition using modal analysis. In: Eurographics 2009, vol. 28(2), pp. 47–67 (2009) 3. Magnusson, A., Ristinmaa, M., Ljung, C.: Behaviour of the extensible elastica solution. Int. J. Solids Struct. 38, 8441–8457 (2001) 4. Oldfather, W.A., Ellis, C.A., Brown, D.M.: Leonhard Euler’s elastic curves. Isis 20(1), 72–160 (1933) 5. Sachkov, Yu.L., Levyakov, S.V.: Stability of inflectional elasticae centered at vertices or inflection points. Proc. Steklov Inst. Math. 271, 177–192 (2010) 6. Shamir, A.: Segmentation and shape extraction of 3D boundary meshes. In: Eurographics 2006, vol. 25, pp. 137–149 (2006)

Chapter 1

Overview of Shapes and Stiffness

We begin by identifying some basic deformation modes in structural members, the mechanisms involved are made tangible through some FE computer experiment results. It is one thing to have a sense about a deformation mechanism of a structure but converting this into a concrete model requires rigorous mechanics principles. This is provided by the principle of virtual work (PoVW) and the associated stationary principles, in conjunction with the concept of strain energy. Real structures with continuously distributed properties are known as continuous systems, we develop two extensions of the PoVW for continuous systems: one is the strong formulation which develops the differential equations governing the deformed shape and makes a clear statement about all the boundary conditions (BCs); the other is the Ritz direct method which uses assumed deformed shapes that need to satisfy only the geometric BCs. Both are used extensively in the following chapters to create models of differing sophistication. It is worth mentioning that they are complementary to each other and not simply alternative methods; this is illustrated through the examples. A key analytical tool, given the name spectral analysis, is also introduced in this chapter. This analysis is intimately connected to the discrete properties of a structure, in particular the stiffness properties; the process of discretizing a structure and extracting the spectral information is developed.

1.1 Deformed Shapes of Simple Slender Members The common types of loading of slender members result in stretching, bending, and twisting. The differences between these deformations are striking which consequently have had a profound influence on the history of the development of structures. Our emphasis here is on establishing some fundamental load-

© Springer Nature Switzerland AG 2020 J. F. Doyle, Spectral Analysis of Nonlinear Elastic Shapes, https://doi.org/10.1007/978-3-030-59494-7_2

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1 Overview of Shapes and Stiffness

0.

5.

10. .

15.

20.

Fig. 1.1 Members under linear loading. For each member, the left-hand side is fixed. (a) Loading cases. (b) Responses. The responses for cases II and III are similar

deformation behaviors so as to better understand the functioning of these slender members. A theme running through this volume is how shape (or geometric configuration) affects the load bearing capability of a structure. We begin with a comparison of load types and then consider the effect of different cross section configurations.

1.1.1 Comparison of Types of Loads Figure 1.1a shows members with square cross section of size [a × a] and length L = N a where N = 1 → 20. The left-hand side is fixed and the right-hand side is loaded either axially or transversely with the same magnitude of load. Each member is modeled using the Hex20 solid element [6] with a mesh of size [10 × 2 × 2]. The x-axis is oriented along the length. Figure 1.1b shows a comparison of the computed displacements. What is striking is that the transverse loading produces very large displacements, in fact, three orders of magnitude larger than the axial loading. In terms of a structural analysis, we say the member is very stiff as regards axial load, and very flexible as regards transverse loads. Figure 1.2a shows the same members but with different types of end couples. The couple is a more complicated action than the linear loads, but the dominant deformations are that the Px a couple (about the z-axis) produces a transverse deflection v, whereas the other two couples produce a rotation about the x-axis giving either a v or w displacement of the outer points. Figure 1.2b shows a comparison of the computed displacements plotted on the same scales as Fig. 1.1b; the level of displacement lies between those of the latter figure. The Pz a couple gives an almost linear relation for the different lengths, whereas the Px a couple shows a quadratic like behavior but not as strong as for the Py or Pz transverse loads. Perhaps nonintuitively, the Px a couple produces a

1.1 Deformed Shapes of Simple Slender Members

9

0.

5.

10.

15.

20.

Fig. 1.2 Members under couple loading. For each member, the left-hand side is fixed. (a) Geometry. (b) Responses. The responses for cases V and VI are similar

Fig. 1.3 Examples of members designed to maximize axial actions so as to increase stiffness, (a) I-beam in bending. (b) Box-beam in torsion. (c) Roman viaduct

deflection an order of magnitude larger than the Py a couple; the reason for this is subtle and we need further developments to fully explain it, but essentially, the Px a couple produces a bending action similar to the bending action in Fig. 1.1a. We see that for a given member, the different loads can produce different actions with greater or lesser levels of displacement or deflections. That is, if a load is supported by a bending action, the deflections are large, but if it is supported by an axial action, then the displacements are small. We can turn this around and pose the structural problem: what is the best way to support a transverse load but with minimal deflections? The solution is to shift more of the support action onto axial actions. The classic example is that of the I-beam shown in Fig. 1.3a where the flanges (the top and bottom material) are predominantly in axial action, and because they are separated from each other and tend to maximize the support couple. Another example is the box-beam construction to enhance torsional stiffness that is used to great effect in the design of metal aircraft wing structures. This has a similar principle as the I-beam but the deformations involved are the shear strains. As we proceed, we introduce the concept of membrane actions, these are analogous to the axial actions but they apply to 2D distributed material like the flange or thinwalls. It is not obvious here but made clear later that while the thin-wall is put in membrane shear the principal stress action is completely analogous to the axial actions of 1D members.

10

1 Overview of Shapes and Stiffness

A final example from antiquity is the Roman viaduct: these massive structures needed to support their own weight plus whatever was transported on top (water in the case of aqueducts), but in addition they also needed to span the terrain. They did this by transferring the load to the arch stones, which then transferred it through compression to the vertical columns. The flying buttresses of Gothic cathedrals used a similar principle [7]. To summarize, the history of efficient structural design is to shift bending support actions over to axial support actions. As we discuss particular structures (arches, cylinders, and the like) we point this out. This is relevant for us because as a nonlinear deformation develops, typically the shape changes so as to support the load through axial actions. The broad categories of structures discussed look different but many of their underlying actions are similar in principle and based on the actions shown in Figs. 1.1 and 1.2. Therefore the remainder of this section develops the modeling of these members in more depth.

1.1.2 Deformation Distributions and Degrees of Freedom Let us look more closely at the deformation distributions in the members under the different loading conditions. Each case considered is the same length of square cross section so that its shape is a [L × h × b] = [10a × 2a × 2a] block. The reported deformations are obtained using the Hex20 solid element with a mesh of size [10 × 4 × 2]. Figure 1.4a shows the deformed shape of the member loaded as case I in Fig. 1.1 except that the resultant force is distributed as a uniform pressure (traction) on the cross section. The analysis is linear but the deformed shape shown is exaggerated so as to more clearly see the consequences of the linear analysis. Think of the grid in Fig. 1.4a as simply scribed lines on the member, then we can estimate the axial strain from the axial displacement data u(x) according to

Fig. 1.4 Axial deformation of a member. (a) Exaggerated deformed shape. (b) Strain distribution along the length. (c) Strain (circles) and stress (squares) distributions on the cross section at midlength

1.1 Deformed Shapes of Simple Slender Members

strain =

change of length original length

or

11

xx ≈

un+1 − un−1 ∂u ≈ xn+1 − xn−1 ∂x

where the subscript n is the enumerated node (grid line intersection) number along the centerline. Figure 1.4b shows the strain distribution computed this way (squares) compared to the strain computed directly by SDsolid (circles); they are the same. Figure 1.4c shows that the stress and strain are distributed uniformly on the cross section. Suppose the axial strain is of most importance to us, then we get a nice concise representation of it by writing u(x, y, z) = [x/L] uL = g(x) uL ,

xx =

∂u = [1/L] uL ∂x

That is, the displacement field and the strain field are written in terms of a single parameter uL , the displacement at the loaded end, plus a characteristic displacement distribution g(x). The parameter is called a degree of freedom (DoF) and characterizes the magnitude of the deformation. While uL is a discrete number, its use in conjunction with g(x) allows the reconstruction of the complete (although usually approximate) displacement and strain fields. This is a very important facility that we utilize as we develop models for the deformation of complex structures. Figure 1.5a shows the deformed shape of the member loaded as case IV in Fig. 1.2. The analysis is linear but the deformed shape shown is exaggerated so as to more clearly show the consequences of the linear analysis. For example, each point on the centerline moves only vertically, points along the top move right-to-left, while those on the bottom move left-to-right. Consequently, the top is in compression, while the bottom is in tension; the strain distribution on the cross section is shown in Fig. 1.5b and is essentially linearly distributed. Each cross section appears to rotate and remain plane (except the last plane which has the distributed point loads); this is often referred to as “plane sections remain plane.” The x-displacement and strain can then be written as

Fig. 1.5 Bending deformation of a member. (a) Exaggerated deformed shape. (b) Strain distribution on the cross section. (c) Slope information extracted two different ways: open symbols are case IV, full triangles are case II

12

1 Overview of Shapes and Stiffness

u(x) = −yφz (x) ,

xx (x) =

∂u ∂φz = −y ∂x ∂x

where φz is the rotation about the z-axis. Again, thinking of the grid in Fig. 1.5a as simply scribed lines on the member, we can estimate the rotation from the displacement information by one of two ways φz ≈ −

ut − ub h

or

φz ≈

vn+1 − vn−1 ∂v ≈ xn+1 − xn−1 ∂x

where ut and ub are the top and bottom axial displacements, respectively; v is the transverse deflection of the centerline and the subscript n is the enumerated node (grid intersection) number along the centerline. Figure 1.5c shows a comparison of the slope computed the two ways as the empty circles and squares; except for the end, they are essentially identical and therefore the strain is related to the second derivative of the transverse deflection. Suppose the axial strain is of most importance to us, then similar to the axial loading case we get a nice concise representation of it by writing φz (x, y, z) = [x/L] φL = g(x) φL , u(x, y, z) = −yφ(x) = −y[x/L] φL = −yg(x) φL so that v(x, y, z) = −y[ 12 x 2 /L] φL ,

xx =

∂φz ∂u = −y = −y[1/L] φL ∂x ∂x

Again, we have written the displacement and strain fields in terms of a single parameter φL , the rotation about the z-axis at the loaded end. This, clearly, is a DoF in the sense introduced earlier but it does not completely characterize the deformation state. For example, loading case II of Fig. 1.1 shows that a transverse load can also produce a transverse deflection (and hence rotation) and this would be true even if the end rotation is zero. Figure 1.5c shows the rotation as the triangle symbols; the distribution is a higher order function. What this means is that to characterize the deformation we actually need two DoFs—the rotation φL and the displacement vL (the details of this assertion are developed more fully in Sect. 1.3). In terms of these we can write    2  x x3 x2 x3 v(x, y, z) = 3 2 − 2 3 vL + − 2 + 3 LφL = g3 (x) vL + g4 (x) LφL L L L L     x x2 x x2 φz (x, y, z) = 6 2 − 6 3 vL + −2 2 + 3 3 LφL L L L L     1 x 1 x xx (x, y, z) = −y 6 2 − 12 3 vL − y −2 2 + 6 3 LφL L L L L

1.1 Deformed Shapes of Simple Slender Members

13

Fig. 1.6 Twisting deformation of a member. (a) Exaggerated deformed shape. (b) End view of the rotations of the cross sections. (c) Full view of the deformed shape of the x–y back outer surface

It is clear that this behavior is more complex than that of the axial loading, but the main point is that, nonetheless, the field distributions can be characterized by relatively few parameters or DoF. Figure 1.6a shows the deformed shape of the member loaded as case V in Fig. 1.2. As in the case of axial loading and bending, the analysis is linear but the deformed shape shown is exaggerated so as to more clearly see the consequences of the linear analysis. Exaggerated deformed shapes can be very misleading especially when there are rotations involved. For example, Fig. 1.6b clearly shows the rotation of each cross section, but it also implies that the diagonals of the cross section expand along the length which, in fact, is not the case. Figure 1.6c is a full view of the x–y back outer surface. Superficially, it may look like it is bending, but on closer inspection we see that all cross sections, on average, remain vertical (parallel to the left fixed edge) and that it is a shearing action that is occurring. The shear strain is the change of angle between the initially perpendicular grid lines, and we can estimate it from the transverse displacement data v(x) according to γxy ≈

vn+1 − vn−1 ∂v ≈ xn+1 − xn−1 ∂x

where the subscript n is the enumerated grid node. Figure 1.7a shows the strain distribution computed this way (squares) compared to the strain computed directly by SDsolid (circles); they are different. Furthermore, Fig. 1.7b shows a strain comparison at midlength and we see that the actual strain distribution peaks at the center and goes to zero at the corners; our estimate of the shear strain has it constant on the cross section. The reason for the discrepancy can be seen in Fig. 1.7c; there actually is an induced axially displacement. This is referred to as warping of the cross section.

14

1 Overview of Shapes and Stiffness

Fig. 1.7 Distributions on the outer face of a twisted bar. (a) Strain distribution along the length. (b) Strain distribution on the cross section. Model result is uniform. (c) Axial displacement on the cross section showing that there is some warping

The warping makes an additional contribution to the shear strain according to warping:

γxy =

∂u , ∂y

total:

γxy =

∂u ∂v + ∂x ∂y

From Fig. 1.7c, we can see that this adds a rather complicated distribution to the original simple model. It is thus clear that the twisting problem is more complicated than it first seemed. Sections 2.2 and 2.3 develop a refined model of torsion of rectangular bars that more accurately accounts for the warping effects. For the present, we stay with the simple formulation because, as it turns out, it correctly describes the behavior of shafts with circular or near circular cross sections. Suppose the shear strain resulting from the axial rotation is of most importance to us, then similar to the axial loading case, we get a nice concise representation of it by writing φx (x, y, z) = [x/L] φL = g(x) φL This gives the displacements fields v(x, y, z) = −zφx (x) = −z[x/L] φL ,

w(x, y, z) = +yφx (x) = +y[x/L] φL

so that the shear strains are γxy =

∂φx ∂v = −z = −z[1/L] φL , ∂x ∂x

γxz =

∂φx ∂w = +y = +y[1/L] φL ∂x ∂x

Again, we have written the displacement and strain fields in terms of a single parameter φL , the rotation at the loaded end. So far we have focused on the deformation, now consider the stress. The stress distributions in Fig. 1.4c shows that σyy (and σzz ) is zero and therefore the stress state is uniaxial computed simply as σxx = Px /area. Because the material behavior

1.1 Deformed Shapes of Simple Slender Members

15

is linear, we can write the uniaxial constitutive relation as σxx = Exx ,

yy = zz = −νxx

where E is the Young’s modulus and ν the Poisson’s ratio. The strain distributions in Fig. 1.5b are nonuniform with respect to y, but they nonetheless are related through Poisson’s ratio yy = −νxx . Indeed, if we consider the block as made of long strips, then each strip is in a state of uniaxial stress. The shear strain distributions in Fig. 1.7b are also nonuniform on the cross section, but each elemental cube of material has the constitutive relation τxy = Gγxy ,

τxz = Gγxz

where G is called the shear modulus. The stress behavior of our structural members is adequately characterized by these two constitutive relations, such relations appropriate for general stressed bodies are developed later. To summarize our analyses: the situations considered show that in the general case the distributions of displacement, strain, and stress can be rather complicated; however, they also show that it is possible, within a reasonable approximation, to replace these distributions with simpler representations in terms of fewer parameters or degrees of freedom (DoFs). For the slender bar, we have identified six such DoF uI , vI , wI ;

φxI , φyI , φzI

corresponding to net displacement and rotations, respectively, at each cross section identified with subscript I . That is, the distributions with respect to y and z are assumed to be of simple form, while the distributions with respect to x are captured by having the DoF specified at multiple sections along x. Thus the unknowns in any given problem are reduced to just the unknown discrete DoF distributed throughout the structure. This concept is further developed in our spectral analyses where the DoF are associated with complete shapes and not just a particular deformation.

1.1.3 Thin-Walled Cross Sections The previous examples consider the member as having a square cross section so that there is symmetric behavior about the y and z axes. In practice, however, it is common to find the cross-sectional material distributed in a preferential manner. For example, consider the first cross section shown in Fig. 1.8a; when h H , this is referred to as a thin-walled cross section and when h ≈ H , it is referred to as a solid cross section. The thin-walled material can be distributed in different ways; the two most common are as open cross sections typified by the I-beam in Fig. 1.8a, and as closed cross sections typified by the box-beam and cylinder in Fig. 1.8b. When the open

16

1 Overview of Shapes and Stiffness

Fig. 1.8 Thin-walled cross sections. (a) Open sections: plate, I-beam, split box-beam. (b) Closed sections: box-beam, circular cylinder, multicelled tube

Fig. 1.9 Characterization of torsional cross sections

cross section is simple it is usually called a plate; consequently, another name for the thin-walled sections is folded plate structure. The biggest differences observed between the cross sections are in their torsional behavior, therefore, we give a word about terminology which can be confusing when cross-sectional shapes change. For torsion, we consider that all cross sections are solid; however, they can be simply connected (no cut-outs) or multiply-connected connected (one or more cut-outs). According to Fig. 1.9, a solid simply connected cross section highly contoured in whatever manner to form a thin-walled structure (think of an I-beam or C-channel) is called an open section. The cut-out in a multiply-connected cross section can be manipulated in many ways, but when it is done to produce a thin-walled structure (think of a cylinder or tube), it is called a closed section. Both of the cross sections originate from the solid cross section, the thin-walled aspect is what makes the big difference. A minor clarification point is that by this terminology a square cross section is an open section; however, it is not thin-walled and hence the open or closed designation does not apply. We begin by contrasting the Py and Pz loading of the flat plate in Fig. 1.8a. The Py load produces strain distributions given by m xx ∝ −y ,

m yy ∝ +y ,

m γxy ∝ [1 − 4y 2 /H 2 ]

The normal strain distributions are similar to the distributions in Fig. 1.5b; the parabolic shear distribution reflects the fact that the shear is zero on the top and bottom surfaces. Of significance is that the distributions do not depend on z—they are constant through the thickness. This type of behavior is referred to as membrane

1.1 Deformed Shapes of Simple Slender Members

17

behavior and designated with the superscript m. The Pz load, on the other hand, produces strain distributions given by f

f

xx ∝ −z ,

yy ∝ +z ,

f

γxz ≈ 0

with negligible variation in y. This type of behavior where the normal strains are zero on the midplane (z = 0) is referred to as flexural behavior and designated with the superscript f . For the same applied traction (Py /H h and Pz /H h), the ratio of maximum axial strains is f

m xx ≈ 10xx

when h = H /10. As indicated earlier, flexure is not an efficient way to support loads. The plate is also inefficient at supporting torques. For example, a Tx torque applied at the end gives the predominant strain f

γxy ∝ z and thus is zero at the middle surface. This is a flexural action. Sections 2.2 and 2.3 develop our formal models for thin-walled structures and elaborate on the significant differences between the membrane and flexural actions. The bending efficiency of the thin plate can be improved considerably by giving it some structural depth. For example, Fig. 1.10a shows the plate curved about the x-axis and supporting a moment about the z-axis. The structural depth has increased from h to nearly half the radius. The neutral axis (NA) is where the bending strain is zero, and the nearly horizontal and equidistant contours show that the bending strain depends primarily on the distance from the neutral axis. The I-beam illustrates a very effective way of increasing the structural depth for a given amount of material. The increased structural depth does not improve the torsional efficiency, the contours of Fig. 1.10b are essentially the same as for the flat plate—just curved to conform to the shape of the plate maintaining the zero shear strain at the middle surface. This is also true of the other open sections in Fig. 1.8b. The way to improve the torsional stiffness is to make the cross section closed as in the box-beam and cylinder. In so doing, the shears become of one sign, that is, they change to a membrane action. This too is developed formally in Sect. 2.2.

Fig. 1.10 Effect of cross section geometry. (a) Bending strain. (b) Torsion shear strain

18

1 Overview of Shapes and Stiffness

The transverse load bearing capability of a member can be improved by allowing the so-called cable action to contribute. A cable is a slender axial member used to support tensile loads. Because of its flexibility, it is unable to support either compression or bending. However, when constructed of high-strength steel wires (typically the wires are twisted together to form a strand much like a rope and multiple strands then form the cable) they have a tensile strength four or five times greater than that of structural steel [9]. Therefore, if only loaded in tension it behaves identical to an axial member. Where the cable is unique among structural members is that when it is designed to support transverse loads, it adjusts its shape to what is called the funicular shape (funiculus is the Latin for rope). That is, the shape is a function of the loading and not something set a priori; it is a graphic illustration of how nonlinear deformations typically adjust to maximize axial contributions. Figure 1.11 shows some FE generated deflected shapes for transversely loaded cables. When it is uniformly loaded, as with gravity, it forms a parabola, this shape changes as the direction of loading changes. Observe how the concentrated loads cause the cable to form a shape consisting of straight line segments. Also, although both loads are the same, the left load causes a deeper deflection because it has a greater moment about the left support than the other force has about the right support. In this type of application, the length of the cable is typically longer than the span with the length being dictated by the required sag—the vertical distance of the point from the chord connecting the supports. The structural depth of a cable is its sag. A cable supporting transverse loads is statically determinate so that once the sag is set by the designer all cable tensions and support reactions can be computed without recourse to the deformation. Typically, the stretching of the cable is negligible because it is associated with the axial strain xx = σxx /E and the strain is less than the yield strain which is 0.01 at most, The I-beam and cable are good examples of transverse loads being supported by axial behaviors but their mechanisms are very different: the I-beam is passive, the cable induces a nonlinear effect by changing its shape. The cable is our archetypal example illustrating the connection between shape and load for our thin-walled structures. But keep in mind that thin-walled structures can also support compressive loads and it is this aspect that primarily influences our later nonlinear analyses.

Fig. 1.11 Deflection of a cable with initial sag. The ends are fixed

1.1 Deformed Shapes of Simple Slender Members

19

1.1.4 3D Continuous Solids Not all structural behaviors fall neatly into the categories of axial, bending, and so forth. For these other cases, we need to apply more general relations. To that end, we now summarize the main relations for 3D solids. These are the ones used by the Hex20 modeling in SDsolid. The loads on a general body cause the body to deform such that each point (in a small volume) has the displacement fields u(x, y, z) ,

v(x, y, z) ,

w(x, y, z)

The small volume is in a deformed state and has a set of strains {  } = {xx , yy , zz , γxy , γyz , γxz }T Note that there are only six independent components of strain because the shear components are related through γxy = γyx , γxz = γzx , γyz = γzy . The description of strain and deformation is fundamentally nonlinear but we initially approach them as linear and add the modifications later in Sect. 3.2. When the deformations (strains and displacements) are small, the strains are related to the displacements by xx =

∂u , ∂x

yy =

∂v , ∂y

γxy =

∂u ∂v + , ∂y ∂x

γyz =

∂w ∂v + , ∂z ∂y

zz = γzx =

∂w ∂z ∂w ∂u + ∂x ∂z

(1.1)

We sometimes make use of the indicial notation where the subscripts i = 1, 2, 3 are associated with the axes x, y, z, respectively. Thus the strain-displacement relations are written compactly as ij =

1 2

 ∂u

i

∂xj

+

∂uj  ∂xi

This is a tensorial relation where the engineering shear strain is related to the tensorial shear strain by, for example, γxy = 212 . Consider a typical small volume taken from a loaded body. This volume is under the action of the system of stresses {σ } = {σxx , σyy , σzz , τxy , τyz , τxz }T Note that there are only six independent components of stress because the shear components are related through τxy = τyx , τxz = τzx , τyz = τzy . When the applied loads are in overall static equilibrium, the stress fields are also in equilibrium and these are described by partial differential equations such as

20

1 Overview of Shapes and Stiffness

 ∂σij + ρbib = 0 j ∂xj

∂τxy ∂τxz ∂σxx + + + ρbxb = 0 ∂x ∂y ∂z

e.g.

(1.2)

where ρbxb is the body force. If a free body cut is made through a stressed body, the exposed area has tractions acting on it. These tractions (designated ti ) are related to the stresses by  fi = ti ≡ lim σij nj j A→0 A

or

⎧ ⎫ ⎡ ⎤⎧ ⎫ σxx τxy τxz ⎨nx ⎬ ⎨ tx ⎬ = ⎣ τxy σyy τyz ⎦ ny t ⎩ ⎭ ⎩ y⎭ tz τxz τyz σzz nz

(1.3)

where ni is the orientation of the area. We restrict ourselves to materials that are elastic. By elastic we mean that the stress–strain curve is the same for both loading and unloading. It is assumed that the materials behave according to Hooke’s law; for three dimensional stress systems, Hooke’s law takes the form   E (1 − ν)xx + ν(yy + zz (1 + ν)(1 − 2ν)   E (1 − ν)yy + ν(zz + xx σyy = (1 + ν)(1 − 2ν)   E (1 − ν)zz + ν(xx + yy σzz = (1 + ν)(1 − 2ν)

σxx =

τxy = Gγxy ,

τyz = Gγyz ,

τxz = Gγxz

(1.4)

where E is the Young’s modulus, ν the Poisson’s ratio, and G = E/2(1 + ν) the shear modulus. These stress–strain relations can be summarized in the matrix forms {  } = [ C ]{σ } ,

[ C ] = [ D ]−1

{σ } = [ D ]{  } ,

where [ D ] is given by ⎡

1−ν ν ν ⎢ ν 1−ν ν ⎢ ⎢ E ν 1−ν ⎢ ν [ D ]= ⎢ 0 0 (1 + ν)(1 − 2ν) ⎢ 0 ⎢ ⎣ 0 0 0 0 0 0

1 2

0 0 0 −ν 0 0

1 2

0 0 0 0 −ν 0

1 2

⎤ 0 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ −ν

(1.5)

The invertibility of [ C ] or [ D ] restricts the Poisson’s ratio to lie between −1 < ν < 12 . Plane stress is usually applicable to thin-walled structure where it can be reasonably assumed that

1.2 Modeling Continuous Structures

σzz ≈ 0 ,

21

σxz ≈ 0 ,

σyz ≈ 0

The strain–stress relations immediately become xx =

1 [σxx − νσyy ] , E

yy =

1 [σyy − νσxx ] , E

γxy =

1 τxy G

Inverting these relations gives   σxx = E ∗ xx + νyy ,

  σyy = E ∗ yy + νxx ,

τxy = Gγxy (1.6)

where, in this context, we introduce the recurring modulus E ∗ = E/(1 − ν 2 ). The equilibrium equations reduce to ∂τxy ∂σxx + + ρbxb = 0 , ∂x ∂y

∂τxy ∂σyy + ρbyb = 0 ∂x ∂y

(1.7)

where ρ is the mass density. Again, keep in mind that the body forces ρbib could be inertia contributions such as ρ u. ¨

1.2 Modeling Continuous Structures Equilibrium is usually thought of in terms of “the sum of the forces equals zero.” This is not a suitable form for our purpose and we recast the equilibrium equations in the form of a variational principle known as the principle of virtual work (PoVW). We have two main choices for applying PoVW to continuous systems. The first is known as the strong formulation and results in a set of differential equations plus the associated BCs. These are then integrated analytically or numerically. The second is known as the Ritz method (sometimes called the weak form) and results in a set of algebraic simultaneous equations which typically are solved numerically. What they both have in common is that they originate from expressions for the strain energy. The Ritz method discretizes it by using assumed elastic shapes, whereas the strong form computes the shapes directly through differential equations. They are complementary to each other and together provide powerful tools for understanding elastic shapes.

1.2.1 Virtual Work Formulation of Equilibrium We begin by establishing the strain energy for some common structural members. For simplicity, the material behavior is assumed to be linearly elastic. The work done by a force is the integral of the force over the displacement. For a simple

22

1 Overview of Shapes and Stiffness

Fig. 1.12 Deformation of a rod. (a) Stretching. (b) Stress–strain behavior. (c) Stressed infinitesimal element showing strain

spring which has the force-deflection relation P = Ku, we have  spring:

B

W=





dW =

P du =

Ku du = 12 Ku2 = U

0

The symbol U is used to denote the strain energy. The simple spring is the archetypal member that has elastic strain energy. This inherently is in discrete form because it relates the energy to just the end behaviors. For continuous systems we deal with stress and strain instead of force and displacement. With reference to Fig. 1.12, the work done by the σxx component of stress during an increment of strain dxx is 

B

W=  =





dW =

P du =

0

[σxx yz]x dxx

 σxx dxx V =

dU = U

where V is the element of volume and we identify this internal work as the strain energy. Hence the strain energy increment for the whole body is obtained by allowing the typical volume to become infinitesimal and then integrating over the total volume. That is,  U=

B

 σxx dxx dV

0

V

Because the material is linear elastic then the strain energy takes the integrated form  U=

1 2

 σxx xx dV = V

1 2

V

2 Exx dV

(1.8)

For the specific case of the axial member where the stress distribution is uniform on the cross section of area A as shown in Fig. 1.4c, the member axial force at any cross section is

1.2 Modeling Continuous Structures

23

 F (x) =

σxx dA = EA

∂u ∂x

and the strain energy has the form  U=

rod:

1 2

L

EA

 ∂u 2 ∂x

0

 dx =

1 2

L

0

F2 dx EA

(1.9)

It is interesting to note that when we apply this expression to the deformation of Fig. 1.4, we get that U=

1 2

EA 2 u L L

In comparison to a spring, this has the effective stiffness of K = EA/L. This is our first example of how the parameters of a continuous system can be related to structural concepts such as stiffness. The normal stresses in a beam in bending form a uniaxial stress system although the stresses (and strains) do vary on the cross section as shown in Fig. 1.5b. The member moment is  M(x) =

σxx y dA = −EIzz



∂ 2v , ∂x 2

Izz =

y 2 dA

and the strain energy can then be computed directly from Eq. (1.8) as  U=

1 2

 σxx xx dV =

V

1 2

 V

2 Exx dV =

1 2

y2E V

 ∂ 2 v 2 ∂x 2

dV

Performing the integration on the cross section gives  beam:

U=

1 2

L

EIzz 0

 ∂ 2 v 2 ∂x 2

 dx =

1 2

L 0

M2 dx , EIzz

 Izz ≡

y 2 dA

(1.10)

where Izz is the second moment of area, sometimes called the moment of inertia. For a rectangular section [b × h] and a circular section (diameter D), this gives Irect =

3 1 12 bh ,

Icirc =

4 π 64 D ,

respectively. With reference to Fig. 1.13, there are two types of shear situations: simple shear of a block, and torsion shear of a shaft of circular cross section where the member load are, respectively,

24

1 Overview of Shapes and Stiffness

Fig. 1.13 Two types of shear situations. (a) Simple shear of a block. (b) Torsion shear of a shaft of circular cross section



∂v , T (x) = τxy dA = GA ∂x

V (x) =





∂φ τxθ rdA = GJ ∂x

2

 J=

,

r 2 drdθ

The strain energies are  U=

block:

 τxy γxy dV =

1 2

1 2

 U=

shaft:

1 2

GA 0

 τxθ γxθ dV =

1 2



L



L

GJ 0

∂v ∂x

2

∂φ ∂x

 dx =

1 2

2 1 2

V2 dx GA

L

T2 dx GJ

0

 dx =

L

0

(1.11) where J = π D 4 /32 is the polar moment of area. For a 3D stressed body where the initial state is the unloaded unstressed state, we have the strain energy  U=

1 2

 [σxx xx + σyy yy + τxy γxy + · · · ] dV =

V

{σ }T {  } dV

1 2

(1.12)

V

Note that there are products such as σyy dyy but not σyy dxx . The strain energy expression for linear elastic, plane stress conditions, reduces to  U =

1 2

[σxx xx + σyy yy + τxy γxy ] dV V

 UM =

1 2

A

 =

1 2

A

  2 2 E ∗ h xx + yy + 2νxx yy dxdy +  h 2 2 σxx + σyy − 2νσxx σyy dxdy + E

 1 2

 1 2

A

A

2 2 Ghγxy τxy dxdy

h 2 τ dxdy G xy

(1.13)

1.2 Modeling Continuous Structures

25

where A is the area of the plate and h is the thickness. The through the thickness integration to get UM is performed on the assumption that the stresses and strains are uniform. The differential equations of equilibrium are given by Eq. (1.2). Integrating these is usually a formidable task and therefore we seek an alternative formulation. This alternative formulation has the merit of dealing with scalar entities and avoids the necessity of free body diagrams. Reference [6] gives a detailed derivation of the method with a number of example problems; here we outline just the main points. Let ui (xio ) be the displacement field which satisfies the equilibrium equations in the volume V . On the surface A, the surface traction ti is prescribed on the portion At and the displacement ui prescribed on Au . Consider a variation of displacement δui (we call this the virtual displacement), then u¯ i = ui + δui where ui satisfy the equilibrium equations and the given BCs. Thus, δui must vanish over Au but be arbitrary over At , that is, δui must satisfy the geometric constraints of the problem. Let δ We be the external virtual work done by the surface traction ti and the body force bi (which could include inertia effects); that is, δ We =

  i

ti δui dA +

At

  i

ti δui dA +

 

Au

i

ρbi δui δV

(1.14)

V

The Au integral term is zero hence we can replace At in the first integral with A on  the understanding that δui cannot be varied over the portion Au . Substitute ti = j σij nj into the work expression, use the integral theorem in the form [5]  V

∂Tj k dV = ∂xi

 ni Tj k dA A

and the equilibrium conditions from Eq. (1.2) to get, in turn, δ We =

  ij

A

ij

V

i

ρbi δui dV

V

  ∂ [σj i δui ]dV + ρbi δui dV ij V ∂xj i V         ∂σj i ∂ui dV + ρbi + δui dV σj i δ = ij V ij V ∂xj ∂xj     = σij δ ij + ωij dV + 0

=

 

σj i δui nj dA +

 

where ij and ωij are, respectively, the symmetric and antisymmetric decomposition of the displacement gradient. The contraction (summation over both subscripts [5])

26

1 Overview of Shapes and Stiffness

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 1.14 Effect of pretension on the deflection of a cable. (a) Sag for differing amounts of pretension. (b) Deflections when there is significant pretension. (c) Large rotations and stretching of a cable member

of a symmetric and antisymmetric tensor is zero, then the term with ωij disappears giving δ We =

  ij

σij δij dV = δ U

or

δ W = δ We − δ U = 0 (1.15)

V

This is the principle of virtual work (PoVW). Although not of immediate concern, the statement of virtual work in this form is valid for nonlinear large deformation problems. Phrased formally, the principle of virtual work states that a deformable body is in equilibrium if the total virtual work is zero for every independent kinematically admissible virtual displacement. The phrase “kinematically admissible” means that geometric BCs cannot be violated and contiguous points remain contiguous (no folds or gaps can form) during the virtual displacement. The symbol δ is interpreted as meaning a variation and Eq. (1.15) as a variational principle. We find it useful to cast the PoVW into different forms more appropriate for particular applications. Two sample illustrations of their application are given, but their full development occurs throughout the remaining chapters. Example 1.1 Figure 1.14a shows the FE generated deflection results for a cable under gravity loading and pretensioned by differing amounts. Estimate the strain energy and deflected shape when the pretension is large. The FE results were generated using the Hex20 element with one element through the thickness and 40 through the length. This gave an element aspect ratio of 10:1. While it is possible to develop an element that has no flexural stiffness, in reality all cables have some measure of bending stiffness. For example, the cables used on the Golden Gate Bridge are almost 1 m meter in diameter. Here, the cross

1.2 Modeling Continuous Structures

27

section was specified as h = 1.27 mm (0.05 in) and b = 25.4 mm (1.0 in), therefore this was more like a ribbon than a cable. Figure 1.14a shows how the pretension was applied: the second boundary is on rollers to allow the action of the force. The pretension ranged over F¯o = (0.0, 0.2, 0.4, . . . , 12.8)Wo where Wo is the weight of the cable. Note that ρ was increased so as to enhance the gravity effects because the analysis performed was fully nonlinear. It is clear that the pretension reduces the sag; indeed, for the results of Fig. 1.14b the displacement of the rollers was about 0.04% of the length L. Keeping in mind that for the largest pretension, all deflections are small, then the axial force is essentially constant throughout the cable so that the horizontal reaction is F¯o and the vertical reaction is Ro = 12 ρALg. On the assumption that no flexural moments are generated, then moment equilibrium about the free body cut at x gives F¯o v − Ro x + ρAxg 12 x = 0

or

v(x) =

 ρAg  − xL + x 2 ¯ 2Fo

This model is shown as the full line in Fig. 1.14b; the model does a very good job in capturing the deflection of the cable. The axial force has the resolved components in the global coordinates of Fx = F¯o cos φ ≈ F¯o = F ,

Fy = F¯o sin φ ≈ F¯o φ = F¯o

∂v =V ∂x

The Fy component is the shear similar to that of the block of Fig. 1.13a. To estimate the energy with pretension, consider the cable segment shown in Fig. 1.14c where the dotted line indicates the centerline. The segment of original length x has a deformed length of s given by s =



 2 [x + uo + u]2 + v 2 = x [1 + o + xx ]2 + ηxx

where o = uo /x ,

xx = u/x ,

ηxx = v/x

and o is the prestraining. Expand to second-order terms so that 2 s/x ≈ 1 + [o + xx ] + 12 [o + xx ]2 + 12 ηxx

Our main assumption, in essence, is that 1 2 [o

2 + xx ]2 12 ηxx

or

u/x v/x

or

u v

That is, the transverse deflections are much larger than the axial displacements. The strain approximation is then

28

1 Overview of Shapes and Stiffness 2 ¯xx = s/x − 1 ≈ o + xx + 12 ηxx

The strain energy is related to the strain squared giving 2 2 2 2 2 = [o + xx ]2 + [o + xx ]ηxx + 14 ηxx ≈ o2 + 2o xx + xx + [o + xx ]ηxx ¯xx

We now conceive of the loading situation as having a pre-existing stress and strain state and we are interested in the additional stresses and strains caused by additional loads. The additional squared strain is then approximated as 2 2 2 ¯xx − o2 ≈ xx + o ηxx

Section 3.2 gives a discussion of the neglected nonlinear terms. The strain terms on the right lead to the strain energies UM and UG , respectively, given by   ∂u 2 F2 dx dx = 12 ∂x EA  2   ∂v V2 1 1 ¯ U G = 2 Fo dx = 2 dx ∂x F¯o 



UM = cable:

1 2

EA

(1.16)

where F¯o = EAo . We thus see that the pretension enters the strain energy expression similar to a stiffness per unit length such as EA. More details on large deflection are developed in Sect. 3.2.

1.2.2 Euler Method This strong formulation is best illustrated through an example. Consider the rod segment of length L shown in Fig. 1.15. Let the Young’s modulus and crosssectional area vary along the length but nonetheless assume the behavior is adequately modeled as a 1D rod. That is, the strain energy is approximated as  U=

1 2



L

EA 0

∂u ∂x

2 dx

 where A = dydz is the cross-sectional area. The virtual work of the loads shown in Fig. 1.15b is  δ We = 0

L

 qu δu dx + F0 δu0 − FL δuL = 0

L

L  qu δu dx + F δu

The vertical bar notation is used to indicate boundary values.

0

1.2 Modeling Continuous Structures

29

Fig. 1.15 Rod with arbitrary cross section. (a) Cross section and coordinate system. (b) Notation for end loads and displacements

We now show how these energy and work terms can be used to derive a set of governing differential equations. The PoVW gives  δU −δW =0

L

⇒

EA 0

L  ∂u  ∂u   δ − qu δu dx − F δu = 0 0 ∂x ∂x

where δ as an operator is similar to a derivative and braces indicate the extent of the space integration—note that the variation is inside the integration and is a function of x. Use integration by parts so as to have all terms multiplied by a common variation δu. Integration by parts uses the relationship 

b a

dg b dg dx = f f  − dx dx a

 a

b

df g dx dx

(1.17)

Observe that terms evaluated at the boundary are generated. Interchange the partial differentiation and the variation  ∂u ∂ ! " δ = δu ∂x ∂x and use integration by parts to get L#

 0

 $ ∂u ∂u EA δ dx ∂x ∂x

L#

 ⇒

− 0

  $ ∂ ∂u ∂u L EA δu dx + EA δu 0 ∂x ∂x ∂x

The virtual work expression becomes  0

L #

 $   L ∂  ∂u ∂u  − EA − qu δu(x) dx + EA − F δu = 0 ∂x ∂x ∂x 0

To obtain the governing differential equation and BCs, we use the argument that δu is arbitrary at all points interior to the integration region, and therefore the integrand

30

1 Overview of Shapes and Stiffness

must be zero, giving   ∂u ∂ EA + qu = 0 ∂x ∂x This is the governing differential equation. This equation is traditionally derived by taking a differential free body diagram [3]. It is also required that the other terms be zero, separately. Thus we must have x = 0, L :

EA

∂u = F = specified ∂x

or

u = specified

These are the BCs. When we apply the PoVW in the strong forms, we need to identify two classes of BCs called essential and natural BCs. The essential BCs are also called geometric BCs because they correspond to prescribed displacements. The natural BCs are implied in the choice of strain energy function used to represent the problem and impose constraints on the higher space derivatives of the shape. The governing relationships for the structural quantities in the rod may now be summarized as Axial displacement : Axial force : Loading :

u = u(x) ∂u , Stress : ∂x ∂u  ∂F ∂  EA =− qu = − ∂x ∂x ∂x F = +EA

σxx = E

∂u ∂x (1.18)

It is seen from these that the displacement (and hence deformed shape) is the quantity that connects them together. That is, the displacement function u(x) can be viewed as the fundamental unknown from which all the other quantities can be determined. Typically, the loading qu (x) is specified and therefore the last equation becomes a differential equation that must be integrated to find the unknown displacement u(x). This gives rise to constants of integration that are determined from the BCs. This procedure is demonstrated in the example problem to follow. We can do a similar analysis for the beam. In summary, we have for the deformation u(x, ¯ y, z) = −y

∂v , ∂x

 ∂ 2v  v(x, ¯ y, z) = v(x)+ 12 ν y 2 −z2 , ∂x 2

w(x, ¯ y, z) = νyz

∂ 2v ∂x 2

Because this approximately leads to a uniaxial stress state, then xx =

∂φz ∂ 2v ∂ u¯ = −y = −y 2 , ∂x ∂x ∂x

σxx = Exx = −yE

The approximate strain energy expression is

∂ 2v , ∂x 2

others = 0

1.2 Modeling Continuous Structures

31

Fig. 1.16 Long slender beam in bending. (a) Axes at centroid. (b) Deflection of a plane about two axes. (c) Applied loads

 U=

1 2

 σxx xx dV =

1 2

V



L

Ey 2 A

o

∂ 2v ∂x 2

2

 dAdx =

1 2



L

EIzz o

∂ 2v ∂x 2

2 dx

 where Izz ≡ A y 2 dA and EIzz is called the flexural stiffness. The virtual work of the applied loads indicated in Fig. 1.16c is 

L

δ We = 0

L L   qv (x)δv dx + Mz δφz  + Vy δv  0

0

All the relationships for the structural quantities may now be summarized for the beam as Displacement : Slope : Moment : Shear : Loading :

v = v(x) φz =

∂v ∂x

Mz = +EIzz

∂ 2v , ∂x 2

Stress :

∂M ∂ 2v  ∂  EIzz 2 = − ∂x ∂x ∂x   2 2 ∂V ∂ v ∂ qv = + 2 EIzz 2 = − ∂x ∂x ∂x

σxx = −yE

∂ 2v ∂x 2

Vy = −

(1.19)

These are the Bernoulli–Euler beam equations. It is seen from these that the deflected shape v(x) can be viewed as the single unknown of interest; all other entities are obtained by differentiation. In contrast to the rod, there are higher order differential relationships among the entities. Typically, the distributed load is specified and therefore we treat the last equation as a differential equation that must be integrated to get the unknown distributions. Figure 1.17 shows the possible BCs for the right end of a beam where Vy is a transverse shear force and Mz is the moment. BCs involving the deflection and

32

1 Overview of Shapes and Stiffness

Fig. 1.17 Possible BCs for a beam; the semicolon separates the two virtual work contributions

slope (v, φz ) are the geometric BCs; BCs involving the shear and moment (Vy , Mz ) are the natural BCs. A simple model for the twisting of a member can be derived based on the strain energy and virtual work of the loads given by  U=

1 2



L

GJ 0

∂φx ∂x

2



L

δ We =

dx ,

0

L  qφ δu dx + Mδφx  0

Using our variational principle similar to the rod, it leads to the summary of equations Twist :

φx = φx (x)

Torque :

Mx = +GJ

Loading :

qφ = −GJ

∂φx , ∂x

Stress :

∂ 2 φx ∂Mx =− ∂x ∂x 2

τxr = Gr

∂φx ∂x (1.20)

The quality of the model hinges on the estimate of GJ ; this is developed in Sects. 2.2. and 2.3. The strong formulation has some drawbacks primarily associated with efforts to solve the differential equations. Consequently, it is mostly restricted to linear (or linearized) systems and to systems with few variables. As might be expected therefore, it does not generalize very well and is not suited for general structures. However, when a successful solution is achieved, this solution can be immensely insightful because it contains the complete information about the actual structure. Example 1.2 Determine the displaced shape for a rod with a uniformly applied load. The rod is fixed at one end and either fixed or free at the other as shown in Fig. 1.18. We begin with the fixed–fixed case. This problem is statically indeterminate, which means that equilibrium alone is insufficient to solve the problem. Thus to complete a solution, we must also impose the compatibility conditions and satisfy

1.2 Modeling Continuous Structures

33

0.0

0.2

0.4

.

0.6

0.8

1.0

Fig. 1.18 A rod with uniformly distributed axial load. (a) Geometry and properties of cases with different BCs: bottom is fixed–fixed, top is fixed-free. (b) Distributions. Circles are FE generated results, continuous lines are the strong formulation results

the material relation. In the present strong formulation, these are achieved by integrating the governing equation and satisfying all of the BCs. Because the loading is specified, we can take the loading equation as the starting point. That is, EA

∂ 2u = −qu = −qo = constant ∂x 2

This is integrated twice to give EA

∂u = −qo x + c1 ∂x

EA u = − 12 qo x 2 + c1 x + c2 where c1 and c2 are constants of integration. To determine these constants, it is necessary to impose additional conditions. For the present problem, we know that the BCs are at

x=0:

u = 0 = c2

at

x=L:

u = 0 = − 12 qo L2 + c1 L + c2

This gives two equations for two unknowns allowing us to solve for the coefficients. They are c1 = 12 qo L, c2 = 0 and the displacement distribution is EAu(x) = 12 qo x[L − x] The corresponding force and stress distributions are F (x) = 12 qo [L − 2x] ,

σxx (x) = 12 qo [L − 2x]/A

34

1 Overview of Shapes and Stiffness

These distributions are shown plotted in Fig. 1.18b as the continuous lines. Notice that while the internal stress goes from positive to negative, the displacement is always positive. Also shown are the FE generated results using the Hex20 element; except for near the boundaries, the two modelings give excellent agreement. It is worth noting that if all the end nodes in the FE mesh (except at the center) are relaxed to be free to move in y and z, then σyy is zero everywhere and the two modelings are identical. The BCs just imposed are examples of geometric BCs where displacements are specified. The other case in Fig. 1.18a has a natural BC at the right end. That is, the BCs are at

x=0:

u = 0 = c2

at

x=L:

F = 0 = EA

∂u = −qo L + c1 ∂x

The displacement and member force distributions are EAu(x) = 12 qo x[2L − x] ,

F (x) = qo x[L − x]

The displacement is shown in Fig. 1.18b labeled as “free.” It is clear that imposing the zero force condition has entailed a change of shape, not just at the boundary but over the whole rod. For comparison, we now add the solutions for some other cases with uniform properties: fixed–fixed: fixed-free: end load:

EAu(x) = 12 qo x[L − x] EAu(x) = 12 qo x[2L − x] EAu(x) = Po x

(1.21)

Note that the end load case has a natural BC where F (L) = EA∂u/∂x = Po .

Example 1.3 Find the deflected shape of the fixed–pinned beam shown in Fig. 1.19a. The applied load per unit length, wo , is uniformly distributed. The loading is constant and given as qv (x) = −w(x) = −wo ; therefore we take this as the starting point. That is, EIzz Integrate to obtain

∂ 4v = qv = −wo ∂x 4

1.2 Modeling Continuous Structures

35

0.0

0.2

0.4

.

0.6

0.8

1.0

Fig. 1.19 Uniformly loaded beams. (a) Beam with pinned end. (b) Distributions for beam with pinned end. (c) Beam with spring support

EIzz

∂ 3v = −wo x + c1 ∂x 3

∂ 2v = − 12 wo x 2 + c1 x + c2 ∂x 2 ∂v = − 16 wo x 3 + 12 c1 x 2 + c2 x + c3 EIzz ∂x

EIzz

1 EIzz v = − 24 wo x 4 + 16 c1 x 3 + 12 c2 x 2 + c3 x + c4

(1.22)

We now impose the BCs. At the fixed end, both the deflection and rotation (slope) are constrained to be zero. At the pinned end, only the deflection is constrained to be zero; in other words, it is free to rotate, which in turn means there is no restraining moment. The four conditions are expressed as at x = 0 : v = 0 ,

∂v =0 ∂x

at x = L : v = 0 ,

Mz = EIzz

∂ 2v =0 ∂x 2

Note how the natural BC at x = L of zero moment is actually a constraint on the second derivative of the deflection but a higher order constraint than associated with the geometric BCs. These give, respectively, 0 = c4 1 0 = − 24 wo L4 + 16 c1 L3 + 12 c2 L2 + c3 L + c4

0 = c3 0 = − 12 wo L2 + c1 L + c2

After solving for the coefficients, we find the deflected shape to be 1 wo x 2 [2x 2 − 5xL + 3L2 ] EIzz v(x) = − 48

36

1 Overview of Shapes and Stiffness

Other entities of the solution can be obtained by differentiation. This gives the slope, moment, and shear distributions as 1 wo [8x 3 − 15x 2 L + 6xL2 ] EIzz φ(x) = − 48 1 Mz (x) = − 48 wo [24x 2 − 30xL + 6L2 ] 1 Vy (x) = + 48 wo [48x − 30L]

These are shown plotted in Fig. 1.19b; each distribution is normalized to unity. Other quantities such as stress and strain are easily obtained from these. For comparison, the corresponding solutions for other cases are fixed–fixed:

1 wo x 2 [2x 2 − 4xL + 2L2 ] EIzz v(x) = − 48

fixed–pinned:

1 EIzz v(x) = − 48 wo x 2 [2x 2 − 5xL + 3L2 ]

pinned–pinned:

1 EIzz v(x) = − 48 wo x[2x 3 − 4x 2 L + 2L3 ] 1 EIzz v(x) = − 48 wo x 2 [2x 2 − 8xL + 12L2 ]

fixed-free:

EIzz v(x) = − 16 Po x 2 [x − 3L]

cantilever, end load:

(1.23)

Both the fixed–fixed and pinned–pinned cases have symmetric deflection distributions. The maximum deflection of the fixed–pinned case lies between the maximum for these two. Let us finish this example by considering the effect of elastic constraints. Specifically, replace the pinned BC with an axial spring as shown in Fig. 1.19c. The left BCs are the same as before, the right still has a zero moment but the vertical displacement is resisted by an elastic force from the spring. A FB (free body) in the vicinity of the end gives −V (L)−αv(L) = 0

% & 1 − −(−wo L+c1 ) −β(− 24 wo L+ 16 c1 + 12 c2 /L)

or

where β = αL3 /EI . This gives the coefficients c1 =

1+

5 24 β 1 + 13 β

wo L ,

1 + 1β w o L2 c2 = − 2 24 1 + 13 β

and the displaced shape as 1 wo x 4 + EI v(x) = − 24

wo L  1 1+

3 3 5 6 (1 + 24 βL )x 1 β 3

− 12 ( 12 +

2 1 24 β)x L



1.2 Modeling Continuous Structures

37

Fig. 1.20 Uniformly loaded beam resting on an elastic foundation

It is easy to verify that when β → ∞ we recover the fixed–pinned case, while β → 0 recovers the fixed-free case. Thus, for an arbitrary spring value, the shape lies somewhere between these two cases. Example 1.4 A simply-supported beam rests on an elastic foundation as shown in Fig. 1.20. Assuming that the foundation material can be modeled equivalent to a distributed linear elastic constraint, establish the structure of the solution and the necessary BCs to affect a solution. The foundation acts as a distributed spring with resistive distributed force of amount qv x = −Kv x v, where Kv has the meaning of a spring stiffness per unit length. The governing differential equation is modified to become EI

∂ 4v = qv = −wo − Kv v ∂x 4

or

EI

∂ 4v + Kv v = −wo ∂x 4

(1.24)

This is an inhomogeneous equation with constant coefficients. Unlike the single spring of the previous example problem, here the distributed spring enters the differential equation itself; this has some significant consequences which we explore. The particular solution is simply vp = −wo /Kv . For the homogeneous equation, assume solutions of the form v(x) = Ceαx

or

v(x) = Ce−ikx

√ where i = −1 is the complex “i.” We generally choose the latter form because it is more suitable for the spectral analysis of problems we introduce later. Substitute this into the governing equation to get  EI k 4 + Kv ]Ce−ikx = 0 This is nontrivially true only if −Kv k = , EI 4

'

−Kv k =± = ±i EI 2

'

Kv , EI



k = ±(1 ± i)β ,

where use is made of the relations ±i = e±iπ/2 ,

√ √ ±i = e±iπ/4 = [1 ± i]/ 2

Kv β≡ 4EI

1/4

38

1 Overview of Shapes and Stiffness

Fig. 1.21 A very long beam resting on an elastic foundation. (a) Loading and displacement distribution. (b) Free body near the load point

There are four roots, so the total homogeneous solution is v(x) = Ae−ik1 x + Be−ik2 x + Ce−ik3 x + De−ik4 x     = Ae−iβx + Be+iβx e−βx + Ae−iβx + Be+iβx e+βx     = c1 cos βx + c2 sin βx e−βx + c3 cos βx + c4 sin βx e+βx Using the real-only form, the total solution is     wo v(x) = c1 cos βx + c2 sin βx e−βx + c3 cos βx + c4 sin βx e+βx − (1.25) Kv The presence of the elastic foundation has complicated the solution shape in that it now depends on the sine and cosine functions plus exponentials; note, however, that there are still only four constants of integration and hence there must be four BCs to be satisfied. Indeed, the BCs for this problem are identical to those in the absence of the foundation. That is, at x = 0 : v = 0, Mz = EI

∂ 2v = 0; ∂x 2

at x = L : v = 0, Mz = EI

∂ 2v =0 ∂x 2

This establishes the four equations necessary to solve for the four coefficients in terms of the known loading wo . It is definitely worth highlighting that the solution shape has sinusoidal components even though the beam itself is straight. This is hinting at some important behaviors concerning coupled deformations that are addressed in the next chapter. To illustrate the sinusoidal effect, consider a very long beam resting on an elastic foundation with a concentrated applied load as shown in Fig. 1.21a. The problem is symmetric, hence we need to consider only one half of the beam. The deflection to the right of the load must diminish at large distances, therefore the solution cannot contain the exponential e+βx and consequently reduces to   v(x) = c1 cos βx + c2 sin βx e−βx ,

β≡

 K 1/4 v 4EI

1.2 Modeling Continuous Structures

39

The BCs at the load point are ∂v = 0, ∂x

Vy = −EI

∂ 3v = 12 Qo ∂x 3

The first follows from symmetry, while the second follows from the free body shown in Fig. 1.21b. The BCs lead to 0 = βc2 − βc1 ,

1 2 Qo

= −EI [2β 3 c1 + 2β 3 c2 ]

Solving for the coefficients then gives the solution v(x) =

 −Qo  cos βx + sin βx e−βx 3 8EIβ

The deflection is shown as the continuous line in Fig. 1.21a. The term inside the square brackets exhibits multiple zero crossings extending to infinity. This bracketed term is modulated by the exponential decay outside. Consequently, the dominant feature of the displacement is that it is localized to the load point, first going to zero where cos βxc + sin βxc = 0

or

βxc = 2.355 ,

1/4  xc = 2.355 4EI /Kv

For a beam of square cross section, this reduces to 1/4  xc = 1.789 h E/Kv This shows that the indent is relatively insensitive to a change of beam modulus or spring stiffness; for example, a change of 16 in E causes only a change of indent of 2. More generally, however, it indicates that constraints affect the range of action of a BC or a load. This crucially important topic is the subject of the next chapter.

1.2.3 Ritz Method Good expositions and background material on the Ritz method can be found in References [1, 10]. It is a weak formulation that deals with discretized unknowns. We first illustrate how a continuous system can be discretized and recast the PoVW for discretized unknowns. Consider the transverse deformation of the pretensioned cable in Fig. 1.22a where a concentrated load is applied at position a from the left end. It is known that the exact solution corresponds to a bilinear distribution of displacement as can be inferred from Fig. 1.11c; however, we do not use this information in our solution

40

1 Overview of Shapes and Stiffness

♦

♦

0.0

0.2

0.4 . 0.6

0.8

1.0

Fig. 1.22 Fixed–fixed pretensioned cable with a concentrated load. (a) Geometry. (b) Ritz functions showing satisfaction of geometric BCs. (c) Displacement distributions; dashed line is the exact solution, full line is the Ritz two-term solution

procedure. To use the Ritz method, we must start with a compatible displacement field or deformation shape—this is achieved by imposing the geometric BCs or geometric constraints. For this problem, they are at x = 0,

v = 0;

at x = L,

v=0

A usually effective scheme for constructing Ritz functions (which are functions that satisfy the geometric BCs) is to begin by assuming a polynomial form for the displacement distribution v(x) = a0 + a1 x + a2 x 2 + a3 x 3 + · · · where the coefficients aI are yet to be determined. Imposing the BCs leads to, respectively, 0 = a0 ,

0 = a0 + a1 L + a2 L2 + a3 L3 + · · ·

Solving for a0 and a1 in terms of the other coefficients then gives the displacement representation (and implicitly the deformed shape) v(x) = a2 [x 2 − xL] + a3 [x 3 − xL2 ] + · · · = g2 (x)a2 + g3 (x)a3 + · · · that satisfies the BCs. Indeed, each of the functions gI (x) separately satisfies the BCs and therefore is individually acceptable Ritz functions. It is also clear from Fig. 1.22b that they are in some sense “shape functions” but not generic (or fundamental) in that both are essentially the same—they both are transverse with zero at each end. We pursue the topic of generic shapes in the next two sections because they illuminate the meaning of stiffness and are part of our spectral analysis. The strain energy of the problem is

1.2 Modeling Continuous Structures

 U=

1 2

0

L

 ∂v 2 dx = F¯o ∂x

41

 1 2

L

 2 F¯o a2 [2x − L] + a3 [3x 2 − L2 ] + · · · dx

0

= U (a22 , a2 a3 , a32 , . . .) The energy is quadratic in the coefficients aI . We have thus discretized the energy into a finite set of unknowns which we now refer to as generalized DoFs. Let us defer the explicit integration until later and consider the changes necessary to the PoVW when the unknowns are discrete. Let the deformation of the structure be described in terms of N independent parameters u1 , u2 , . . . , uN , and the strain energy represented by U = U (u1 , u2 , . . . , uN ) The elasticity relations could be nonlinear in which case the strain energy is not necessarily quadratic in the DoF. By the chain rule for differentiation, the virtual strain energy can be written as δ U = δ U (u1 , u2 , . . . , uN ) =

N ∂ U ∂U ∂U δu1 + δu2 + · · · = δuI I ∂uI ∂u1 ∂u2

The virtual work of the applied loads can be written as δ We = P1 δu1 + P2 δu2 + · · · =

N I

PI δuI

where PI are the generalized forces (some of which could be zero) conjugate (acting parallel) to the displacements uI . These lead to the PoVW being rewritten as  N  ∂ U δ W = δ U − δ We = − PI δuI = 0 I ∂uI Because this must be true for each independent variation δuI , we conclude that ∂U − PI = 0 , ∂uI

I = 1, 2, . . . , N

(1.26)

In this way, the single virtual work statement becomes N simultaneous equations. Equation (1.26) is our new statement of static equilibrium and is the one that is used (in slightly different forms) in the remainder of the book for discretized systems. It has an interpretation roughly analogous to a resultant force so that for equilibrium the resultant force in each direction is zero; keep in mind, however, that there are N such resultant generalized forces. We can interpret the derivative term as a force (called the elastic force) so that equilibrium is

42

1 Overview of Shapes and Stiffness

FI − P I = 0 ,

FI ≡

∂U ∂uI

(1.27)

The beauty of this statement is that it is about the structure as a whole and not just about a small subset free body. That is, regardless of how complex the structure is in terms of shape, material distributions, and loads, this is the statement of its equilibrium conditions. Returning to the cable problem, there are N generalized DoF (a2 , a3 , a4 , . . .), hence there are N generalized loads PaI , one corresponding to each DoF. To obtain these loads, set the virtual work of the generalized loads equal to the virtual work of the actual loads. That is,   Pa2 δa2 + Pa3 δa3 +. . . = Po δv(x = a) = Po [a 2 −aL] δa2 +[a 3 −aL2 ] δa2 +· · · We therefore conclude that Pa2 = Po [a 2 − aL] ,

Pa3 = Po [a 3 − aL2 ] ,

...

It is important to realize that the single applied load Po has led to many generalized loads. Equilibrium of the problem is described by Eq. (1.26) ∂U − P aI = 0 , ∂aI

I = 2, 3, . . .

This leads to, on performing the indicated differentiations, 

L

  F¯o a2 [2x − L] + a3 [3x 2 − L2 ] + · · · [2x − L] dx − Po [a 2 − aL] = 0

0



L

  F¯o a2 [2x − L] + a3 [3x 2 − L2 ] + · · · [3x 2 − L2 ] dx − Po [a 3 − aL2 ] = 0

0

Performing the required integrations then gives ⎧ 2 ⎫ ⎡ ⎤⎧ ⎫ a − aL ⎪ 10L3 15L4 . . . ⎪ a2 ⎪ ⎪ ⎨ ⎬ ⎬ ⎨ ¯ Fo ⎢ 15L4 24L5 . . . ⎥ a 3 2 3 = Po a − aL ⎣ ⎦ ⎪ ⎪ 30 .. .. .. . . ⎪ ⎩ ⎭ ⎭ ⎩ .. ⎪ . . . . .

or

[ K ]{ u } = { P}

It is typical in these problems to obtain a set of simultaneous equations. We can interpret the system matrix as a stiffness matrix [ K ] (this is elaborated on in the next two sections), note that it is symmetric which is always the case when it is derived from the strain energy function as done here. Solving this system when only one term is used in the expansion gives

1.2 Modeling Continuous Structures

43

a2 =

Po a 2 − aL F¯o L L2

This Ritz analysis, therefore, yields the approximate solution v(x) =

Po [a 2 − aL][x 2 − xL] F¯o L3

In the special case of a = L/2, the maximum deflection (also at x = L/2) is vmax =

Po L 3 , 4F¯o 4

vexact =

Po L 4F¯o

This underestimates the maximum by a sizable amount. The first example problem discusses the inclusion of more terms in the displacement representation. To summarize: the Ritz method assumes a representation of the displacement field in terms of undetermined parameters. The number of parameters is reduced by imposing that all geometric BCs (geometric constraints) are satisfied. Equilibrium is then established through Eq. (1.26) where the reduced undetermined coefficients are treated as the generalized DoFs. This powerful method forms the basis of the FE formulation of structural problems with distributed elasticities. An important consideration is the selection of the Ritz functions gI (x) and we leave that discussion to later. But, if properly set up, as the number of functions are increased (N becomes large) we approach the exact solution, that is, the strong formulation solution as presented previously. Selecting efficient functions may not be easy; fortunately, many problems closely resemble other problems that have been solved before, and the literature is full of examples that can serve as a guide. It must also be kept in mind that these functions need only satisfy the essential BCs and not (necessarily) the natural BCs. The point being emphasized is that it is usually easy to guess the deformed shape of a structure and the Ritz method then provides the analytical framework for computing all the mechanics quantities of interest. For practical analyses, this is a significant point and largely accounts for the effectiveness of the displacement-based finite element analysis procedure. Our spectral analysis method introduced in Sect. 1.4 generates Ritz functions automatically through a vibration eigenanalysis. An especially positive attribute of the Ritz method is that it is equally applicable to linear and nonlinear problems alike as long as appropriate energy expressions are available. We make significant use of this in the nonlinear analyses beginning with Chap. 3. Example 1.5 Continue the introductory worked example using extra DoFs. When two terms are included in the expansion then

44

1 Overview of Shapes and Stiffness

Fig. 1.23 A beam with uniformly distributed load. (a) Geometry. (b) Some distributions using a one-term Ritz solution

# $ # $ Po a2 −6La(3L − 5a)(L − a) = a3 10a(L − 2a)(L − a) F¯o L5 and the displacement is v(x) =

 Po  2 3 2 −6La(3L−5a)(L−a)[x −xL]+10a(L−2a)(L−a)[x −xL ] F¯o L3

For the special case when a = L/2, we get the interesting result that a2 = −

P 3 , 4F¯o L 4

a3 = 0

This Ritz analysis yields the same result as the one-term expansion. What has happened is that the special case a = L/2 makes the deformed shape symmetric; however, the second term (in the expansion) is nonsymmetric and so the system of equations determined that its contribution is zero. We would get an improved result by including the term associated with a4 because a4 [x 4 − xL3 ] = a4 x[x − L][x 2 + xL + L2 ] is a symmetric shape. The a3 term makes a useful contribution when a = L/2 as shown in Fig. 1.22c. Example 1.6 Use the Ritz method to determine an approximation for the deflection shape of the beam shown in Fig. 1.23. Assume the distributed load acts downward. To use the Ritz method, we must start with a compatible displacement field. This is achieved by imposing the geometric BCs. For this problem they are at x = 0,

v = 0,

∂v = 0; ∂x

at x = L,

Assume the deflection can be represented by a polynomial

v=0

1.2 Modeling Continuous Structures

45

v(x) = a0 + a1 x + a2 x 2 + a3 x 3 + a4 x 4 + · · · Imposing the BCs leads to 0 = a1 ,

0 = a0 ,

0 = a0 + a1 L + a2 L2 + a3 L3 + a4 L4 + · · ·

Solving for a0 , a1 , and a2 in terms of the other coefficients leads to the deflection representation v(x) = a3 [x 3 − x 2 L] + a4 [x 4 − x 2 L2 ] + · · · that satisfies the geometric BCs. Indeed, each of the functions separately satisfies the BC’s and therefore is individually acceptable Ritz functions. The coefficients a3 , a4 , . . . are our generalized DoFs. The strain energy of the beam is  U = 12



L

EI 0

∂ 2v ∂x 2

2

 dx =

1 2

L

 2 EI a3 [6x − 2L] + a4 [12x 2 − 2L2 ] + · · · dx

0

= U (a32 , a3 a4 , a42 , . . .) We defer the integrations until later. The virtual work done by the actual loads is  δ We =

L

qv (x)δv(x) dx 0

 =

L

0

= wo

  [−wo ] δa3 [x 3 − x 2 L] + δa4 [x 4 − x 2 L2 ] + · · · dx



1 4 12 L δa3

+

2 5 15 L δa4

+ ···



Corresponding to each generalized DoF a3 , a4 , . . ., there is a generalized force Pa3 , Pa4 , . . .. Let the work done by these forces be equivalent to the actual virtual work. That is, Pa3 δa3 + Pa4 δa4 + · · · = wo



1 4 12 L δa3

+

2 5 15 L δa4

+ ···



Because the variations are arbitrary, we conclude that 1 4 L , Pa3 = wo 12

2 5 Pa4 = wo 15 L ,

...

We are now in a position to utilize our PoVW in the form of Eq. (1.26). Consider a one-term solution, then

46

1 Overview of Shapes and Stiffness

∂U = ∂a3



L

EI a3 [6x − 2L][6x − 2L] dx = EI 4L3 a3

0

and equilibrium is given by EI 4L3 a3 −

4 1 12 wo L

=0

or

a3 =

wo L 48EI

The deflected shape is given by v(x) =

wo L 3 [x − x 2 L] 48EI

At x = L/2, the deflection is negative indicating that the beam sags. An important point to keep in mind is that while the fundamental problem is reduced to a set of discretized unknowns, the solution is nonetheless continuous over the domain. This is evident in the deflection function just stated, but we also have wo L ∂v = [3x 2 − 2xL] ∂x 48EI

slope:

φ(x) =

strain:

xx (x) = −y

stress: σxx

∂ 2v wo L [6x − 2L]y =− 2 48EI ∂x wo L = Exx (x) = − [6x − 2L]y = M(x)y/I 48I

These are shown in Fig. 1.23b. Observe that the stress (and hence moment) distribution is predicted to be linear with a nonzero value at x = L. The actual natural BC at x = L is that the moment is zero. It can generally be expected that the natural BCs are violated when the solution is not fully converged. This solution is approximate; the exact solution can be obtained by adding the a4 term to the deflection representation. While, in general, the Ritz method is approximate it is capable of giving very accurate results by either a judicious choice of functions or by including enough functions. The former is difficult to generalize, while the latter increases the computational effort. The FE method formalizes the process of automating the increase of the number of Ritz functions. The spectral shapes developed in this book illustrate another way of automating the process.

1.3 Structural Stiffness and Its Spectral Properties The stiffness property of a single spring is easy to grasp, but when multiple springs or multiple elastic members are interconnected to form a structure, the concept of stiffness is more elusive to grasp. This is compounded when the deformation

1.3 Structural Stiffness and Its Spectral Properties

47

Fig. 1.24 Fixed–fixed pretensioned cable with two concentrated loads. (a) Geometry. (b) Displacement distributions with two DoFs showing satisfaction of the geometric BCs and geometric compatibility between segments. (c) Separated Ritz functions showing their local compact support

behaviors are nonlinear. In fact, the concept of stiffness is truly to be understood only in a nonlinear structural context. Because stiffness is a fundamentally important concept in structural analysis, this section is an introduction to some of its basic aspects; however, its fuller elucidation is postponed until the nonlinear analyses of Sect. 3.1.

1.3.1 Discrete Stiffness of Structures Our idea of a structure is an object comprising many components. The assemblage of these components must obey certain rules arising from the mechanics of the situation. The remaining chapters develop these rules in good detail for a variety of component types; here we wish to concentrate on a particular structural system and emphasize the mechanics principles that are applicable for that system when the components are assembled into a structure. Consider the fixed–fixed cable with two loads shown in Fig. 1.24a. This problem could be solved in the identical manner as done in association with Fig. 1.22, instead, we want to introduce the idea of a structure as an assemblage of components (called elements). To this end, rather than using shape functions that span the whole region, we use piece-wise specified functions. The displaced shape in Fig. 1.24b is given by the representation 0 44 4R 2 Eh3

or

√ z2 > 42 3 Rh

A cylinder is considered thin-walled when R > 20h [5], substituting this into the inequality then gives √ z2 20 > 42 3 R R

or

z > 23 R

In other words, the effects of the boundary conditions are confined to within one radius of the ends which is clearly significant in the case of short cylinders as discussed here but would not be significant for long cylinders or tubes. A final point worth mentioning is that most cylinders are given spherical caps instead of flat caps so as to smooth the transition and minimize the bending effects. A relevant exception is vertical storage tanks which typically are flat bottomed (so as to sit flatly on the ground) and therefore reinforcement is used to reduce the bending stresses. Example 2.6 Figure 2.18 shows a FE deflection result for a free-free cylinder with diametrically opposed point loads. What is quite surprising is that the deformation is

112

2 Shapes with Coupled Deformations

♦

.

1.0

♦

Fig. 2.18 Exaggerated deformed shape of a cylinder of length 2Lo with two point loads. (a) Side view. (b) End view

0.0

♦

0.0

1.0 .

2.0

♦

0.0

1.0

2.0 .

3.0

4.0

Fig. 2.19 Cylinder with two point loads. Symbols are FE generated data; circles are for θ = 0, π , squares are for θ = ± 12 π . Continuous lines are model results. (a) L = Lo . (b) L = 2Lo

predominantly global with only small local load effects. Construct a model to help explain this result. The MRT/DKT shell element [5] was used with 20 modules along the length and 64 in the hoop direction. The question of BCs is very important because six rigid body modes must be suppressed and this can be done in a variety of ways. An analysis of this cylinder problem (as done in Ref. [19], for example) need not directly confront the issue of BCs, but FE methods must because they are displacement based. The FE global BCs imposed here are that, in the plane of the load, we set θ = 0, π :

{0, 1, 0, 1, 0, 1} ,

θ = ± 12 π :

{1, 0, 1, 1, 1, 1}

The second of these is necessary to prevent a rigid body displacement in y. Setting the axial displacements to zero at the load points is an arbitrary choice but Fig. 2.19a shows that there is significant warping (axial displacement w) of the cross section which if (inadvertently) prevented changes the nature of the deformed shape. For example, if we set w = 0 at all four quarter points, the radial deflection is reduced by more than an order of magnitude. Furthermore, the cross-sectional shape has a

2.2 Deflections of Thin Curved Plates and Shells

113

higher order sinusoidal shape. That warping occurs is indicative of shear actions being in play. Some axial displacement might be anticipated because of Poisson’s ratio effect, but this is estimated to be much smaller than what is shown; in addition, it would be distributed so that the ends move in opposite directions which is not the case here. Because the axial displacement is uniform along the length, some other mechanism is at play. That the effect of a localized load is transmitted to the whole structure is indicative that the deformation is mostly unconstrained and inextensible—think of a cantilever beam with a transverse load applied close to the fixed end. Let us use this as our working assumption and follow the consequences. Inextensible means that all membrane strains are zero; from Eq. (2.15) we have that ss =

∂v u + = 0, R ∂s

zz =

∂w = 0, ∂z

γsz =

∂w ∂v + =0 ∂z ∂s

It is more convenient to work with the angle so that these conditions become u + v,θ = 0,

w,z = 0,

Rv,z +w,θ = 0

The warping is symmetric about the vertical and horizontal axes which suggests taking w(θ, z) = w(θ ) = 1, cos(2θ ), cos(4θ ), · · · One set of solutions is to take w = 0 and the other displacements as low order sinusoids. Using inextensibility, we get that w1 = 0 ,

v1 = an sin(nθ ) ,

u1 = −an n cos(nθ )

These are essentially the same as obtained for the arch in Sect. 2.1. Another solution set is to take w = bn cos(nθ ), then using inextensibility leads to w2 = bn cos(nθ ) ,

v2 =

z nbn sin(nθ ) , R

z u2 = − n2 bn cos(nθ ) R

This gives a linear distribution in z for the radial displacement in agreement with Fig. 2.19. For convenience of modeling and discussion, we place z = 0 at the center of the cylinder. Our assumed displacement solution is the sum of these two sets. Because of the restrictive behavior in z, the strain energy expression of Eq. (2.16) simplifies considerably to  UF =

1 2

D [u,θθ −v,θ ]2 Rdθ dz + R4

 1 2

D¯ [2u,θz −v,z ]2 Rdθ dz R2

The axial displacement does not appear in this, but it has affected the allowable form for the other two displacements. Substituting for these and integrating, we get

114

UF =

2 Shapes with Coupled Deformations

1 2

# 2 D  3 n − n an2 + R4

1 3

 2 L2 2 $ D¯  3 4 2 2 2n − n bn2 LR2π n −n bn 2 + D R

This gives rise to the two stiffnesses # 2 $ D  3 K11 = v 4 n − n Vo , R

# 2 L2 D¯  2 $ D  4 2 3 1 Vo K22 = 4 n − n 3 R 2 + D 2n − n R

where Vo = LR2π is the material volume of half a cylinder. The stiffness matrix is diagonal. The virtual work of the two loads is δ We = −Po δu(0, c) − Po δu(π, c)   c = −Po −δan n cos(n0) − δbn n2 cos(n0) R   c −Po −δan n cos(nπ ) − δbn n2 cos(nπ ) R Nonzero work is done for sinusoids with n = 2, 4, · · · , this was already anticipated from the warping discussion. The two force contributions are then the same giving Pa = Po 2n, Pb = Po 2n2 c/R. Our PoVW gives two equilibrium equations K11 an = Po 2n ,

K22 bn = Po 2n2 c/R

The equations are uncoupled and easily solved. The radial and axial displacements are given by u(θ, z) = u1 + u2 = −an n cos(nθ ) −

z 2 n bn cos(nθ ) R

w(θ, z) = w1 + w2 = bn cos(nθ ) These model results are plotted as the continuous lines in Fig. 2.19a for n = 2. This model captures the center of the cylinder displacements quite well; interestingly, these displacements are unaffected by the location of the load c. Furthermore, if the loads are placed at c = 0, then all displacements are uniform along the length. The model captures the essence of the behavior of the eccentric load, but its magnitude is off a little. It might be thought that, because of the linear distribution along the length, a longer cylinder (with all else remaining the same) would give a larger displacement at the ends. This is not what happens and the model makes the reasons clear. First note that the generalized loads are unaffected by the cylinder length, and therefore we need to only focus on the stiffnesses. The K11 stiffness increases with cylinder length (through Vo ) and therefore the average deflection decreases with length. The eccentric load effect is governed by K22 and has a L2 contribution in addition to the

2.2 Deflections of Thin Curved Plates and Shells

115

L contribution from Vo . Therefore, this stiffness rises very rapidly with length and the eccentric load contribution diminishes. Both effects are observed in Fig. 2.19b for the longer cylinder. Example 2.7 Figure 2.20 shows some FE deflection and stress results for the twisting of an elliptical tube fixed at one end. The data are sampled at z = L/2. Construct a model to help explain this result. The tube was constructed by taking the cylinder of Fig. 2.18 and remapping the coordinates so that x → ×1.5, y → ×0.5, leaving the z coordinate unaffected. Two cases of end-plate are considered: one is very stiff, the other uses the parameters of the shell. A cylindrical shell under similar end conditions is unaffected by the end-plate. The FE analysis also shows that except in the immediate vicinity of the boundaries, the responses are predominantly membrane with the kinematic behaviors u(s, z) = 0 ,

v(s, z) = v(z) = φˆ z zR ,

w(s, z) = 0

where φˆ z is the twist per unit length. The shear stress is the only significant stress and is uniform in both the hoop and length directions. The membrane energy of the shell reduces to    2 2 2 1 1 1 ˆ UM = 2 Gh[v,z +w,s ] dsdz = 2 Ghφz R 2π R dz = 2 GJ φˆ z2 dz where J = 2π R 3 h. This is the elementary torsion modeling. For the elliptical tube, significant flexural actions are generated (especially for the flexible end-plate) but these are mostly confined to the ends. Away from the ends, the behavior is mostly membrane actions. where the membrane energy reduces to 

E ∗ hw,2z dsdz +

 Gh[v,z +w,s ]2 dsdz

1 2

♦

1 2

♦

UM =

♦

-180. -90.

0. .

90.

180.

♦

-180. -90.

0. .

90.

180.

Fig. 2.20 Axial twisting of an elliptical tube at midlength. Circles use a rigid end-plate, triangles use a flexible end-plate. (a) Twisting and warping. (b) Shear and axial stress

116

2 Shapes with Coupled Deformations

Fig. 2.21 Torsion of thin-walled tubes. (a) Geometry. (b) Contrast in shear stress distribution between open and closed cross sections

The leading term is the warping, the second term is the shear as if the cylinder is unrolled to be flat. Now consider that the cross section of the cylinder is distorted: what changes in the stress and strain distributions? Such cross sections are of interest here. A hollow shaft is an example of a multiply-connected cross section. The torsional behavior of these structures are difficult problems to solve in general and numerical methods must be resorted to in order to effect a solution. There is one situation that is relatively easy to solve [18] and that is when the wall thickness is thin. Bars and the like with thin-walled closed sections are called tubes, and the area enclosed is called a cell. When multiple such cavities are present, this is called a multicelled structure. Here we focus on single-cell tubes. Consider the general thin-walled tube shown in Fig. 2.21a; the contour of the cross section is arbitrary as is the thickness, but assume both are uniform along the length. Note that when the thickness changes with position around the cross section, that is, h = h(s), a useful quantity to introduce is the shear flow defined as q ≡ τxs h = c1

(2.20)

This quantity is constant on the whole cross section contour regardless of the thickness variation. The strain energy for a tube segment of length L reduces to  U=

1 2

 2 h ds τxs

=

1 2

(τxs h) ds/ h = 2

1 2 c1

ds h

where the integration is taken around the contour of the wall of the tube. The torque caused by the stress τxs on an arc segment of length ds is dTx = aτxs h ds = aq ds = ac1 ds where a is the perpendicular distance from the origin to the line of action of τxs and q as shown in Fig. 2.21a. A simple geometric construction based on triangles with base ds shows that dA = 12 a ds, from which we get that

2.2 Deflections of Thin Curved Plates and Shells

 Tx =

 A¯

2q dA =

A¯

117

2c1 dA = 2q A¯ = 2c1 A¯

where A¯ is the area enclosed by the centerline of the tube. The total virtual work of our problem is δ W = δ U − Tx δ φˆ x L =



L 2c1 2G

-

 ds ¯ ˆ − 2Aφx L δc1 = 0 h

from which it is concluded that c1 = 2GA¯ φˆ x /

-

ds h

We can now summarize the torsion relations as Tx q ds q= , Tx = GJ φˆ x , , φˆ x = h 2A¯ 2GA¯

J = 4A¯ 2 /

-

ds h

(2.21)

These equations are usually referred to as St. Venant torsion. The shear stress distribution is obtained from τxs (s) = q/ h(s) and is constant through the thickness. On the assumption that the FE cross section can be treated as elliptical, then A¯ = π ab and we get the model results τsz =

Tz q = , ¯ h 2Ah

φz = φˆ z L/2 =

q 2GA¯

-

τsz ds = h 2GA¯

ds

Both of these are shown as the continuous lines in Fig. 2.20. It is clear that the model represents the average behavior but the particulars of the actual BCs do make a difference. Therefore, not surprisingly, it represents the average axial behavior (the warping and consequent axial stress) as zero. Example 2.8 A curved plate has a uniform radial applied load, determine the deflected shape. We find it convenient to describe the plate as a segment of a circular cylinder but with BCs appropriate for the full cylinder. The ends are radially constrained but free axially, and along the straight edges u = 0 and v and w are free. That v is free makes the plate different than the extruded arch. These BCs were specifically chosen so that we could work directly with the governing system of Eq. (2.17), that is, we obtain a strong formulation solution. We take the solution in the form u(s, z) = uo sin(ks) sin(mz) ¯ , w(s, z) = wo sin(ks) cos(mz) ¯

v(s, z) = vo cos(ks) sin(mz) ¯

118

2 Shapes with Coupled Deformations

For an infinitely long cylinder, the wavenumbers k and m ¯ are continuous functions but for the finite cylinder they are discretized as k = n¯ = n2/R and m ¯ = mπ/L. Because the differential equations are complicated, the system matrix is also complicated. It is therefore instructive to separate it into its different energy contributions. To make the patterns more apparent, we introduce the wavenumber ko ≡ 1/R; this makes all terms in the matrices comparable (that is, wavenumber of dimension 1/length). The membrane contributions are ⎡

⎤ ko2 −kko −νko m ¯ ⎦, E ∗ h ⎣ −kko k 2 νk m ¯ ¯ νk m ¯ m ¯2 −νko m

⎡

⎤ 00 0 Gh ⎣ 0 m ¯ 2 km ¯⎦ ¯ k2 0 km

with the DoF arranged as {uo , vo , wo }T . It is noteworthy that the membrane shear affects only v and w, but the normal stresses affect all three displacement components. The flexural contributions are ⎡

⎤ k4 + m ¯ 4 + 2νk 2 m ¯ 2 −k 3 ko − νkko m ¯2 0 D ⎣ −k 3 ko − νkko m ¯ 2 k 2 ko2 0⎦ , 0 0 0

⎡

⎤ 4k 2 m ¯2 −2kko m ¯2 0 D¯ ⎣ −2kko m ¯ 2 ko2 m ¯2 0⎦ 0 0 0

The flexural behavior involves only the u and v displacements and the axial behavior is absent. In other words, the coupling is only that similar to the arch—if the shell had a curvature in the z-direction we would expect a coupling in that direction also. Thus, the shapes and stiffnesses are the same as already presented for the cylinder, simply multiply each matrix by RαL/4 which is the weighted area of the plate. The virtual work of the generalized loads must equal the virtual work of the actual loads so that we have  δ We = Pnm δunm n,m    = qu (θ, z) δu(θ, z) Rdθ dz = qo δunm sin(nθ ¯ ) sin(mz) ¯ Rdθ dz n,m

The generalized loads evaluate to   ¯ − 1][cos mb ¯ − 1]/n¯ m ¯ α L = 4qo /n¯ m ¯, Pnm = qo [cos na

n, m = 1, 3, 5, · · ·

Because the load distribution is symmetric, only some of the generalized loads contribute to the deflection. Using our PoVW, we get the coupled equations for the coefficients ⎫ ⎧ ⎫ ⎤⎧ ⎡ ⎨ unm ⎬ RαL ⎨1⎬ 4q o ⎣ Knm ⎦ vnm = (2.22) 0 ⎩ ⎭ 4 n¯ m ¯ ⎩ ⎭ wnm 0

2.2 Deflections of Thin Curved Plates and Shells

119

♦

♦

♦

♦

♦

0.

30.

.

60.

90.

♦

0.0

0.5

1.0

.

Fig. 2.22 Displacement distributions for a curved plate under uniform radial loading. Circles are FE data, continuous lines are model results. (a) Hoop distributions. (b) Axial distributions

0.5 # of terms 1 4 9 16 25

u/uFE 1.13495 0.99411 1.00129 0.99813 1.00006

0.0

0.0

0.5

Fig. 2.23 Additional aspects to the solution. (a) Convergence behavior for the central radial displacement. (b) Amplitude ratios testing the inextensibility condition. Circles are vo /uo , triangles are wo /vo

The matrix [ K ] is the sum of the matrices given earlier but without the inertia contribution. The equations are coupled because the deformation (represented by separate u, v, w functions) is coupled, but it is important to realize the equations are not coupled across the subscripts n, m; thus only a [3 × 3] system must be solved irrespective of the number of coefficients used. The inset table in Fig. 2.23 shows an example of the convergence behavior; it is quite rapid with four terms yielding sufficient accuracy. The referenced FE results used a mesh twice as dense in both directions as that shown in Fig. 2.51. Figure 2.22 shows sample distributions using four terms; the comparisons are uniformly good. Of particular note is that the hoop distribution for u shows a flat top indicating a strong contribution from the n = 3 term. Although we reduced the system of equations to [3 × 3], it is worthwhile to explore if it can be reduced even further. For example, can we impose inextensibility? The inextensibility condition requires that the membrane strains be zero; from Eq. (2.15) this leads to u + v,θ = 0 ,

w,z = 0 ,

Rv,z +w,θ = 0

120

2 Shapes with Coupled Deformations

Substitute our solution form to get the amplitude ratios vo 1 = , uo n¯

wo = 0 ,

−

1 wo 1 = vo mR ¯ n¯

Figure 2.22 shows that while the axial displacement is small it is not insignificant. Figure 2.23 is a plot of the left-hand side against the right-hand side for each amplitude ratio; if the assumption is reasonable, then all data should fall on the arrow. It is seen that this is not the case. In fact, while we check extensibility in a number of situations, it seems to be the case that it is not a particularly good approximation.

2.2.2 Flat Plate Equations We now specialize the cylinder equations to the flat plate. These equations form the basis for the very interesting nonlinear deformed shapes covered over the next three chapters. Here we highlight the differences between the flexural and membrane actions. The flat plate equations are recovered by setting R → ∞,. (ξ, s, z) → (z, −y, x), and (u, v, w) → (w, −v, u). With the plate lying in the x-y plane, this leads to the strains and curvatures xx =

∂u ∂x

κxx =

∂ 2w ∂x 2

yy =

∂v ∂y

κyy =

∂ 2w ∂y 2

γxy =

∂u ∂v + ∂x ∂y

2κxy = 2

∂ 2w ∂x∂y

(2.23)

The flexural strains are ij = −κij z where z is measured from the middle surface. The membrane energy reduces to  ) UM =

1 2

∗

)

E h A



2

∂u ∂x

+

∂v ∂y

2

∂u ∂v + 2ν ∂x ∂y

*



∂u ∂v + + Gh ∂y ∂x

2 * dx dy (2.24)

and the flexural energy to  ) UF =

1 2

) D

A

∂ 2w ∂x 2



2

+

∂ 2w ∂y 2

2

*  2 2 * ∂ 2w ∂ 2w ∂ w + 2ν 2 + D¯ 2 dx dy ∂x∂y ∂x ∂y 2 (2.25)

2.2 Deflections of Thin Curved Plates and Shells

121

Fig. 2.24 The x-face of a stressed plate. (a) In-plane displacements and stresses. (b) Flexural displacements and stresses

Fig. 2.25 Flat plate FE generated contour results for some generic loadings. The left side is fixed, others are free. (a) Contours of membrane displacements under axial load. (b) Contours of flexural displacement and rotation under end transverse moment. (c) Contours of flexural displacements and rotation under end twist moment

Note that the flexural problem is uncoupled from the membrane problem but the membrane problem has coupled u and v displacements (Fig. 2.24). It is of value to relate the behavior of a flat plate to the archetypal behaviors introduced in Sect. 1.1. There is no need to do all six cases, so we focus on just case I axial, case IV bending, and case VI twisting. FE contour results for a plate of size [8a × 2a × h] with h = a/10 are shown in Fig. 2.25. In each case, the left edge is completely fixed, and a traction distribution is specified along the right edge. Let us begin with case I depicted in Fig. 1.1. A tx = constant traction is specified so that there is no out-of-plane deflections and (at least away from the fixed edge) the contours in Fig. 2.25a suggest we have the displacements u(x, ¯ y) ≈ u(x) ,

v(x, ¯ y) ≈ −yνu,x (x)

The membrane strain energy simplifies considerably to  UM =

1 2

  E ∗ h u,2x +ν 2 u,2x +2νu,2x (−νu,x ) dxdy =

 1 2

EAu,2x dx

Not unexpectedly, the strain energy is that of a uniaxially stressed member.

122

2 Shapes with Coupled Deformations

Case IV of Fig. 1.2 applies a moment at the end, in the present case this is achieved by applying a distributed moment my (y) = constant and the contours are shown in Fig. 2.25b. The equispaced and almost parallel φy contours indicate that the plate is bending like a beam. The φx contours are equispaced in y indicating a linear distribution of strain that is zero along the centerline. The action, called anticlastic bending, arises from the Poisson’s ratio effect: the top of the plate is in compression and therefore expands in y, the bottom is in tension and therefore contracts in y, the difference between the two gives a curvature. We have the expected behaviors that u(x, ¯ y) ≈ −zφy (x) = −yw,x (x) v(x, ¯ y) ≈ −zφx (x, y) = zνyφy (x) = zνyw,xx (x) which gives the strains xx = −zw,xx ,

yy = zνw,xx

There is also a shear strain related to w,xxx but this is small and therefore neglected. Substituting into the energy expression and integrating with respect to y and z gives  UF =

1 2

  1 3 h w,2xx +ν 2 w,2xx +2νw,xx (−νw,xx ) dx = E ∗ 12

 1 2

EI w,2xx dx

This is the strain energy for a simple beam in bending. Case VI of Fig. 1.2 applies a moment at the end, but it causes a twisting action. In the present case this is achieved by applying a distributed moment mx (y) = constant and the contours are shown in Fig. 2.25c. The contours of w(x, y) with zero along the centerline make the twisting obvious. The equispaced and almost parallel φx contours indicate that the displacement is linear in y. We have the expected behaviors that w(x, ¯ y) ≈ yφx (x) ,

u(x, ¯ y) = −zw,x ≈ −yzφx ,x ,

v(x, ¯ y) = −zw,y ≈ −zφx

which gives the strains xx = −yzφx ,xx ,

yy = 0 ,

γxy = −2zφx ,x

Substituting into the energy expression and integrating with respect to y and z gives  UF =

1 2

1 3 1 3 h 12 b [φx ,xx ]2 dx E ∗ 12

 +

1 2

1 3 h b[2φx ,x ]2 dx G 12

The leading term is called secondary warping and is typically negligible. The second term is very significant for our later applications, it represents the St. Venant torsion of a strip with cross section b >> h. It can be put in the form of Eq. (1.11), that is,

2.2 Deflections of Thin Curved Plates and Shells

123

Fig. 2.26 Shear stresses generated on the cross section of a thin plate. (a) Transverse loading. (b) In-plane loading. (c) Twist loading

 UF =

1 2

GJ = 13 h3 b

GJ φx ,2x dx ,

once we realize that the effective torsional stiffness GJ is different. Three shearing actions can be identified when the loading cases of Figs. 1.1 and 1.2 are particularized to the flat plate: case II bending due to a transverse load, case III bending due to in-plane load, and case VI twisting due to axial moment. These are shown in Fig. 2.26. The second case is a membrane action similar to that in cylinders and tubes under axial torque; we refer to the other two as flexural shears. The effect of these shears is developed further in Sect. 2.3 dealing with thin-walled open section structures. We now present additional results separately for the membrane and flexural actions.

Analysis I: Membrane Behavior Consider the PoVW in the form    δ UM − δ We = E ∗ h [xx + νyy ]δu,x +[yy + νxx ]δv,y dxdy  +

  Gh [γxs ]δ(u,y +v,x ) dxdy −

 [qu δu + qv δv] dxdy = 0

The terms in small brackets in conjunction with the moduli give stresses. After integration by parts, we get for the u and v variations, respectively, − σxx ,x h − τxy ,y h − qu h = 0 ,

−σyy ,y h − τxs ,x h − qv h = 0 (2.26)

These are the same stress equilibrium equations as first given in Eq. (1.7). These equilibrium equations can be satisfied directly (assuming for the moment that there are no body forces) by taking

124

2 Shapes with Coupled Deformations

σxx = ψ,yy ,

σyy = ψ,xx ,

τxy = −ψ,xy

where ψ is called a stress function and sometimes more specifically, the Airy stress function. Although the so-derived stresses are in equilibrium, the corresponding strains are not necessarily compatible. To see this, consider the normal strains given in Eq. (2.23): there are three strains (associated with three stresses) but there are only two displacements. Hence, when stresses and strains are computed via a stress function, additional constraints must be imposed on them, these are the compatibility conditions. For example, the collection of derivatives xx ,yy +yy ,xx −γxy ,xy = u,xyy +v,yxx −u,yxy −v,xxy = 0 shows that the three strains are not independent. The strains in terms of the stresses are given by, for example, xx = [σxx − νσyy ]/E = [ψ,yy −νψ,xx ]/E When substituted into the compatibility relation we get that ∇ 2∇ 2ψ = 0 ,

∇2 =

∂ ∂ + 2 2 ∂x ∂s

This is called the biharmonic equation. It is homogeneous when there are no body forces. When body forces are present we derive them from a potential according to qu = − V,x ,

qv = − V,y

For example, gravity loading in the negative y-direction has V = gy. Equilibrium is satisfied by taking σxx = ψ,yy + V ,

σyy = ψ,xx + V ,

τxy = −ψ,xy

We must add to the homogeneous solution for the stress function the particular solution obtained from ∇ 2 ∇ 2 ψp = −(1 − ν)∇ 2 V References [1, 18] have many examples of using stress functions for various plane problems. Example 2.9 A stretched member with free lateral sides has a lateral contraction due to the Poisson’s ratio effect. When this contraction is restrained, significant stresses can then be generated. The effect this has on the shape enters through the BCs. Construct a model to help explain the effects.

2.2 Deflections of Thin Curved Plates and Shells

125

♦

♦

Fig. 2.27 Plate fully constrained at one end. (a) Geometry and loading. L = 32 H . (b) Contours of stresses when qu = 0

♦

♦

0.0

0.5

.

1.0

1.5

-0.5

0.0

0.5 .

1.0

1.5

Fig. 2.28 Plate fully constrained at one end. (a) Stress distributions along centerline. (b) Stress distributions close to the fixed edge

Figure 2.27b shows some stress contours for the thin plate. The dominant stress is the σxx component (not shown), but there are also significant σyy and τxy components. Figure 2.28 shows the distributions against both x and y. The y distribution is for the column of nodes which is one element away from the fixed end. What is striking for the y-distribution is that (except immediately at the free edge) σyy is predominantly constant and τxy is predominantly linear. The FE results were generated using the MRT element [5] acting under plane stress conditions. To construct a model, begin by expanding the deformations in a Taylor series and retain one additional term in each expansion to obtain the approximate displacements u(x, ¯ y) ≈ u0 (x) ,

v(x, ¯ y) ≈ v1 (x)y/H

In this form, u0 (x) has the meaning of average axial displacement. We say that the two displacement fields u(x, y), v(x, y) are represented by two generalized DoFs u0 (x), v1 (x). The variational principle is used to derive the governing equations for these DoFs. This use of our PoVW is sometimes called the Ritz semi-direct method

126

2 Shapes with Coupled Deformations

because it leads to simultaneous differential equations rather than simultaneous algebraic equations; it is a hybrid strong formulation. The normal and shear strains corresponding to these deformations are ¯xx =

∂u0 ∂ u¯ = , ∂x ∂x

¯yy =

v1 ∂ v¯ = , ∂y H

γ¯xy =

  ∂ v¯ ∂v1 y ∂ u¯ + = ∂y ∂x ∂x H

Assume the thin plate is in plane stress, then the corresponding stresses are obtained from the displacements by σxx σyy τxy

   ∂ u¯ ∂ v¯ v1 ∗ ∂u0 +ν =E +ν =E ∂x ∂y ∂x H     ∂ u¯ ∂u0 ∗ ∂ v¯ ∗ v1 +ν =E +ν =E ∂y ∂x H ∂x     ∂ v¯ ∂v1 y ∂ u¯ + =G =G ∂y ∂x ∂x H ∗



where E ∗ = E/(1 − ν 2 ). Observe that σyy is predominantly constant on the cross section and τxy is linear as per our comments about Fig. 2.28. The assumption used to obtain the rod model of Eq. (1.18) essentially is that v1 = −νH ∂u0 /∂x. If that is done here, then σyy is zero contrary to Fig. 2.28. Thus our current model says that σyy is constant on the cross section and ignores what happens on the free lateral boundaries. Substitute the strains into the strain energy and integrate on the cross section, to get  U=

1 2

L

    E ∗ A u0 ,2x +v12 /H 2 + 2νu0 ,x v1 /H + GA 13 v1 ,2x dx

0

where the subscript comma means partial differentiation. The virtual work of the applied distributed load qu (x) and end loads is 

L

δ We = 0

L L   qu (x)δu0 dx + Q0 δu0  +Q1 δv1  0

0

where the generalized end forces Q0 and Q1 are associated with the generalized DoF u0 and v1 , respectively. The PoVW in the form δ U − δ We = 0 is then L#



  E ∗ A [u0 ,x +νv1 /H ] δu0 ,x +E ∗ A v1 /H 2 + νu0 ,x δv1

0

+GA



1 3 v1 δv1 ,x

$



L

dx − 0

L L   qu (x)δu0 dx + Q0 δu0  +Q1 δv1  = 0 0

0

2.2 Deflections of Thin Curved Plates and Shells

127

Use integration by parts so that, for example, L  ∂ 2 u ∂u0 ∂δu0 ∂u0 0  δu0 dx dx → δu0  − 0 ∂x ∂x ∂x ∂x 2   L ∂δu0 ∂v1  dx → v1 δu0  − δu0 dx v1 0 ∂x ∂x



with the other variations yielding similar sort of terms. The final expression for the PoVW is    L# ∂ 2 u0 ∂v1 1 − E ∗ A 2 − νE ∗ A − qu δu0 ∂x H ∂x 0 $   2 ∂ v1 v1 ∂u0 1 δv1 dx − 13 GA 2 − E ∗ A 2 − νE ∗ A ∂x h ∂x H    L  L ∂v1   ∗ ∂u0 ∗ v1 1 − Q0 − E A δu0  − Q1 − 3 GA δv1  = 0 − νE A 0 0 ∂x H ∂x We now use the argument that because δu0 and δv1 are independent variations both in the integration region and at the boundary limits, that the terms in square brackets must be zero. This leads to the governing equations E∗A

∂ 2 u0 ∂v1 1 = −qu + νE ∗ A 2 ∂x H ∂x

∂ 2 v1 1 3 GA ∂x 2

− E∗A

v1 ∂u0 1 =0 − νE ∗ A ∂x H H2

(2.27)

with the associated BCs (at each end of the plate) # u0 ;

Q0 = E ∗ A



v1 ∂u0 +ν ∂x H

$ ,

# v1 ;

Q1 = 13 GA

∂v1 ∂x

$ (2.28)

The first equilibrium equation can be obtained by substituting for u¯ and v¯ into the Navier’s equation, but this is not so for the second equilibrium equation. Similarly for the BCs. Furthermore, the BCs are not recognizable as our “usual” BCs for a rod and therefore worth elaborating on. There are two BCs at each end: either the DoF u0 (v1 ) or the corresponding generalized force Q0 (Q1 ) are specified. The leading term for Q0 resembles the axial force term in Eq. (1.18) except that it is for plane stress. Therefore, Q0 is strongly associated with the resultant axial force. However, the second term for Q0 is new and we have not seen anything comparable to it before. The expression for Q1 resembles part of the shear strain term in Eq. (1.1), and therefore, suggests it is strongly associated with a shear force. Note, however, that it is not the overall

128

2 Shapes with Coupled Deformations

resultant shear force on the cross section because that is zero; it is actually a shear moment. Equations (2.27) and (2.28) are our strong formulation of the problem posed in Fig. 2.27a; to effect a solution, the governing equations must be integrated subjected to satisfying the prescribed BCs. Consider the situation where qu (x) = 0. Integrate the first equation to get E∗A

∂u0 v1 + νE ∗ A = c1 ∂x H

(a)

and substitute into the second equation to get ∂ 2 v1 1 3 GA ∂x 2

− EA

v1 ν = c1 H H2

(b)

This has constant coefficients and therefore has exponentials as solutions. Solving gives v1 (x) = c2 e

−βx

+ c3 e

+βx

νH c1 , − EA

1 β=± H

'

3EA GA

Substitute this into Equation (a) and integrate to get u0 (x) =

 ν  −βx 1 c2 e c1 x + c4 − c3 e+βx + Hβ EA

(2.29)

The coefficients are determined from the BCs. There are four BCs: at x = 0 the two displacements are zero, while at x = L the axial force is equal to the applied resultant force and the shear force is zero. Utilizing Eq. (2.28) for the natural BCs, the four conditions become u0 = 0 =

ν [c2 − c3 ] + c4 Hβ

v1 = 0 = c2 + c3 −

νH c1 EA

Q0 = Po = σo A = c1   Q1 = 0 = 13 GA −βc2 e−βL + βc3 e+βL Solving simultaneously gives c1 = Po ,

c2 = Po

νH 1 , EA 1 + e

c3 = Po

νH e , EA 1 + e

c4 = −Po

ν2 1 − e βEA 1 + e

2.2 Deflections of Thin Curved Plates and Shells

129

where e = exp(−2βL). The stress reconstructions are shown as the continuous lines in Fig. 2.28, the model seems to catch the essential features of the FE generated distributions. The stresses (rather than uo and v1 ) are plotted because these are generated objectively by the FE analysis and because they give a better indication of how the natural BCs affect the distributions. It is interesting to consider the limit of large L, that is, the length of the member is much larger than its depth; the results in Fig. 2.27 are specifically for L = 32 H . Noting that c3  exp(−βL) → 0, we get the solution u0 (x) = Po

 ν 2  −βx Po e x −1 + βEA EA

v1 (x) = Po

 νH  −βx e −1 EA

Q0 = Po

which is close to the elementary rod solution for large x. However, the complex stress state near x = 0 still persists and is σxx (x, y) = σo

σyy (x, y) = νσo e−βx

σxy (x, y) = −νσo

y −βx  e 3EA/GA H

The additional stresses are on the order of Poisson’s ratio times the axial stress; this is a significant factor. Example 2.10 Consider the plate in Fig. 2.28 with a distribute shear traction ty = τo [H 2 − y 2 ] at the right end. Determine the stress and displacement distributions. At x = L, we have that   ∂ 2ψ ty = τo H 2 − y 2 = τxy = − ∂x∂y Integrate twice to get   ψ = −τo xyH 2 − xy 3 /3 + f (x) + g(y) where f (x) and g(y) are functions of integrations. The polynomials for f (x) and g(y) must exceed x 1 or y 1 to give nonzero stresses. Assume that the highest power in the polynomials does not exceed x 3 or y 3 ; in this way, the stress function automatically satisfies the homogeneous biharmonic equation. Take the representation as     ψ = Axy + Bxy 3 + a2 x 2 + a3 x 3 + b2 y 2 + b3 y 3 At y = ±H , tx = 0 = τxy = −φ,xy = A − 3BH 2 ,

ty = 0 = σyy = φ,xx = [2a2 + 6a3 x]

130

2 Shapes with Coupled Deformations

These must be true for any x, hence A = 3BH 2 , a2 = 0, a3 = 0. At x = L, tx = 0 = σxx = φ,yy = 6BLy + [2b2 + 6b3 y]     ty = τo H 2 − y 2 = τxy = −φ,xy = −A − 3By 2 = τo H 2 − y 2 These must be true for any y, so that b2 = 0, b3 = −BL and A = −τo H 2 , B = τo /3. Thus, the stress solution is σxx = 2τo [x − L]y ,

σyy = 0 ,

  τxy = τo H 2 − y 2

At this stage, we have a stress field that satisfies equilibrium and compatibility, and the tractions on three sides of the plate. In order to guarantee that this is indeed the solution, we must also satisfy the BCs along the side at x = 0. But what are the traction conditions? This is a mixed BC problem and we must invoke some information about the displacements. To obtain the displacements, we must integrate the strain/displacement relations. Thus, from the normal strains   Eu(x, y) = 2τo x 2 y/2 − xyL + f1 (y) , v(x, y) = f2 (x) where f1 (y) and f2 (x) are functions of integrations. The displacements must also satisfy the shear strain/displacement relation, hence substitute and regroup in terms of only x and y. The separate groups must be equal to a constant (λ, say), therefore integration gives the separate functions f1 (y) and f2 (x). We finally get for the displacements   Eu(x, y) = 2τo x 2 /2 − xL + h2 − y 2 /3 y − λy + c1 Ev(x, y) = τo [L − x/3]x 2 + λx + c2 where λ, c1 , c2 are unknowns. These contribute a rigid body motion. If we look at the displacements at x = 0, we have   Eu(0, y) = 2τo h2 − y 2 /3 y − λy + c1 ,

Ev(0, y) = c2

The horizontal displacement is nonzero; this is not what we want for the fixed end condition. Our solution is not the exact solution for the fixed cantilever beam problem; the simple stress function polynomial is not capable of representing the singular stress behavior at the fixed end where y = ±H . The solution, however, is the exact solution if the tractions at x = 0 were specified as tx = +2τo Ly ,

ty = −τo [H 2 − y 2 ]

2.2 Deflections of Thin Curved Plates and Shells

131

Note that if these tractions were specified otherwise, then global equilibrium is probably violated. The above solution gives a good approximation to the cantilever beam problem (i.e., when L is much greater than H ) because it satisfies the exact traction conditions top and bottom, and as can be verified, satisfies an approximate version of the tractions in the form of resultants on the ends. In fact, this is a very useful approach to obtaining practical solutions: satisfy some of the traction conditions exactly, and the others approximately in the form of resultants. If the region of interest is remote from these latter boundaries, then the solution will be quite insensitive to the specific distributions of the applied tractions. This is known as St. Venant’s Principle.

Analysis II: Flexural Behavior Sometimes it is convenient to use the flexural energy in the form  UF =

1 2

Ug =

1 2

2  D ∇ 2 w dx dy + Ug , )

 D2(1 − ν)

∂ 2w ∂x∂y

2

−

∇2 =

∂2 ∂2 + ∂x 2 ∂y 2 *

∂ 2w ∂ 2w dx dy ∂x 2 ∂y 2

(2.30)

where the additional term is the so-called Gaussian curvature term [5]. A reason for separating the energy terms as done here is that in many problems, the Gaussian curvature term is either zero or close to zero; this generally happens when the boundaries are restrained. The governing equation for flexure reduces to 

∇ 2∇ 2 =

∂2 ∂2 + 2 2 ∂x ∂y

2

∂4 ∂4 ∂4 +2 2 2 + 4 4 ∂x ∂x ∂y ∂y (2.31) The type of BCs (for an edge located at x = constant) are to be chosen from the first five of the summary D∇ 2 ∇ 2 w = qw ,

=

Displacement : w = w(x, y, t) Slope : φy = Bending Moment : Mxx Twisting Moment : Mxy

∂w ∂x

 ∂ 2w ∂ 2w = +D +ν 2 ∂x 2 ∂y  2  ∂ w = +D(1 − ν) ∂x∂y 

132

2 Shapes with Coupled Deformations



Transverse Shear : Vxz

∂ 3w ∂ 3w = −D + (2 − ν) ∂x 3 ∂x∂y 2

Loading : qw = +D∇ 2 ∇ 2 w



(2.32)

The corresponding expressions for the y-face are obtained by permuting x and y. Note that Poisson’s ratio ν enters the moment and shear relations and acts to couple the gradients in x to those in y. Also, once the resultants have been solved for, then the stresses are obtained from, for example, σxx = −Mxx z/Ip with Ip ≡ h3 /12. Example 2.11 Figure 2.29 shows some FE generated results for a simply-supported rectangular plate with a uniformly applied transverse load qo . Construct a model to explain the deformed shape. The first step in constructing the model is to obtain appropriate Ritz functions. For 1D systems, the simplest approach is to assume a polynomial and then impose the geometric BCs. This becomes somewhat unwieldy in the 2D case. Instead, it is sometimes simpler to use a product function. For rectangular problems, for example, we assume w(x, y) = agx (x)gy (y) where gx (x) satisfies the BCs on the x =constant edges and gy (y) satisfies the BCs on the y =constant edges. This can then be generalized for an arbitrary number of terms by   w(x, y, t) = gx (x)gy (y) a00 + a10 x + a20 x 2 + a01 y + a02 y 2 + a11 xy + · · · Consider, for example, a rectangular plate [a × b] simply supported on all edges. Represent the x-behavior as

Fig. 2.29 Uniformly loaded simply-supported rectangular plate. Circles and contours are FE generated data, continuous lines are simple model results. (a) Contours of out-plane-displacement. (b) Displacement distribution along x-centerline. (c) Displacement distribution along y-centerline

2.2 Deflections of Thin Curved Plates and Shells

133

w = ao + a1 x + a2 x 2 The geometric edge conditions are that w = 0 at x = 0 and x = a, this leads to w = a2 [−ax + x 2 ] While the natural BC of zero moment need not be considered, it is worth noting that the assumed solution does not satisfy this condition. A similar function is obtained for the y-behavior. Put both of them together to get w(x, y, t) =

  16w1  2 2 xa − x yb − y a 2 b2

where w1 is the deflection at the center of the plate. Noting that ∇ 2w =

     ∂ 2w ∂ 2w 16w1  2 2 (−2) yb − y + xa − x (−2) + = ∂x 2 ∂y 2 a 2 b2

we compute the strain energy term as U = 12 Dw12

 2ab  4 5 2 2 a + 3 a b + b4 256 4 4 15a b

The contribution from the Gaussian curvature term is zero. The virtual work of the applied pressure is  δ We =

qo qw δw(x, y) dx dy = δw1 16 2 2 a b

= δw1 16

 

x x2 − 2 a a



y y2 − 2 b b

 dx dy

qo ab 36

The PoVW then gives w1 =

a 2 b2 5qo a 2 b2 384D a 4 + 53 a 2 b2 + b4

For the plate with b = a/2 we get  w

1 1 2 a, 2 b



= 0.529wo ,

wo =

16qo a 4 Dπ 6

This is a −15% difference in comparison to the FE results. A strong-form solution is given in Ref. [5] and a spectral shape solution is developed in Sect. 3.4 that turns out to be the same as the exact solution reported in Ref. [19].

134

2 Shapes with Coupled Deformations

2.2.3 Shallow Curved Plates It is worth noting that the results for the cylinder are also applicable to some curved plates—specifically those, as seen from Fig. 2.10, that can be periodically extended to form a cylinder. A curved plate which is an extrusion of an arch with pinned BCs is not such an example. We elaborate on the difference presently. For a general curved plate or cylinder, the complex branches of the spectrum relations must also be considered. These are quite complicated (see Ref. [4] for some examples) and therefore not pursued here; besides, our recommended spectral shapes approach automatically implements arbitrary BCs. Figure 2.14b shows the curved plate coordinates and notation. In this notation, the strains and curvatures are xx = u,x ,

ss = v,s +w/R ,

κxx = w,xx ,

γxs = u,s +v,x

κss = −v,s /R + w,ss ,

(2.33)

κxs = −v,x /R + 2w,xs

The membrane and flexural energies are, respectively,    UM = 12 E ∗ h u,2x +(v,s +w/R)2 + 2νu,x (v,s +w/R) dxds  + 12  UF =

1 2

Gh [u,s +v,x ]2 dxds   D w,2xx +(−v,s /R + w,ss )2 + 2νw,xx (−v,s /R + w,ss ) dxds 

+ 12

D¯ [(v,x −2Rw,xs )/R]2 dxds

The curved plate problem, just as for the cylinder, has a total of three unknowns. Constructing a model thus requires the selection of three displacement functions with two independent variables. This is too many for our purpose, therefore we explore some simplifications and their implications. For some of the arch problems, we made the assumption that the flexural behavior is dominated by the transverse deflection. A similar assumption for curved plates says that the v related terms can be ignored in UF . That is, we approximate the flexural energy as     2 2 1 1 UF ≈ 2 D w,xx +w,ss +2νw,xx w,ss dxds + 2 D¯ [2w,xs ]2 dxds The virtual strain energy terms are  δ UM = E ∗ h [[xx + νss ]δu,x +[ss + νxx ]δ(v,s +w/R)] dxds  +

Gh [γxs δ(u,s +v,x )] dxds

2.2 Deflections of Thin Curved Plates and Shells

135

 δ UF =

D [(w,xx +νw,ss )δw,xx +(w,ss +νw,xx )δw,ss ] dxds  +

D¯ [(2w,xs )2δw,xs ] dxds

The membrane terms in small brackets in conjunction with the moduli give stresses. After substitution into our PoVW and integration by parts, we get for the u and v variations, respectively, − σxx ,x h − σxs ,s h − qu h = 0 ,

−σss ,s h − σxs ,x h − qv h = 0 (2.34)

These are the same stress equilibrium equations as first given in Eq. (1.7) and are identical to those of Eq. (2.26). In this form, they do not explicitly contain the transverse displacement and superficially the membrane problem is uncoupled from the transverse displacement. However, we get for the w variation (after using D¯ = D(1 − ν)/2) D[w,xxxx +2w,xxss +w,ssss ] − qw h + σss h/R = 0

(2.35)

(The σss contribution came from δ UM and is equivalent to the β1 Rm term in Eq. (2.9).) This is the flat plate model of Eq. (2.32) but has explicit coupling with the membrane stress and therefore the overall problem remains coupled. This equation is to be compared to Eq. (2.10) for the arch and we see that the effect of the curvature is like a distributed load. We can use a stress function to reformulate the membrane problem. In this case, compatibility of the strains becomes xx ,ss +ss ,xx −γxs ,xs = w,xx /R which shows that the in-plane strains are connected to the out-of-plane displacement and initial curvature. When the strains are replaced with stresses, and using (assuming no body forces) σxx = ψ,yy ,

σss = ψ,ss ,

τxs = −ψ,xs

we get that ∇ 2 ∇ 2 ψ = Ew,xx /R ,

∇2 =

∂ ∂ + 2 2 ∂x ∂s

Again, it leads to a biharmonic equation, but it is inhomogeneous. Body forces can be accounted for as done for the flat plate. To summarize, we can write the governing equations in the form D∇ 2 ∇ 2 w + ψ,xx h/R = qw h ,

∇ 2 ∇ 2 ψ − Ew,xx /R = 0

(2.36)

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2 Shapes with Coupled Deformations

which makes the coupled nature of the problem obvious. This formulation of the curved plate and cylinder problem is typically referred to as Donnell’s model [3]. While any stress function gives an equilibrated stress state, only those satisfying the biharmonic equation also gives a compatible strain field which is a significant restriction on the allowable distribution forms. It should be pointed out that a good attribute of our usual Ritz approach is that the stresses and strains are automatically compatible because we begin with displacement functions. We postpone application of the Donnell model until Sect. 3.4 where a spectral analysis is used to show its good performance in comparison to that of the more exact equations. Consequently, we make use of it in the buckling and postbuckling shell problems in Sects. 4.3 and 4.3, respectively.

2.3 Long Structures with Open Cross Sections We now consider aspects of a more general treatment of thin-walled open section structures within the context of their connection to shells. To fix ideas, consider the segment of a thin-walled shell predominantly oriented along the z axis as shown in Fig. 2.30. A generally applied load causes both rotation and translation of the cross section. When the load is applied transversely through the shear center only a translation occurs. That is, the arbitrarily positioned load produces both bending and torsional twisting with respect to the shear center. An axial load applied at the centroid produces axial membrane stresses, but applied off the centroid also produces bending actions. A torque applied about the longitudinal axis produces only a twist about the shear center which is why the shear center is also sometimes called the center of twist. These structures therefore have coupled deformation primarily arising from the loading. Thus, in one sense, these are not really coupled deformation problems in that a choice of axes and smart specification of loading can give uncoupled responses that

Fig. 2.30 Open section shell structures. (a) Coordinate description. (b) Translation and rotation of the cross section. (c) Cross-sectional geometry referred to the centroid

2.3 Long Structures with Open Cross Sections

137

can subsequently be superposed. We treat them as coupled for two reasons: first, the spectral shapes are inherently coupled, and second, the buckling problems of open sections treated in Sect. 4.4 are also coupled. The modelings we consider basically correspond to the upper right of Fig. 1.9, the lower right has already been considered as part of the discussions of tubes. It is worth reiterating that the shear stress distribution for open sections is very different from that of closed sections and accounts for the large difference in behaviors. It is the differences discussed in connection with Fig. 2.26. We therefore begin with a discussion of shear stress in thin-walled sections and then develop our general model for open sections.

2.3.1 Shear Stress in Open Cross Sections Shear stresses arise from nonuniform loading, be it membrane, bending, or twisting; Fig. 2.26 illustrates the three types. The beam models presented thus far give estimates of the resultant shear force (related to the third space derivative of the deflection) even though the key assumption of the models that “plane sections remain plane” implies that the shear strain (and hence shear stress) is zero. We now try to resolve this contradiction and give an estimate of the shear stress distribution. In the first analysis, we invoke differential equilibrium to estimate the shear stress distribution but essentially retain the zero shear strain assumption. In the second analysis, we introduce a shear deformation mechanism into the model so that the shear stress has a definite relationship to the shear strain. The third analysis considers twisting.

Analysis I: Shear Stress from Equilibrium To make the discussion simple, we restrict ourselves to bending only about the zaxis for a beam with a cross section typified by the I-beam in Fig. 2.31 and oriented along x. The governing equations derived for the beam model show that Vy = −

∂Mz , ∂x

∂Vy + qv = 0 ∂x

From these it is clear that a variable moment or a distributed load causes resultant shear forces. We must make a distinction between the origin of the shears be they forces or strains. The present models say that the shearing action arises due to the need to satisfy equilibrium and not due to the deformation. This is why there is no strain energy associated with the shear forces. Apparent inconsistencies like this arise quite frequently in developing structural models because, fundamentally, these models are only approximations of the true 3D behavior of the component.

138

2 Shapes with Coupled Deformations

Fig. 2.31 Shear stresses in beams. (a) Geometry of I-beam. (b) Vertical shear stress distribution in web and horizontal shear stress distribution in right flanges. (c) Shear stress distributions in C-channel

Nonetheless, this does not prevent us from extracting very useful information from the simplified approximate model as we now show. The shear resultant Vy is some integrated average of the corresponding shear stress; we estimate the shear distribution consistent with these resultants by specifically invoking the differential equations of equilibrium. Consider the 3D equilibrium Equation (1.2) written for the x-direction ∂τxy ∂σxx ∂τxz + + =0 ∂x ∂y ∂z The axial bending stress has the distribution σxx = (Mz /Izz )y. Substitute this into the equilibrium equation and replace the x-derivative of the moment with the shear to get −

Vy ∂τxy ∂τxz + =0 y+ Izz ∂y ∂z

It is clear from this how a transverse force can generate both shear stresses on the face. That is, the single vertical shear force in Fig. 2.31a produces a τxy shear stress in the web and a τxz shear stress in the flange. Integrate the equilibrium equation with respect to z to get −

Vy ∂ yb + Izz ∂y



z+

z−

z+  τxy dz + τxz  − = 0 z

The width b = z+ − z− , in general, is a function of y. We impose the condition that τxz is zero on the two lateral surfaces (i.e., z− and z+ ). Introduce the quantity  qy = τ¯xy b =

z+

z−

τxy dz

2.3 Long Structures with Open Cross Sections

139

where τ¯xy has the interpretation of an average shear stress across the section and the quantity qy is called a shear flow. For convenience, we drop the “bar” notation but retain the notion that the shear stress is an average. Integrating the equilibrium equation with respect to y gives qy =

Vy Izz



y

yb(y) dy =

o

Vy Qz (y) Izz

where Qz is the first moment of area. Reversing the sequence of integrations gives a similar expression for τxz in the flange but it is the Qy (z) moment of area that is used. For a cross section that is a simple [b × h] rectangle with the origin at the centroid, we have that  y    Vy 1  2 Qz (y) = yb(y) dy = 12 b y 2 − h2 /4 , qy = τxy b = b h /4 − y 2 2 Izz −h/2 The stress distribution is parabolic as indicated in the web sections of Fig. 2.31b. It is interesting to note that the maximum shear stress is 1.5 times the average stress given by τ = V /A. The moments of area for the I-beam shown in Fig. 2.31 are I = 2Af



1 2H

+ 12 hf

2

+

3 1 12 hw H ,

  Qf = hf 12 14 (H + 2hf )2 − y 2     Qw = 12 H + 12 hf Af + hw 21 14 H 2 − y 2

Af = Wf hf b = Wf b = hw

for the τxy stress. The moments of area change continuously on the section but there is an abrupt change of stress at the junction of the web and flange because of the change of width b. There is also a distribution of τxz stress; the appropriate Q is Qf = 14 (H + hf )



1 2 Wf

 −z ,

b = hf

Note that this is linear in the distance from the vertical centerline. The distributions of τxy and τxz are shown in Fig. 2.31b. Example 2.12 Figure 2.32 shows FE generated bending and twisting results for a thin-walled open section C-channel fixed at one end and loaded vertically along the side wall. Of especial curiosity is the twisting action; although the transverse load acting vertically forms a clockwise moment about the centroid, the rotation is counterclockwise. Explain this result. Figure 2.32a shows partial Hex20 meshes; what is not shown is the end plate and joint fillers. The mesh has two elements through the thickness and full integration is

140

2 Shapes with Coupled Deformations

♦

♦

♦

-0.25

♦

-0.50

0.00

.

Fig. 2.32 Deformation of a C-channel not loaded through the shear center. (a) Deformed shape (exaggerated) under transverse along the vertical wall. (b) Rotation of end cross section for different vertical load locations. (c) Distributions of membrane shear stress for different vertical load locations

used. The end plate has a stiffness ×10 000 that of the channel material. To explain the counterclockwise rotation in Fig. 2.32a, we need to consider where the load acts relative to the shear center. Figure 2.32b shows the net end rotation as the vertical load position is changed. (The load was applied via a rigid frame construction and x = 0 is located at the web.) It can be seen that there is a position x/W ≈ − 38 at which the rotation is zero; this is to the left of the vertical web. This result indicates that if the vertical load is applied through this point then there is no rotation of the cross section. As discussed in connection with Fig. 2.31, the vertical force generates shear flows in the web and flanges. Figure 2.32c shows the shear stress distributions for three load positions relative to the shear center: empty squares to the left (x/W = − 12 ), empty circles to the right (x/W = − 14 ), and full circles at the shear center (in plotting the distributions, the C-channel centerline is the zero stress reference). When the load acts through the shear center, the shear stress distribution in the flange is almost linear. In fact, all stress distributions resemble those portrayed for the I-beam and C-channel in Fig. 2.31. We can estimate the maximum shear stress in the flange (which is the same as the minimum in the web) as τzx =

Vo Q , Ib

Q = W hH /2 ,

I=

1 12



6hW H 2 + hH 3



and b = h. Thinking of Fig. 2.32c as a free body where the arrows represent resultant shear forces, we see that they form a resultant torque; for zero twist, this must be counterbalanced by the torque due to Vo ; thus by taking moments about the web position we can estimate the shear center from eVo = 2





1 1 2 τzx W h 2 H

or

e=

3W 2 h = 37 W 6hW + hH

2.3 Long Structures with Open Cross Sections

.

141

♦

.

0.0

0.5 .

1.0

Fig. 2.33 Shearing of a block, the left side is fixed, others are free. Circles are FE data, lines are model results. (a) Displacement contours. (b) Transverse displacement of the centerline

While close, this is not exactly the same as that obtained from Fig. 2.32b. The main reasons for the discrepancy are the parabolic distribution in the web and the effect of restrained rotation due to the end-plate (this is explained presently in Analysis III); a longer member with thinner walls would probably show better agreement with the simplified modeling. This example shows that care must be exercised when the elementary models of bending and twisting are applied to thin-walled open sections. In fact, generally speaking, these members are best modeled as folded plate shells as described in Ref. [5] or using solid elements as done here.

Analysis II: Shear Stress from Shear Deformation The shear deformation in a beam is significant when the beam is deep relative to its length. This is a very important consideration, therefore, to amplify on it, let us take a closer look at the deformation of the block shown in Fig. 1.13a. Figure 2.33 shows some FE generated results for the block in plane stress and loaded uniformly along the right side which is unconstrained. The block can be modeled as in simple shear or in bending with deflections given by, respectively, vs (x) =

Po x, GA

vb (x) =

 Po  1 Lx − 16 x 3 , 2 EI

vc (x) = vs + vb (2.37)

These are shown plotted in Fig. 2.33b; it is clear that neither displacement dominates and that the block is experiencing both actions. Our objective here is to develop a beam model that relates the shear stresses to the shear strains and therefore can predict the total deformation. Specifically, consider the rectangular beam of length L, depth H , and thickness h, shown in Fig. 2.34; we are interested in developing a beam model when H is large relative to h and sizable in comparison to L. We use the results of Fig. 2.33 to motivate our developments. Another preliminary point, when h is small relative to H (and L), the beam can be regarded as a plate in a state of plane stress, and the w(x, y, z) displacement can

142

2 Shapes with Coupled Deformations

♦

♦

Fig. 2.34 Timoshenko beam with applied distributed and end loads

♦

♦

Fig. 2.35 FE generated distributions on the mid-length cross section for a beam of size [H × 2.5H ]. (a) Displacements; empty circles are for unconstrained right side, full circles are for constrained right side. (b) Stresses

be eliminated in the formulation by using the plane stress Hooke’s law after setting σzz = σxz = σyz = 0 in Eq. (1.4). That is, the beam is of thin-walled type under membrane actions; the resulting beam model is called the Timoshenko beam model. References [4, 6] give details of the derivation of the model, here we summarize the main points. The membrane displacements are approximated as u(x, ¯ y) ≈ −yφ(x) ,

v(x, ¯ y) ≈ v(x) + y 2 21 ν

∂φz ∂x

so that the stresses are essentially σxx = −yE

∂φz , ∂x

  ∂v τxy = G −φz + , ∂x

others = 0

These displacements and stresses are in substantial agreement with the distributions shown in Fig. 2.35 except that the shear stress should be parabolic. The strain energy is approximated as     2 2 1 1 U = 2 [σxx xx + τxy γxy ] dV = 2 Exx dV + Gγxy V

V

2.3 Long Structures with Open Cross Sections

143

Substitute for the strains to get the strain energy as  2  *   L  H /2 ) ∂v 2 2 ∂φz 1 + G φz − U = 2 Ey b dydx ∂x ∂x o −H /2 2 2 *    L) ∂φ ∂v z = 12 + GAK2 φz − EIzz dx ∂x ∂x o We have added an adjustable parameter K2 to the shear strain energy. If the applied surface tractions and end loads on the beam are as shown in Fig. 2.34, then the virtual work of these loads is 

L

δ We = o

L L   qv (x)δv dx + Mz δφz  + Vy δv  0

0

The resulting governing equations are    ∂v ∂ GAK2 − φz = −qv ∂x ∂x     ∂v ∂ ∂φz EIzz + GAK2 − φz = 0 ∂x ∂x ∂x

(2.38)

These are the Timoshenko equations for a beam. The associated BCs (at each end of the beam) are specified in terms of any pair of conditions selected from the following groups:  v = specified

or

GAK2

φz = specified

or

EIzz

 ∂v − φz = Vy = specified ∂x

∂φz = Mz = specified ∂x

(2.39)

The BCs are specified similar to that for the elementary beam model—the difference is the relations between the resultants and the deformations. Thus a zero shear force at the boundary constrains the deformation so that φz = ∂v/∂x. Let us dwell on the BCs for a while. Figure 2.36 shows some distributions from the 2D continuum modeling of a plate similar to Fig. 2.33 but of size [L × 12 L] fixed at the left with a uniform distributed load on the top. Two BCs are considered: case I has the right side completely free, and case II has the right side on rollers—it can move vertically but not rotate. The processed data is just for the 2D modeling where the actual BCs imposed are case I: tx = σxx = E ∗ [u,x +νv,y ] = 0 , case II: u = 0 ,

ty = τxy = G[u,y +v,x ] = 0 ty = τxy = G[u,y +v,x ] = 0

2 Shapes with Coupled Deformations ♦

♦

144

♦

♦

0.0

0.2

0.4

.

0.6

0.8

0.0

1.0

0.2

0.4

.

0.6

0.8

1.0

Fig. 2.36 FE processed data for the centerline distributions approaching the right boundary. (a) [v,x −φ] combine to give a zero shear. Continuous line is φ, dashed line is v,x . (b) φ,x is related to the applied moment

where ti are components of the traction vector. These are imposed for each node along the right vertical side. The Timoshenko beam model deals in terms of resultants or averages on the cross section. The corresponding imposed BCs are case I: M = EI φz ,x = 0 , case II: φ = 0 ,

Vy = GA[v,x −φz ] = 0 Vy = GA[v,x −φz ] = 0

In both models, for both cases, the shear BC is imposed in the form of a natural BC. In the case of the Timoshenko model this involves the rotation which also appears as a geometric BC for case I. We conclude definitely that the geometric and natural BCs both involve constraints on the allowable deformation and there is no simple separation of the type of constraints. This is emphasized in Fig. 2.36a where the natural BC imposes that [v,x −φz ] = 0, but leaves the individual values unspecified, whereas case I imposes the same condition, but in addition, forces φz = 0 and therefore changes the distributions of both v,x and φ individually. When we substitute the Timoshenko model assumption that u = −yφz into the first BC for case I, we get −yφz ,x +νv,y = 0. Thus, in general, a free edge does not result in φz ,x = 0 as shown in Fig. 2.36b. This goes back to the imposition that yy = −νxx in order to satisfy zero tractions on the transverse sides. The nature of the applied loads affects this assumption. Integrating the Timoshenko equations is straightforward and for future reference, we state results for a beam with a distributed load qv (x) = qo being constant and a beam with imposed end displacement vo and rotation φo at x = L. fixed–fixed:

qo 24EIzz qo φz (x) = 12EIzz v(x) =

   x 4 − 2x 3 L + x 2 L2 + γ L2 −x 2 + xL   2x 3 − 3x 2 L + xL2

2.3 Long Structures with Open Cross Sections

pinned–pinned:

fixed except vo :

qo 24EIzz qo φz (x) = 12EIzz v(x) =

v(x) =

145

   x 4 − 2x 3 L + xL3 + γ L2 −x 2 + xL   2x 3 − 3x 2 L + xL2

vo L3 [1 + γ ]

  3Lx 2 − 2x 3 + γ L2 x ,

γ ≡

  vo 2 6Lx − 6x L3 [1 + γ ]   φo x 3 − 12 x 2 L(2 − γ ) − 12 γ L2 x fixed except φo : v(x) = 2 L [1 + γ ]   φo 3x 2 − xL(2 − γ ) φz (x) = 2 L [1 + γ ]

12EIzz GAL2

φz (x) =

(2.40)

In both distributed load cases, the deflection is the simple sum of the Bernoulli– Euler deflection plus a shear related deflection and the rotation is unaffected. It is interesting that the shear deflection is the same in each case which means that, proportionally, the shear has more consequence in the fixed–fixed case. We see that the refined models help us extend our understanding of shapes to go beyond just those of the elementary models. As shown here, we can capture the broad shape of a 2D plate structure using two functions v(x) and φz (x). This is a further elaboration of the flat plate membrane analyses done in Sect. 2.2. Two final points are worth noting. The first is that once the moment and shear force resultants are obtained, the distributions of stress are obtained as previously discussed in Analysis I. The second is that the adjustable parameter K2 is typically of order unity and Ref. [4] suggests K2 = π 2 /12 based on a dynamic analysis.

Analysis III: Torsion of Open Sections Elementary torsion theory is strictly for members with circular cross sections; because of our interest in structural members such as I-beams, C-channels, and the like, we need to develop a torsion modeling ability for arbitrary cross sections such as shown in Fig. 1.8. Our first objective is to establish the effective torsional stiffness per unit length, GJ , and estimates of the stress distributions for different cross section types. This is then followed by estimating the deformed shapes under twisting loads. We already looked at the torsion of the plate in Fig. 2.37a in Sect. 2.2 and found, in connection with Fig. 2.25c, that the stresses and torque-rotation relation are given by, respectively, τxy = −2Gφˆ x z ,

τxz = 0 ,

Tx = GJ φˆ x ,

J ≡ 13 bh3

(2.41)

146

2 Shapes with Coupled Deformations

Fig. 2.37 Bars with thin-walled rectangular cross sections. (a) Geometry of cantilevered plate-like strip. (b) Sample open thin-walled cross sections

The torsional stiffness is exact in the limit of h/b → 0 and only off by +6% when h/b = 0.1. The stress distribution is shown schematically in Fig. 2.26c. Note that it is zero along the centerline and this contrasts with the membrane shear as discussed in Analysis II. The strain displacement relations are γxy =

∂u ˆ , − φˆ x z = −2φz ∂y

γxz =

∂u + φˆ x y = 0 ∂z

Integrate either one of these to get the warping displacement u(y, z) = −yzφˆ where the constant of integration is set to zero because the axial displacement is zero at the origin. The extremities have the maximum displacement with neighboring corners moving in opposite directions. An estimate of the torsional stiffness properties of rolled sections such as angles, channels, I-beams, and the like as shown in Fig. 2.37b can be obtained simply by summing the stiffness of each segment even if curved. Thus, GJ =

 i

GJ =



1 Gi bi h3i i3

(2.42)

This essentially infers that there are no reinforcing effects between the different attached sections. Obviously this is not true if the thickness of each section is relatively large compared to the length of the segment. It is also not true if the collection of segments forms a closed cross section as discussed for tubes in the previous section. There is also a membrane mechanism for supporting torques (this is the case illustrated in Fig. 2.20 for torsion of a tube) which we now elaborate on but for open sections. The origin of this is typically the need to support applied transverse loads. Consider the FE generated results for the twisting (about the length axis) deformation of the I-beam shown in Fig. 2.38. The two flanges experience a shearing

2.3 Long Structures with Open Cross Sections

147

Fig. 2.38 Twisting deformation of an I-beam. (a) Exploded view of deformed shape. (b) End view of rotation along centerline

action that causes a rotation of the cross section. We are interested in the effect that the fixity of the left end has on the amount of rotation generated in the rest of the structure. Three fixity cases are considered corresponding to the extent that the left cross section is restrained. The FE modeling used the MRT/DKT shell element [5]. The first case fixes three nodes closest to the x-centerline (this is the case shown in Fig. 2.38). It is clear that the centerline does not displace but there is a rotation about the centerline. Figure 2.39a shows this rotation (as case I) along the length. After an initial large rate of rotation, it seems that the rate of rotation is then approximately constant. This rate of rotation is estimated as GJ

∂φx = Tx = Vo H , ∂x

J = G[2W + H ] 13 h3

This is the St. Venant torsion contribution. The moment is constant along the length, which means the rotation is linear and this is shown as the full line in Fig. 2.39a. The rotation estimate is low. The second case fixes the nodes at the centerline plus the nodes at the intersections of the flanges and web. In other words, the web is globally restrained from rotation but the web and flange edges are mostly shear free. The rotation distribution is shown in Fig. 2.39a as case II. This restraint reduces the initial large rate of rotation but after that it seems that the rate of rotation is approximately the same as for the first case. The final case, which is the one of particular interest here, fixes all the nodes on the left end. Figure 2.39b shows the rotations induced. Two aspects are significant. First, the level of rotation is an order of magnitude smaller than the other two cases. Reduced rotations are expected, obviously, because there are more restraints but the degree of reduction is notable. This leads to the second (and subtly more important) point: the initial rate of rotation is zero. That is, the rotation only becomes significant

148

2 Shapes with Coupled Deformations

0.04

0.3

0.03

0.2

0.02

0.1

0.01

.1

.1

0.4

0.0

0.00

♦

0.00

0.25

0.50 .

0.75

1.00

♦

0.00

0.25

0.50 .

0.75

1.00

Fig. 2.39 Rotations along an I-beam. Circles are FE generated data, continuous lines are model results. (a) Essentially unrestrained. (b) Completely restrained

over distance away from the fixed end. This contrasts significantly with our first case. The obvious question now to ask is: what are the significant mechanics behind this change of stiffness for the rotational behavior? To answer this question, think of the top flange as a cantilevered beam with an end load Vo as shown in Fig. 2.38a. Then, from the elementary beam model of Eq. (1.19), we have EIf

∂ 3v∗ = −Vo ∂x 3

1 where If = 12 hW 3 is the flange second moment of area. But the transverse deflection is related to the global rotation by v ∗ = φx 12 H , and because the two shears form a torque, we have that

Tf = Vo H = − 12 H 2 EIf

∂ 3 φx ∂ 3 φx = −EC , w ∂x 3 ∂x 3

Cw = 12 H 2 If =

2 3 1 24 hf H W

The coefficient Cw is called the warping constant. This torque must be added to the St. Venant torsion contribution so that the total torque supported is given by Tx = GJ

∂ 3 φx ∂φx − ECw ∂x ∂x 3

or

ECw

∂ 4 φx ∂ 2 φx − GJ = qm (2.43) ∂x 4 ∂x 2

where qm is a distributed applied torque. The rotation is governed by a higher order differential than for the simpler St Venant torsion. It is important to realize that the nature of the associated shears on the cross section is different for the two torque contributions; for the restrained warping case it is membrane as in Fig. 2.26b; whereas the free-to-warp shear is flexural/twisting as shown in Fig. 2.26c. They both, consequently, have different strain energies given by, respectively,

2.3 Long Structures with Open Cross Sections



 U = c

1 2

ECw

∂ 2 φx ∂x 2

149

2

 U = f

dx ,

1 2



∂φx GJ ∂x

2 dx

This is an illustration of some of the complexity of thin-walled behaviors. All the relationships for the structural quantities may now be summarized (under the condition of uniform properties) as Twist :

φx

Warp :

u

Axial stress : Torque : Loading :

= φx (x) = 2A¯ s

∂φx ∂x 2

∂ φx σxx = 2A¯ s ∂x 2 Tx = GJ

∂ 3 φx ∂φx − ECw ∂x ∂x 3

qφ = ECw

∂ 4 φx ∂ 2 φx − GJ ∂x 4 ∂x 2

(2.44)

The full explanation for these terms is given subsequently, here we are interested in the torque-rotation relation. However, it is worth commenting that each of these relations in their own way is imposing constraints on the allowable twisted shape. As shown in the example problem to follow, the general homogeneous solution for the rotation is φz (x) = c1 e−βx + c2 e+βx + c3 x + c4 ,

β=



GJ /ECw

(2.45)

Based on the FE results for the beam in Fig. 2.38, we impose the BCs x=0:

φx = 0 ,

∂φx = 0; ∂x

x=L:

∂ 2 φx =0 ∂x 2

plus the condition on the end torque given in Eq. (2.43). Solving for the coefficients gives Vo H 2  GJβ 1 − e−βL β + ECw β 3 2e−βL   c2 = −c1 e−2βL , c3 = c1 β 1 + e−2βL , c1 =

  c4 = −c1 1 − e−2βL

A plot of the rotation is shown as the full line in Fig. 2.39b. The comparison with the FE results is quite good. The role of β is to have the zero slope condition penetrate deeper along the beam. In summary, nonuniform torsion occurs if any cross section is not free to warp or if the torque varies along the length of the bar. As a consequence, the warping

150

2 Shapes with Coupled Deformations

varies along the length generating axial stresses. In addition, the rate of change of the angle of twist is no longer constant. We therefore conclude that for solid (but not thin-walled) sections the nonuniform torsion is not significantly different from that of pure torsion although there are local effects at constrained ends. There is a significant exception to this conclusion and that is when the cross section is of the thin-walled open type as discussed here. This is because the axial stress causes significant shearing effects. Example 2.13 An I-beam is completely fixed at one end and has a rigid plate at the other end with an applied axial torque. Figure 2.40a shows the FE generated results for the rotation along the length of the beam. Elementary torsion modeling gives that φx =

Tx x GJ

and is also shown in the figure. Explain why there is a very large discrepancy between the elementary modeling and the FE results. The FE results were generated with a mesh similar to that of Fig. 2.38 that used the MRT/DKT shell element [5]. The web and flange thickness of the beam are the same so that the torsional stiffness and constant are, respectively, J = 13 h3 [2W + H ] ,

Cw =

2 3 1 24 hH W

The shear center is at the centroid. The loading equation, with a uniform distributed torque, becomes ECw

∂ 4 φx ∂ 2 φx − GJ = mo 4 ∂x ∂x 2

or

∂ 2 φx ∂ 4 φx − β2 = mo /ECw = m∗o 4 ∂x ∂x 2

♦

♦

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

.

Fig. 2.40 Torsion of an I-beam. (a) Rotation of the cross section. (b) Variation of the first and second derivatives of the twist

2.3 Long Structures with Open Cross Sections

151

with β 2 = GJ /ECw . Integrate this twice to get ∂ 2 φx − β 2 φx = c1∗ x + c2∗ + 12 m∗o x 2 ∂x 2 The homogeneous solution comprises the hyperbolic functions so that the total solution is φx (x) = c1 cosh βx + c2 sinh βx + c3 x + c4 + 12 m∗o x 2 We impose the BCs to determine the coefficients. At x = 0, there is no rotation and no warping, hence, φx = 0 ⇒ 0 = c1 + c4 ∂φx = 0 ⇒ 0 = c2 β + c3 ∂x At x = L, there is no warping but there is an applied torque, hence ∂φx = 0 ⇒ 0 = c1 β sinh βL + c2 β cosh βL + c3 ∂x ∂ 3 φx ∂φx − ECw ⇒ To = GJ [c1 β sinh βL + c2 β cosh βL + c3 ] ∂x ∂x 3   −ECw c1 β 3 sinh βL + c2 β 3 cosh βL

To = GJ

Solving directly gives c1 =

−To 3 β EC

w

1 − Ch Sh

c2 =

−To 3 β EC

w

c3 =

To 2 β EC

w

c4 =

To 3 β EC

w

1 − Ch Sh

where Ch ≡ cosh βL and Sh ≡ sinh βL. As a result, the rotation distribution is φx (x) =

  [1 − cosh βL] sinh βx To x+ [1 − cosh βx] − GJ β sinh βL β

(2.46)

The leading term is the elementary solution. This result is shown plotted in Fig. 2.40a; it matches the FE results quite accurately. A couple of points are worth noting about this solution. The maximum rotation is about one-sixth that of the elementary model. This increase in stiffness is at the expense of generating axial stresses in the beam. The elementary model gives a constant rate of twist, but as seen in the figure, the rate of twist actually varies significantly along the length. As the length of the beam is increased, the middle region for the rate of twist becomes flat and the axial stress is then confined to the ends.

152

2 Shapes with Coupled Deformations

The controlling parameter is . βL =

/ ' 0 1 3 0 G h [2W + H ] GJ h L 24G 3 1 = = 1 ECw HH E E 24 hH 2 W 3

Larger βL gives smaller boundary effect or penetration. Increasing the thickness of the flange and web (h) has the same effect as increasing the length (L) of the beam, that is, the elementary model is achieved except for at the ends. Similarly, a beam of smaller cross section (H ) also tends toward the elementary solution. Example 2.14 A drawback of the strong formulation solutions is that they do not lend themselves to easy interpretation. For example, for the I-beam problem, the results in Fig. 2.40a clearly show that the end constraints increase the structural stiffness, but this is difficult to see in the solution of Eq. (2.46). Construct a model to help explain the various contributions to the solution of the I-beam problem. Assume a polynomial solution for the twist and impose that at x = 0 :

φx = 0,

∂φx = 0, ∂x

at x = L :

∂φx =0 ∂x

This leads to the 1DoF representation   φx = a3 − 32 Lx 2 + x 3

or

 2  x x3 φ x = φo 3 2 − 2 3 L L

where φo is the rotation at the end. The strain energy contributions are 

 U=

1 2

GJ

∂φx ∂x

2



 dx +

1 2

ECw

∂ 2 φx ∂x 2

2 dx =

1 2

GJ 2 12 1 ECw 2 φ + φ 12 L o 10 2 L3 o

The only applied load is the torque at x = L, so that the virtual work is δ We = To δφo and To is the generalized load. The equilibrium equation and solution immediately become   2   GJ 12 ECw x To x3 3 + 12 φ = T , φ (x) = − 2 o o x GJ 12 ECw L 10 L3 L2 L3 + 12 L 10 L3 When φx (x) is superposed in Fig. 2.40a, it is indistinguishable from either the strong solution or the FE results.

2.3 Long Structures with Open Cross Sections

153

We therefore conclude that the simpler model seems to capture the essential behaviors of the I-beam. Consequently, the elastic stiffness is given by KE =

EhH 2 W 3 1 ∂2 U GJ 12 ECw G[2W + H ]h3 4 + + = 12 = L 10 L 10 2 ∂φo2 L3 L3

For the case H = W , KE =

  GH h3 12 E H 2 H 2 10 1+ L 10 G h2 L2 24

For a given depth of beam, thinner material or shorter lengths increase the restrained warping effect. For the parameters of Fig. 2.40, the ratio of the two stiffnesses is about 1:4 with the restrained warping being larger.

2.3.2 General Open Sections Figure 2.30 indicates two sets of axes, a global set (x, y, z) that is at the centroid C of the cross section and associated with the displacements (u, v, w); and a local set (n, s, z) at the mid-surface and oriented with respect to the shell and associated with the displacements (un , vs , w). Under general loads, the cross section rotates and warps (displaces in z). Our modeling is basically a membrane model for shells (local bending effects are neglected) except that we allow torsional shears (the shear stress τxy indicated in Fig. 2.26c) and just discussed in Analysis III. The three aspects considered are: the warping displacement, the strain energies, and the governing equations.

Aspect I: Warping Displacement We make the kinematic assumptions that the motion of any cross section entails a rigid body translation (u, v) and rotation (φz ) of the cross section; these are the same for all points on the cross section and therefore are a function only of z. We also assume there is an axial warping (w) which is a function of z and also s in the circumferential direction. As a consequence of these assumptions, the tangential displacement of a point on the surface is vs = −u sin θ + v cos θ + rφz where θ is the orientation of the shell normal at the point as shown in Fig. 2.30c and r is the distance to the shear center. The shear strain is

154

2 Shapes with Coupled Deformations

γsz =

∂vs ∂w ∂u ∂v ∂φz ∂w + = − sin θ + cos θ + r + ∂z ∂s ∂z ∂z ∂z ∂s

Integrate this strain over an arc segment of length s to get 

s o

 s  s  s ∂u ∂v ∂w ∂φz sin θ ds + cos θ ds + ds + ds γsz ds = − r ∂z ∂z ∂z o o o o ∂s  s    ∂u s ∂v s ∂φz s = dx + dy + 2 d A¯ + dw ∂z o ∂z o ∂z o o 

s

= [x − xo ]

∂φz ∂u ∂v + [y − yo ] + 2A¯ s + [w − wo ] ∂z ∂z ∂z

where A¯ s is the area enclosed by the origin and the segment as shown in Fig. 2.30c and is a function of position, that is, A¯ s = A¯ s (s). For an open section under distributed end torques, the shear stress τsz is zero along the centerline of the wall and hence the integral of γsz is zero. For an open section under transverse load, there is a membrane shear stress; in the following, we invoke elementary beam modeling and assume that shear arises from equilibrium and not from the deformation. In some of the example problems we show the consequence of this assumption and replace it with the Timoshenko beam modeling. Thus, for open sections, the warping is given by w = wo − [x − xo ]

∂φz ∂u ∂v − [y − yo ] − 2A¯ s ∂z ∂z ∂z

(2.47)

This warping is constant across the wall thickness but varies on the cross section as a whole because A¯ s = A¯ s (s).

Aspect II: Strain Energies The axial strain is zz =

∂wo ∂ 2 φz ∂w ∂ 2u ∂ 2v = − [x − xo ] 2 − [y − yo ] 2 − 2A¯ s 2 ∂z ∂z ∂z ∂z ∂z

(2.48)

The first derivative term of wo is the axial-stretching term, the second derivative terms of u and v are beam-bending terms (about two axes), the final contribution arising from the rotation term is the warping displacement (and hence strain) discussed in connection with Fig. 2.38. Assume that the normal stress state is predominantly uniaxial so that σzz = Ezz , then the strain energy is

2.3 Long Structures with Open Cross Sections

155

 U =

1 2

V

 =

1 2

V

2 Ezz dV



∂wo ∂ 2 φz ∂ 2u ∂ 2v E − [x − xo ] 2 − [y − yo ] 2 − 2A¯ s 2 ∂z ∂z ∂z ∂z

2 dV

This gives rise to integrals such as 





[x − xo ] dV

[x − xo ]2 dV

V

V



[x − xo ]A¯ dV

[x − xo ][y − yo ] dV V

V

The first type is zero because the global axes are at the centroid. We make the specification that these axes are also principal (bending) axes so that the third type of integral vanishes and the second type is the moment of inertia. By computing the integral for A¯ s with respect to the centroid, the fourth integral also vanishes. The strain energy then simplifies to  2 2  2 2    ) ∂ u ∂ v ∂wo 2 + EIyy + EIxx U = EA 2 ∂z ∂z ∂z2 L *  2 2 ∂ φz ∂wo ∂ 2 φz +ECw + ESw dz ∂z ∂z2 ∂z2 1 2

We recognize the first three terms as strain energies for a rod and a beam bending about two axes, respectively. The fourth and fifth terms are new (although mentioned in connection with the I-beam of Fig. 2.38) and gives rise to the interesting effects observed in thin-walled structures. Because the energies are uncoupled so too are the governing equations and hence the deformations. We achieved this by specifying the axes as being principal (bending axes) and using the centroid and shear center ideas. Keep in mind, however, that for arbitrary loads and axes (e.g., an FE analysis) the deformations appear coupled. We illustrate this in the example problems. The coefficients Cw and Sw are defined by, respectively,  Cw ≡ c

4A¯ 2s h ds ,



2A¯ s h ds

Sw ≡ c

They are called the warping constant and first sectorial moment, respectively [14]. Some values for Cw for typical cross sections are shown in Fig. 2.41; the notations used are illustrated in Fig. 2.31. The torsional properties of additional cross sections are given in Ref. [17] and an example of computing Cw for a general flat-faceted cross section is given in Ref. [11]. Secondary torsion of a rectangular cross section 1 3 3 h H . This is typically negligible. Therefore, of of size [h × H ] has Cw = 144 practical note, cross sections that have a single intersection of straight segments, such as those in Fig. 2.42, also have a negligible Cw ; the reason they are negligible

156

2 Shapes with Coupled Deformations

Fig. 2.41 Torsional properties of some open sections. (a) I-beam. (b) C-channel. (c) Circular arc of total angle 2α and radius R

Fig. 2.42 Cross sections with negligible warping constant Cw

is that by choosing the origin (for the integration) at the intersection, then all area contributions are zero. The rotation of the cross section causes a torsional shear distribution through the thickness of the wall given by γsz = −2

∂φz n ∂z

where n is the distance from the centerline as illustrated in Fig. 2.21. This is the St. Venant torsion contribution. The contribution to the strain energy is  U=

1 2

V



 2 Gγsz dV =

1 2

GJ L

∂φz ∂z

2

 dz ,

J =

+h/2  S

−h/2

o

4n2 dn ds = 13 Sh3

where S is the total length of the cross section. In the case of compound open sections such as those in Figs. 2.37 and 2.42, S is the sum of each segment length. This is essentially the same result we already established in Eq. (2.42).

Aspect III: Governing Equations Let there be distributed loads qu (z), qv (z), qφ (z) where the third of these is a distributed moment about the z axis. Also let there be end loads so that the total virtual work of the loads is

2.3 Long Structures with Open Cross Sections

 δ We = o

L

157

L L L  ∂φz L    qu δu+qv δv+qφ δφz dz+Px δu +Py δv  +· · · +Tz δφz  +Qz δ  o o o ∂z o

Combine this with the strain energies and using our variational principle (noting that variations are taken only with respect to δu, δv, δφz ) then leads to the uncoupled equations   ∂ ∂ 2u EIyy 2 = qu ∂z2 ∂z   ∂ ∂ 2v EIxx 2 = qv ∂z2 ∂z     ∂ ∂φz ∂ ∂ 2 φz ECw 2 − GJ = qφ ∂z ∂z ∂z2 ∂z

(2.49)

We recognize the first two as simple bending about the x and y axes. The fourth order differential equation for torsion is essentially Eq. (2.43) already derived for the I-beam. The BCs for the first two equations are the same ones for beam bending as given in Sect. 1.2. The BCs for the third equation are stated in terms of   ∂φz ∂ 2 φz ∂ ECw 2 = Tz = specified GJ − ∂z ∂z ∂z

or

φz = specified

∂ 2 φz = Qz = specified ∂z2

or

∂φz = specified ∂z

ECw

at either end; a summary of the BCs is given in Eq. (2.44). Possible sets of boundary conditions are simple: fixed:

u=v =w=0, u = v = w = 0,

∂ 2u ∂ 2v ∂ 2w = 2 = 2 =0, ∂z2 ∂z ∂z ∂v ∂w ∂u = = = 0, ∂z ∂z ∂z

∂ 2 φz =0 ∂z2 ∂φz φz = 0, =0 ∂z

φz = 0,

The simply-supported condition allows the cross section to warp. Because of Eq. (2.47), we see that the fixed boundary requires ∂φz /∂z = 0 in order to force the zero warping. This latter condition could be achieved by using relatively stiff end plates. Because of Eq. (2.48), we see that imposing zero second derivative on φz imposes zero zz and hence zero σzz . These explain the BCs used for the I-beam associated with the results in Fig. 2.40, Example 2.15 Use an FE analysis to assess the adequacy of Eqs. (2.49) to describe the bending and twisting behavior of thin-walled beams of open section.

158

2 Shapes with Coupled Deformations

0.0

0.2

0.4

.

0.6

0.8

1.0

Fig. 2.43 Deformation of a uniformly loaded I-beam with totally fixed ends. (a) Mesh and properties. (b) Deflection distribution of the beam centerline

Figure 2.43a shows the mesh for a uniformly loaded I-beam with totally fixed ends. The beam is modeled as a folded plate shell using the MRT/DKT element [5]. The transverse load is a line loading applied along the top and bottom of the web. Figure 2.43b shows the deflection (as open circles) sampled along the middle line. Also shown in the figure are the results taken from Eq. (1.23) (which use Eq. (2.49)) and are labeled as “BE.” The comparison is quite poor. The Timoshenko beam model takes shearing deformations into account in deep beams. Results from this model are shown labeled as “T1” in the figure and are taken from Eq. (2.40). While this shows an improvement there is still a marked difference. We can get improved Timoshenko results by realizing that the shear deformation comes from just that portion of the cross section experiencing significant shear stresses. In the case of the I-beam, as designed, it is the web that supports most of the shear and the flanges contribute little (it is the reverse for an H-beam). Let the Timoshenko model be modified so that the area in the GA term is just the web area; this is shown labeled as “T2” in the figure and is indistinguishable from the FE results. It must be concluded that Eq. (2.49) is not especially good at describing the bending behavior of open thin-walled beams when there is significant shearing actions. This is essentially a membrane plate problem and generally should be treated as such. Figure 2.44 shows FE generated results for the twisting of a thin-walled open section C-channel fixed at one end with a rigid plate at the other end. Figure 2.44a shows partial Hex20 meshes; what is missing is the end plate and joint fillers. The mesh has two elements through the thickness and full integration is used. The end plate has a stiffness ×10,000 that of the channel material. The channel has a clockwise couple applied to the end plate and Fig. 2.44b shows that the rate of twist varies along the length. Significantly, the total twist is a factor of 0.029 smaller than the elementary modeling given by Eq. (1.20). The comparison model is based on the governing equation

2.4 Spectral Analysis of Coupled Deformations

159

Fig. 2.44 Twisting deformation of a C-channel. (a) Deformed shape (exaggerated) under resultant clockwise torque (b) Distribution of twist along the axis. Squares are Hex20 FE results, triangles are MRT/DKT shell FE results, continuous line is model result

ECw

d 4 φz d 2 φz − GJ = qm 4 dz dz2

taken from Eq. (2.49). This is integrated in the manner of the I-beam example associated with Fig. 2.40 and results are shown in Fig. 2.44b. The comparison is good showing that the restrained warping imposed by the fixed end and the rigid end plate severely restricts the amount of rotation. It is concluded that if thin-walled open sections are to be analyzed using the strong formulation, then Eq. (2.49) must be amended to include the effect of shear on the bending modeling. Realistically, however, open cross sections should be modeled (at least) as folded plate shell structures.

2.4 Spectral Analysis of Coupled Deformations When studied closely, the spectral shapes of Figs. 1.29 and 1.30 are quite informative as regards the hierarchies of stiffnesses established. This attribute carries over to the plate and shell structural systems but it is necessary to take the extra dimensions into account; this is what we illustrate here. To make the presentations simple, we choose BCs that lead to relatively simple spectral shapes but the principles in play are straightforwardly generalized for the other BCs used in conjunction with FE analysis to get the general spectral shapes. The BCs are of the periodic BC type introduced in Sect. 2.1.

160

2 Shapes with Coupled Deformations

2.4.1 Spectral Decomposition of Coupled Deformation Shapes The orthogonality of the spectral shapes for a system with coupled deformations is slightly more involved than for a simple system. We use the arch of Fig. 2.2 to illustrate the various aspects involved. The orthogonality conditions arise from the kinetic and strain energies because they give rise to the mass and stiffness matrices, respectively. These energies are expressed as  T =

1 2

  ρA u˙ 2 + v˙ 2 ds



 EA EI 2 1 [u,θ +v,θθ ]2 dθ U = [u,θ −v] dθ + 2 R R3   F2 M2 ds + 12 ds = 12 EA EI 1 2

Figure 2.45a shows the distributions of u(θ ) and v(θ ) as determined by a linear vibration eigenanalysis. The arch was modeled with 128 frame elements; this number is a compromise between modeling the arch adequately for its geometry, and accurately performing the various volume integrals associated with the energies. It is probably clear that the symmetric and antisymmetric shapes are orthogonal to each other; however, unlike the simple sinusoids, two symmetric shapes such as v2 (θ ) and v6 (θ ) are not orthogonal to each other. Therefore, the mass matrix associated with these two modes is fully populated. Orthogonality arises from the

♦

♦

♦

♦

♦

♦

♦

♦

♦

♦

♦

0.

30.

.

60.

90.

♦

0

30

.

60

90

Fig. 2.45 Spectral analysis of an arch. Bottom three shapes are antisymmetric, top three are symmetric, and the horizontal arrows are the zero references. (a) Spectral shapes distributions. Circles are v(θ), triangles are u(θ). (b) Member load distributions (scaled for plotting). Circles are M(θ) or φ,s , triangles are F (θ) or u,θ −v

2.4 Spectral Analysis of Coupled Deformations

161

total kinetic energy so that while the mass matrices associated with u˙ and v˙ are separately fully populated, the sum of the two yield a diagonal mass matrix. Similarly, the various space derivatives of u(θ ) and v(θ ) are not orthogonal but the combinations giving F (θ ) and M(θ ) summed in the form F 2 /EA + M 2 /EI are orthogonal. With the mass and stiffness computed as M˜ I I =



ρA[uI uI + vI vI ] ds ,

K˜ I I =

 [FI FI /EA + MI MI /EI ] ds

where the summation is over the element values and the subscript I refers to a particular shape, then the frequencies reported by the FE eigenanalysis are the same as ωI2 = K˜ I I /M˜ I I . Furthermore, most FE eigenanalyzers normalize shapes so that M˜ I I = 1, consequently, the stiffness values can be taken directly from the frequencies squared. As an illustration of using spectral shapes, let us revisit the problem given in Fig. 2.2b for a point load on an arch. For this discussion, the shapes are expressed in curved beam coordinates, but our general scheme developed in Sect. 3.4 for nonlinear problems uses a global coordinate system. The displacements and member loads are represented by u(s) = F (s) =

 I

aI uI ,



I

aI F I ,

v(s) =



aI vI  M(s) = aI MI I

I

Observe that there is only one generalized DoF (aI ) associated with a given spectral shape and that we consider the displacement distributions and their various space derivatives as constituting “the shape.” The virtual work of the applied load is δ We =

 I

PI δaI = Po δv(θ = α/2) =

 I

 δaI vI α/2

This leads to  PI = Po vI α/2 ,

aI = Pm /K˜ I I

Figure 2.46 shows the reconstructions using just the symmetric modes 2, 4, and 6; the quality is quite good. It is of value to compare the present solution with that given in Fig. 2.13. First, the distribution of F (θ ) does not suffer the zero values at the boundaries. Second, there were no simultaneous equations to be solved. Third, using the same collection of spectral shapes, solutions for other loading conditions can be easily generated simply by computing the appropriate generalized loads PI . Finally, the same procedure is easily applied to other arch problems simply by recording the appropriate spectral shapes for that problem with the appropriately imposed geometric BCs.

162

2 Shapes with Coupled Deformations ♦ ♦

0.0

0.5

♦

♦

♦

♦

1.0

.

0.0

0.5 .

1.0

Fig. 2.46 Reconstructions for an arch under central radial loading. Circles are FE data, continuous lines are model results, and the horizontal arrows are the zero references. (a) Displacement distributions. (b) Member load distributions

The quality of the results is affected by the number of shapes used, therefore, let us consider the various contributions in a systematic way. First, we do a spectral decomposition of the deformed shape to identify the relative contribution of each of the shapes. The shape information is represented in the form u(si ) =

 I

aI uI (si ) ,

v(si ) =

 I

aI vI (si )

The subscript i enumerates the element centroidal values. We form the least squares error function 2  2 $  #   ui − Error = aI uI (si ) + vi − aI vI (si ) i

I

I

Minimizing this with respect to the choice of parameters aI leads to 

A˜ I N aN = dI

(2.50)

where   uI (si )vI (si ) + uI (si )vI (si ) i   ui uI (si ) + vi vI (si ) dI =

A˜ I N =

i

This is a diagonal system of equations because of the orthogonality of the shapes. A useful consequence of this is that the same response amplitudes for each mode are obtained irrespective of the total number of modes used in the decomposition. In other words, there is not a question of convergence as regards each of the separate modes. Furthermore,  A˜  is essentially the mass matrix and if treated as such (i.e.,

2.4 Spectral Analysis of Coupled Deformations

163

include ρA in the error definition), then has only ones on the diagonal; consequently, aI = dI . An obvious measure of relevance of a particular shape is the identified value of aN . Because this is determined from the shape itself, it tends to mask the contributions of particular shapes to, for example, the axial force distribution. It is therefore prudent to have a collection of measures depending on the entities of interest. For the symmetric load case, the two significant quantities worth monitoring are the deflection and axial force (given as the collection of shapes u,s −v/R) at the center given by, respectively, vc = aN v (sc ) ,

Fc = aN F (sc )

The relative amplitudes of the three possibilities are given by (using just the symmetric modes) 2 4 6 7 8 11 aN : 1.000 0.074 0.034 −0.015 0.008 −0.004 0.002 vc : 1.000 0.047 0.031 0.008 0.007 0.003 0.002 Fc : 0.300 −0.230 1.000 −0.230 0.044 −0.032 0.072 where they are ratioed to their maximum value. The three measures are different but they do show that beyond the first four or so shapes, there is convergence in the displacement but additional terms are required for the full convergence of the axial force. These conclusions are reflected in Fig. 2.46. The reason shape #6 is dominant for the axial force is shown in Fig. 2.45, it has an almost uniform force distribution.

2.4.2 Spectral Shapes of Flat Plates The flexural behavior of a flat plate is not an example of a coupled deformation but it is a necessary preliminary to understanding the coupled behaviors of curved plates and shells, Figure 2.47a shows some spectral shapes for a simply-supported rectangular plate; the dimensions are the same as shown in Fig. 2.29. The pattern of shapes is an increasing number of half-waves in the length and transverse directions. Labeling these numbers as [n, m], respectively, then Fig. 2.47b shows the sequence of numbers against frequency. The FE data form patterns which we call deformation modes; we now explain these patterns. The data plotted in this manner is called a spectrum relation [4, 6] similar to that established for slender members in Sect. 1.4. The difference here is that there are multiple deformation modes, that is, we consider the collection of points for m = 1 (one half-wave in the transverse direction) as different than m = 2 or 3 with more half-waves. While in one sense each individual data point is a vibration mode shape,

164

2 Shapes with Coupled Deformations

Fig. 2.47 Simply-supported flat plat. (a) Sample out-of-plane shapes labeled as [n, m]. (b) Spectrum relations collated for the different deformation modes. Symbols are the FE generated data and continuous lines are the modeling results

it is the set of points we wish to distinguish. All the data in Fig. 2.47b are from a single FE run, but are collated according to their [n, m] shape. The significance of the mode designation is that if the plate where doubled in length, all the m = 1 data would plot along the same line although there would be twice as many data points. In other words, the plate exhibits periodic BCs. Let us continue with slightly more general BCs and assume that the edges x = 0, x = a can be arbitrary but the other two edges are simply supported. The spectral EoM can be established from Eq. (2.31) by setting qw → qw − ρhw¨ giving 

∂2 ∂2 + D ∂x 2 ∂y 2



 ∂2 ∂ 2w ∂2 w + ρh + = qw ∂x 2 ∂y 2 ∂t 2

Represent the homogenous solution as ˜ sin(my) ¯ eiωt w(x, y, t) = w(x) ˜ sin(mπy/b) eiωt = w(x) This removes y and t from the governing equation leaving 

  d 2    d2 2 2 2 2 w˜ = 0 , + β −m ¯ − β −m ¯ dx 2 dx 2

 β 2 = ω ρh/D

The coefficients of the ODE are constant, hence assume solutions of the form e−ikx giving the characteristic equation 

−k 2 − m ¯ 2 + β2

  −k 2 − m ¯ 2 − β2 = 0

This leads to the four spectrum relations

2.4 Spectral Analysis of Coupled Deformations

 k1 (ω, m) ¯ = ± β2 − m ¯ 2 = ±αm ,

165

 k2 (ω, m) ¯ = ±i β 2 + m ¯ 2 = ±α¯ m

The second spectrums are wholly imaginary, while the first spectrums start imaginary-only but have a cut-on frequency where they become real-only. Reference [4] has a plot of the full spectrums; Fig. 2.47b shows the real parts plotted as the continuous lines. The agreement with the FE data is very good. There are no FE results for m = 0; this mode is the same as for a simple beam and is suppressed by the lateral BCs. We can write the total solution (or spectral shape) as w(x, ˆ y) = [c1 cos(αx) + c2 sin(αx) + c3 cosh(αx) ¯ + c4 sinh(αx)] ¯ sin(my) ¯ (2.51) Written in this form, the shape has a good deal in common with that of a beam given by Eq. (1.36), the significant difference being the variation in y. We could proceed to solve for the coefficients ci using the arbitrary BCs at the ends. The results would be similar to those obtained for the beam in Sect. 1.4. If all the BCs are simply supported, then we get that c1 = c3 = c4 = 0 and the shapes are as shown in Fig. 2.47a. Spectral shapes for circular plates are given in Ref. [5]. These too have patterns to them but their analytical description requires the use of Bessel functions [15] and therefore not pursued here. Example 2.16 Solve the simply-supported rectangular plate problem stated in Fig. 2.29 using spectral shapes. Compare the results where the BCs are changed. Motivated by the FE results, we can assume a shape is described by w(x, y) =

 n,m

wnm sin(nπ x/a) sin(mπy/b) =

 n,m

wnm sin(nx) ¯ sin(my) ¯

Because the boundaries have zero displacement, the Gaussian contribution to the strain energy is zero. The strain energy and stiffness evaluate, respectively, to U=

1 2

 nm

 2 2 n¯ 2 + m Dwnm ¯ 2 14 ab ,

2  K˜ nm = D n¯ 2 + m ¯ 2 14 ab

The stiffness matrix is diagonal because the spectral shapes are orthogonal. The virtual work of the generalized load must equal the virtual work of the actual load so that  δ We = Pnm δwnm n,m    = qw (x, y) δw(x, y) dxdy = qo δwnm sin(nx) ¯ sin(my) ¯ dxdy n,m

The generalized loads evaluate to

166

2 Shapes with Coupled Deformations

  Pnm = qo [cos nx ¯ − 1][cos my ¯ − 1]/n¯ m ¯ a b = 4qo /n¯ m ¯,

n, m = 1, 3, 5, · · ·

Because the load distribution is symmetric, only some of the generalized loads contribute to the deflection. Using our PoVW, we get that the deflection at an arbitrary point is given by w(x, y) =



Pnm n,m K ˜

sin(nx) ¯ sin(my) ¯

nm

The comparison with the FE results of Fig. 2.29 is almost exact using just n, m = 1, 3, 5, 7. This result is the same as reported in Ref. [19] as the exact solution. For completeness, we state the result for the kinetic energy and mass as T = 12 ρh 14 ab ,

M = 12 ρh 14 ab

The mass is the same irrespective of the shape. Now consider a rectangular plate similar to that in Fig. 2.29 but with fixed BCs on all sides, and subjected to a central point load. The FE eigenanalysis gives the stiffness and responses of the doubly-symmetric shapes as K˜ I I = {6.2, 20.5, 77.9, 155.7, 207.4, 232.1, 337.2, 560.3, · · ·} ωo2   I 12 a, 12 b = {343, 316, −306, −290, 300, 292, 297, 289, · · ·} h   I 14 a, 12 b = {205, −288, 124, −193, −233, 281, −151, 111, · · ·} h where ωo2 = D(π/b)4 ab/ρabh. The generalized loads are PI = Po I



1 1 2 a, 2 b



so that the response at an arbitrary point is w(x, y) =

  Po I 12 a, 12 b I (x, y) IK ˜ mm



Reconstructions for the two points are shown in Fig. 2.48. Convergence at the load application point is rather slow for two reasons. First, each modal contribution is positive resulting in the monotonic convergence, and second, the spread in stiffness values (essentially dependent on [n¯ 2 + m ¯ 2 ]2 ) is not very distinct because of the changing n and m. This can be seen in Fig. 2.47b where the m = 3 frequencies are interspersed among the higher m = 1 frequencies. Convergence for other points, as typified by the quarter point, is quite rapid requiring three or so spectral shapes.

2.4 Spectral Analysis of Coupled Deformations

167

Fig. 2.48 Flat plate with fixed BCs and a central point load. Convergence of displacement with respect to number of modes

1

3

5

7

9

11 13

This solution highlights an interesting aspect of shapes and stiffness known as Maxwell’s reciprocity. Let the load be applied at ( 14 a, 12 b) and displacement reconstructed at ( 12 a, 12 b). The result is  w

1 1 2 a, 2 b



=

    Po I 14 a, 12 b I 12 a, 12 b IK ˜ mm



which is the same as for the load applied at ( 12 a, 12 b) and displacement reconstructed at ( 14 a, 12 b). This comes about because the stiffness matrix is symmetric. In other words, the spectral responses at two (or any number) points can be used to form a reduced stiffness matrix connecting just those points. This is a well known result in vibration modal analysis [6].

2.4.3 Circular Cylinders and Curved Plates The governing differential equations for a curved plate and a cylinder are the same, consequently we expect the spectral analysis to be the same. The boundaries can be different but as seen from Fig. 2.10, we can view the cylinder as a periodic extension of the curved plate. Therefore, in the present discussion we concentrate on the cylinder because the results are easier to present, the example problem considers the curved plate. Figure 2.49 shows an example spectral shape for a cylinder. While it looks complicated, thinking in terms of the cylinder being unfolded as a flat plate, we see the same pattern of half-waves in the length and transverse (hoop) directions. The FE shell global BCs imposed at each end are that {u, v, w; φx , φy , φz }g = {0, 0, 1; 1, 1, 1}

168

2 Shapes with Coupled Deformations

♦

20.

♦

Fig. 2.49 Spectral shape [2, 10] for a cylinder radially constrained at the ends and free axially

15.

.

10.

5.

0.

X

♦

0.0

1.0

2.0 .

3.0

4.0

0.0

2.0

.

X

4.0

X

♦

6.0

Fig. 2.50 Spectral analysis of a circular cylinder. Symbols are FE generated data, lines are model results. (a) Complete cylinder equations. (b) Modeling of cylinder as a shallow plate. The symbol “x” indicates a cut-on frequency

Note that the axial DoF is unconstrained and warping can be observed in the side view of the figure. These conditions are equivalent to periodic BCs (pBCs) along the length, the BCs are already periodic in the hoop direction because the shape is continuous through 2π . The cylinder was meshed with 24 modules along the length and 64 in the hoop direction, using the MRT/DKT shell element [5]. The halfwaves in the hoop and length directions are designated here as [n, m], respectively. Figure 2.50 shows a plot of the number of half-waves against frequency as the data symbols. The meaning of k2R is the number of half-waves in the hoop direction, the meaning of m is the number of half-waves in the length direction. The stiffnesses are the same as those developed in connection with Fig. 2.22, we need to only add the inertia which is simply

2.4 Spectral Analysis of Coupled Deformations

169

⎡

⎤ 100 ρhω2 ⎣ 0 1 0 ⎦ 002 The EVP becomes   ¯ · ] − ω2 ρh[ · ] { u } = 0 E ∗ h[ · ] + Gh[ · ] + D[ · ] + D[

(2.52)

with the DoF arranged as { u } = {uo , vo , wo }T . When presented in this way, each matrix can be viewed as a stiffness matrix (the last involving the mass is called a dynamic stiffness). The characteristic equation is quartic in k 2 and cubic in ω2 ; hence in either case to find the roots k(ω) or ω(k) requires using numerical methods. The simple scheme adopted here is: for a range of frequencies and a particular m, a scan over real-only k is performed and the zeros of the determinant recorded. Figure 2.50 shows the results for the first five m modes plotted as the continuous lines. The comparison with the FE results is good. Keep in mind that because of the repeated roots, a total of 80 FE roots are involved. The significant difference in comparison to the flat plate spectrum relations of Fig. 2.47 is the absence of modes in the lower left corner. This is a direct consequence of the stiffening due to the curvature which raises the stiffness of the small n modes. The higher modes (m = 2, 3) associated with the membrane dominated actions cut on at approximately 4 kHz and 5.5 kHz. The EVP also provides the amplitude ratios between {uo , vo , wo }T ; we refer to the resulting shapes as synthesized spectral shapes. At this point, it is worth commenting on the advantage that the spectral shapes (obtained from an FE analysis, say) have over the current representation: it is that they automatically contain the correct amplitude ratios. That is, the shape is a complete shape as illustrated in Fig. 2.51 and not a synthesized shape. Consequently, there is only one generalized DoF for each shape. When these are used instead of the synthesized shapes for the static problem of Fig. 2.22, the results are indistinguishable from those shown in the figure. More book keeping is involved, but once set up, general curved plate problems can be tackled. In fact, basing the computer book keeping on a shell element (the MRT/DKT triangular element is a good choice because of its inherent Tessellation) general thin-walled structures can be conveniently analyzed. We now apply the curved plate model of Eq. (2.36) to the full cylinder and assess its quality through a spectral analysis. We know that the transverse displacement is a sinusoid in both directions, and because all derivatives are even powers, then the stress function must be of a similar form. That is, assume w(x, s, t) = wo (t) sin(nx) ¯ sin(ms) ¯ ,

ψ(x, s, t) = ψo (t) sin(nx) ¯ sin(ms) ¯

where n¯ = nπ/L m ¯ = mπ/a = m/2R. The inertia enters as qw h = −ρhw¨ so that the vibration EVP becomes

170

2 Shapes with Coupled Deformations

Fig. 2.51 First six spectral shapes for a curved plate. Order is lower left to upper right. (a) End view showing transverse displacements. (b) Plan view showing in-plane displacements



¯ 2 ]2 −n¯ 2 h/R [n¯ 2 + m 2 ¯ 2 ]2 +n¯ E/R [n¯ 2 + m

#

$ wˆ o =0 ψˆ o

Setting the determinant to zero, we get ρhω2 =

2  n¯ 4 Eh 2 ¯2  2 + D n¯ + m 2 R n¯ 2 + m ¯2

Figure 2.50b shows a comparison of the frequencies with those of the FE analysis. The agreement is quite good except for small wavenumber. In particular, the cut-on frequencies (indicated by the “x” symbols in the figure) are off significantly. This comes about because for m → 0, we get ρhω2 =

  π4 h2 R 2 4 Eh Eh 4 1 + + D n ¯ = n R2 R2 12(1 − ν 2 ) L2 L2

For n = 1, the second term is of order 10−5 and therefore negligible unless n is large. Although the model has its deficiencies, we nonetheless find it useful because of its simplicity. In fact, we make particular use of it in modeling the nonlinear behavior of cylinders. Example 2.17 Figure 2.52a shows the first six spectral shapes of a shallow curved plate. All boundaries are flexurally simply supported, the remaining boundary DoFs allow axial movement. The mesh shown uses the MRT/DKT shell element [5] and the parameters of the case are taken from Ref. [20]. Construct some models to explain the shapes. Because the BCs are nearly periodic, the shapes are readily identified according to their number of half-waves [n, m]. The sequence of antisymmetric and symmetric shapes is

2.4 Spectral Analysis of Coupled Deformations

171

1.0

6

♦

7

♦

Fig. 2.52 Shallow curved plate with simply-supported flexural BCs. First six spectral shapes starting lower left

5 .

4 0.5 .

3 2 1 0

♦

0.0

0.5

1.0

.

1.5

2.0

0.0

2.5

♦

0.0

0.5

1.0

.

Fig. 2.53 Shallow curved plate. (a) FE generated spectrum relations for the different modes. Empty circles are mode 1 results, full circles are mode 4 results, lines are model results. (b) Comparison of FE frequencies with model results. Squares are antisymmetric shapes, circles are symmetric shapes, the straight line is for perfect agreement

2 [1,2] [2,2]

4

6 [3,2]

[1,1] [2,1]

[3,1] [4,1]

8 [4,2]

10

12

14 [5,2]

(1,3) [5,1] (2,3) (3,3)

[6,1]

The symmetry referred to is for the transverse deflection in the hoop direction, that is, m = odd. The modes in parentheses are higher symmetric modes. The purely axial modes appear at 30, 86, and 103. The spectral shapes are correlated with frequency in Fig. 2.53a. An interesting feature of the spectrum relation is that in comparison to Fig. 2.47b for the flat plate, the low [n, 1] modes are shifted higher in frequency than the low [n, 2] modes. This is an artifact of the stiffening effect due to curvature. Figure 2.14b shows the curved plate coordinates and notation. The strains are

172

2 Shapes with Coupled Deformations

♦ ♦

♦

♦ ♦

♦

♦ ♦

♦

♦ ♦

0.

2.

4.

6.

8.

♦

0.

2.

4.

6.

.

8.

0.

2.

4.

6.

8.

Fig. 2.54 Antisymmetric deflection shapes of a shallow curved plate at x = L/2. Continuous lines are model results and arrows indicate the zero reference. (a) Transverse displacements. (b) In-plane displacements. (c) In-plane hoop strains

xx = u,x , κxx = w,xx

ss = v,s +w/R ,

γxs = u,s +v,x

κss = −v,s /R + w,ss ≈ w,ss

κxs = −v,x /R + 2w,xs ≈ 2w,xs

The linear membrane and flexural strain energies for the curved plate are  UM =

1 2

  E ∗ h u,2x +(v,s +w/R)2 + 2νu,x (v,s +w/R) dxds 

+ 12

Gh [u,s +v,x ]2 dxds

 UF =

1 2

D [w,xx +w,ss ]2 dxds

where the Gaussian contribution to the flexural energy is set to zero. The curved plate problem has a total of three unknowns similar to the cylinder. Let us consider the antisymmetric modes first. Figure 2.54a shows the FE data for the displacements sampled at x = L/2; the transverse displacement is represented well by w(x, s) = wo sin(mπ s/a) sin(nπx/L) = wo sin ms ¯ sin nx ¯

m = 2, 4, · · ·

n = 1, 2, · · ·

which is shown as the continuous line with m = 2. If the deformation is inextensible, then v,s = −w/R

or

v(x, s) = (wo /mR)[cos ¯ ms ¯ − 1] sin nx ¯

Figure 2.54b shows that v(x, s) deteriorates for the higher modes; not only is the amplitude off but the shape itself changes with the formation of extra zero crossings.

2.4 Spectral Analysis of Coupled Deformations

173

What this means is that the higher modes are not inextensible. Figure 2.54c shows the hoop strains; there is a definite cos(π s/a) form to them. Although the average hoop strain is zero, there is extensibility occurring (although small for n = 1). To account for extensibility, we therefore must have at least two DoF. As a simplified two-DoF model, assume the displacements can be represented by w(x, s) = wo sin ms ¯ sin nx ¯

v(x, s) = (vo /mR)[cos ¯ ms ¯ − 1] sin nx ¯

u(x, s) = 0

The axial displacement is assumed to not contribute on the assumption that the modes are predominantly flexural (as justified by Fig. 2.52) and therefore mostly involve the coupling between w(x, s) and v(x, s). This displacement representation gives a hoop and shear strain of, respectively, ss =

1 [wo − vo ] sin(ms) ¯ sin nx ¯ , R

γxs = −

vo n¯ [cos ms ¯ − 1] cos nx ¯ Rm ¯

The distribution of the hoop strain in s is not of the form we would prefer, but we see that to the extent the DoFs wo and vo are different, then we get a membrane hoop strain contribution. Unlike the arch, we see the appearance of a membrane shear; this contribution is definitely worth tracking. The membrane energy reduces to  UM =

∗



E h[w/R + v,s ] ds dx +

1 2

2

1 2

Gh[v,x ]2 ds dx

Substitute the assumed shape and integrate to get # UM =

1 2

$ E∗h 2 2 1 Gh aL/4 [w − v ] + 3[v n/ ¯ m] ¯ o o o 2 R2 R2

On its own, this gives a stiffness matrix of [KM ] =

  aL E∗h 1 −1 , R 2 −1 1 + 3γ2 4

γ2 =

Gh n¯ 2 E∗h m ¯2

with the DoF arranged as { u } = {wo , vo }T . This is a rather interesting result because it resembles the element stiffness for a rod given by Eq. (1.32) if γ2 = 0. In other words, the two displacements wo and vo have an elastic constraint between them and the elasticity is E ∗ h/R 2 ; if the radius is very large, the two displacements are uncoupled. Note that the shear has no coupling effect and for the cases of n and m tested γ2 ranges from 0.02 to 0.58. However, if γ2 = 0, then the lower mode (the one of interest) of just the membrane problem has a zero frequency. Therefore, it is the shearing action that specifically contributes the membrane stiffness; this should be clear by realizing that det[KM ] ∝ 3γ2 . The flexural strain energy has no contribution from the axial displacements and therefore not affected by its omission. By assumption it also does not depend on v

174

2 Shapes with Coupled Deformations

so that the form given earlier integrates to    2 ¯2 +m ¯ 4 aL/4 = 12 Dwo2 n¯ 2 + m ¯ 2 aL/4 UF = 12 Dwo2 n¯ 4 + 2n¯ 2 m This gives a diagonal contribution so that the total stiffness is estimated as [ K ]=

    2 aL E∗h 1 ¯ 2 ]2 0 aL −1 [n¯ + m + D 0 0 4 R 2 −1 1 + 3γ2 4

The primary coupling is due to the membrane actions, and the flexural action of a flat plate is then superposed on top of that. The kinetic energy is  T =

1 2

  ¯ 2 aL/4 ρh[u˙ 2 + v˙ 2 + w˙ 2 ] dsdx = 12 ρh w˙ o2 + v˙o2 3/(mR)

giving the mass matrix 

1 0 [ M ] = ρh 0 3/(mR) ¯ 2



aL 4

The mass matrix is diagonal with the second mass being orders of magnitude smaller than the first. The results of the [2 × 2] vibration EVP are shown in Fig. 2.53b as the square symbols for the lower of the two modes; note that the second mode of the model is related to some very high membrane dominated mode (of the two-DoF simplified system) completely unrelated to the current modes being discussed and therefore is not plotted. The agreement is good with the FE generated data for most of the frequency range. In other words, our simplified model captures the essential stiffness contributions. The symmetric modes for the arch did not exhibit an m = 1 mode (see Fig. 1.30, for example), whereas, in contrast, the first four symmetric modes for the shallow plate have m = 1. The displacement distributions for the higher modes are shown in Fig. 2.55. The higher modes have a radial displacement approximated by w(x, s) = wo sin(mπ s/a) sin(nπ x/L) = wo sin ms ¯ sin nx, ¯ m = 1, 3, · · · n = 1, 2, · · ·

The displacement behaviors are opposite to the antisymmetric modes in Fig. 2.54 in that the correspondence improves with increasing mode. If the deformation is inextensible, then v(x, s) = (wo /mR)[cos ¯ ms ¯ + co ] sin nx ¯ This cannot satisfy the BCs that v = 0 at both straight edges. Let us add a linear function in s so that the displacement and hoop strain become, respectively,

175 ♦

2.4 Spectral Analysis of Coupled Deformations

♦

♦ ♦

♦

♦ ♦ ♦

♦

♦

♦ ♦

0.

2.

4.

6.

8. 0.

2.

4.

6.

8.

0.

2.

4.

6.

8.

Fig. 2.55 Symmetric deflection higher modes of a shallow curved plate. Continuous lines are model results. (a) Transverse displacements. (b) In-plane displacements. (c) In-plane hoop strains. Dashed line is the simple model mean value

v(x, s) = (wo /mR)[cos ¯ ms ¯ − 1 + 2s/a] sin nx ¯ ,

ss = (2wo /Ra) sin nx ¯

This satisfies the BCs but in doing so generates a uniform hoop strain (at each x = constant cross section) and is shown plotted as the dashed line in Fig. 2.55c. It seems to represent the average hoop strain reasonably well. Thus the symmetric modes generate significant hoop strains that alternate in sign along the length. We could improve the hoop representation by adding another shape in s as done for the arch but that requires three DoF because we must include v(s) for the shear. Just as for the antisymmetric modes, we need at least two DoFs to represent the extensibility. As a simplified 2D model, assume the axial displacement does not contribute and the displacements can be represented by w(x, s) = wo sin ms ¯ sin nx, ¯

v(x, s) = (vo /mR)[cos ¯ ms ¯ − 1 + 2s/a] sin nx, ¯

u(x, s) = 0

The membrane energy integrates to UM =

1 2

  aL E∗h  2 ¯2  2 2 aL 1 Gh n + w v − 2w v (1 − γ ) + v (5/3 − 2γ ) o o 1 1 o o o 2 R2 m 4 4 R2 ¯2

where γ1 = (2/a m) ¯ 2 . On its own, this give a stiffness matrix of   E∗h 1 −1 + γ1 ab [KM ] = 2 −1 + γ1 1 + γ3 4 R

γ3 =

Gh n¯ 2 (5/3−2γ1 ) = γ2 (5/3−2γ1 ) E∗h m ¯2

For the cases of n and m tested, γ1 is constant at 0.4, while γ3 ranges from 0.02 to 0.64. Again, the shear has no coupling effect but it does affect the strength of the membrane contribution.

176

2 Shapes with Coupled Deformations

The flexural strain energy and stiffness integrate to ¯ 2 ]2 aL/4 , UF = 12 Dwo2 [n¯ 2 + m

[KF ] = D

 2  ¯ 2 ]2 0 aL [n¯ + m 0 0 4

These are the same as for the antisymmetric shapes. The kinetic energy and mass are, respectively, 

T =

1 ˙ o2 2 ρh[w

+ (5/3 − 16/π

2

)v˙o2 ]aL/4 ,

1 0 [ M ] = ρh 0 5/3 − 16/π 2



aL 4

The primary mass is the same as for the antisymmetric shapes, the secondary mass is different but this has little effect because it is very small. The stiffness results are shown in Fig. 2.53b as the circle data points; the comparison is reasonable and follows the behavior of the antisymmetric shapes. Again, we conclude that our simplified model captures the essential stiffness contributions. The main coupling effect for the symmetric modes occurs through the parameter γ1 so that in relation to Fig. 2.53a, the membrane action alone sets a minimum frequency ratio around 0.5 kHz, that is, with no flexural behavior whatsoever. The origin of the coupling comes from w v,s → wo vo (−1 + γ1 ) R and γ1 originated with the 2s/a term in the displacement expression for v(x, s). That, in turn, arose from imposing zero v displacement BCs. Thus the primary coupling is due to the membrane actions, and the flexural action of a flat plate is then superposed on top of that.

2.4.4 Spectral Shapes of Open Sections There are an enormous variety of structures with open shapes. To help focus on generic behaviors, we consider a particular open structure that of a C-channel. Open structures have both local and global shapes; this distinction is somewhat arbitrary because it is connected with the type of model being used. We try to make this clear. Figure 2.56 shows the first six (non-rigid body) spectral shapes for a rectangular C-channel freely supported and with no end plates. We can think of the flanges and web as essentially flat plates. The flange and web dimensions are similar to that of Figs. 1.29 and 2.47, therefore, the shapes in Fig. 2.56 should be read in conjunction with these two figures; the differences in shapes are because of the effective BCs. Our idea of a global mode is that the shape in some sense resembles one of the elementary structural member models, for example, a beam or a shaft. In this sense, the first shape in Fig. 2.56 is twisting and global. The second and sixth are uniform

2.4 Spectral Analysis of Coupled Deformations

177

Fig. 2.56 Spectral shapes for a rectangular C-channel without end plates. Order is lower left to upper right. (a) End view. (b) Side view

Fig. 2.57 Spectral shapes for a rectangular C-channel with end plates (hidden). Order is lower left to upper right. (a) End view. (b) Side view

along the length and are referred to as being deformational in contrast to the long structures of the previous section which have rigid cross sections. The others are local plate modes where the flange-as-a-plate is free on three sides and fixed on one of the long sides the web-as-a-plate is free on the two ends and fixed on the two long sides. The higher spectral shapes are similar to these but with more half-waves, the free edge plate modes look like wrinkling. Figure 2.57 shows the effect of adding stiff end plates. (The end-plate was given negligible inertia so that only its stiffness affected the shapes.) There are no uniform modes and no obvious global modes. All the shapes seem to be plate-type modes with the fourth and sixth indicating a slight hint of twisting. It is notable that the side views of the flanges have shapes similar to that of a clamped–clamped beam [6] again highlighting the very special nature of the sinusoids in representing the elastic shape of structures. We conclude that the flange, because it has a free edge, dominates the flexibility. It turns out, however, that the C-channel has a significant amount of mode coupling (because inertia acts through the centroid and not the shear center) which we elaborate on in the example problem to follow. In both cases, the web acts as a stiff backbone and while it does exhibit plate modes, because its effective BCs are clamped on the long sides the stiffnesses are

178

2 Shapes with Coupled Deformations

relatively high and the shapes are not observed in the ones shown. When looking at complex structures, we see that it is (conceptually) possible to decompose behaviors into their subcomponent modes. The simpler case is when there are no modal interactions, the complicated one involves mode interactions, but in both cases, the concept of mode facilitates the analysis. Example 2.18 A C-channel, clamped at both ends, has a transverse load applied along the web at the one-third length location. Figure 2.58 shows that the responses are coupled. Use a spectral analysis to model the responses. As mentioned at the beginning of Sect. 2.3, using the concepts of shear center and principal axes, all static analyses of long structures can be presented as examples of uncoupled deformations. But because inertia acts through the centroid and not the shear center, the spectral shapes have coupled deformations. We illustrate how this affects the static analysis of long structures. We want to focus on the global modes, hence rather than deal with a folded plate model of the channel (as shown in Figs. 2.56 and 2.57), we find it more insightful to work directly with an FE model that incorporates the modeling assumptions of the long structure. Much of the following FE modeling is taken from Ref. [12]. The strain energy is given by  U=

1 2

 EIxx v,2zz dx +

 GJ φz2 ,z dx +

1 2

1 2

ECw φz ,2zz dx

where x is the stiffer principal axis. Because v(z) and φz (z) have second derivatives in their energy expressions, a reasonable choice for the element interpolation functions are those given in Eq. (1.30). That is,

♦

♦

♦

♦

♦

♦

♦

♦

♦

♦

♦

♦

♦

♦

0.

20.

40. .

60.

80.

Fig. 2.58 Loading of a C-channel. (a) Displacement and member load distributions. Circles are FE results, continuous lines are spectral reconstructions. (b) First six spectral shapes as the continuous lines. Dashed lines are the higher stiffness shapes that are complementary to the lower stiffness shapes

2.4 Spectral Analysis of Coupled Deformations

v(x) =

4 i

gi (z)vi ,

179

φz (z) =

4 i

gi (z)φzi

with the DoF chosen as { v } = {v1 , v1 ; v2 , v2 }T ,

  T {φz } = {φz1 , φz1 ; φz2 , φz2 }

The prime indicates a space derivative. Integration leads to the standard stiffness for the beam behavior given by Eq. (1.31), the torsional stiffness is given by ⎡

⎤ 36 3L −36 3L 2 2⎥ GJ ⎢ ⎢ 3L 4L −3L −L ⎥ + ECw [ kT ] = 30L ⎣ −36 −3L 36 −3L ⎦ 30L 3L −L2 −3L 4L2

⎡

⎤ 12 6L −12 6L ⎢ 6L 4L2 −6L 2L2 ⎥ ⎢ ⎥ ⎣ −12 −6L 12 −6L ⎦ 6L 2L2 −6L 4L2

where L is the length of the element. The warping contribution has the same matrix form as for bending stiffness. The bending and twisting combined can be represented by the [8 × 8] matrix ) * [ kB ] [ 0 ] [ k ]= [ 0 ] [ kT ] The bending and twisting stiffnesses are not coupled. This is assembled in the usual manner done for structural elements. Figure 2.58a shows the computed distributions when the channel is loaded along the web at the one-third length point. Similar distributions are obtained using the folded plate model with the important difference that, just as with the I-beam in Fig. 2.43, there is a discrepancy in the displacement because deep-beam membrane shear effects are not taken into account. It would take us too far afield to account for this deficiency at present. To obtain the spectral shapes we need to do a linear vibration analysis. To this end, the kinetic energy is written as  T =

1 2

ρA[v˙ + (e + c)φ˙ z ]2 dz +

 1 2

ρIc φ˙ z2 dz

We get consistent mass matrices by using the same interpolation functions as used for the stiffness. This results in   A[ C ] (e + c)A[ C ]T [M ]= (e + c)A[ C ] ((e + c)2 + Ic )[ C ] ⎡ ⎤ 156 22L 54 −13L 2 13L −3L2 ⎥ ρL ⎢ ⎢ 22L 4L ⎥ [ C ]= 54 13L 156 −22L ⎦ 420 ⎣ −13L −3L2 −22L

4L2

180

2 Shapes with Coupled Deformations

The mass matrix couples the bending and twisting deformations even if a lumped mass model is used. The consequences of this are rather subtle which we elaborate on. Figure 2.58b shows the first six computed shape distributions as the continuous lines. They exhibit the expected pattern of increasing the number of zero crossings for the higher modes. That v(z) and φz (z) appear in tandem is indicative of the deformation coupling. What is surprising is that some of the even higher stiffness shapes exhibit shapes similar to the lower modes; these are shown as the dashed lines in the figure. These complementary shapes have the mode number pairs (1, 3), (2, 7), (4, 10), (5, 13), (6, 17) While we generally associate a simpler shape (fewer zero crossings) with a lower stiffness, this is not necessarily true as shown here. The spectral shapes are coupled deformation shapes but the complementary shapes are not coupled modes; all modes are uncoupled because they are orthogonal computed through an eigenvalue analysis. The next three chapters explore what happens when these modes are coupled through nonlinear deformations. The element interpolation functions were used to compute the higher spatial derivatives so that the complete information stored for each shape is v = {v, v,z , v,zz v,zzz } ,

φ = {φz , φz ,z , φz ,zz φz ,zzz }

These are stored as nodal values for each element. We can use the collection of shapes in Fig. 2.58b to do a spectral decomposition of the shape in Fig. 2.58a. The particular form of Eq. (2.50) becomes % & ρA[uI + (e + c)φI ][uN + (e + c)φN ] + ρIc φN φI = δI N i % & ρA[ui + (e + c)φi ][uI + (e + c)φI ] + ρIc φi φI dI =

A˜ I N =

i

where δI N is the Kronecker delta. The results are shown in Fig. 2.59. Because the individual shapes have different magnitudes, the circles are aN normalized Fig. 2.59 Spectral decomposition of the deformed shape of a loaded C-channel

1

5

9

.

13

17

21

Explorations

181

to |v + (e + c)φ|. The first four modes are dominant encompassing symmetric and antisymmetric modes, although we also see a significant mode #7 which is complementary to mode #2. The continuous lines in Fig. 2.58a are the reconstructions having used our PoVW to determine aN . The displacements and moment are in excellent agreement with the FE results, the torque suffers from the absence of small wavelength terms.

Explorations 2.1 Analyze a problem similar to that of Fig. 2.4 but have the gravity action acting horizontally. • Construct a model for comparison with the text solution. • Do an FE analysis for additional comparisons. 2.2 A rectangular plate has fixed BCs and a uniform transverse load. • Develop a simple one-term Ritz solution. • Compare the results to an FE analysis. 2.3 A rectangular plate has simply-supported BCs and a uniform transverse load. It is proposed to represent the transverse displacement by w(x, y) = w1 [a 3 x − 2ax 3 + x 4 ][b3 y − 2by 3 + y 4 ]

• Determine what geometric and natural BCs the assumed displacement satisfies. • Use the function to develop a one-term Ritz solution for the deflection of the plate • Compare the solution to an FE analysis. • How much improvement is there over the one-term solution given in connection with Fig. 2.29? 2.4 A straight frame member has uncoupled axial and flexural behaviors. The axial action has constant strain distribution and the flexural has linear strain distribution. Because curved members can be modeled adequately using many straight elements the purpose of this problem is to develop a curved element formulation based on constant axial strain and linear flexural strain distributions. • Show that the following representations give the required distributions u(θ ) = a1 sin θ + a2 cos θ + a3 + a5 θ + a6 θ 2 v(θ ) = a1 cos θ − a2 sin θ + a4 + 2a6 θ

182

2 Shapes with Coupled Deformations

There are six parameters (generalized DoF) which is the same number as for a plane frame member. • Show that the element shape functions are as shown plotted in the figure, • Choose a standard problem and compare its performance to that of many straight elements.

♦

♦

♦

♦

♦

♦

♦

♦

♦

References 1. Boresi, A.P., Chong, K.P.: Elasticity in Engineering Mechanics. Elsevier, New York (1987) 2. Crandall, S.H., Dahl, N.C., Lardner, T.J.: An Introduction to the Mechanics of Solids. Wiley, New York (1972) 3. Donnell, L.H.: A new theory for the buckling of thin cylinders under axial compression and bending. ASME Aeronautical Eng. 56:795–806 (1934) 4. Doyle, J.F.: Wave Propagation in Structures. Springer, New York (1989), 2/E (1997) 5. Doyle, J.F.: Nonlinear Analysis of Thin-walled Structures: Statics, Dynamics, and Stability. Springer, New York (2001) 6. Doyle, J.F.: Nonlinear Structural Dynamics using FE Methods. Cambridge University, Cambridge (2015) 7. Kuo, S-R., Yang, Y-B.: New theory of buckling of curved beams. ASCE Eng. Mech. 117(8):1698–1717 (1991) 8. Langhaar, H.L., Boresi, A.P.: Snap-through and postbuckling behavior of cylindrical shells under the action of external pressure. Bull. Eng. Exp. Stn. 54(59) (1957) 9. Leissa, A.W.: Vibration of shells. In: NASA SP-288 (1973) 10. Markus, S.: Mechanics of Vibrations of Cylindrical Shells. Elsevier, New York (1988) 11. Megson, T.H.G.: Aircraft Structures. Halsted Press, New York (1990) 12. Mei, C.: Coupled vibrations of thin-walled beams of open section using the finite element method. Int. J. Mech. Sci. 12:883–891 (1970) 13. Naghdi, P.M., Berry, J.G.: On the equations of motion of cylindrical shells. J. Appl. Mech. 21(2):160–166 (1964) 14. Oden, J.T.: Mechanics of Elastic Structures. McGraw-Hill, New York (1967) 15. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes. Cambridge University, Cambridge (1986), 2/E (1992) 16. Reissner, E.: Stress and displacement of shallow spherical shells. J. Math. Phys. 25(1):80–85 (1946) 17. Timoshenko, S.P., Gere, J.M.: Theory of Elastic Stability. McGraw-Hill, New York (1963)

References

183

18. Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity. McGraw-Hill, New York (1970) 19. Timoshenko, S.P., Woinowsky-Krieger, S.: Theory of Plates and Shells. McGraw-Hill, New York (1968) 20. Ventzel, E., Krauthammer, T.: Plates and Shells. CDC Press, New York (2001) 21. Yoo, C.H., Lee, S.C.: Stability of Structures: principles and applications. Elsevier, Amsterdam (2011)

Chapter 3

Nonlinear Elastic Shapes

When load levels are low, the structural behavior is linear. An increasing load generally generates nonlinearities in two ways. Either the material response becomes nonlinear (e.g., plasticity is induced) or the deflections become large so that the strain-displacement relation is nonlinear. We are concerned exclusively with this latter case. Most engineering structures are purposely built to be stiff so that typically the only source of nonlinear displacements is associated with the out-of-plane deflections in thin-walled flexible structures as discussed in Chap. 2. The situations of primary concern to us in structural statics (especially when analyzing light-weight structures) are when the deflections and rotations are large but the strains are small (that is, the strains are still within the elastic limit); then the local rigid body motions are the significant contributors to the nonlinear behavior. These nonlinear deflections get supported primarily by the generation of in-plane membrane stresses. In a general sense, the ability to support these loads without excessive deformations is called stiffness. This chapter begins by elaborating further on the concept of stiffness introduced in Sect. 1.3 and how, in a nonlinear situation, it is affected by load. The notion of a change in shape—not just that there is a deformed shape slightly different than the initial shape, but that the shape is different, is introduced here. This is highly nonlinear and gets its full expression over the next two chapters that associate this change with an instability in the structural behavior. Here, the focus is on establishing some basic aspects that are required by these two chapters.

3.1 Stiffness of Nonlinearly Deformed Structures There are some structures that receive their significant stiffness properties from pretension. A simple example already considered is a cable: while the axial properties are similar to that of a rod member, the transverse stiffness properties © Springer Nature Switzerland AG 2020 J. F. Doyle, Spectral Analysis of Nonlinear Elastic Shapes, https://doi.org/10.1007/978-3-030-59494-7_4

185

186

3 Nonlinear Elastic Shapes

arise only after tensioning (either by sagging or by pretensioning). Another example is an inflatable (gossamer) structure; the pressure induces membrane stresses which gives the stiffness. There are also situations of pre-existing loads which affect the subsequent response to loads: for example, there could be the dead weight load of a building, and it is required to determine the additional responses due to winds. The situations where the applied loads themselves cause a change of stiffness properties during their application are highly nonlinear and require a fully nonlinear analysis as discussed here. Much deeper discussions are given in Refs. [1, 4, 6]. The concept of stiffness is intuitively obvious but analytically rather elusive for a general loaded structure. Reference [9] focusses on this in good detail therefore here we limit ourselves to the ideas of immediate use for plates and shells. First, we introduce the notion of a structural system that includes certain types of loads; that is, loads that can be considered part of the system itself. These are called conservative loads and the system can be represented by a total potential energy function and equilibrium is viewed as achieving a stationary value of this function. We consider the effect of changing shape on stiffness; in this way, we get the connection, in a generic sense, between deformed shape and stiffness.

3.1.1 Equilibrium and Equilibrium Paths A system is conservative if the work done in moving the system around a closed path (in force-displacement space) is zero. For engineering structures, we need to distinguish between the material behavior of the system and the load behavior applied to the system. An elastic material (even if nonlinear) is an example of a conservative system; elastic-plastic material behavior is nonconservative because energy is dissipated through the plastic work. Even if the system material behavior is conservative, the applied loads could be nonconservative. Follower loads such as the reaction force from a jet nozzle and the pressure on a deforming plate are examples of nonconservative loads, but they seem to arise mostly in dynamic problems [9, 10] and are therefore not considered in this book. Here, we specifically rewrite the principle of virtual work for conservative systems (both materials and loads). The PoVW is a statement of equilibrium and can be written as δ U − δ We = 0

or

δU −

 I

PI δuI = 0

where PI are the generalized loads corresponding to the generalized DoF uI . For a conservative system this can be written as    δ U− PI uI = 0 I

or

  δ U + V = 0,

 V ≡− PI uI (3.1) I

where V is called the potential of the applied loads. The quantity ≡ U + V is called the total potential energy of the system and the statement δ = 0 is

3.1 Stiffness of Nonlinearly Deformed Structures

187

called the Principle of Stationary Potential Energy. Essentially what was done is that the applied (generalized) loads are associated with a potential V (analogous to the internal elastic forces computed from the strain energy) and computed as derivatives according to PI = −

∂V , ∂uI

FI =

∂U ∂uI

Thus, for a totally conservative system, both the internal and external loads are derived from a potential. For discrete systems, we have  ∂  U+ V =0 ∂uI

for I = 1, 2, . . . , N

(3.2)

Equilibrium is thus seen as achieving an extremum of the total potential energy. As an illustration, let us compute the equilibrium paths for the simple mechanical structure shown in Fig. 3.1 under the action of the applied loads Po and Qo . An equilibrium path is a collection of equilibrium states that are changed by a controlling variable, in this case the loads Po and Qo . The initial configuration has the rigid bar vertical and the spring at its natural length. The roller ensures that the spring is always horizontal. Because the bar is rigid, there is only one free DoF. We use the angle off the vertical, φ, as our single generalized DoF and allow it to vary in the range ±π . If the applied load were only vertical, it would be reasonable to assume that there is no rotational motion of the bar. This can be an erroneous assumption for nonlinear problems because of their sensitivity to small load and geometry imperfections; we elaborate on this significant point as we proceed and the load Qo helps us to delineate the possibilities. The restraints on the horizontal and vertical displacements are u = L sin φ ,

v = L cos φ − L

1.0 0.0 .

♦

-1.0 -1.0

-0.5

0.0 .

0.5

1.0

Fig. 3.1 Mechanical system with rigid bar and spring. (a) Disturbed equilibrium state. (b) Possible equilibrium states for different load levels; dashed lines indicate states negative stiffness

188

3 Nonlinear Elastic Shapes

The total potential energy is given by

= 12 K1 u2 − [−Po ]v − [Qo u] = 12 K1 L2 sin2 φ + Po L[cos φ − 1] − Qo L sin φ The equilibrium configurations are found by setting  ∂  = K1 L2 cos φ − Po L sin φ − Qo L cos φ = 0 ∂φ Figure 3.1b shows some equilibrium paths for different values of Qo which we now explain. First consider the case when Qo = 0. The resulting nonlinear equilibrium equation has two sets of solutions or equilibrium paths. It is worth emphasizing that although there is only one DoF, there are multiple solutions; this is in the nature of nonlinear problems. The first set of equilibrium paths (or primary loading paths) is given by I:

sin φ = 0

or

φ = 0, ±π, ±2π, . . .

These correspond to when the bar is in a vertical alignment and therefore is independent of the spring values. The other path (the secondary loading path) is given by II:

Po = K1 L cos φ

or

cos φ = Po /K1 L

This solution path exists only for loads in the range −1 ≤ Po /K1 L ≤ +1 because of the allowable range on the cosine function. Both solutions (equilibrium paths) are shown plotted in Fig. 3.1b; path I is the heavy full line, path II is the heavy dashed line. We see that the solutions intersect at Po /K1 L = 1 thus causing a singular point where two separate equilibrium paths have the same state meaning that there are two separate possibilities for satisfying the equilibrium conditions. This astounding situation causes an instability which is elaborated on presently. Now consider when Qo is small and applied before Po . This causes a small initial rotation, and the equilibrium path becomes   Po = K1 L sin φ − Qo cos φ/ sin φ This is shown plotted as the fine full and dashed lines in Fig. 3.1b. There is no intersection of equilibrium paths, and the maximum load achieved has decreased. The point of maximum load is called a limit point; the stiffness (as judged by the slope ∂Po /∂φ) is positive up to the limit point and negative beyond. We have more to say about limit points and stiffness later but we observe from Fig. 3.1b that the shape of the structure changes under the load Qo (it moves off the vertical) and we use this opportunity to expand on the meaning of equilibrium.

3.1 Stiffness of Nonlinearly Deformed Structures

189

-3.

-2.

-1.

0.

Fig. 3.2 Topology of the energy space. (a) Disturbed equilibrium states of a ball. I and III: stable equilibriums, II: unstable equilibrium. (b) Potential energy curve changes when loads are applied to the system

In one sense, equilibrium is simply that “the sum of forces is zero.” But equilibrium is deeper than that; for example, we can wonder how sensitive the equilibrium is to small disturbances, is it like the proverbial house of cards or is it rock solid? Surely this consideration too is part of the general concept of equilibrium. Look at the equilibrium situations shown in Fig. 3.2 where the black dot represents a ball and the line represents a surface it is resting on. In both cases, the black ball is in equilibrium as can be easily verified by drawing a free body. Now consider displacing the ball a small amount away from the equilibrium position— this is indicated by the white ball. What happens when the displacing agent is removed? In Fig. 3.2a case I, the ball rolls back toward the original position; if there is dissipation in the system, it oscillates but eventually settles down to the original position. We therefore say that the original equilibrium position (and not the disturbed position!) is stable. In Fig. 3.2a case II, the ball rolls away from the original position; we therefore say that the original equilibrium position is unstable. If either the valley or the peak is flat (a plateau, say) the ball stays put and therefore we say that the original equilibrium is neutral. For case II, the ball can become a case I or III, either way, we say that case II is “unstable in the small” but it is clear from the figure that it is “ stable in the large.” This is a reminder that we must be aware of how large the relevant deformations are. What Fig. 3.2a gives us is a sense as to what happens after the instability; it could go to nearby state III or far away state I, the latter exhibiting a more energetic transition. To help formalize our concept, consider the structure of Fig. 3.1 in a loaded state under Po and Qo , the deformation is φ. Now change the deformation by the small amount φ b without changing the loads; this is like what we did in deriving the PoVW. The total deformation is φ¯ = φ + φ b Expand the total potential for small additional deformation and use the small parameter  to track the different orders. With the approximations

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3 Nonlinear Elastic Shapes

    cos(φ + φ b ) ≈ cos φ 1 − 12  2 φ b2 + · · · − sin φ φ b + · · ·     sin(φ + φ b ) ≈ sin φ 1 − 12  2 φ b2 + · · · + cos φ φ b + · · · we get ¯ ≈

 sin2 φ + Po L[cos φ − 1] − Qo L sin φ   + K1 L2 cos φ sin φ − Po L sin φ − Qo L cos φ φ b   + K1 L2 (cos2 φ − sin2 φ) − Po L cos φ + Qo L sin φ 12  2 φ b2 + · · · 1

2 K1 L

2

= 0 + 1 φ b  + 2 φ b2 12  2 + · · ·

(3.3)

We recognize the second bracketed term 1 as associated with the equilibrium condition and is therefore zero. Thus, at the current deformation φ, the total potential is quadratic in φ b ; for small φ b , if the energy increases the system is stable, but if it decreases it is unstable. The border between the two is when 2 = 0, this is called neutral equilibrium.

2 in conjunction with 1 not only give us information about the equilibrium state, but also inform us of its stability condition. Consider the case when Qo = 0, then 2 = K1 L2 − Po L and for Po > K1 L its sign is negative. We can therefore declare such equilibrium states as unstable. When Qo > 0, 2 is zero at the limit point thus indicating that the limit point is a neutral equilibrium point, higher order

i terms would then inform us of the stability condition. The next subsection shows that 2 is associated with the nonlinear stiffness of the structure. While the second variation of the total potential establishes the stability of the system, it is how the potential (and hence the second variation) varies with the applied loads that leads to the interesting instability behavior in structural analyses. Figure 3.2b shows the changing topology as loads are applied to a system: at zero load there are two equilibrium states, but the one on the right gets annihilated at load level Pc . The next example problem summarizes some of the complexities involved when the load changes; this is a modification of the problem in Fig. 3.1 but with the spring slanted and not on rollers. Example 3.1 Determine the equilibrium paths for the structure shown in Fig. 3.3. Because the bar is rigid, then the restraints on the horizontal and vertical displacements give u = L sin φ ,

v = L − L cos φ

where φ is measured off the vertical and both u and v are in the same sense as the applied loads Po and Qo , respectively. From this we get the current length of the spring as

3.1 Stiffness of Nonlinearly Deformed Structures

191

1.0 0.8 .

0.6 0.4 0.2 0.0 -0.50

-0.25

0.00 .

0.25

0.50

Fig. 3.3 Postbuckling behavior of a simple mechanical system. (a) Geometry. (b) Load and stiffness for a = 2L. Solid line id for Qo = 0

l=



L2 cos φ 2 + (a + L sin φ)2 =



lo2 + 2aL sin φ ,

lo =



a 2 + L2

The total potential energy is given by

= 12 K1 [l − lo ]2 − Po v − Qo u = 12 K1 [l − lo ]2 − Po L[1 − cos φ] − Qo L sin φ The equilibrium configurations are found by setting   lo ∂

= K1 1 − aL cos φ − Po L sin φ − Qo L cos φ = 0 ∂φ l √ The linearized version of this equation with Qo = 0 and 1/ 1 + x ≈ 1 − 12 x + 38 x 2 is  K1

 aL lo2

−

 a 2 L2  φ aL − P L φ=0 o lo4

This equilibrium equation has two equilibrium paths given by  I:

φ = 0;

II:

Po = K1

 aL a 2 L2 − 4 φ a lo2 lo

The first corresponds to when the bar is in the vertical alignment and therefore is independent of the spring stiffness. The other solution depends on φ making the path asymmetric in the sense that the load-deflection behavior is different to the left and to the right. It is worth emphasizing that although there is only one DoF, there are multiple solutions; this is essentially what is meant by Euler’s concept of alternate nearby equilibrium states.

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3 Nonlinear Elastic Shapes

To elaborate, we see that the solutions intersect at Po = K1 La 2 / lo2 = Pc thus causing a singular point where two separate equilibrium paths have the same state. This means that there are two separate possibilities for satisfying the equilibrium conditions. The singular point that occurs due to the intersection of two equilibrium paths is called a bifurcation singular point. We can state Euler’s original bifurcation view of the loading process: as Po is increased, the primary solution (φ = 0) is stable until the critical point Pc = K1 La 2 / lo2 is reached where there are now two possible equilibrium solutions to the same given loads. The primary path is unstable beyond this point and a further increment of load combined with any disturbance (a small transverse load Qo , for example) causes it to move and seek another equilibrium position. Where it goes depends on Qo . Ascertaining where it goes requires a full nonlinear analysis. The nonlinear version of the bifurcated path is    lo  aL cos φ − Qo L cos φ /L sin φ Po = K1 1 − l This is shown plotted in Fig. 3.3b for different values of Qo . The postbuckling stability depends on whether the system is nudged to the left or to the right and therefore is referred to as an asymmetric bifurcation. (We can point specifically to the third term in the expansion of the square root as contributing this asymmetric effect.) If the system is nudged to the right, there are no nearby equilibrium states and the bar snaps to a point vertically downward. If the system is nudged to the left, the load can continue to increase. Therefore, we need to address the question if the bifurcation is stable or unstable. It seems that if a softening mechanism can occur it will occur, then the nonstable path is likely to be chosen. If the stable path is forced to occur (through a negative load imperfection Qo ) we could say the path is stable but this is true only because of the forcing constraint. On this “stable” path, at φ ≈ −45o , the load can no longer increase. This is called a limit point instability. For small imperfection Qo > 0, the structure changes from exhibiting a bifurcation to exhibiting a limit point instability.

3.1.2 Concept of Structural Stiffness Revisited Section 1.3 introduced the concept of structural stiffness within a linear context, here we extend it to cases involving nonlinear deformations. A few new aspects arise not anticipated from the linear discussions. These are mainly connected with the loading equation and how the applied loads can significantly affect the structural stiffness. We use the nominally simple spring-only structure shown in Fig. 3.4a to illustrate the ideas. The springs have their natural length in the configuration shown and remain linear elastic throughout the deformation. In the following, we draw the distinction between the natural stiffness of the individual springs versus the structural stiffness

3.1 Stiffness of Nonlinearly Deformed Structures

193

Fig. 3.4 Types of nonlinear structural stiffness behaviors. (a) Spring system with nonlinear stiffness effects. (b) Force-deflection behaviors of various systems: small dots is linear, full line is hardening (cable), empty circles is softening (pendulum), full circles is nonsymmetric softening in compression (arch)

of the assembled springs. When discussing the stiffness of structures, it is important to have a particular DoF in mind; first consider a vertical deflection v1 . The change of length of the vertical spring is simply L1 = v1 , the change of length of a horizontal spring due to the deflection is L2 =



L2 + v12 − L ≈ L 12

v12 L2

where L is the original length. The total strain energy of the springs and the virtual work of the applied load are, respectively,   v2 2 v4 UE = 12 K1 v12 + 2 12 K2 L 12 12 = 12 K1 v12 + 14 K2 L2 14 , L L

δ We = Pv δv1 = P1 δv1

For small vertical deflections, the horizontal springs do not contribute any stiffness; it is only after so-called second-order effects (represented by [v1 /L]2 ) are significant, that we see their contribution and this contribution is nonlinear. Note that the spring itself behaves linearly (the force relation is F = KL, always) it is its structural contribution that is nonlinear. Substitute into Eq. (1.26) to get the equilibrium condition K1 [1 + αv12 ]v1 = P1 ,

α = (K2 /K1 )/L2

A plot of the force against deflection is shown in Fig. 3.4b as the continuous heavy line. The plot is symmetric in that the magnitude of the forces generated on the downswing are the same as those generated on the upswing. It also stiffens in both directions in that it requires a larger force increment to achieve a given deflection

194

3 Nonlinear Elastic Shapes

increment as the load level gets higher. This is the “cable effect” which we now quantify. Because we deal with load increments and displacement increments, a measure of structural stiffness is given by KT =

∂P1 = K1 [1 + 3αv12 ] ∂v1

which is usually referred to as the tangent stiffness because it is the slope (tangent) of the force-deflection curve. This stiffness is obviously nonlinear with the K2 springs (α) not contributing at small deflections. This example is intended to show the distinction between structural stiffness and member (spring) stiffness; it is clear they are not synonymous, one can be linear while the other nonlinear. We can gain additional insight into this by considering the internal elastic force. This elastic force and corresponding stiffness are obtained from the strain energy according to F=

∂ UE = K1 [1 + αv12 ]v1 , ∂v1

KE =

∂F ∂ 2 UE = = K1 [1 + 3αv12 ] ∂v1 ∂v12

which is the same as the structural tangent stiffness. We thus associate, in general, the concept of structural stiffness as arising from the second derivative of the strain energy with respect to the DoFs or the change of elastic force with respect to the DoFs. This concept is presently generalized. Parenthetically, the nonlinear behavior exhibited by the springs in Fig. 3.4a is a common form of nonlinearity in structural systems because it is associated with the nonlinear deflections of beams, panels, and plates with the additional stiffening coming from the membrane stretching identical to that of the horizontal springs. The other examples of stiffening behaviors shown in Fig. 3.4b are the simple pendulum (open circles) and the arch (closed circles); the arch stiffens on the upswing but softens on the downswing and this can lead to an instability (significant changes of shape for small changes of load) which is discussed extensively in the next two chapters. Returning to the original problem, suppose the common point is given a horizontal displacement u1 by a load Q1 acting in the horizontal direction. Then the energy and stiffness are, respectively, UE = K2 u21 + 18 K1 L2

u41 , L4

KT =

u2 ∂ 2 UE = 2K2 + 32 K1 12 2 L ∂u1

Obviously, this stiffness is different than the one previously obtained. When both loads are applied simultaneously there are two equilibrium equations (e.g., P1 = f (u1 , v1 ), Q1 = g(u1 , v1 )), there are also stiffness derivatives such as

3.1 Stiffness of Nonlinearly Deformed Structures

∂P1 ∂ 2 UE = , ∂u1 ∂v1 ∂u1

195

∂Q1 ∂ 2 UE = ∂v1 ∂u1 ∂v1

In this way, the stiffness must be represented as a matrix with nontrivial off-diagonal terms and therefore is more difficult to interpret. One role of spectral analysis that we have been preparing in Sects. 1.3, 1.4, and 2.4, is to extract meanings from the stiffness matrix; this is illustrated in the example problem to follow. We now develop a more general concept of structural stiffness. Consider the quasi-static loading of a structure: at each stage of loading, the structure achieves a new equilibrium state—the succession of equilibrium states is the loading path as indicated earlier. The equilibrium of the structure is determined by Eq. (1.27) 0 = { P} − {F } ,

{F } =

 ∂U  ∂u

For this discussion, we use matrix notation because it helps to focus on ideas. We are interested in the response when these equilibrium states are disturbed slightly by “poking” (Ref. [17] has the wonderful credit of a photograph experimentally showing a finger poking a structure to ascertain its stability). Conceive of the total applied load and response as made up of the two parts { P} = { Po } + {Q} ,

{ u } = {uo } + { ξ }

(3.4)

That is, there is the primary response {uo } due to the primary load { Po }, and the smaller perturbation response { ξ } due to the poke load {Q}. The size of the perturbation is tracked by the small parameter . Substitute this decomposition into the equilibrium equation to get 0 = { Po (uo + ξ )} + {Q} − {F(uo + ξ )} We take that it is possible for the applied load { P} to be a function of the deformation (e.g., gravity on a pendulum). Because { ξ } is a small perturbation, we can expand the elastic force {F } to give {F(uo + ξ )} ≈ {F(uo )} +

 ∂F  { ξ } + . . . = {F(uo )} + [KE (uo )]{ ξ } + . . . ∂uo

where [KE (uo )] is the elastic stiffness at the current state of deformation {uo }. A similar expansion for the loads gives { P(uo + ξ )} ≈ { P(uo )} +

 ∂P  { ξ } + . . . = { P(uo )} + [KG (uo )]{ ξ } + . . . ∂uo

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3 Nonlinear Elastic Shapes

where [KG (uo )] is the geometric stiffness at the current state of deformation {uo }. Substitute these expressions for {F } and { P} into the equilibrium equation and group terms according to different powers of  to get the two equations 0 : 1 :

0 = { Po } − {F(uo )} [ KT ]{ ξ } = {Q}

(3.5)

where [KT (uo )] is the total stiffness of the structure given by [ KT ] = [KE ]+[KG ]. It is computed explicitly according to [KT I J ] =

 ∂F   ∂ P   ∂ 2 U   ∂ P   ∂ 2  I I E I − = − = (3.6) ∂uJ ∂uJ ∂uI ∂uJ ∂uJ ∂uI ∂uJ

This relation makes clear how the stiffness has contributions arising out of the elasticities of the systems (the springs, for example) and contributions arising from the loads (gravity, for example). The elastic stiffness is always symmetric as seen from its double derivative form, but the geometric stiffness can be nonsymmetric depending on the particular type of load. We exclusively deal with symmetric geometric stiffness matrices and ignore follower loads; Ref. [10] has an interesting long discussion of the appropriateness of follower forces in static analyses. The symmetric geometric stiffness is associated with conservative loads which accounts for the final form in Eq. (3.6). We see that the response due to the poke load is that of a linear system with a stiffness [ KT ]. How the stiffness changes is governed by the first of equations (3.5); as the deformation unfolds, the total stiffness can change, indeed, it can even become singular in some situations—these are the situations focussed on in Chaps. 4 and 5. In the special case when the poke load is applied to the unloaded structure, then the perturbation displacements are the actual displacements and we have the linear governing system [ KT ]{ u } = {Q} The total stiffness is computed at the zero-load state, i.e., it is just the elastic stiffness. In the special case when Q(t) is a ping load (short duration) applied to the deformed structure, we get the free transient response governed by [ M ]{ u¨ } + [ KT ]{ u } = 0 where inertia terms have been added. Note that it is the total stiffness at the current state of deformation that is used. This shows that a spectral analysis (e.g., an eigenvibration analysis) gives information about the current stiffness and is much more efficient and general than the poke-load method.

3.1 Stiffness of Nonlinearly Deformed Structures

197

Equation (3.5) leads to a very elementary strategy for solving nonlinear equations—simply let the poke load be an increment in the applied load so that the increment in displacement can be computed and updated from [ KT ]{u} = {P } ,

{ u } = { u }o + {u}

(3.7)

This equation is referred to as the loading equation. After each increment of load, the overall geometry is updated so that a new total stiffness can be computed. The updating of the geometry is the key part of the nonlinear analysis. This scheme is not very efficient but it forms the basis for the better schemes [6]. We mention it here not so much that it can solve nonlinear problems but because, in conjunction with Eqs. (3.5) and (3.6), it nicely explains the connection between the change of elastic shape (the accumulated sum of displacement increments), the total potential energy function (formed from the updated shape), equilibrium through the first derivative of this function (giving the elastic force F), and stiffness through its second derivative. Example 3.2 Determine the stiffness behavior along the equilibrium paths for the structure shown in Fig. 3.1. The second variation of the potential energy determines the structural stiffness. That is, KT =

∂ 2

= K1 L2 [cos2 φ − sin2 φ] − Po L cos φ + Qo L sin φ = KE + KG ∂φ 2

The load potential V = Po L[cos φ − 1] − Qo L sin φ has led to a load-dependent stiffness with the geometric contribution given by KG = −Po L cos φ + Qo L sin φ. It is noteworthy that the stiffness expression is the same as the 2 term in Eq. (3.3). As before, first consider the case with Qo = 0. For the equilibrium path φ = 0, this becomes I:

KT = K1 L2 − Po L

Hence, when Po > K1 L this equilibrium path has negative stiffness. The main point to be made here is that in the nonlinear analysis of structures, applied loads can have a significant effect on the structural stiffness. Thus, if the load is reversed and acting up, the stiffness is always positive and increasing with load. For the equilibrium paths φ = ±π where the bar is in the hang-down orientation, the stiffness becomes KT = K1 L2 + Po L This is positive for positive load as expected because the bar is being pulled in tension. If the load is reversed, then the stiffness can be made negative. For the second equilibrium path with Po = K1 L cos φ, we get II :

KT = −K1 L2 sin2 φ = −K1 L2 [1 − (Po /K1 L)2 ]

198

3 Nonlinear Elastic Shapes

This is always negative, however, this path exists only for the narrow load range shown in Fig. 3.5 because the cosine function has the range −1 < cos φ < 1. An interesting exercise is to conjecture what would happen if the bar is placed in the horizontal orientation φ = π/2 under no load; Fig. 3.1b indicates that it would be in an equilibrium state. In fact, the figure indicates that there are (at least) three equilibrium states (φ : − 12 π, 0, + 12 π ) for no applied load; this is something not encountered in the linear analysis of structures. The case where Qo = 0 is handled in the same way and Fig. 3.5 shows it plotted as the dashed line for angles φ > 0. We see that the system is stable up to the limit point (where Po reaches a maximum) and unstable thereafter until the negative load begins to decrease. To make the connection to linear analyses, consider the situation where the angle off the initial vertical is small. A Taylor series expansion of the total potential energy gives

≈ 12 K1 L2 [φ + · · · ]2 + Po L[[1 − 12 φ 2 + · · · ] − 1] − Qo L[φ − · · · ] ≈ 12 K1 L2 φ 2 − Po L 12 φ 2 − Qo Lφ Note that it is necessary to retain second-order terms in the cosine function otherwise Po would have no contribution to the total stiffness. The equilibrium paths and stiffness are found by setting ∂

= [K1 L2 − Po L]φ − Qo = 0 , ∂φ

KT =

∂ 2

= K1 L2 − Po L ∂φ 2

The load Qo does not have a contribution to the linearized total stiffness. If Po = K1 L, then the equilibrium equation becomes [ 0 ]φ = Qo and the angle is indeterminate. The linearized analysis correctly identified the singular solution as well as the change of stiffness with respect to load, but this was achievedonly Fig. 3.5 Stiffness behavior. Dashed line is for Qo = 0. Normalizations are: Pc = K1 L, Ko = K1 L2

3.1 Stiffness of Nonlinearly Deformed Structures

199

by retaining second-order terms in the potential for the load; this is something we return to often as we do approximate nonlinear analyses. Example 3.3 Determine the nonlinear stiffness matrix for the spring system shown in Fig. 3.4. Let the displacement at the common point be (u1 , v1 ), therefore the stiffness matrix is of size [2 × 2]. The linearized equilibrium equation is  # $ # $ K1 + K2 0 u1 Px = Py 0 K3 v1 showing that the linearized stiffness matrix is diagonal and remains diagonal (and the same) regardless of the loading. That is, both displacement components can be generated but only when both components of load are active. What we are specifically interested in here are coupling effects arising from the stiffness matrix being deformation dependent. Begin with the K1 spring, the stretched length is L1 = L



(u1 /L)2 + (1 + v1 /L)2 − 1 = l1 − 1

The strain energy is simply U = forces are Fu =

1 2 2 K1 L1

∂U = K1 [1 − 1/ l1 ]L(u1 /L) , ∂u1

=

1 2 K1 [l1

Fv =

− 1]2 L2 so that the elastic

∂U = K1 [1 − 1/ l1 ]L(1 + v1 /L) ∂v1

The stiffness is the derivative of the elastic forces leading to [ K ]1 =

  K1 (u1 /L)2 + (l1 − 1)l1 (u1 /L)(1 + v1 /L) (u1 /L)(1 + v1 /L) (1 + v1 /L)2 + (l1 − 1)l1 l13

Note the presence of the coupling, it is zero if the displacement is vertical only. The K11 stiffness is zero when there is no deformation, that is, the single vertical spring has no horizontal stiffness. After a v1 deformation, l1 changes and horizontal stiffening occurs—this is the cable effect, a stretching in one direction causes stiffening in the transverse direction. The other two springs are treated similarly. Note that the change of lengths is L2,3 = L



This leads to the stiffnesses

(1 ± u1 /L)2 + (v1 /L)2 − 1 = l2,3 − 1

200

3 Nonlinear Elastic Shapes

  K2 (1 + u1 /L)2 + (l2 − 1)l2 (1 + u1 /L)(v1 /L) [ K ]2 = 3 (v1 /L)2 + (l2 − 1)l2 /L (1 + u1 /L)(v1 /L) l2   K3 (1 − u1 /L)2 + (l3 − 1)l3 −(1 − u1 /L)(v1 /L) [ K ]3 = 3 −(1 − u1 /L)(v1 /L) (v1 /L)2 + (l3 − 1)l3 /L l3 When the two springs are the same, the off-diagonal terms combine to give zero stiffness, that is, there is no coupling as expected. The total stiffness behavior is shown in Fig. 3.6a for two slightly different loading cases: one is a Py only load, the other is a Py plus a small Px component. For convenience, the plots were made taking all Ki the same. There are a number of points worth noting. First, all plots are nonlinear and for the v1 -only case, there is a difference between displacing up versus down. When the load is vertical only, there is no coupling, that is, the K12 component is zero. The slightly off-vertical load (which can be considered as an imperfect vertical load) does exhibit coupling. In general, therefore, a fully populated stiffness matrix is usually generated. This is an important observation because we tend to think of stiffness as something associated with the structure, i.e., the particular arrangement of springs. What we are developing here is how the stiffness is profoundly affected by the load. We have a challenge as to how to interpret the contribution of the coupling components of stiffness; more specifically, in relation to Fig. 3.6a, does an increasing K12 imply an increasing stiffness of the structure? does a negative K12 imply the structure is less stiff? We must look at the spectral properties (as initially introduced in Sect. 1.3) of the stiffness matrix to answer these questions. First, to see these coupling effects more clearly, consider a second-order expansion of the strain √ energy, that is, replace the square root expressions for li with the approximation 1 + x ≈ 1 + x/2. This gives the approximate energies

♦

♦

 U1 ≈ 12 K1 (v1 /L)2 + (v1 /L)(u1 /L)2 ]L2 ,

2.0

.

1.0

0.0

-1.0

♦

-0.5

0.0

0.5 .

1.0

-0.5

0.0

0.5

1.0

.

Fig. 3.6 Total stiffness behavior of three connected springs for two slightly different loading regimes. Dotted line is vertical only applied load, full line is for slightly off-vertical applied load. (a) Component stiffnesses. (b) Spectral values of stiffness

3.1 Stiffness of Nonlinearly Deformed Structures

201

 U2,3 ≈ 12 K2,3 (u1 /L)2 ± (u1 /L)(v1 /L)2 ]L2 Differentiate to get the stiffnesses  [ K ]1 = K1

 v1 /L u1 /L , u1 /L 1

 [ K ]2,3 = K2,3

1 ±v1 /L ±v1 /L ±u1 /L



The coupling terms depend on the level of deformation (and hence load) and therefore the spectral properties are different at each load level. Now, although not necessary, imagine a single mass located at the junction in Fig. 3.4; we can write the spectral EVP as # $  K12 uˆ 1 K11 − ω2 M =0 K12 K22 − ω2 M vˆ1 Setting the determinant to zero gives 2 ω4 M 2 − ω2 (K11 + K22 )M + K11 K22 − K12 =0

On solving the quadratic equation in ω2 , we get  2 =K ˜ 11 , K˜ 22 2ω2 M = K11 + K22 ± (K11 − K22 )2 + 4K12 There are two principal (or extremum) values of stiffness because there are two DoF. The two values of stiffness are shown plotted against load in Fig. 3.6b. At zero deformation, the (a) and (b) plots coincide because there is no coupling (K12 = 0). Under deformation, the spectral analysis gives us the lowest and highest values of stiffness. The sign of K12 does not matter when the matrix is symmetric. For fixed K11 and K22 , an increasing K12 implies that one extremum increases while the other decreases, it seems a situation where both increase is not possible. In fact, we see that it is possible for the lower one to go to zero even though both K11 and K22 remain positive. In other words, coupling decreases one of the principal stiffness values. This is a more refined monitor of the stability condition of a system than simply observing the displacement behavior and we utilize it regularly in the next two chapters. In this connection, note that we get information about both modes (or all modes in a general case), whereas the displacement information is restricted to just that induced by the particular loading. The coupling referred to earlier is the coupling of the spectral shapes. At zero deformation, the shapes are simply either a horizontal displacement, or a vertical displacement. Under deformations, these couple so that the shapes are combination shapes. From the EVP, we have the amplitude ratio given by uˆ 1  K12  =− vˆ1 I K11 − ωI2 M We leave a fuller discussion of coupled spectral shapes until Chaps. 4 and 5.

202

3 Nonlinear Elastic Shapes

3.2 Large Deflections of Thin-Walled Structures We now look at the nonlinear behavior of straight beams and flat plates. Similar kinematical assumptions as in Sect. 2.2 are used, but with a large strain measure that takes large rotations (without stretching) into account. What makes these problems interesting but difficult to solve, is that the membrane and flexural actions become coupled under load; this is an aspect we focus on. First an appropriate measure of strain must be developed. The concept of stress introduced earlier is adequate for nonlinear analyses; however, because we need the stress and strain to be related to the same configuration for constitutive purposes (either deformed or undeformed, here the latter is chosen), we also introduce a new measure of stress that is conjugate to the large strain measure. It must be said, however, that the usual measure of stress is adequate for thin-walled structures because automatically a local coordinate system is used and the strains are not large. We introduce the new stress here for completeness and is an integral part of the Hex20 nonlinear solid modeling [9].

3.2.1 Modeling Large Deformations of Solids In the nonlinear examples discussed in the previous section, the springs exhibited both large stretching and large rotations and we were careful to use the axial stretching L = L − Lo as our measure of the deformation. In extending the nonlinear analysis to continuous systems, the measures of strain we have used thus far, such as, xx =

∂u , ∂x

γxy =

∂v ∂u + ∂x ∂y

are no longer sufficient for general application. The primary reason for this is that they do not take rigid body rotations (which do not result in strains) adequately into account. Our first priority is to introduce an appropriate measure of large strain. As a body deforms, various points translate and rotate. The easiest way to distinguish between deformation and the local rigid body motion is to consider the change in distance between two neighboring material points. Strain is a measure of this stretching of the material points within a body; that is, it is a measure of the relative displacement without rigid body motion. There are many measures of strain in existence, the one developed here uses Lagrangian variables, i.e., the reference state is the undeformed state. With reference to Fig. 3.7a, let the two material points before deformation have coordinates (x o ) and (x o + x o ), respectively; and after a deformation in the plane have the coordinates (x, y) and (x + x, y + y). The new and original positions are related to the displacements by

3.2 Large Deflections of Thin-Walled Structures

203

Fig. 3.7 Large deformations. (a) Stretching and rotation of a line element in the plane. (b) Large deformations in a general solid

x = xo + u ,

y = y o + v;

x = x o + u ,

y = y o + v

The distance between these initial and final neighboring points is given by s 2 = [x o + u]2 + v 2 Only in the event of stretching or straining is s 2 different from x o2 . That is, s 2 − x o2 = 2x o u + u2 + v 2 is a measure of the relative displacements that is insensitive to rotation. We can use this to define a measure of strain valid for large deformation that includes large rotations. Let Exx ≡

 ∂u 2  ∂v 2  2 2 s 2 − x o2 u ∂u 1 u 1 v 1 = + + = + + x o 2 x o2 2 x o2 ∂x o 2 ∂x o ∂x o 2x o2

The leading term is the elementary strain measure, the additional nonlinear terms account for the rotations. The space derivatives are with respect to the original positions irrespective of how much deformation occurs. Our measure of strain is easily generalized to multidimensions. With reference to Fig. 3.7b, let two material points before deformation have coordinates (xio ) and (xio + dxio ); and after deformation have the coordinates (xi ) and (xi + dxi ) where i, j range over 1, 2, 3. The initial and final distances between these neighboring points are given by dso2 =

 i

dxio dxio = (dx1o )2 + (dx2o )2 + (dx3o )2

and   ds 2 = dxi dxi = i

∂xm ∂xm o o dxi dxj , ij m ∂x o ∂x o i j

 ∂xi  dxi = dxjo = Fij dxjo o j ∂x j j

204

3 Nonlinear Elastic Shapes

respectively where Fij is called the deformation gradient. Only in the event of stretching or straining is ds 2 different from dso2 . That is, ds 2 − dso2 =

   ∂xm ∂xm  o o − δ [2Eij ]dxio dxjo ij dxi dxj = o o ij m ∂x ∂x ij i j

is a measure of the relative displacements. It is insensitive to rotation as demonstrated in Ref. [9]. The strain measure 2Eij ≡

 ∂xm ∂xm − δij , m ∂x o ∂x o i j

2[ Eij ] ≡

 ∂x T  ∂x  m m − I  ∂xio ∂xjo

(3.8)

is a symmetric tensor of the second order and is called the Lagrangian strain tensor. The tensorial nature of the measure (i.e., that it has particular transformation properties) is established in an example problem. Sometimes it is convenient to deal with displacements and displacement gradients instead of the deformation gradient. These are obtained by using the relations o xm = xm + um ,

∂xm ∂um = + δim o ∂xi ∂xio

The Lagrangian strain tensor Eij can be written in terms of the displacement as Eij =

1 2

 ∂u

i ∂xjo

+

∂uj  ∂um ∂um  + m ∂x o ∂x o ∂xio i j

(3.9)

Typical expressions for Eij in expanded nonindicial notation are  ∂u 2  ∂v 2  ∂w 2  + + ∂x o ∂x o ∂x o  ∂u 2  ∂v 2  ∂w 2  + 12 + + ∂y o ∂y o ∂y o  ∂u ∂u ∂u ∂v ∂v ∂v ∂w ∂w  = + + + + ∂y o ∂x o ∂x o ∂y o ∂x o ∂y o ∂x o ∂y o ∂u ∂x o ∂v = ∂y o

Exx = Eyy 2Exy

+

1 2

(3.10)

Note the presence of the nonlinear terms in the square brackets. Also note that the tensorial shear strain and the engineering shear strain are related by γxy = 2Exy . We have chosen to use the Lagrangian variables for the description of a body with finite deformation. For consistency, we need to introduce a measure of stress also referred to the undeformed configuration. Whether the deformation is large or not, the virtual strain energy in terms of the Cauchy stress is the same as given by Eq. (1.15), that is,

3.2 Large Deflections of Thin-Walled Structures

δU =



205

 δij =

σij δij dV ,

i,j

V

∂δuj ∂δui + ∂xj ∂xi

This is not surprising because the stress is already in the deformed state and the strain increments δij are small. Note, however, that the space derivatives are with respect to the deformed geometry and the volume is the deformed volume. We now restate this relation in terms of the original (undeformed) geometry; that is, we make the equivalence between the virtual work referred to the two different configurations. As shown in Ref. [9], the increments of strain in the two configurations are related by  δmn =

 ∂x o T  ∂x o  j i [δmn ]= [δEij ] ∂xm ∂xn

o

∂xio ∂xj δEij , i,j ∂xm ∂xn

xi =xio +ui

or

We also have that the change of volume is computed as dV = J o dV o ,

J o = det

 ∂x  ∂x o

where J o is the Jacobian of the deformation computed as the determinant of the deformation gradient. Substitute these relations into the virtual strain energy expression to get δU =

 i,j



 Vo

σijK δEij

dV = o

Vo

{σ K }T {δE} dV o

(3.11)

where terms other than δEij are grouped to form the Kirchhoff stress tensor given by ≡ Jo

 ∂x −1  ∂x (−1T ) j i [σ ] , mn o ∂xm ∂xno

[ σij ] =

1  ∂xi  K  ∂xj T [σmn ] o J o ∂xm ∂xno (3.12)

The meaning of [∂xi /∂xjo ]−1 is the ij component of the inverse of the matrix [∂xi /∂xjo ] and the superscript (−1T ) means the transpose of the inverse of the matrix. Significant examples of elastic behavior under large strains are those associated with rubber-like materials and soft-tissue biological materials [1, 4, 8]. In fact, polymeric materials in general fall into this category. This book does not have the space to explore the modeling and shapes of structures made from these materials. The situations of primary concern to us in structural analyses (especially when analyzing light-weight thin-walled structures) are when the deflections and rotations are large but the strains are small (that is, the strains are still within the elastic limit); then the local rigid body motions are the significant contributors to the nonlinear behavior. As shown in Ref. [9], the Lagrangian strain measure removes these rigid

206

3 Nonlinear Elastic Shapes

body terms as does the Kirchhoff measure of stress. Thus, for small strains (but large deflections and rotations) the Hooke’s law can be written as {E } = [ C ]{σ K } ,

{σ K } = [ D ]{E } ,

[ C ] = [ D ]−1

where [ D ] can be inferred from Eq. (1.5). In these, the tensors are arranged into vectors as [δEkl ] = [3 × 3] → {δE} = {6 × 1} = {δE11 , δE22 , δE33 , 2δE12 , 2δE23 , 2δE13 }T K K K K K K T [ σklK ] = [3 × 3] → {σ K } = {6 × 1} = {σ11 , σ22 , σ33 , σ12 , σ23 , σ13 }

While the relationship is linear between stress and strain, this is actually a nonlinear constitutive relation because of the nonlinear measures of stress and strain used. The strain energy in a linear elastic body (as just described) undergoing large deformations is given by  U=

1 2



Vo

 K K K K K K σxx Exx + σyy Eyy + σzz Ezz + σxy 2Exy + σxz 2Exz + σyz 2Eyz dV o

 =

1 2

Vo

 {σ K }T {E } dV o =

1 2

Vo

{E }T [ D ]{E } dV o

(3.13)

These expressions are similar to those already presented, the significant difference is the nonlinear measures of stress and strain. Thin-walled flexible structures such as plates and shells are typically modeled as being in a state of plane stress. Under this circumstance, the strain energy reduces to U=

1 2

   K K K Exx + σyy Eyy + σxy 2Exy dx o dy o dzo σxx

=

1 2

   2 2 2 dx o dy o dzo + Eyy + 2νExx Eyy ] + G4Exy E ∗ [Exx

(3.14)

where E ∗ = E/(1 − ν 2 ). These take additional special forms when particular assumptions about the variation of strains through the thickness are made. We presently introduce some of these special forms. Example 3.4 Give physical interpretation to the components of the Lagrangian strain tensor. Consider two deformed line elements ds and d s¯ as shown in Fig. 3.8; they are initially perpendicular to each other. Let E1 be the extension per unit original length of the element, that is,

3.2 Large Deflections of Thin-Walled Structures

207

Fig. 3.8 Deformation of two initially perpendicular line elements

E1 =

ds − dso dso

or

ds = (1 + E1 )dso = λ1 dso

where λ1 is called a stretch. For this line element, we also have ds 2 − dso2 = 2E11 dSo2 and combining with the above yields E11 =

ds 2 − dso2 = E1 + 12 E12 2dso2

or

E1 =

 1 + 2E11 − 1

There are similar relations for line elements originally in the x2o and x3o directions. The components E11 , E22 , and E33 are called the normal components of strain. The new length of a line segment at an arbitrary angle θ in the undeformed configuration is obtained from   ds 2 − dso2 = 2 E11 dx1o dx1o + E21 dx1o dx2o + E12 dx2o dx1o + E22 dx2o dx2o Realizing that dx1o = dso cos θ and so on, and that the strain of this arbitrary line segment is ds 2 − dso2 = 2Eθθ dso2 leads to Eθθ = E11 cos2 θ + 2E12 cos θ sin θ + E22 cos2 θ

(3.15)

where use is made of E12 = E21 . This gives us the transformation rule for the normal components of strain—they transform as second-order tensors [19]. Consider now two initially perpendicular lines which after deformation have components designated by dxi and d x¯i . Denoting the angle between these vectors by φ12 and taking the dot product, we obtain ds d s¯ cos φ12 =

 i

dxi d x¯i =



∂xi ikm ∂x o k

 ∂xi ∂xi ∂xi o dxko d x¯m = dx1o dx2o o i ∂x o ∂x o ∂xm 1 2

which can readily be rewritten in terms of the Lagrangian strain tensor as ds d s¯ cos φ12 = 2E12 dso d s¯o

208

3 Nonlinear Elastic Shapes

By substituting for ds and d s¯ in terms of the extensions, we get cos φ12 =

2E12 2E12 =√ = sin α12 √ (1 + E1 )(1 + E2 ) 1 + 2E11 1 + 2E22

where α12 = 12 π −φ12 is the change of angle. All the Lagrangian strain components contribute to the change of angle; however, it is only when E12 = 0 that the angle between the two line elements would be preserved, i.e., no shear. The component E12 therefore seems a good measure of the shearing of perpendicular line segments and is physically related to cos φ12 or sin α12 . The new angle between two perpendicular line segments oriented at an arbitrary angle θ is obtained from  ds d s¯ sin α12 =

 i

dxi d x¯i =



 ∂xi ∂xi o o dxko d x¯m = [2Ekm +δkm ]dxko d x¯m o o i,k,m∂x ∂x k,m m k

Expanding this gives  = (2E11 +1)dx1o d x¯1o +2E21 dx1o d x¯2o +2E12 dx2o d x¯1o +(2E22 +1)dx2o d x¯2o ds d s¯ sin α12

Realizing that the undeformed segment lengths are given by dxio = dso {cos θ, sin θ, 0} ,

d x¯io = d s¯o {− sin θ, cos θ, 0}

and that the shear strain of these arbitrary perpendicular lines is   = 2E12 dso d s¯o ds d s¯ sin α12

leads to  = −(E11 − E22 ) cos θ sin θ + E12 (cos2 θ − sin2 θ ) E12

(3.16)

This gives us the transformation rule for the components of shear strain—they, too, transform as components of a second-order tensor. Example 3.5 Use Hex20 elements to model the large deflection, large strain loading of the block shown in Fig. 3.9. One end of the block is fixed while the other end has a distributed traction but the discrete loads do not change direction during the loading history. Thus these applied loads transition from being initially a shear traction to being a mostly normal traction as the orientation of the area changes. The effective nodal loads are computed using the PoVW in conjunction with the interpolation functions [9] to give for the edge and center nodes, respectively, 1 Pe = − 12 tA ,

Pc = + 13 tA

3.2 Large Deflections of Thin-Walled Structures

209

1.0

.

0.5

0.0

-0.5

Load 0.

10.

20.

30.

40.

Fig. 3.9 Large deformation of a cantilevered block. (a) Initial and deformed shapes using five Hex20 elements. (b) Displacements against load. Dashed lines are for top nodes

where t is the applied traction and A is the area. The x axis is along the length and the y axis is the transverse loading direction. Figure 3.9b shows the displacements at the four corner end nodes. It is expected that the w displacements be nearly zero because (initially, at least) the axial stress (and hence strains) are zero at the end and hence there is no Poisson’s ratio effect. The other displacements show very large values with a definite stiffening effect at the large loads. The discrete data points indicate that the analysis was performed in a load incremental fashion. As part of the nonlinear incremental modeling developed implicitly in Ref. [9], the total stiffness is constructed as a combination of the elastic stiffness and geometric stiffness [ KT ] = [KE ] + [KG ]. The element elastic and geometric stiffness matrices coded in subroutines «stiffE_HEX20» and «stiffG_HEX20», respectively, of module «nonl324.for» for the SDsolid program. This is also our choice for solving nonlinear problems involving plates and shells. To reiterate, there are times when the reduced shell FE models such as the MRT/DKT element [6] are very useful, other times these reduced models are questioned and a higher order model is required. This is what the Hex20 element provides. This problem illustrates how transverse loads end up being supported by axial actions when the deformations become large. Example 3.6 The cable shown in Fig. 3.10a does not have any pretension. Show how the generation of axial forces produces a transverse stiffness allowing the cable to support the transverse applied load. This is a companion problem to that of Fig. 1.14. It is a nonlinear problem because the (transverse) stiffness arises only after deformation has begun to occur. We first establish the strain energy in the cable because with that in hand, a straightforward application of the Ritz method then leads to an estimate of the deflection behavior even though the problem is nonlinear. As seen presently, this

210

3 Nonlinear Elastic Shapes ♦

0.05 0.04 0.03 .

♦

0.02 0.01 0.00

.

♦

0.00

0.05

0.10

0.15

Fig. 3.10 Deflection of a fixed–fixed cable due to a uniform transverse load. Circles are FE generated data. (a) Deflected shape and axial force distributions. (b) Load-deflection curve where the normalization factor is qR = EA/Lo

is not the same energy expression as derived earlier in Sect. 1.2 for the pretensioned cable because there is no pretension in this problem. The strain in the cable is given by Exx =

∂u + ∂x o

 1 2

∂v ∂x o

2

As a first approximation, assume that the axial strain can be neglected so that the strain energy becomes  U=

1 2



 Lo

2 EAExx

dx = o

1 2

Lo

EA 14

∂v ∂x o

4 dx o

where Lo is the original length of the cable. The virtual work of the applied distributed load is   o o o qv (x )δv(x ) dx = qo δv(x o ) dx o δ We = Lo

We are now in a position to estimate the stiffness behavior. Taking a Ritz approach, let the assumed deflected shape be given by v(x o ) = v1 [x o Lo − x o2 ]/L2o = v1 g1 (x o ) where v1 = 4vc with vc being the deflection at the center. This deflected shape satisfies the geometric BCs of zero deflection at each end. Substitute into the energy expression and integrate to get U = 12 EA

v14 Lo , 20L4o

δ We = P1 δv1 = qo 16 Lo δv1

3.2 Large Deflections of Thin-Walled Structures

211

Equilibrium is then given by v3 ∂U − P1 = EA 1 4 Lo − qo 61 Lo = 0 ∂v1 10Lo

or

qo =

6 EA v13 10 Lo L3o

This relation is shown plotted as the dashed line in Fig. 3.10b; the comparison with the nonlinear FE results is reasonable, at least it captures the initial behavior of zero stiffness followed by strong stiffening. The stiffness is computed as KT =

∂2 U 3 EA v12 = 10 Lo L2o ∂v12

This shows a stiffness increasing quadratically with deflection as well as a dependence on the axial (rod) stiffness term EA/Lo . The axial force is given by  ∂v 2 F¯ = EAExx = EA 12 = EA 12 v12 [Lo − 2x o ]2 ∂x o This is shown in Fig. 3.10a as the dashed curve. It contrasts considerably with the FE generated results which exhibits an almost constant distribution of axial force. The problem with the axial force lies in neglecting the u,x contribution, because any higher approximation of v(x o ) would always have a zero derivative at the center. As an improved displacement representation, let u(x o ) = u2 sin(2π x o /Lo ) = u2 sin(n¯ 2 x o ) ,

v(x o ) = v1 sin(π x o /Lo ) = v1 sin(n¯ 1 x o )

The axial displacement is zero at both ends and at the center. The strain energy evaluates to  2  1 U = 2 EA u2 n¯ 2 cos(n¯ 2 x o ) + 12 (v1 n¯ 1 cos(n¯ 1 x o ))2 dx o   = 12 EA u22 n¯ 22 21 + u2 n¯ 2 v12 n¯ 21 41 + 14 v14 n¯ 41 81 Lo The virtual work of the applied distributed load is  qo δv1 sin(n¯ 1 x o ) dx o = qo 2/n¯ 1 δ We = Lo

There are two equilibrium equations u2 :

EAu2 n¯ 22 21 Lo + 12 EAn¯ 2 v12 n¯ 21 41 Lo = 0

v1 :

EAu2 n¯ 2 v1 n¯ 21 41 Lo + 12 EAv13 n¯ 41 81 Lo = qo 2/n¯ 1

212

3 Nonlinear Elastic Shapes

The first of these gives u2 = −

n¯ 21 2 v 4n¯ 2 1

This is like a geometric constraint. On substituting this into the second equation, we get qo =

1 ¯ 51 v13 , 16 EALo n

KT =

3 ¯ 51 v12 16 EALo n

The load-deflection relation is shown plotted in Fig. 3.10b as the full line. There is good agreement with the FE generated results. The axial force is   F¯ (x o ) = EA u2 n¯ 2 cos(n¯ 2 x o ) + 12 (v1 n¯ 1 cos(n¯ 1 x o ))2 = 14 EAn¯ 21 v12 and is shown plotted in Fig. 3.10a as the full line. Surprisingly, it is constant and in good agreement with the FE data. This example problem shows how directly the Ritz method can be applied to nonlinear problems; the only strict requirements are that an estimate of the strain energy be available, and the assumed deformed shape satisfies the geometric BCs.

Example 3.7 Use the strong formulation to model the cable problem of Fig. 3.10. Consider a segment of cable with end axial forces Fo , FL and shear forces Vo , VL where L is the length of segment. The total potential is 



=

1 2

2 dx EAExx

−

L [qu u + qv v] dx − [F u + V v]o

where, for convenience, it is understood that the initial configuration is the reference state, i.e., the superscript o for original configuration is understood on all relevant quantities. The strain is Exx = u,x + 12 v,2x so that the PoVW becomes 

 δ =

L [qu δu + qv δv] dx − [F δu + V δv]o = 0

EAExx [δu,x +v,s δv,x ] dx −

Using integration by parts, we get 

 −

[(EAExx ),x +qu ]δu dx −

[(EAExx v,x ),x +qv ]δv dx

L L + [EAExx − F ]δuo + [EAExx − V ]δv o = 0 The integrals give the governing equations −(EAExx ),x −qu = 0 ,

−(EAExx v,x ),x −qv = 0

3.2 Large Deflections of Thin-Walled Structures

213

When qu = 0, the first equation has the result that (EAExx ),x = 0

EAExx = constant = F¯o

⇒

Consequently, we can write the second equation as (F¯o v,x ),x = −qv

F¯o v,xx = −qv

or

This is, in fact, the governing equation for the linear behavior of a cable; the difference is that we interpret x as actually x o . For our problem qv = qo , therefore integrating the governing equation twice and imposing the zero displacement BCs give F¯o v = 12 qo [Lo x − x 2 ] Interestingly, this is the same shape as the simple Ritz function of the previous example problem. Our solution is only implicit because we do not know the relation between F¯o and qo . But because F¯o is constant, then it is equal to its average, giving 1 F¯o = Lo

 EAExx dx =

1 Lo

 EA[u,x + 12 v,2x ] dx =

1 Lo

 EA[ 12 v,2x ] dx

The last form is because u is zero at the integration limits. Substitute for v,x from v,x = 12 (qo /F¯o )[Lo − 2x] and integrate to get 1 EA 12 (qo /2F¯o )2 31 L3o F¯o = Lo

or

F¯o3 =

2 2 1 24 EALo qo

Additionally, evaluating the solution at the center, we get F¯o v1 = 18 qo L2o . Replace qo with v1 so that we finally get EA 8 F¯o = 2 v12 Lo 3 In comparison to the previous solution, we have 1 8 L2o 3

vs

n¯ 21 4

or

8 = 2.67 3

vs

π2 = 2.47 4

This is a nice comparison and the plot would superpose on those of Fig. 3.10 quite closely.

214

3 Nonlinear Elastic Shapes

3.2.2 Straight Beams and Flat Plates Superficially, the block in Fig. 3.9 looks like a cantilever beam; however, the associated modeling is that of a fully 3D continuum. In other words, none of the modeling would change even if the initial shape is significantly altered. In this subsection, we are specifically concerned with developing nonlinear models for slender beams and thin plates. That is, we a priori impose kinematic constraints on the allowable deformations. We develop two distinct models for the strain energy, one for the membrane actions, the other for the flexural actions.

Action I: Membrane Let the coordinate system for the plate be as shown in Fig. 2.14c. Because the plate is thin, the displacements for the membrane action can be represented by u(x ¯ o , y o , zo ) ≈ u(x o , y o ) ,

v(x ¯ o , y o , zo ) ≈ v(x o , y o ) ,

w(x ¯ o , y o , zo ) ≈ w(x o , y o )

There is no zo dependence because the strain is assumed uniform through the thickness. Furthermore, while there is transverse deflection, this is not associated with any flexural straining. Substitute into Eq. (3.10) and regroup terms to get the large deflection measure of strain as 2 Exx = u,x + 12 [u,2x +v,2x +w,x ],

2 Eyy = v,y + 12 [u,2y +v,2y +w,y ]

2Exy = u,y + v,x + [u,x u,y + v,x v,y + w,x w,y ]

(3.17)

where the subscript comma indicates partial differentiation and the subscripted variable refers to the original configuration, i.e., xio . The nonlinear contributions are in square brackets. These resemble the 3D strains given earlier, the difference is that there is no Ezz and associated shears Exz , Eyz in the formulation because the plate is modeled as being in plane stress. Now that all differentiations have been performed, we no longer need to distinguish the undeformed and deformed geometries and can drop the superscript “o.” The strain energy for a plate in plane stress is given by Eq. (3.14), when the strains are substituted and integrated through the thickness, we get UM =

1 2

 

 2 2 2 dxdy + Eyy + 2νExx Eyy ] + Gh4Exy E ∗ h[Exx

(3.18)

This expression, when written in terms of displacements, is rather lengthy and we leave it to later after some additional approximations are introduced. The membrane (axial) beam equations can be obtained by setting w = 0 and assuming uniaxial stress. The resulting expression reduces to

3.2 Large Deflections of Thin-Walled Structures



UM =

1 2

 2 EAExx dx =

1 2

 =

1 2

215

 2 EA u,x + 12 u,2x + 12 v,2x dx

(3.19)

  EA u,2x +u,3x +u,x v,2x + 14 u,4x + 12 u,2x v,2x + 14 v,4x dx

The coupling between the u and v displacements is evident. Sometimes, especially for beams, we find it useful to use the approximation v,s ≈ φz so that the full strain expression for the beam becomes Exx = o + 12 [o2 + φz2 ] ,

o = u,x ,

φz = v,x

But it should be realized that it is not generally true as shown next.

Action II: Flexure of Beams Modeling the nonlinear flexural behavior is more complicated than that of the membrane action, we therefore first introduce the major ideas in the context of a slender beam deforming in the plane. The main kinematical assumption is that the flexural actions do not cause any stretching action of the middle surface because the stretching is already accounted for through the membrane action modeling. To explain, consider the plane deflection of a beam; Fig. 3.11a shows the deflection of the centerline, as well as that of a point off the centerline. Let s be the distance of a material point along the beam; the assumption is that the point always has the same value of s because this middle line is assumed to be inextensible, this is the elastica assumption. A point originally at position x o = s, y o = 0, moves to a location (x ∗ , y ∗ ) a distance s along the elastica. The inset triangle shows that sin φ =

dy ∗ , ds

cos φ =

dx ∗ , ds

 dy ∗ 2 ds

+

Fig. 3.11 Deformation of an elastica. (a) Geometry of the centerline

 dx ∗ 2 ds

=1

216

3 Nonlinear Elastic Shapes

The third equation is the inextensibility condition. By differentiating the first two and a little manipulation, we get that φ,s = x,∗s y,∗ss −y,∗s x,∗ss In this form, φ,s is related to only the behavior of the centerline. Now consider a point initially at (x o , y o ) and moves to a position (x, y). This movement entails a displacement of the centerline plus a rotation of the (original) vertical. The new position is given by x = x ∗ − y o sin φ ,

y = y ∗ + y o cos φ

Differentiating gives the deformation gradients x,s = x,∗s −y o φ,s cos φ = x,∗s [1 − y o φ,s ] , y,s = y,∗s −y o φ,s sin φ = y,∗s [1 − y o φ,s ]

Realizing that s and x o are the same, then the Lagrangian strain component in terms of the deformation gradients becomes 2Exx =

∂x ∂x ∂y ∂y + o o − 1 = [x,∗s (1 − y o φ,s )]2 + [y,∗s (1 − y o φ,s )]2 − 1 o o ∂x ∂x ∂x ∂x

Expanding and regrouping terms, we get ∗2 o ∗2 ∗2 o2 ∗2 ∗2 2 2Exx = [x,∗2 s +y,s −1] − 2y [x,s +y,s ] φ,s +y [x,s +y,s ] φ,s

The leading term is zero because the centerline is assumed inextensible, the third term is neglected because the beam is slender (y o is considered small) and we want a linear distribution of strain on the cross section. The second term simplifies to −2y o φ,s because of inextensibility. Noting that flexural strain is related to curvature by xx = −yκxx , we then have  2 UF = 12 EI κxx dx (3.20) κxx = φ,s = 1/r , This is our key result for the curvature. The relation resembles the linear modeling results reported in Sects. 1.2 and 2.1, but we should not interpret φ,s as being synonymous with v,ss , as we now demonstrate. The position of the centerline is given by x∗ = xo + u = s + u ,

y∗ = yo + v = 0 + v

so that our final expression for the curvature, expressed in alternate forms, is κxx = φ,s = x,∗x y,∗xx −y,∗x x,∗xx = [1 + u,s ]v,ss −v,s u,ss = [1 + u,x ]v,xx −v,x u,xx

3.2 Large Deflections of Thin-Walled Structures

217

Although we associate flexural actions with the transverse deflections, these relations show that there is coupling with the axial displacements. The reason is: under large deflections, an initially straight beam becomes a curved beam and as amply demonstrated in Sect. 2.1, curved beams have coupled deformations. We also have the slope relationships tan φ =

v,s y,∗s = , ∗ x,s 1 + u,s

φ,s =

v,ss v,ss = cos φ 1 + u,s

Just as for the linear analysis of the initially curved beam (see, for example, Eq. (2.4)), the nonlinear slope involves a coupled deformation.

Action III: Approximate Nonlinear Behavior of Plates Our approximation for the large deflection of a thin plate is given by u(x ¯ o , y o , zo ) ≈ u(x o , y o ) − z

∂w , ∂x o

v(x ¯ o , y o , zo ) ≈ v(x o , y o ) − z

∂w ∂y o

w(x ¯ o , y o , zo ) ≈ w(x o , y o ) The first example problem discusses this approximation in the context of beams. The z = zo dimension is measured relative to the center plane of the plate. Substitute into Eq. (3.10) and regroup terms to get the large deflection measure of strain E¯ xx = Exx − zκxx ,

E¯ yy = Eyy − zκyy ,

2E¯ xy = 2Exy − z2κxy

where the overhead bar is used to denote the total (membrane plus flexural) strain. Consistent with the thin plate assumption, higher powered terms in z are neglected. The Eij terms are as already given in Eq. (3.17), the approximate curvatures are given by κxx = w,xx +[u,x w,xx +v,x w,xy ] ,

κyy = w,yy +[v,y w,yy +u,y w,xy ]

2κxy = 2w,xy +[u,y w,xx +v,x w,yy +u,x w,xy +v,y w,xy ]

(3.21)

The nonlinear contributions are in square brackets. The strain energy for a plate in plane stress is given by Eq. (3.14) using the total strain E¯ ij . When the strains are substituted into the energy expression, we get z0 , z1 , 1 3 and z2 terms; these integrate to h, 0, and 12 h , respectively. Labeling the h terms as membrane and h3 as flexural, we get a separation of the energies as U = UM + UF where the membrane energy is as already given in Eq. (3.18), the flexural energy is

218

3 Nonlinear Elastic Shapes

UM =

1 2

UF =

1 2

 

 2 2 2 dxdy + Eyy + 2νExx Eyy ] + Gh4Exy E ∗ h[Exx

   2 2 2 ¯ xy dx dy + κyy + 2νκxx κyy ] + D4κ D[κxx

(3.22)

and D¯ = Gh3 /12 as before. While we have managed to separate the energy expressions, a large deflection plate problem involves simultaneous membrane and flexural actions because of the nonlinear coupling of the displacements in the strain expressions. Thus separation of the energy expressions does not imply separation of the membrane and flexural displacements because, for example, Exx contains u, v, w as does κxx . If all nonlinear terms in u and v are neglected as small (this situation approximately prevails, for example, when all sides of a plate or panel are constrained), then we recover the so-called von Karman plate strains [14] given by 2 Exx = u,x + 12 w,x ,

κxx = w,xx ,

2 Eyy = v,y + 12 w,y ,

κyy = w,yy ,

2Exy = u,y + v,x + w,x w,y

2κxy = 2w,xy

(3.23)

The flexural contributions are those of the linear model and the membrane strains have contributions similar to that of a cable. That is, large out-of-plane deflections w(x o , y o , zo ) generate significant in-plane strains Eij (x o , y o ). We make the additional approximation for the membrane strains that w,x = −φy , w,y = +φx . At this stage it is worth our while to introduce some unifying terminologies. The significant nonlinear effects reside in the membrane strains. The x-component of the membrane strain and strain energy density are, respectively, Exx = o + 12 v,2x ,

∗ 2 2 UM = Exx = o2 + o v,2x + 14 v,4x

We label the first energy contribution, obviously, the linear contribution. We call the third contribution the “cable energy” because, as per Fig. 3.10 and associated discussions, this energy term always contributes a positive stiffness related to the deflection squared. Most, if not all, interesting nonlinear phenomena arise from the middle term. This term arises from an interaction between the axial deformation and the transverse deformation. A descriptive label is the interaction energy, but more commonly it is called the geometric stiffness contribution; as we proceed we use both terms interchangeably. Because stiffness is related to strain energy, this term can profoundly affect the structural stiffness and hence the structural response. The example problems to follow explore cases where the significant displacement is the out-of-plane displacement, although, as demonstrated, coupling with the in-plane displacements can be crucial even if these displacements are small. Example 3.8 Up to this point for the flexure of beams, we have assumed that y o is small (the beam is slender) but we have not restricted φ, all slope interactions have been expressed in terms of sin φ and cos φ. Our result is thus valid even for very

3.2 Large Deflections of Thin-Walled Structures

219

large deflections and rotations. Follow the consequences of assuming the rotations are small. Assume a displacement representation as u(x ¯ o , y o ) = u(x o ) − y o φ(x o ) ,

v(x ¯ o , y o ) = v(x o )

which is commonly done, for example, in Ref. [6]. Then, from the beginning, the modeling is restricted to small rotations (and implicitly, small deflections). Furthermore, the modeling requires that φ and v be treated as independent DoF (as done for the Timoshenko beam model in Sect. 2.3) or be related through the constraint φ = v,x arising from “plane sections remain plane.” Let us follow this latter condition and assume u(x ¯ o , y o ) = u(x o ) − y o v,x (x o ) ,

v(x ¯ o , y o ) = v(x o )

The terms in the Lagrangian strain can be collected as 2 ] − y o [1 + u,x ]v,xx + 12 y o2 v,2xx Exx = u,x + 12 [u,2x +v,2x +w,x

The leading term is the correct membrane strain, the third term can be discarded because the beam is slender, and we see that the second term contains an approximation for the curvature that is only partially reflected in Eq. (3.21). An example problem to follow quantifies this approximation. The conclusion is that the approximation is good providing the rotation are not very large. Example 3.9 Figure 3.12a shows the large deflection (not exaggerated) of a beam under transverse distributed loading. Figure 3.12b shows the FE generated strain and curvature. The right end is on rollers purposefully to exaggerate the flexural actions. Examine the applicability of the nonlinear measures of membrane and flexural strains. The beam was modeled with 128 2D frame elements, and the nonlinear FE analysis used the implicit corotational scheme. The primary data collected were the displacements and these were postprocessed to give the various space derivatives. The strain data were then compared to the strain data generated directly by the FE analysis. Figure 3.13 shows the processed data for the first and second derivatives. These data are plotted against x o and the derivatives are with respect to x o . Observe that the FE generated values for φ are not the same as the first derivative of v which is especially noticeable near the right end but is also true to the left of the inflexion point. In fact, as shown earlier, we have that tan φ =

∂v 2 ∂u  ∂y ∗ 2 ∂x ∗ = o 1+ o o o ∂x ∂x ∂x ∂x

220

3 Nonlinear Elastic Shapes

.

1.

0.

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 3.12 Large deflections and rotations of a transversely loaded beam. (a) Geometry and deformed shape. (b) Nonlinear measures of strain and curvature. Circles are direct FE results, lines are processed results, dashed lines are contributors, and horizontal arrows are the zero reference

1.

♦

♦

.

♦

♦

.

1. 0.

♦

0. ♦

0.0

0.2

0.4

.

0.6

0.8

1.0

0.0

0.2

0.4

.

0.6

0.8

1.0

Fig. 3.13 Processed FE data. Horizontal arrows are the zero references. (a) Axial behavior. (b) Transverse behavior

In the present case, ∂u/∂x o is negative and sizable, making the slope larger than ∂v/∂x o . The nonlinear axial strain is given by Exx =

 ∂u 2  ∂v 2 ∂u + 12 + 12 = u,x + 12 u,2x + 12 v,2x o o ∂x ∂x ∂x o

Figure 3.12b shows that u,x and 12 v,2x are of comparable magnitude and both are three orders of magnitude larger than the strain. Consequently, the sum of the three contributors gives a rather noisy estimate of Exx . This is the same consequence encountered in relation to Fig. 2.6.

3.2 Large Deflections of Thin-Walled Structures

221

The nonlinear curvature is given by  ∂u  ∂ 2 v ∂v ∂ 2 u − = [1 + u,x ]v,xx −v,x u,xx κxx = 1 + o ∂x ∂x o2 ∂x o ∂x o2 In this particular case, the distributions of u,x v,xx , and v,x u,xx are similar with the magnitude of the latter about twice that of the former. The combined nonlinear contribution has only a small contribution to the overall curvature. Consequently, it is a common practice to neglect these nonlinear contributions. This is an approximation we frequently make in the following example problems and analyses.

Example 3.10 Consider a fixed–fixed beam loaded as shown in Fig. 3.14a. The linear modeling indicates that no axial forces are generated but the FE generated nonlinear results show that an axial force, almost uniformly distributed along the length, more than five times the applied force is generated. Construct a model to explain what is happening. This is like the earlier cable problem of Fig. 3.10 except that flexural actions are present. Let us, therefore, begin by constructing a simplified model to ascertain the severity of the nonlinearity. We use the strain energy separated as  UM =

1 2

 EA[u,2x

+u,x v,2x

+ 14 v,4x

] dx ,

UF =

1 2

EI v,2xx dx

These are taken from Eq. (3.19); we have neglected the nonlinear flexural contributions plus some higher nonlinear membrane terms. Assume the problem is dominated by the transverse deflection so that we can write the deformations as v(x) = vo 16[x 2 /L2 − 2x 3 /L3 + x 4 /L4 ] ♦

u(x) ≈ 0 ,

10.

.

5.

0.

♦

0.

1.

2.

3.

Fig. 3.14 Large deflections and rotations of a transversely loaded beam. (a) Geometry and properties. (b) Load-deflection curves. Circles are FE results, squares are pinned–pinned BCs, lines are model results, and QR = 500(EI /L3 )h. (c) Mohr’s circle for stiffness

222

3 Nonlinear Elastic Shapes

where vo is the deflection of the beam center. The latter satisfies the geometric BCs of zero displacement and slope at each end and therefore is a suitable Ritz function. Substitute into the energy expressions to get UM = 18 EAvo4 I4 /L3 ,

UF = 12 EI vo2 I1 /L3

where I1 = 204.8 and I4 = 34.92. The virtual work of the applied load is δ We = Pv δvo = Qo δv(x = 12 L) = Qo δvo

⇒

P v = Qo

Application of our PoVW then gives EI vo I1 /L3 + 12 EAvo3 I4 /L3 = Qo

or

& % Qo = EI vo I1 + 12 EAvo3 I4 /L3

The leading term is the linear solution. Figure 3.14b shows the nonlinear results as the dashed line. It has the general trend of the FE results although off. Most important, however, is the significant difference with the linear modeling shown as the fine straight line. We conclude that this problem is dominated by the nonlinearities, and in particular by the cable effect associated with v,4x . That is, the solution shows that the beam behaves like a cable but with additional flexural stiffness. The total stiffness is KT =

& ∂2 % UF + UM = EI I1 /L3 + 32 EAvo2 I4 /L3 2 ∂vo

The linear and nonlinear contributions to the stiffness are comparable when vo ≈ h/2. Our model gives the axial force as F = EA[u,x + 12 v,2x ] = EA128[2x/L − 6x 2 /L2 + 4x 3 /L3 ]2 vo2 /L2 This is zero at the ends and the center and therefore obviously not uniform along the length; however, it is always positive and the peak value is about eight times the applied load so that its average is about that of the FE results. The axial displacement distribution is such that it is zero at the ends and at the center (because of no horizontal force). The simplest representation satisfying these is u(x) = uo [x/L − 3x 2 /L2 + 2x 3 /L3 ] It turns out, however, that the derivative of this function is orthogonal to v,2x and there is no coupling strain energy (u,x v,2x ) contribution. The next simplest function is sin(4π x/L), this has a total of five zeroes. The strain energies are UM = 12 (EA/L)I2 u2o + 12 (EA/L2 )I3 uo vo2 + 18 (EA/L3 )I4 vo4 , UF = 12 (EI /L3 )I1 vo2

3.2 Large Deflections of Thin-Walled Structures

223

where I1 = 204.8 ,

I2 = 78.96 ,

I3 = −27.74 ,

I4 = 34.92

There are two equilibrium equations given by Fv = (EI /L3 )I1 vo + (EA/L2 )I3 uo vo + 12 (EA/L3 )I4 vo3 = Qo Fu = (EA/L)I2 uo + 12 (EA/L2 )I3 vo2 = 0 The second equation can be solved explicitly to give uo = − 12 (I3 /I2 )vo2 /L This shows that uo is a second-order effect the same as occurred for the horizontal springs in Fig. 3.4. Substitute for uo into the first equilibrium equation to get Qo = (EI /L3 )I1 vo + 12 (EA/L3 )[−I32 /I2 + I4 ]vo3 This is shown in Fig. 3.14b as the heavy continuous line; there is an improvement in the correlation with the FE results. Adding axial extensibility has softened the structural stiffness somewhat (there is more deformation for a given load level) and this is worth exploring in a little more depth because it arises from the coupling of the transverse deflections and the axial displacements. The stiffness matrix is given by [ K ]=



    ∂ 2 U  EI I1 0 EA I3 uo /L + 32 I4 vo2 /L2 I3 vo /L = 3 + I2 0 0 I3 vo /L ∂uI ∂uJ L L

The coupling of the deformations is tracked through the energy integral parameter I3 . This appears in the stiffness matrix in two places: the diagonal stiffness K11 and the off-diagonal stiffness K12 . Consider an initially diagonalized system, as shown in Fig. 3.14c an off-diagonal contribution expands the radius (regardless of sign); consequently one stiffness eigenvalue must decrease. We can confidently say that the elastic coupling of separate elastic systems results in a softening of the combined system. This conclusion goes beyond our immediate beam problem and has great significance for the stability analyses of the next two chapters. Finally, although the force-deflection curve improved by the addition of extensibility, the axial member force distribution remains poor. The reason for this is explained in Sect. 3.4 using spectral analysis. Example 3.11 A thin plate of size [a × b × h] has simple supports at x = 0, x = a and a uniform transverse applied load. Contrast the nonlinear stiffness behavior when the other two sides are either simply supported or free.

224

3 Nonlinear Elastic Shapes

Our simple assumption is that the boundary constraints prevent any sizable inplane displacements from occurring. That is, we assume there is only a transverse displacement state accompanied by bending strains related only to the curvature of the plate. We can write the total displacements as u(x, ¯ y, z) = 0 − zw,x ,

v(x, ¯ y, z) = 0 − zw,y ,

w(x, ¯ y, z) = w(x, y)

so that the strains from Eq. (3.10) become 2 − zw,xx , Exx = 12 w,x

2 Eyy = 12 w,y − zw,yy ,

2Exy = w,x w,y − 2zw,xy

When these strains are substituted into the strain energy expression, we get z0 , z1 , 1 3 and z2 terms; these integrate to h, 0, and 12 h , respectively. Labeling the h terms as 3 membrane and h as flexural, we get for the energies UM =

1 2

UF =

1 2

   E ∗ h 14 [w,4x +w,4y +2νw,2x w,2y ] + Gh[w,2x w,2y ] dxdy  

 2 2 2 ¯ + w,yy + 2νw,xx w,yy ] + D[4w ] D[w,xx ,xy dx dy

It is quite curious and interesting that the single transverse displacement w(x, y) causes both membrane and flexural actions, the difference is in the order of the space derivatives with the membrane terms being lower order derivatives. Furthermore, the flexural action on its own is linear (in that there are only quadratic energy terms) but the problem as a whole is nonlinear because both membrane and flexural actions occur simultaneously and the membrane energy is nonlinear. Note that the product term w,x w,y in the membrane energy is not just associated with the shear strain but also occurs due to the product Exx Eyy . In contrast, the mixed derivative term w,xy is exclusive to the flexural shear behavior. It is instructive to compare the relative contributions of each of the terms in our energy expressions. The two cases of interest have the deflection representations ss-ss: free-free:

w(x, y) = w1 sin(nπ x/a) sin(mπy/b) w(x, y) = w1 sin(nπ x/a)

The number of half-waves in the x and y directions are set by n and m, respectively. We expect m and n to be unity but the present form is retained because it is the form used in the plate buckling analyses in Chap. 4. Because the free-free case can be recovered from the ss-ss case by disappearing the (mπ/b) terms, we concentrate on the ss-ss case. Performing the integrations gives UM =

1 2



  9   9  + Ghw14 n¯ 2 m ab E ∗ hw14 n¯ 4 + m ¯ 4 + 2ν n¯ 2 m ¯2 ¯2 256 256

3.2 Large Deflections of Thin-Walled Structures

UF =

1 2

225

  1   ¯ 12 n¯ 2 m Dw12 n¯ 4 + m ¯ 4 + 2ν n¯ 2 m ¯ 2 + Dw ¯ 2 ab 4

where n¯ = nπ/a and m ¯ = mπ/b. It is possible to simplify the flexural energy expression by replacing G with E/2(1 + ν); this is not done here because it is useful to separately track the contribution of the shear strain term. The potential of the applied load is  V=−

qw (x, y)w(x, y) dxdy = −qo w1

4 4 ab = −qo w1 ab n¯ m ¯ nmπ 2

Equilibrium is achieved through  ∂  UM + UF + V = 0 ∂w1 This results in qo =

9  9 nmπ 2  ∗ 3  4 E hw1 n¯ + m + Ghw13 n¯ 2 m ¯ 4 + 2ν n¯ 2 m ¯2 ¯2 8 64 64  1    ¯ 12 n¯ 2 m ¯ 4 + 2ν n¯ 2 m ¯ 2 (mπ/b)2 + Dw ¯2 2 +Dw1 n¯ 4 + m 2

which is shown plotted as the full line labeled I in Fig. 3.15b; it shows a strong nonlinear effect. In fact, the linear solution on this plot is essentially flat (i.e., very little stiffening effect). We conclude that nonlinear membrane effects predominate when the transverse deflections are on the order of the thickness of the plate and the plate edges are constrained. When the two opposite edges are free, the load-deflection relation reduces to

6.

.

4. 2. 0. 0.

2.

4. .

6.

8.

Fig. 3.15 A uniformly laterally loaded plate with different boundary conditions. (a) Geometry. (b) Load-deflection curve for the center point. The FE generated results are the squares (Hex20 elements) and triangles (MRT/DKT elements), and the continuous lines are the model results. The load is normalized by qo = E × 10−6

226

3 Nonlinear Elastic Shapes

qo =

 nπ 4 1  nπ 2  ∗ 3  nπ 4 9 E hw1 + Dw1 8 a 64 a 2

where it is the y = 0, y = b edges that are free. This is shown plotted as the full line labeled II in Fig. 3.15b; it too shows a strong nonlinear effect but with larger deflections. This result is essentially the same as for the cable in Fig. 3.10 but with the addition of the flexural behavior. However, here as for the cable, the flexural contribution is negligible. When we switch a and b then we have the solution where it is the shorter x = 0, x = a edges that are free. This is shown plotted as the full line labeled III in Fig. 3.15b; it is indistinguishable from the ss-ss solution thus showing that it is the membrane behavior of the shorter side that dominates the stiffness. Also shown in Fig. 3.15b are the FE generated results using the MRT/DKT shell element [6] and Hex20 solid element [9]; there is a nice agreement between the model and these FE results. The stiffness is obtained from KT =

 ∂2  U = A + B w12 + U M F ∂w12

from which it is clear that it has constant and w12 terms. For large deflections, it is the latter terms that dominate.

8.

♦

Example 3.12 The thin plate shown in Fig. 3.16 is simply supported on three sides and is free on the fourth. It has an applied transverse load that is uniformly distributed on the face. The FE generated results for the transverse deflection shown in Fig. 3.16b exhibit a very strong nonlinear behavior. Construct a model to explain the contributions to this nonlinear behavior.

6.

.

4. 2. 0.

♦

0.

1.

2.

.

3.

4.

5.

Fig. 3.16 A plate with one free edge and loaded laterally with a uniform pressure. (a) Geometry. (b) Load-deflection curve for the top central point

3.2 Large Deflections of Thin-Walled Structures

227

In the FE modeling, the simply-supported BCs at the ends allow contraction in the y-direction but none in the x-direction and the BCs is like periodic BCs in x. That is, if the plate is extended in x but the same BCs are imposed for each segment of length a and the load direction alternates, then the same results are obtained. Looking ahead, the buckling shapes of Fig. 4.23 help to visualize what happens. Let us begin by assuming the problem is dominated by the flexural deformation of the plate and that the deflected shape has a sinusoidal shape. That is, u(x, y) = 0 , v(x, y) = 0 , w(x, y) = w1 sin

 nπ x  y a

b

= w1 sin

 nx ¯ y a

b

, n=1

where w1 is the transverse displacement at the top midpoint. The strain energy using Eqs. (3.22) and (3.23) but neglecting the nonlinear terms, integrates to UM = 0 ,

UF =

1 2

  nπ  1  1  1 3 E ∗ w12 n¯ 4 ( )4 + Gw12 2n¯ 2 2 ab 12 h a 6 b

The potential of the applied load is  V=−

po w(x, y) dxdy = −po w1

1 ab n¯

Equilibrium is then given by   ∂  1 1 1 3 1 UF + V = E ∗ w1 n¯ 4 + Gw1 2n¯ 2 2 12 h − po = 0 ∂w1 6 n¯ b or

 2 1 3 2 1 po = nπ E ∗ n¯ 2 + G 2 12 h n¯ w1 6 b

This is shown as the linear solution in Fig. 3.16b; at most, this linear approximation is good for deflections up to one plate thickness, beyond that, the problem is dominated by the nonlinearities. For the nonlinear analysis, it is tempting to stay with the same assumed deformation and just include the nonlinear terms in the strain expressions. When this is done, the normal membrane strains are calculated as 2 = 12 w12 n¯ 2 sin2 (nx) ¯ Exx = 0 + 12 w,x 2 Eyy = 0 + 12 w,y = 12 w12 sin2 (nx) ¯

y2 , b2

1 , b2

average =

1 4

w12 π 2 h2 h2 a 2

average =

1 4

w12 h2 h2 b 2

where the averages are taken along the top free edge. The ratio of the average strains is Eyy /Eyy = (a/π b)2 ≈ 2.5; but we expect this number to be negative and approximately the magnitude of Poisson’s ratio because the free edge must be in uniaxial stress even when the deflections are large. In addition, significant

228

3 Nonlinear Elastic Shapes

shear strains are predicted. Thus, there is something seriously wrong with the strain estimate. The source of the miscalculation is shown in the inset in Fig. 3.16a. As the plate deflects, each constant-x cross section rotates an angle φx (x) and contracts (due to the Poisson’s ratio effect). The assumed displacement has each point moving only in z and therefore the strain is grossly overestimated because it is based on the length of the hypothenuse rather than the rotated height of the plate. Figure 3.16b shows that there is indeed a negative v displacement. From the plot at maximum load, we have that  v w ≈ 4.8 , ≈ −.29 , b = (40h − 0.29h)2 + (4.8h)2 ≈ 39.997h h h Therefore, at least in the midsection, the cross section rotates without significant straining. That is, it is predominantly a rigid body rotation. A revised assumption about the deflected shape is that u(x, y) = 0 ,

v(x, y) = y cos φx − y ,

w(x, y) = y sin φx

which corresponds to a rigid body rotation of each cross section. (A particular form for the rotation φx (x) is assumed presently.) This gives for the membrane strains Exx = 0 + 12 [0 + (−y sin φx φx ,x )2 + (−y cos φx φx ,x )2 ] = 12 y 2 φx ,2x Eyy = [cos φx − 1] + 12 [0 + (cos φx − 1)2 + (sin φx )2 ] = 0 2Exy = 0 − [y sin φx φx ,x ] + 0 − [y sin φx φx ,x ][cos φx − 1] +[y cos φx φx ,x ] sin φx = 0 This is a situation in which the nonlinear terms in both u and v needed to be retained in order to get the correct behavior. Note that even though u(x, y) is assumed zero, we get a significant Exx strain. We could amend the v(x, y) representation to account for the Poisson ratio effect due to this strain; it is simpler, however, to just assume that the state of stress is uniaxial so that the strain energy is given by  UM ≈

1 2

 σxx xx dxdy =

1 2

 2 dxdy = EhExx

1 2

Eh 14 y 4 φx ,4x dxdy

Note that E and not E ∗ is used. Let the rotation have a sinusoidal x-dependence φx (x) =

 nπ x  w w1 1 sin = sin(nx) ¯ , b a b

then the membrane energy evaluates to

n=1

3.3 Structures with Initially Coupled Deformations

UM = 12 Eh

229

3 4 4 n¯ w1 ab 160

Assume the flexural energy and potential of the applied load do not change much due to the nonlinearities, so that the equilibrium condition becomes  ∂  Um + UF + V ∂w1  1 1 1 3 1 3 h − po = 0 = Eh n¯ 4 w13 + E ∗ w1 n¯ 4 + Gw1 2n¯ 2 2 12 80 6 n¯ b or po = nπ Eh

 2 1 3 2 1 3 4 3 n¯ w1 + nπ E ∗ n¯ 2 + G 2 12 h n¯ w1 80 6 b

This is shown plotted as the full line in Fig. 3.16b; the comparison is quite good considering that the problem was reduced to just a single DoF. Clearly the membrane energy has the significant contribution to the stiffening. While the assumed membrane stress state is uniaxial, it is not uniform. In fact, it has a rather complicated distribution estimated by K σxx = EExx = 12 E

  nπ w12 y 2 2 nπ x cos a a 2 b2

This has maximums at the y = b corners and diminishes to zero at the center vertical line and at the bottom. The stress in the free edge is reminiscent of the axial stress distribution for the cable in Fig. 3.10a but as discussed in relation to that figure, the Ritz stress distribution should not be taken too literally when developing approximate models. We can conjecture that addition of the u displacement would improve the results.

3.3 Structures with Initially Coupled Deformations The previous section showed that nonlinear deformations under load can couple initially uncoupled deformations. This situation is compounded when the initial deformations are already coupled. These situations prevail in arches, curved plates, and shells. This is what we discuss here.

230

3 Nonlinear Elastic Shapes

3.3.1 Modeling Nonlinear Behavior of Curved Beams and Plates Figure 3.17a shows the deformed shapes of an arch under point linear and torque loads. It is clear that there is a significant change of shape and therefore a proper nonlinear measure of strain must be used. This is also true for curved plates and shells and is what we concentrate on here. As discussed in relation to the nonlinear straight beam, it is common to assume that the flexural actions do not contribute any nonlinearities. That is, UF is given by Eq. (2.16). We therefore concentrate on establishing the membrane strain energy. Nonetheless, we outline the nonlinear flexural contributions because they are an interesting study of the connection between shape and strain. In Sects. 2.1 and 2.2, we found it convenient to initially use cylindrical coordinates and then transform to the beam coordinates of Fig. 2.3 and the plate coordinates of Fig. 2.14b (or any other notation scheme as desired); we do the same here. Let (ur , uθ , uz ) be the displacements of an arbitrary point originally located at (θ, zo ) as shown in Fig. 3.17b where the curved plate is extruded in the z-direction. The displaced position is x = R cos θ + ur cos θ − uθ sin θ y = R sin θ + ur sin θ + uθ cos θ z = zo + uz

(3.24)

We essentially follow what was done in reference to Fig. 3.7b. The differentials of position are     dx = − (R + ur + uθ,θ ) sin θ + (ur,θ − uθ ) cos θ dθ + ur,z cos θ − uθ,z sin θ dzo     dy = (R + ur + uθ,θ ) cos θ + (ur,θ − uθ ) sin θ dθ + ur,z sin θ + uθ,z cos θ dzo     dz = uz,θ dθ + 1 + uz,z dzo

Fig. 3.17 Large deflections (not exaggerated) of an arch. (a) Loaded arches. (b) Notation for derivation of strain

3.3 Structures with Initially Coupled Deformations

231

These are the deformed components of two lines of initial length dso = Rdθ and d s¯o = dzo . The deformed lengths are   ds 2 = (R + ur + uθ,θ )2 + (ur,θ − uθ )2 + (uz,θ )2 dθ 2   d s¯ 2 = (u2r,z + u2θ,z + (1 + uz,z )2 dzo2 Divide by the initial lengths so as to get the squared stretches, then the normal components of Lagrangian strain are given by  ds 2  − 1 = (ur + uθ,θ )/R ds o   + 12 (ur + uθ,θ )2 + (ur,θ − uθ )2 + (uz,θ )2 /R 2  d s¯ 2    = 12 − 1 = uz,z + 12 u2r,z + u2θ,z + u2z,z o d s¯

Ess =

Ezz

1 2

We recognize the leading terms as the linear expressions developed earlier. The zcomponent is the same as for a flat plate, the θ -component (Ess ) has similarities to that of a curved beam but with axial contributions. The shear strain is related to the change of angle between two initially perpendicular lines. Thinking of ds and d s¯ as vectors, then the angle between them is ds d s¯ d sˆ · d sˆ¯ = dsd s¯ cos φ = cos φ dso d s¯o = 2Esz dso d s¯o dso d s¯o Substituting for the lengths gives d sˆ · d sˆ¯ = [(uθ,z R + uz,θ ) + ur,z (ur,θ − uθ ) + vθ,z (vθ,θ + ur ) + wz,z uz,θ )]dθ dzo When there is no shear, the angle φ remains 90◦ and the cosine is zero. As shown earlier, we take cos φ as a measure of the shear strain so that 2Esz = cos φ =

d sˆ · d sˆ¯ = [(uθ,z + uz,θ /R) + [ur,z (ur,θ − uθ ) + vθ,z (vθ,θ + ur ) dso d s¯ +wz,z uz,θ )]/R

The leading term is the linear term developed earlier. We get the strain for a curved beam by setting ur → −v ,

uθ → u ,

and dropping the uz related terms. This gives

∂ ∂ →R ∂θ ∂s

232

3 Nonlinear Elastic Shapes

Ess = u,s −v/R +

1 2

  (u,s −v/R)2 + (u/R + v,s )2

(3.25)

which can be rewritten to resemble a straight beam through Ess = o + 12 [o2 + φ 2 ] ,

o = u,s −v/R ,

φ = u/R + v,s

(3.26)

For R → ∞ we recover the first of Eq. (3.10) with w = 0. It is not clear, however, to what extent these equations can be approximated similar to the von Karman nonlinear plate equations such as the first of Eq. (3.23). Reference [13] gives Ess ≈ u,s −v/R + 12 [u/R + v,s ]2

(3.27)

Some authors [18] also drop the u/R nonlinear term. Therefore, the first example problem investigates the relative importance of the different nonlinear terms. We get the strains for a curved plate by setting ur → w ,

uθ → −v ,

uz → u ,

∂ ∂ → −R , ∂θ ∂s

z→x

The notation adopted here specifically for curved plates is based on Ref. [13]—it has the merit of being most similar to the notation used for flat plates. The strains are Exx = u,x + 12 [u,2x +v,2x +w,2x ]

(3.28)

Ess = (v,s +w/R) + 12 [(u,s )2 + (v,s +w/R)2 + (v/R − w,s )2 ] Exs = 12 (u,s +v,x ) + 12 [u,x (u,s ) + v,x (v,s +w/R) − w,x (v/R − w,s )] from which it is clear that we recover the flat plate equations as R → ∞. Note that, in getting the shear strain, we also had to take into account the orientation of the axes. The deformed geometry of a curved plate or beam is described in exactly the same way as for an initially straight beam; the difference is that the straining curvature is the difference between the deformed and undeformed curvatures. Thus, from Eq. (3.21) we have that κθθ =

 1 1  1 1 − = x,∗s y,∗ss −y,∗s x,∗ss − = 3 x,∗θ y,∗θθ −y,∗θ x,∗θθ −R 3 r R R R

We use Eq. (3.24) to describe the centerline geometry and this leads to  1  2RB − 2RAθ + A2 + B 2 + AB,θ −A,θ B (3.29) 3 R A = [ur , θ − uθ ] = −v,s R − u, B = [ur + uθ , θ ] = −v + u,x

κθθ =

3.3 Structures with Initially Coupled Deformations

233

This is a complicated expression and therefore difficult to dissect the relative contributions of the different nonlinear terms. However, we concluded that for the grossly deformed shape of Fig. 3.12, the nonlinear flexural terms are relatively small and can be neglected. Our initially curved beams and plates do not have curvatures more severe than Fig. 3.12 and therefore it seems reasonable to neglect the nonlinear flexural contributions. Of course, this must always be checked and Sect. 5.1 has an interesting example illustrating the difference the nonlinear flexural terms can make. For completeness, however, we also state the twist along the axis of a curved plate. It is given by φz =

∂φ = x,∗s y,∗xs −y,∗s x,∗xs ∂z

(3.30)

Reference [13] has a complete derivation of all the nonlinear strain terms. References [11, 12] present the nonlinear equations for 3-D thin-walled curved beams. Example 3.13 Determine the relative importance of the terms in the expression for the Lagrangian strain during the deformation of an arch. Figure 3.17 shows the nonlinearly deformed shape of an arch. It uses 128 straight frame elements (each arch segment is less that 1o ). The corotation scheme is used for the nonlinear analysis [6], so that the rigid body rotation is automatically subtracted from the strain measurement. The data were generated using a load level close to collapse and scanning the nodal values. The FE data were then converted to arch coordinates as discussed in connection with Fig. 2.5. In connection with Eq. (3.25) for the curved beam, let us designate the three contributions to the Lagrangian strain as I: u,s −v/R ,

II:

1 2 [u,s

−v/R]2 ,

III:

1 2 [u/R

+ v,s ]2

and these terms are shown in Fig. 3.18a. The continuous line is obtained directly from the displacement data by approximating the derivative, the circle data are the FE reported strains. There are two remarkable aspects of the figure. The first is that the linear term and the second nonlinear term in Eq. (3.25) (i.e., I and III) are of comparable order; this counters a common intuition in nonlinear analyses that nonlinear terms have a relatively small contribution in magnitude (although very significant effects in the overall consequences). The second remarkable aspect is that both terms are orders of magnitude larger than the actual axial strain. This startling fact highlights the very special nature of the curved beam (also curved plates and shells) problem in that they involve a significant amount of rigid body rotation with very little associated strain. Figure 3.18b shows that the contributions from I and III must be mostly rigid body motions because their addition produces a term that is orders smaller and comparable to the contribution of II and the actual strain. Clearly, a simplistic generalization from straight beams where only the 12 v,2s nonlinear term is considered significant is not justified—each of the other terms, in their own way, contribute.

♦

♦

3 Nonlinear Elastic Shapes

♦

234

♦

♦

♦ ♦

♦

♦

♦

0.0

♦

♦

♦

♦

♦

♦

0.4

0.8 .

1.2

1.6

0.0

0.4

0.8 .

1.2

1.6

0.0

0.4

0.8 .

1.2

1.6

Fig. 3.18 Contributions to the large deflections of an arch. The arrows indicate the zero reference. (a) Direct comparison of terms. All scales are common. (b) Combination effects. All scales are common. (c) Relative comparisons. Top scales are zoomed by 10

This would imply that there is no equivalent to von Karman’s simplification for curved plates and shells. Figure 3.18c compares the relative contribution of each of the separate terms in the nonlinear contributions. We see that u,s and v/R are comparable, but v,s is significantly larger than u/R. Inextensibility typically invokes that u,s = v/R; while this is true on an order of magnitude basis, it is not as strong as the inequality comparison. This is something we track throughout the analyses both here and the subsequent chapters. More or less, our conclusion is that inextensibility is not a good working assumption. The sensitivity of the various nonlinear contributions is intimately related to the nearly inextensible nature of frames and plates. The ability to simplify the nonlinear terms depends on how the membrane actions are to be handled. This is something we spend considerable time discussing especially in the next example and the next two chapters. Example 3.14 Figure 3.17a shows the deformed shape of an arch under a central radial load. Construct a model to explain the shape. We develop a model similar to the one for the linear arch associated with Fig. 2.8; the main difference is the use of the nonlinear membrane strain measure. The model is based on Refs. [2, 15, 16]. The nonlinear hoop strain from Eq. (3.25) is approximated as Ess ≈ u,s −v/R + 12 v,2s = [u,θ −v]/R + 12 v,2θ /R 2 The PoVW becomes 

a

δ U − δ We = 0 = o



a

β1 REss [δu,θ −δv + v,θ δv,θ ] dθ + o

β2 v,θθ [δv,θθ ] dθ

3.3 Structures with Initially Coupled Deformations

235

a a a  a    − F δu − V δv  − Mδv,θ /R  − [qu δu + qv δv] Rdθ o

o

o

o

Integration by parts leads to the governing equations −β1 REss ,θ = qu R ,

−β1 REss − β1 R(Ess v,θ ),θ +β2 v,θθθθ = qv R

and the natural BCs conjugate to the geometric BCs of u, v, v,θ β1 REss − F = 0 ,

−β2 v,θθθ −V = 0 ,

β2 v,θθ −M/R = 0

Just as for the linear case, when qu = 0, we get that the axial force is constant. We can write the governing equation for the transverse deflection in the form v,θθθθ +μ2 v,θθ = −μ2 R + qv R 3 /EI

(3.31)

where F = EAEss and μ2 = −F R 2 /EI . Our point load problems have qv = 0, nonetheless, the problem is still inhomogeneous (as was the linear version of the problem). The general solution is v(θ ) = c0 + c1 θ + c2 cos μθ + c3 sin μθ − 12 Rθ 2 There are four BCs just as for the linear problem; in fact, the specification of the BCs is identical. The arch must be treated as two segments. Measuring θ off the center, we have at θ = 0 :

v,θ = 0 ,

at θ = α¯ :

v = 0,

V = (EI /R 3 )v,θθθ = 12 Po M = (EI /R 2 )v,θθ = 0

where α¯ = α/2 is one half the subtended angle. Substituting and solving gives the coefficients c3 = − 12 Po R 3 /μ3 EI,

c0 = c3 μα¯ + R/μ2 + 12 R α¯ 2 ,

c1 = −c3 μ,

c2 = −c3 Sα¯ /Cα¯ − R/μ2 Cα¯ where Sα¯ = sin μα, ¯ Cα¯ = cos μα. ¯ While Po is known the resulting axial force F is not. We know, however, that the axial strain is constant and therefore equal to its average so that Ess

1 F = = EA R α¯

 0

α¯

 α¯     u α¯ 1 1 2 1 2 u,θ −v + 2 v,θ dθ = − v + dθ + v, θ 2 R α¯ 0 R α¯ 0

236

3 Nonlinear Elastic Shapes

Because of symmetry, u is zero at the limits and the average strain is just a function of v(θ ). This relation gives F as an implicit function of Po . We showed in Sect. 2.1, that a positive v(θ ) distribution gives a compressive axial force, this is not necessarily so when the deflections are large. In anticipation of solving more complicated problems (e.g., more general BCs), let us write the displacement solution as v(θ ) = c3 f1 (θ ) + Rf2 (θ ) f1 (θ ) = μα¯ − μθ − (Sα¯ /Cα¯ ) cos θ + sin θ,

f2 (θ ) = 1/μ2 + 12 α¯ 2 − 12 θ 2

−(1/μ2 Cα¯ ) cos θ Then, the axial force relation becomes 1 F = EA R α¯



α¯ 0

  c3 f1 + Rf2 + 12 [c3 f1 + Rf2 ]2 dθ

25.

0.

♦

♦

where the prime indicates differentiation with respect to θ . The integrals are best performed numerically. Replacing F in terms of μ2 , we then have a quadratic relation between c3 (and hence Po ) and μ. Figure 3.19a shows the two values of Po computed for each compressive value of F , Fig. 3.19b shows how the central deflection depends on the applied load. The limited comparison with the FE data is quite good. Three regimes are identified corresponding to the shallowness of the arch; the length of arch is the same in each case given by a = Rα = Ro αo where the latter parameters are the same as in Fig. 2.2, but the radius and angle change. The shallowest arch shows a monotonically increasing deflection but the other two cases show a limit point beyond which higher loads cannot be supported without

.

♦

♦

-25. -100. -75.

-50. .

-25.

0.

0.0

0.5

1.0

.

1.5

2.0

2.5

Fig. 3.19 Large deflections of a pinned–pinned arch with a central radial load; heavy line α = 90◦ , dashed line α = 10◦ , thin line α = 2.5◦ . (a) Relation between applied load and axial compression, PR = EI /a 2 . (b) Deflection against load, H is the rise of the arch. Circles and squares are FE generated data

3.3 Structures with Initially Coupled Deformations

237

significant change of shape. Note that a deflection of v/H = 2 means that the arch has flipped to the other side. Chapter 5 discusses the behavior beyond the limit point. The importance of this example problem is that it shows we can develop solutions that accurately describe the nonlinear deformations of an arch all the way through its limit point, to its flipped position. The drawback of the solution is that it is mostly implicit and we cannot easily identify such entities as stiffness and, consequently, coupling effects. Therefore, we prefer to emphasize approximate solutions more amenable to identifying these coupling effects. An illustration is given in the next example problem. Example 3.15 The drawback of the strong formulation of the previous example problem is that ultimately the solution became an implicit nonlinear relation between the applied load and the axial force. Construct a Ritz solution that provides more explicit relationships. In Sect. 2.1, we introduced what we called the mixed Ritz method for arch problems. This utilizes the result of the strong formulation that the axial force is constant. It also uses the zero moment natural BC. It has the significant attribute that the arch problem can be presented just in terms of v(θ ) and this facilitates studying the interaction of shapes. Following Analysis III of Sect. 2.1, let the transverse deflection be represented by v(θ ) = b1 sin(n¯ 1 θ ) + b3 sin(n¯ 3 θ ) ,

n¯ j = j π/α

This satisfies the zero displacement and moment end conditions. The flexural energy is   2 1 1 UF = 2 β2 [u,θ +v,θθ ] dθ ≈ 2 β2 [0 + v,θθ ]2 dθ = 12 β2 [n¯ 41 b12 + n¯ 43 b32 ] 12 α and leads to the uncoupled stiffness [KF ] =

  β2 α n¯ 41 0 0 n¯ 43 2

The focus of the remainder of the analysis is on the coupling of the shapes (i.e., b1 and b3 ) engendered by the nonlinear membrane strain. The constant axial strain (force) condition becomes     1 1 Ess = Ess dθ = u,θ −v + 12 v,2θ /R 2 dθ α Rα    1 =− − v + 12 v,2θ /R 2 dθ Rα

238

3 Nonlinear Elastic Shapes

=−

1 2  b1 b3  + + [b2 n¯ 2 + b32 n¯ 23 ] Rα n¯ 1 n¯ 3 4R 2 1 1

The achievement of the approximations is to be able to write the membrane strain (and hence energy) in terms of just b1 , b3 . The membrane energy is  UM =

1 2

 2 EAEss

ds =

1 2

2 2 dθ = 12 β1 R 2 Ess α β1 R 2 Ess

4 1 1 1 1 2 b1 + b3 − 12 β1 b1 α n¯ 1 n¯ 3 R n¯ 1 α 1  + b3 [b12 n¯ 21 + b32 n¯ 23 ] + 12 β1 [b2 n¯ 2 + b32 n¯ 23 ]2 n¯ 3 16R 2 1 1

= 12 β1

This, in turn, gives the stiffnesses   β1 4 1/n¯ 21 1/n¯ 1 n¯ 3 α 1/n¯ 1 n¯ 3 1/n¯ 23   β1 3b1 n¯ 1 + b3 n¯ 21 /n¯ 3 b1 n¯ 21 /n3 ¯ + b3 n¯ 23 /n¯ 1 [KM2 ] = − ¯ + b3 n¯ 23 /n¯ 1 3b3 n¯ 3 + b1 n¯ 23 /n¯ 1 R b1 n¯ 21 /n3   β1 α 3b12 n¯ 41 + b32 n¯ 21 n¯ 23 2b1 b3 n¯ 21 n¯ 23 [KM3 ] = 2b1 b3 n¯ 21 n¯ 23 3b32 n¯ 43 + b12 n¯ 21 n¯ 23 8R 2

[KM1 ] =

These represent different levels of coupling. The negative sign for [KM2 ] means that for positive displacement there is a softening mechanism. The potential of the load is V = −Po v(α) ¯ so that the loading equation becomes   #b $ # 1 $ 1 [KM1 ] + [KM2 ] + [KM3 ] = Po b3 −1 and the coefficients are updated by bi = bi + bi . Figure 3.20a shows the central deflection against load, the model performs quite well in comparison to the FE data. Figure 3.20b shows a set of spectral plots for different accumulative levels of coupling. The dashed horizontal line is the flexural behavior with no coupling for the shape sin(n¯ 1 θ ). The line labeled M1 adds [KM1 ] to the stiffness. This couples the shapes to give one resembling the symmetric shape in Fig. 1.30d which is dominated by the sin(n¯ 3 θ ) shape. The line labeled M2 accumulates [KM2 ] to the stiffness and we see the very strong softening effect. The line labeled M3 has all stiffnesses in the model and is in reasonable agreement with the FE data. It should be pointed out that the FE analysis also reported a lower stiffness corresponding to the first shape in Fig. 1.30d and this shape became critical around

239 ♦

6.

♦

3.3 Structures with Initially Coupled Deformations

4.

2.

0.

♦

0.

2.

4.

6.

♦

0.

2.

4.

6.

Fig. 3.20 Large deflections of a pinned–pinned arch with a central radial load. PR = EAh2 /4R 2 , circles are FE generated data, continuous lines are model results. (a) Central deflection against load. Dashed line is the linear behavior. (b) Spectral stiffness plots for different levels of coupling

♦ ♦

♦

♦ ♦

♦

0.

30.

60.

90.

1

3

5

7

9

11

Fig. 3.21 Large deflections of a pinned–pinned arch with a central load. (a) Displacement distributions for torque loading. Circles are nonlinear FE results, triangles are linear FE results, continuous lines are spectral reconstructions. FE data are thinned for clarity. (b) Spectral decomposition of the displacements

Po = 1.27PR . The reason the deformation results do not exhibit this is postponed to the next chapter and in particular to the meaning of buckling instability. Example 3.16 Figure 3.21a shows some nonlinear displacement distributions for a simply-supported arch with a central torque load. The significant feature in comparison to the linear data is that the transverse deflection (v) has a shift (the center point moves vertically) so that the distribution is no longer antisymmetric. Use spectral analysis to explain this shift. First we do a spectral decompositions of the nonlinear shapes. To that end, the arch was modeled with 128 frame elements, this gives sufficient data for our purpose. Some of the shapes are shown as the fourth column in Fig. 1.30—the

240

3 Nonlinear Elastic Shapes

1.0

.

0.5

0.0

0.0

0.1

0.2

.

0.3

0.4

0.5

Fig. 3.22 Large deflections of a pinned–pinned arch. (a) Some higher spectral shapes for the arch. (b) Nonlinear transverse deflection of the center point. TR = EAh3 /4R 2

shapes alternate as ASASASS. Figure 2.2 shows examples of linearly deformed arches, the shapes are exaggerated for easier viewing; Fig. 3.17a shows examples of nonlinearly deformed arches, the shapes are not exaggerated. Figure 3.21b shows the amplitude spectrums for all four shapes. The Po linear case is dominated by the first symmetric mode (#2) but when loaded nonlinearly to the same load level two significant effects occur. First, the lowest symmetric mode has a larger amplitude and second we see the appearance of a sizable third symmetric mode (#6). There are also hints of the second (#4) and fourth (#7) symmetric modes but no hints of any antisymmetric shapes. The To linear case is dominated by the first antisymmetric mode (#1) with a hint of the second antisymmetric mode (#3). When loaded nonlinearly to the same load level we see the appearance of all shapes with an especially strong third symmetric (#6) mode. Figure 3.22a shows the shapes for modes #4 through #6. Shape #6 has a strong sin(π θ/α) mean to it; this is the first shape shown in the third column of Fig. 1.30 that is suppressed by the curvature of the arch. Actually, it is not quite suppressed but coupled with a higher mode so that its associated stiffness is higher. We conclude that our nonlinear model for the transverse deflection must include the symmetric sin(π θ/α) contribution. In its simplest terms, our problem is analogous to the nonlinear transverse loading of a cantilever beam such as in Fig. 3.9: the first increment of load produces only a transverse deflection, but this generates an axial tensile force so that the subsequent load increment results in an axial displacement to reduce the tension. The coupling is manifested through the nonlinear stiffness matrices and that is what we focus on illustrating. We could use our mixed Ritz method for this problem and eliminate u(θ ) from the formulation but then it is required to segment the integrals into two halves; it is simpler for our present purpose to make the appearance of u(θ ) explicit. We begin by particularizing the energy expressions to that of a uniform arch and replace the hoop coordinate by s = Rθ so that

3.3 Structures with Initially Coupled Deformations

241



  1 1 β1 [u,θ −v]2 + [u,θ −v] v,2θ + 2 v,4θ dθ = UM1 + UM2 + UM3 R 4R   UF = 12 β2 [u,θ +v,θθ ]2 dθ ≈ 12 β2 [0 + v,θθ ]2 dθ (3.32) UM =

1 2

where β1 = EA/R and β2 = EI /R 3 . For illustrative purposes, let us represent the displacements as u(θ ) = a2 [1 − cos n¯ 2 θ ] ,

v(θ ) = b2 sin n¯ 2 θ + b1 sin n¯ 1 θ

where n¯ j = j π/α. The leading term in each expression is the linear representation of Analysis I in Sect. 2.1 and we add the b1 term to account for the nonlinear shift. The results in Fig. 2.11 give us a baseline for the expected fidelity of our reduced model—the nonlinear model comparisons are not going to be better than the linear modeling comparisons. We can now lay out the various contributions to the stiffness matrices. The linear energies are UM1 = 12 β1 [a22 − 2a2 b2 + b22 + b12 ] 12 α ,

UF = 12 β2 [b22 n¯ 42 + b12 n¯ 41 ] 12 α

The corresponding stiffnesses are ⎡

n¯ 22 −n¯ 2 [ K1 ] = β1 ⎣−n¯ 2 1 0 0

⎤ 0 α 0⎦ , 2 1

⎡

⎤ 0 0 0 α [ K5 ] = β2 ⎣0 n¯ 42 0 ⎦ 2 0 0 n¯ 41

with the DoF arranged as {a2 , b2 , b1 }. Consider an applied torque at the center, then the virtual work is given by  δ We = To δφ α/2 = To δ[u + v,θ ]/R = To [2δa2 − n¯ 2 δb2 + 0δb1 ]/R The generalized loads are Pa = 2To /R, Pb2 = −n¯ 2 To /R, and Pb1 = 0. The incremental equation can be presented as ⎫ ⎫ ⎧ ⎤⎧ ⎨a2 ⎬ ⎨ 2 ⎬ T o ⎣ KM1 + KF ⎦ b2 = −n¯ 2 ⎩ ⎭ R ⎭ ⎩ b1 0 ⎡

There is coupling between a2 and b2 coming from the membrane stiffness, this is the arch effect. There is no coupling between b1 and either a2 or b2 and because the applied torque does not have a generalized force associated with that DoF, then a nonzero component of b1 is never generated in a linear analysis. The first load increment of a nonlinear analysis also does not produce a b1 .

242

3 Nonlinear Elastic Shapes

The first nonlinear energy is given by the interactive term UM2 = − 12 β1 [n¯ 22 b1 b2 I122 + n¯ 21 b13 I111 ]α/R + β1 [n¯ 1 n¯ 22 a2 b1 b2 − n¯ 1 n¯ 2 b12 b2 ]α/R This is a relatively simple expression because most trigonometric integrals are zero except for Iij k =

1 α

 sin n¯ i θ cos n¯ j θ cos n¯ k θ dθ

I212 = 0.1333/α ,

I122 = 0.4666/α ,

I111 = 0.3333/α

The corresponding stiffness is ⎤ ⎡ 0 0 0 β1 α ⎣ 2 [KM2 ] = − 0 n¯ 2 b1 I122 n¯ 22 b2 I122 ⎦ R 0 n¯ 22 b2 I122 3n¯ 21 b12 I111 ⎤ ⎡ 0 n¯ 2 b1 n¯ 2 b2 β1 α ⎣ + n¯ 2 b1 −2b1 n¯ 2 a2 − 2b2 ⎦ n¯ 1 n¯ 2 I212 R n¯ 2 b2 n¯ 2 a2 − 2b2 0 Note the presence of a2 and b2 in the third column, this is the interaction terms associated with the increasing deformation. On the second load increment, because a2 and b2 are nonzero, a nonzero b1 is generated. The results are shown in Fig. 3.22b as the dashed line and compared to the FE data; the low-level load response is captured reasonably well but the model result shows a strong singularity which is not present in the FE data. This was also observed in Fig. 3.20b. The third membrane energy term always adds positive stiffness. When added to the total stiffness, this results in the continuous line in Fig. 3.22b. While it does mitigate the singular contribution of [KM2 ], it does so by making the system slightly overly stiff; keep in mind that the plot is for the very small transverse displacement in the center of the arch. At this stage, it is possible to explore the contribution of the other terms in the spectral representation. We do not do that here because we achieved our primary goal of showing how the vertical deflection of the center point is generated.

3.4 Monitoring Changes of Shape and Stiffness As a structure is loaded, two things of significance occur: the members are stressed, and the shape changes. Thus, for a load applied in two increments, say, the initial stress and geometry (and hence stiffness) for the second increment is different

3.4 Monitoring Changes of Shape and Stiffness

243

Fig. 3.23 Organization of spectral shapes. (a) Discretization of arbitrary structures. (b) Description of shell and frame finite elements deforming nonlinearly

from the initial stress and geometry for the first increment. Thus the effect of each increment of load is different. During a nonlinear analysis we would like to be able to monitor these changes of shape and stiffness in a systematic way. This section uses spectral analysis to develop some tools to achieve this.

3.4.1 Spectral Decomposition and Reconstruction A general structure has coupled deformations; we use the 2D frame to illustrate the general formulation required to handle these problems. The formulation is easily extended to arbitrary frame structures. It is also applicable to arbitrary shells with a suitable interpretation of the modal shapes . Consider a general 2D frame such as a multistory frame or a segment of an arch. Irrespective, the structure is described as a collection of straight (or flat) segments as shown in Fig. 3.23b. The spectral shapes give information about the deformed shape as well as various space derivatives originating from the member load distributions. Note that the stored shape information does not change during a nonlinear deformation, it is only the amplitudes aI that change. We represent this information grouped according to geometric (displacements and rotations) and natural (forces and moments or higher space derivatives) as geometric: u = natural: ¯ =

 I

aI uI ,

v=

 I

 ∂ u¯ F¯ aI I , = = I ∂ x¯ EA

aI vI ,

φ=

 I

aI φI

 ∂ 2 v¯ ∂φ M = = = aI κI 2 ∂ x¯ EI ∂ x¯

The space derivatives are local derivatives in accordance with the interpretation of the member loads. A small detail: the data are stored on an element nodal basis (rather than structural nodal) and when necessary centroidal values are obtained by averaging. This facilitates computing the volume integrals associated with the

244

3 Nonlinear Elastic Shapes

energies. Each ×I is viewed as a vector representing the distributions over space. Thus, the orthogonality conditions result in   [FI FJ /EA+MI MJ /EI ] ds= aI2 TI J /EA+TκI κJ /EI  ds K˜ I J = i

M˜ I J  =

 i

ρA[uI uJ + vI vJ ] ds =

 i

i

ρAaI2 TuI uj + TvI vI  ds

where i represents the element and the summation represents the volume integral. At zero load, the shapes are uncoupled and the stiffness and mass matrices are diagonal. The stiffness matrix changes during a nonlinear deformation, but the mass matrix remains the same. It is important to realize that all displacement distributions are modulated by the same DoF aI in contrast to the separate distribution representations used in the Fourier analyses of Sect. 2.1. That is, each I is a complete single shape. Although the spectral decomposition scheme introduced in Sect. 2.4 was done in a linear context, it is nonetheless applicable to nonlinear shapes as already used in Fig. 3.21. It is the reconstruction of entities such as strain, stress, and stiffness, that requires the use of nonlinear measures which we now illustrate. The nonlinear strain energies are  UM =

2  ¯ 2s ds , EA ¯o + 12 ¯o2 + 12 v,

1 2

 UF =

EI [φ,s ]2 ds

1 2

The membrane energy expands to  UM =

  ¯ 2s + 14 v, ¯ 4s + 14 ¯o4 + 12 ¯o2 v, ¯ 2s ds EA ¯o2 + ¯o v,

1 2

The terms are ordered according to a typical small-deflection nonlinear analysis; we find it most convenient to identify the separate energies accordingly. In the following we get a mixture of distribution terms (they retain the amplitudes aN ) and modal terms (they do not involve the amplitudes aN ); the former are enclosed in parentheses for easy recognition. The linear elastic energy, force, and stiffness are, respectively,  UE =

1 2

=

1 2

FEI =

 EA¯o2 ds +

EI φ,2s ds

(3.33)

2 2      EA ds aN N + 12 EI aN κN i

∂ UE = ∂aI

KEI J =

1 2



UE = ∂aI ∂aJ ∂2

i

N

N

     EA aN N I + EI aN κN κI ds N

 i

N



EA[I J ] + EI [κI κJ ] ds

3.4 Monitoring Changes of Shape and Stiffness

245

The inner sums are over the number of shapes and hence gives a single shape in the parentheses, the outer sum is over the number of elements representing the space integration. Because of orthogonality of the shapes, the matrix [KE ] is diagonal. The geometric energy contributions are 

UG =

1 2

¯ 2s ds = EA¯o v,

1 2

  i

N

aN N

 N

aN φN

2 

ds

(3.34)

2     ∂ UG 1   =2 aN φN +2 aN N aN φN φI ds I i N N N ∂aI       2    ∂ UE = aN N [φI φJ ]+ aN φN [I φJ +J φI] ds KGI J = i N N ∂aI ∂aJ

FGI =

The leading stiffness term is recognizable as being of the form F¯o v T v and would be the only term present if the load is considered as pre-existing (i.e., the parenthesis term is not varied). The second stiffness term accounts for the changing load. The nonlinear cable-energy contribution is  U N1 =

1 8

EAv, ¯ 4s ds =

1 8

4     EA ds aN φN i

N

(3.35)

3     ∂ UN 1 EA = 12 aN φN φI ds i N ∂aI 2     ∂ 2 UN 1 EA = = 32 aN φN [φI φJ ] ds i N ∂aI ∂aJ

FN 1I = KN 1I J

This contributes positive stiffness irrespective of the load being tensile or compressive. The other nonlinear terms are treated in the same manner and need not be detailed here because when all nonlinear terms are included a convenient alternative form for the nonlinear stiffnesses is  " ! [Kaa ] = EA ¯o + 12 ¯o2 + 32 v, ¯ 2s [φI φJ ] (3.36) i  " ! EA 3¯o + 32 ¯o2 + 12 v, ¯ 2s [I J ] [Kbb ] = i  " ! EA v, ¯ s +¯o v, ¯ s [I φJ + J φI ] [Kab ] = i

This form shows how the core structure of the stiffness matrices are related to the shapes. The parenthetical terms correspond to the parenthetical terms in the earlier equations. The shapes can be produced automatically from a linear vibration eigenanalysis. In some of the example problems over the next two chapters, up to 200 shapes have been used.

246

3 Nonlinear Elastic Shapes

10.

.

5.

0. 0.

1.

2.

3.

Fig. 3.24 Large deflections and rotations of a transversely loaded beam. (a) Geometry and properties for beams with different BCs. (b) Load-deflection curves. Circles are fixed–fixed BCs, squares are pinned–pinned BCs, lines are model results, and Qr = 500(EI /L3 )h

We view the developments here as a decomposition and reconstruction scheme. That is, the FE analysis generates the actual results (displacements and so on under load), the spectral analysis identifies the significant contributors and reconstruct them in different forms. It is possible, of course, to formulate a nonlinear solver scheme that determines the aI without a nonlinear FE analysis, but the FE solver scheme is so superior in every way there would be no point to having an alternative scheme. We think of the spectral analysis as contributing to the postprocessing of the FE results. Example 3.17 Use spectral analysis to add insight into the nonlinear deformation of the beam shown in Fig. 3.14. We explore two variations on the problem. The first variation uses our general spectral decomposition method on the fixed–fixed beam so as to establish the contributors to the shapes and to determine a sufficient number of spectral shapes for the adequate reconstruction of the distributions. The second variation uses the simply-supported beam to analytically elucidate the various contributing factors; the geometry of both cases is shown in Fig. 3.24a. As seen in Fig. 3.24b, the maximum load level is chosen sufficiently high that the nonlinear terms are the significant contributors. The beam was modeled with 128 frame elements. Normally this would be considered over-refined, but it is done for two reasons. First the associated volume integrals are replaced with finite sums and hence the elemental volume needs to be small, and second, the axial spectral shapes are relatively high in stiffness and a sufficient number of modes need to be generated. For example, the highest axial mode used was actually mode # 42. The nonlinear FE analysis used the corotational scheme to generate the deformed shapes. The circles in Fig. 3.25 show the recorded distributions. That the transverse deflections are more than three times the beam thickness is indicative that this problem is in the large deflection range for structures. A linear eigenvibration analysis was used to generate ten flexural and ten axial shapes. Because the linear problem has uncoupled deformations, this could have

3.4 Monitoring Changes of Shape and Stiffness

247

♦

♦

3.

♦

2.

♦

.

♦

1. 0.

0.0

0.2

0.4

.

0.6

0.8

1.0

0.0

0.2

0.4

.

0.6

0.8

1.0

Fig. 3.25 Large deflections and rotations of a fixed–fixed beam. Circles are FE generated results, continuous lines are identified model reconstructions, and the horizontal arrows are the zero references. (a) Deformation responses. (b) Member distributions

been done as two separate EVPs. Our spectral decomposition scheme identified the amplitudes as F: 1.0000 0 0.0095 0 0.0369 0 −0.0031 0 −0.0057 0 A: 00 0 −0.00436 00 0 0.00084 00 The sequence of modes for both sets of shapes is SASA· · · . It is not surprising that none of the antisymmetric shapes n = 2, 4, · · · contributes to v(x), what is surprising is that only the m = 4, 8, · · · axial modes contribute to u(x). The reconstructions are shown as the heavy continuous lines in Fig. 3.25. Obviously the displacement and rotation distributions are represented well as expected, but the member distributions are also reasonable. The amplitude spectrums allow us to judge how many shapes are required for a reasonable reconstruction. The fine line in Fig. 3.24b going through the circles is the spectral recreated load-deformation history; clearly, the FE results can be duplicated with an adequate collection of shapes. The presentation of the amplitude spectrums is somewhat misleading in that they are displayed according to mode number; if they were displayed according to stiffness (frequency), then the axial modes would be shifted significantly to the right. In other words, we are mixing very high stiffness axial modes with low stiffness flexural modes. However, interaction between the modes is related to their spectral wavelength (analogous to interference effects in optics [7]) and is strongest when the wavelengths are comparable. The fine line in Fig. 3.25b is the axial force computed using EAu,x , clearly this is an inadequate representation. The correct computation is Exx = u,x + 12 v,2s ,

σxx = EExx ,

  F = σxx A = EA u,x + 12 v,2s

248

3 Nonlinear Elastic Shapes

and the figure shows that its average is close to the FE results. We expect that should more shapes be included in the representation then the FE and identified (based on displacements alone) representation would tend toward each other. As discussed in Sect. 1.4 the shapes for the fixed–fixed beam are awkward to work with in analytical (as apposed to numerical) form. We find it convenient therefore to study a similar problem but with easily represented analytical forms for its spectral shapes. Again, as discussed in Sect. 1.4, the spectral shapes for a fixed–fixed rod and a pinned–pined beam are given by u(x) =

 m

um sin m ¯ mx ,

v(x) =

 n

vn sin n¯ n x

where m ¯ = mπ/L and n¯ = nπ/L. The deformations are uncoupled (at zero load) thus we associate with each u(x) mode a corresponding v(x) = 0 distribution, and vice versa and these then are spectral shapes. As done for our earlier model, let us first consider the deformation as dominated by the transverse deflection and explore the effect of multiple modes. That is, assume the deformed shape as u(x) = 0 ,

v(x) = v1 sin n¯ 1 x + v3 sin n¯ 3 x

The linear elastic energy is then  UE =

EI [−v1 n¯ 21 sin n¯ 1 x − v3 n¯ 23 sin n¯ 3 x]2 dx = 12 EI [v12 n¯ 41 + v32 n¯ 43 ] 12 L

1 2

The corresponding elastic forces are FE1 =

∂ UE = EI v1 n¯ 41 21 L , ∂v1

FE2 =

∂ UE = EI v3 n¯ 43 21 L ∂v3

The stiffness is then  [KE ] = EI

n¯ 41 0 0 n¯ 43

 1 2L

This is diagonal as per design. The nonlinear contribution to the energy is  UN =

1 2

EA 14 [v1 n¯ 1 cos n¯ 1 x + v3 n¯ 3 cos n¯ 3 x]4 dx

  = 18 EA 38 v14 n¯ 41 + 48 v13 n¯ 31 v3 n¯ 3 + 64 v12 n¯ 21 v32 n¯ 23 + 0v1 n¯ 1 v33 n¯ 33 + 38 v34 n¯ 43 L The elastic forces and stiffness are

3.4 Monitoring Changes of Shape and Stiffness

{FN } =

1 8 EA

{KN } = 18 EA

249

#3

3 ¯ 4 + 3 v2n 3 ¯ + 3v n 2 2 ¯2 3 1 ¯ 1 v3 n 1 3 2 v1 n 2 1 ¯ 1 v3 n 1 3 3 2n 2v n 2 + 3 v3n 4 v n ¯ n ¯ + 3v ¯ ¯ ¯ 3 3 1 1 3 3 1 1 3 2 2

9

2 ¯4 1 2 v1 n

$ L

 + 3v1 n¯ 31 v3 n¯ 3 + 3n¯ 21 v32 n¯ 23 32 v12 n¯ 31 n¯ 3 6v1 n¯ 21 v3 n¯ 23 L 3 2 3 ¯ 1 n¯ 3 6v1 n¯ 21 v3 n¯ 23 3v12 n¯ 21 n¯ 23 + 92 v32 n¯ 43 2 v1 n

Both expressions are somewhat complicated indicating that it would not be fruitful to explore adding more shapes. The main feature to note is that the stiffness provides coupling between the two shapes. This means that while the spectral shapes do not change during a nonlinear deformation, their combination does change giving rise to a changing resultant shape. We explore this in greater depth in later sections. As the number of shapes is increased the axial force approaches a uniform distribution but with a sharp dip to zero in the center (it has the distribution of v,2s (x)). Now consider the effect of adding an axial shape. In contrast to the fixed–fixed beam, the pinned–pinned beam has a sin(m ¯ 2 x) axial contribution, therefore assume the simple representation v(x) = v1 sin n¯ 1 x ,

u(x) = u2 sin m ¯ 2x

The linear elastic energy is then  UE = 12

 EI [−v1 n¯ 21 sin n¯ 1 x]2 dx+ 12

¯ 2 cos m ¯ 2 x]2 dx EA[u2 m

= 12 EI v12 n¯ 41 21 L+ 12 EAu22 m ¯ 22 12 L The elastic forces and stiffness are {FE } = EI

3 3 ) ) 4 4 * * 4 v1 n¯ 41 1 0 1 L, [ K ] = EI n¯ 1 0 1 L + EA 0 0 1L L + EA E 2 u2 m 0m ¯ 22 2 ¯ 22 2 0 0 0 2

with the DoF arranged as {v1 , u1 }. Again, the stiffness is diagonal by design. The geometric energy is given by  UG =

1 2

¯ 2 cos m ¯ 2 x][v1 n¯ 1 cos n¯ 1 x]2 dx = 12 EAv12 u2 n¯ 21 m ¯ 2 41 L EA[u2 m

The elastic forces and stiffness are $ # 2v1 u2 2 1 1 n¯ 1 m {FG } = 2 EA ¯ 2 4L , v12

 u2 v1 2 1 n¯ m [KG ] = EA ¯2 L v1 0 1 4 

The stiffness has two significant effects. First, it couples the two deformations, that is, a nonzero transverse deflection induces an axial displacement but not vice versa. Second, the axial displacement contributes a stiffness effect to the transverse

250

3 Nonlinear Elastic Shapes

behavior (the K11 term) but not vice versa. If more transverse and axial shapes are added, the structure of the geometric stiffness is decomposed as [KG ] = [KG1 ]+[KG2 ] ,

  [ T ][ 0 ] [KG1 ] = , [ 0 ][ 0 ]

)

[ 0 ][ A ] [KG2 ] = [ AT ] [ 0 ]

*

This form reiterates the fact that the geometric energy contains the interaction or coupling behaviors across different deformation modes. The interactions are related to the integrals of cos nx ¯ and cos 2nx. ¯ While the stiffness of the axial shapes is very high, it is the wavelength that is important. The remaining nonlinear contribution to the energy is  UN =

1 2

EA 14 [v1 n¯ 1 cos n¯ 1 x]4 dx = 18 EA[ 38 v14 n¯ 41 ] L

The elastic forces and stiffness are # 3 3 4$ v n¯ {FN } = 18 EA 2 1 1 L , 0

{KN } = 18 EA

9

2 ¯4 1 2 v1 n

0

 0 L 0

This has no axial contribution specifically because we are using the von Karman approximate strains from Eq. (3.23) instead of the more general form of Eq. (3.17). The two equilibrium equations are 3 EI v1 n¯ 41 21 L + EAv1 u2 n1 ¯ 2m ¯ 2 14 L + EAv13 n1 ¯ 4 16 L = Qo

EAu2 m ¯ 22 21 L + EAv12 n1 ¯ 2m ¯ 2 18 L = 0 There are too few terms to give an accurate result; in particular, as the deflection increases the beam behaves like a cable and the sides straighten, the sin n¯ 1 x on its own is not a good representation. The second equation gives that ¯2 u2 = − 14 v22 n¯ 21 /m While this is small, it has a significant contribution to the axial force which is given by ¯ 2 cos m ¯ 2 x + 12 v12 n¯ 21 cos2 n¯ 1 x] = 14 EAv12 n¯ 21 F (x) = EA[u,x + 12 v,2x ] = EA[u2 m where use is made of cos2 n¯ 1 x = 12 [cos 2n¯ 1 x + 1] = 12 [cos m ¯ 2 x + 1]. The axial term removes the zero at the beam center and, in fact, gives F (x) = constant. Again we point out that in spite of considerable difference in the elastic stiffnesses (EA/L versus EI /L3 ) there is significant interference between the two deformation modes.

3.4 Monitoring Changes of Shape and Stiffness

251

Example 3.18 When the beam in Fig. 3.26 is loaded only with the transverse load Qo , the vertical deflection (exceeding multiple thicknesses) is predominantly linear with load and can usefully be analyzed as such. The horizontal displacement of the movable support is, however, definitely nonlinear. Figure 3.27a shows the distribution of the resulting horizontal displacement when the transverse displacement is about 2.5 h; it has a curious sinusoidal aspect to it. Furthermore, if the axial force is taken as F (x) = EAu,x then we would estimate a large compressive axial force. In fact, the FE results in Fig. 3.27b show that the axial force is predominantly zero (the actual distribution is a small tensile force with a zero in the center). Construct a model to help explain these results. Let us begin by performing a spectral decomposition of the deformed shape to ascertain the significant contributors. The beam was modeled with 128 frame elements and the first ten flexural modes and first ten axial modes documented in the form {u, v, φ; u,x , v,xxx , v,xx }. The space derivatives of the latter three are with respect to the local member coordinates. In this particular case, the axial and flexural modes are uncoupled but the general computational procedure implemented assumes they are coupled. As demonstrated in Sect. 1.4, the asymmetric u(x) and symmetric v(x) spectral shapes are given by, respectively, u (x) = u(x) ˆ = um sin(mπ x/2L) = um sin(mx) ¯ v (x) = v(x) ˆ = vn sin(nπ x/L) = vn sin(nx) ¯

m = 1, 3, 5, · · ·

n = 1, 3, 5, · · ·

The u(x) ˆ shapes are unconstrained at the right end. The amplitude spectrum for a load level giving a transverse displacement of about 2.5h is shown in Fig. 3.26b; the deformed shape is dominated by the first flexural shape (n = 1) and therefore not shown in the figure. The first and third axial shapes dominate the axial displacement. The reconstructions in Fig. 3.27a show that the deformations can be reconstructed quite accurately. This does not apply to the member force and moment distributions. Focussing just on the axial force, Fig. 3.27b shows a growing discrepancy toward the free end, but this was anticipated in situations such as Fig. 2.13.

♦

2

4

6

8 .

10 12 14

Fig. 3.26 Large deflections and rotations of a pinned–pinned beam. (a) Geometry and properties. (b) Amplitude spectrum. The dominant amplitude for v1 is not shown

♦

3 Nonlinear Elastic Shapes ♦

252

♦

2.

.

.

♦

0. ♦

0.00

♦

0.05

♦

0.00

0.25

0.50 .

0.75

1.00

♦

0.00

0.25

0.50 .

0.75

1.00

Fig. 3.27 Nonlinear interactions in a transversely loaded simply-supported beam. Horizontal arrows are the zero reference in each plot, circles are FE data, continuous lines are reconstructions over 20 shapes, dashed lines are reconstructions over three selected shapes. (a) Axial displacement and rotation responses. (b) Axial member loads normalized to Po = EA/1000

Initially the shapes are uncoupled, therefore, consider the interaction between any two of them. Using the membrane energy given by Eq. (3.19) but neglecting u,2x , and the linear flexural energy, we get  UM =

1 2

2  ¯ cos mx ¯ + 12 vn2 n¯ 2 cos2 nx ¯ dx EA um m

  = 12 EA u2m m ¯ 2 12 + um vn2 m ¯ n¯ 2 Imn + 14 vn4 n¯ 4 83 L    1 UF = 2 EI [−vn n¯ 2 sin nx] ¯ 2 dx = 12 EI vn2 n¯ 4 21 L where Imn =



cos(mx) ¯ cos2 (nx) ¯ dx/L. The potential of the loads is

V = −Po u(L)−Qo v(L/2) = −Po um −Qo vn

⇒

Pu = Po .

P v = Qo

This leads to the equilibrium equations     ∂

= EA um vn m ¯ n¯ 2 Imn + 12 vn3 n¯ 4 83 L + EI vn n¯ 4 21 L − Qo = 0 ∂vn   ∂

= EA um m ¯ 2 21 + 12 vn2 m ¯ n¯ 2 Imn L − Po = 0 ∂um It is clear that equilibrium is governed by a set of coupled nonlinear equations. In the case when the horizontal force is absent (Po = 0), we get the interesting result that

3.4 Monitoring Changes of Shape and Stiffness

253

um = −(Imn n¯ 2 /m)v ¯ n2 This is the coupling effect where a transverse displacement (due to load) produces a horizontal displacement even though there is no horizontal load. Furthermore, this relation is independent of material properties and therefore is like a geometric constraint relation. To see this, substitute for u and v into Eq. (3.17) for Exx (neglecting u,2x ) and integrate using a weighting of cos mx ¯ to get  ¯ cos mx ¯ + 12 vn2 n¯ 2 cos2 nx] ¯ cos mx ¯ dx = um m ¯ 12 L + vn2 n¯ 2 Imn L = 0 [um m This is inextensibility imposed in a weighted average sense and shows that the horizontal displacement is a second-order effect usefully neglected in a linear analysis that focusses on the dominant deflection v(x). Here we specifically want to examine this contribution. The strength of the interaction depends on the integral Imn ; consider when n = 1, we get I11 = 0.297 ,

I31 = 0.030 ,

I51 = 0.241 ,

I71 = −0.113

This explains why the second axial mode is weakly represented in the deformed shape and why u(x) in Fig. 3.27a exhibits a higher order sinusoidal effect. Superposed on the plot of Fig. 3.27a as a thin dashed line is the reconstruction using just m = 1 and m = 5; it captures the dominant behavior, but clearly the higher shapes are also needed. The structural stiffness matrix also reflects this coupling through the off-diagonal terms [KT I J ] =



    9 ∂ 2  EA un m ¯ n¯ 2 Imn + vn2 n¯ 4 16 vn m ¯ n¯ 2 Imn 2 EI n¯ 4 12 0 4 = + L L vn m ¯ n¯ 2 Imn m ¯ 2 12 0 0 ∂uI ∂uJ L L3

with the DoF arranged as {vn , um }. When Po = 0 and um is replaced in terms of 2 + 9 ]n 4 2 vn2 in the K22 stiffness, we get the core nonlinear term as [−Imn 16 ¯ vn , the latter coefficient dominates and the stiffness increases with increasing transverse deflection because of vn2 . An eigenanalysis of the [2 × 2] stiffness gives an approximation for the lower eigenvalue as  2 + λ ≈ (EI /L3 ) 12 n¯ 4 + (EA/L) − Imn

9 16

 2 4 2 n¯ 4 vn2 − (EA/L)2Imn n¯ vn

9 The leading term is the flexural contribution, and the 16 is the cable effect contribution. The coupling contributes in two ways, first it changes the membrane term and second it provides an off-diagonal coupling term, both make the transverse stiffness smaller. An almost identical result is obtained for the pinned–pinned beam

254

3 Nonlinear Elastic Shapes

not on rollers, the only difference is that Imn = I21 = 0.25 which is slightly smaller and therefore the structure is stiffer. Figure 3.27b shows the distribution of axial force computed as F = EAu,x ; not only is it orders of magnitude larger than the applied transverse load Qo , but it is even of the wrong expected sign because the transverse load, just as for a cable, should put the beam in tension. We therefore need to address the meaning of the spectral shapes as regards the space-derivative information. Spectral shapes are recorded at zero load in a linear analysis. In a nonlinear analysis, this represents the undeformed shapes. The space-derivative information is therefore, for example, ∂u ∂x o and the like. The spectral information stored is the Lagrangian space-derivative information. This is the same as FE interpolation functions used in nonlinear analyses—the reference structure is always the original (undeformed) structure as shown in Fig. 3.23b. The Hex20 implementation in Simplex uses the measure of axial strain given by Exx = u,x + 12 [u,2x +v,2x +w,2x ] as discussed in Sect. 3.2. Frame and shell programs that use a corotational scheme transform to a local coordinate system which removes the rigid body motion, their computed values use a local (linear) stress–strain relation. As set up, the spectral method is a global Lagrangian method and therefore for the beam case we calculate the member axial force, in general, as  2  2  2   ∂u 1 ∂u 1 ∂v 1 ∂w F (x) = EA + + + 2 ∂x o 2 ∂x o 2 ∂x o ∂x o The result of this is shown in Fig. 3.27b, certainly it gives a diminished axial force oscillating about zero. Example 3.19 The circles in Fig. 3.28 show the FE generated results for the nonlinear deformation of an arch under a central point load. The load puts the arch in tension. Use a spectral analysis to identify the various contributors to the results. The arch was modeled with 128 frame elements, this gives sufficient data for the identification. Figure 3.29 shows the amplitude spectrum for five load levels where only the symmetric shapes were used. The top plot is for aM and are ratioed to the first mode which is the largest. This spectrum shows typical behavior in that the lower modes dominate and beyond the fifth or so mode the contributions are negligible. It also shows that shapes #2 through #4 grow in significance relative to the first shape.

3.4 Monitoring Changes of Shape and Stiffness

255

♦

♦ ♦

4.

♦

♦

2. .

.

3. 2. 1. 0.

0.00

0.25

0.50 .

0.75

0.

1.00

0.00

0.25

0.50 .

0.75

1.00

Fig. 3.28 Large deflections and rotations of a pinned–pinned arch under central point radial load. Circles are FE generated results, continuous lines are identified model reconstructions, and horizontal arrows are the zero references. (a) Deformation responses. (b) Member distributions

0

10

20

30

40

♦

♦

♦

♦

50

0

10

20

30

40

50

Fig. 3.29 Response spectrums for five load levels. (a) Central transverse load using only symmetric shapes. (b) Central torque load using all shapes

A reconstruction based on the first five symmetric shapes gives reasonable distributions for all quantities except the axial force which exhibits many zero crossings. Furthermore, a reconstruction of the derivative u,s does not correlate with u(s). The axial behavior is not accounted for adequately using just five shapes. We get a different picture of the situation by monitoring the contributions to the axial force; the lower plot in Fig. 3.29a shows the amplitude spectrum for the force at the center of the arch ratioed to shape #6. This shows much more activity across a wider range of shapes; in fact, the largest contribution comes from shape #28. We identify shapes #: 3, 15, 22, 28, 34, 39, 44, and 48, as significant. The force distributions associated with these is well represented by u(x) ∝ sin(2mπ x/L) ,

F (x) ∝ cos(2mπ x/L)

3 Nonlinear Elastic Shapes

6. 4. .

Fig. 3.30 Nonlinear response of a simply-supported arch to a transverse central load. Circles are FE data, continuous lines are spectral results. The reference load is Pr = 50EA(h/R)3

♦

256

2. 0.

♦

0.0

1.0 .

2.0

with m = 0, 1, 2, · · · . These are the longitudinal modes as if the curved beam were straightened out to form a straight rod with fixed end conditions.√ Reference [5] shows that the curved beam has a cut-off frequency given by ωc = E/ρ/R and at frequencies about twice this, the longitudinal behavior is very close to that of a straight rod. In the present case, this happens at about shape #5. We therefore have the interesting situation quite similar to what happened with respect to Fig. 3.25 for the straight beam: the flexural shapes needed to be supplemented with higher stiffness axial shapes. Although the curved beam has axial displacements these are insufficient to represent the axial behavior during a nonlinear deformation. The reason comes back again to the nonlinear relation between axial force, axial displacement and transverse displacement given by F /EA = u,s −v/R + 12 [u/R + v,s ]2 = ss + 12 φ 2 This does not appear in a linear analysis; indeed, linear analyses often invoke inextensibility where u(s) is specified according to u,s = v/R, that is, there are no axial displacements in our sense of causing axial strains. Figure 3.30 shows that the limited collection of spectral shapes is capable of recreating the nonlinear deformation behavior. As already laid out, the quality of the distributions depends on the number of terms but we have established the basic principle that the spectral shapes can recreate the full nonlinear behavior. It is interesting that ignoring the geometric energy stiffens the structure. We finish this example by looking at the nonlinear torque loading of the arch shown in Fig. 3.17. Because the geometry and geometric BCs are unchanged, we can use the same collection of spectral shapes to identify the significant contributors. The top plot in Fig. 3.28b shows the coefficient amplitudes ratioed to the first load value. While only antisymmetric shapes participate in a linear deformation, all shapes, symmetric and antisymmetric, participate in the nonlinear deformation. Interestingly, the antisymmetric modes hardly change their amplitude ratio, but there is a significant growth in the amplitudes of the symmetric shapes. The right-hand side of the plot exaggerates the higher mode amplitudes from which we can see that shapes #:9, 10, 11. 15, 22 are significant contributors.

3.4 Monitoring Changes of Shape and Stiffness

257

The lower plot in Fig. 3.29b shows the amplitude spectrum for the axial force contribution at the center of the arch. In contrast to the symmetric loading case, this shows very little change with load level. The reason is that the axial force generated is very small at all load levels. The identified higher stiffness shapes are the same as those identified earlier.

3.4.2 Monitoring Nonlinear Deformations Figure 3.31 shows a simple frame structure with a horizontal load applied at the right; the left is fixed, the right is on rollers. Clearly, there are significant changes of shape under the indicated compressive load. The maximum load is different in each case but surprisingly, it is the top row (case III) that supports the most load. Superficially, we can say that each has a different stiffness that inversely correlates with the height H of the initially vertical member. The situation, however, is much more interesting than that as a spectral analysis helps to reveal. Figure 3.32 shows how the stiffness changes with load. Initially, when there is no load, it is true that the stiffness correlates with H but this changes with load. All would show an initial increase with tension load and all do show an initial decrease with compression (because the horizontal members are put into compression), but case I achieves a minimum and then increases. The minimum is the second shape from the right in Fig. 3.32a. Case III on the other hand goes directly to a zero stiffness at the maximum load. Case II is especially interesting because the first two modes go to zero stiffness nearly simultaneously. Furthermore, unlike case III, (FE numerical) convergence difficulties were encountered for zero stiffness implying that the state is highly unstable. The next two chapters are devoted to the situation where there is complete loss of stiffness because this clearly is of great structural significance, the remainder of this section considers some modelings to show the connection between changes of

Fig. 3.31 Changes of shape under load. (a) Case I, H = Ho . (b) Case II, H = 34 Ho . (c) Case III, H = 12 Ho

3 Nonlinear Elastic Shapes ♦

258

♦

0.

20.

40.

60.

Fig. 3.32 Spectral analysis of a simple frame. Circles are case I with H = Ho , squares are case II with H = 34 Ho , triangles are case III with H = 12 Ho

Fig. 3.33 Three situations illustrating change of stiffness due to a transverse load. Top row is spectral plots (for two mode) for continuous versions of the problems, bottom row are simple models using only springs. Dashed lines represent deformed shapes. (a) Straight beam. (b) Arch. (c) Straight beam with initial axial stress

stiffness and shape driven by loads. This is an elaboration of the ideas introduced in Sect. 3.1 and therefore we continue to use mechanical models to express the points but supplement them with nonlinear FE results for continuous systems. But the main point is that the spectral analysis identifies behaviors in the FE results that are not obvious from just the displacements and stress. The role of the modeling is to explain these identified behaviors. For illustrative purposes, we consider three cases taken from some of the earlier example problems. and dissect their behavior by considering simpler situations involving just two springs as shown in Fig. 3.33. The first is when the springs are initially horizontal at their natural length and we show the effect that transverse stretching has; the second is when the springs are given an initial orientation and we show the stiffness effect of this orientation. In the third case the springs are initially axially stretched, and we show the effect on the transverse displacement as a complement to the first case. We finish with a linearized analysis of the second case so as to better illustrate the geometric contribution to the stiffness.

3.4 Monitoring Changes of Shape and Stiffness

259

Case I: Second-Order Stiffness Effects We begin with the initially unstretched springs in Fig. 3.33a. When loaded transversely, they generate axial tension which in combination with the large transverse deflection produces a resolved transverse component of force that supports the transverse load. The change of length of the springs is given by L = L − Lo =



L2o + vo2 − Lo ≈ 12 vo2 /Lo

The approximation is based on the assumption that the transverse deflection vo is smaller than the spring length Lo and allows us to avoid having to carry square root terms. The strain and axial force in the spring are then given by xx = [L − Lo ]/Lo ≈ 12 vo2 /L2o ,

F¯ = KL = Kxx Lo = 12 KLo vo2 /L2o

The total potential energy for the two springs and load is

= 2 12 K L2 − Po vo = 14 KL2o

vo4 − Po vo L4o

This is essentially the same result as obtained in connection with the cable in Fig. 3.10. The equilibrium equation and tangent stiffness are therefore ∂

v3 = KLo o3 − Po = 0 , ∂vo Lo

KT =

∂ 2

vo2 = 3K ∂vo2 L2o

(3.37)

The tangent stiffness KT is shown plotted in Fig. 3.34a as the curve F¯o = 0. Observe that initially, the connected springs have no (transverse) stiffness, it is only after deflections have occurred and axial forces are generated does the system begin to have transverse stiffness. This is also indicated in the spectral plots of Fig. 3.33a.

Case II: Effect of Initial Orientation Consider the simple mechanical model in Fig. 3.33b where the initial geometry (Lo , αo ) is for P = 0 and the springs are unstretched. The problem is symmetric so assume that the only DoF is the vertical displacement v. From geometry, we get that the axial displacement and strain in each spring are u¯ =



L2o + 2vLo sin αo + v 2 − Lo ,

The total potential for the problem is

¯ = u/L ¯ o

260

3 Nonlinear Elastic Shapes

Fig. 3.34 Stiffness behaviors. (a) Stiffness changes for initially horizontal springs. Full line is for F¯o > 0, dashed line is for F¯o = 0. (b) Stiffness changes with deflection for the arch. The dashed line is the approximate model

⎡. v

= 2 12 K u¯ 2 − P v = KL2o ⎣ 1 + 2 sin αo + Lo



v Lo

2

⎤2 − 1⎦ − P v

The equilibrium path is ⎤

⎡  ⎢ v ⎢ ∂

⎢1 − . = 2KLo sin αo + ∂v Lo ⎢ ⎣

1 v 1+2 sin αo + Lo



v Lo

⎥ ⎥ ⎥ Lo − P = 0 2⎥ ⎦

The structural stiffness is   [sin αo + v/Lo ]2 ∂ 2

1 KT = , + = 2K 1 − 1 + ¯ ∂v 2 [1 + ¯ ]3/2

¯ = u/L ¯ o

This is shown plotted in Fig. 3.34b as the full line. Although the expression is complicated, it plots as almost linear; which is also reflected in the spectral plots of Fig. 3.33b. This suggests that we can usefully replace it with a polynomial expression. Noting that 1 ≈ 1 − γ x + 12 γ (γ + 1)x 2 + · · · (1 + x)γ we get the approximations P ≈ 2KLo

 v  3 v2 2 , sin2 αo + sin α cos α o o Lo 2 L2o

3.4 Monitoring Changes of Shape and Stiffness

261

  v KT ≈ 2K sin2 αo + 3 sin αo cos2 αo Lo The approximation for the stiffness is shown as the dashed line in Fig. 3.34b, it is indistinguishable from the exact result. At zero deflection, the stiffness has the value KT = 2K sin2 αo . Let the initial height be vo so that sin αo ≈ vo /Lo , we see that the stiffness is less than that of Eq. (3.37); in other words, the stiffness is not just due to the geometry. The difference between the two is that at the height vo one is stressed and the other is not. We now look at this aspect of stiffness.

Case III: Initial Stiffness Due to Prestress Let the springs be initially stretched and have an initial pretension force F¯o before the transverse load Po is applied. That is, as shown in Fig. 3.33c, each spring is given an initial strain of o = L/Lo so that the initial force is F¯o = KL = Ko Lo and the stretched length under transverse load becomes   L= (Lo + L)2 + vo2 =Lo 1 + 2o + o2 + (vo /Lo )2 ≈ Lo [1 + o + 12 (vo /Lo )2 ] The total strain energy is then given by  U = 2 12 K[L − Lo ]2 = K o +

1 2

vo2 2 2 vo2 vo4 2 2 2 1 ¯ L = K L + F L + K o o o o o 4 L4 Lo L2o L2o o

The potential of the load is the same as in case I so that the stiffness becomes KT =

∂ 2

vo2 F¯o = 3K +2 = KE + KG 2 2 Lo ∂vo Lo

(3.38)

This is shown plotted in Fig. 3.34a as the full line F¯o > 0. The pretension has given the spring system some initial transverse stiffness that is related to F¯o /Lo as shown in Fig. 3.34a and this is also reflected in the spectral plots of Fig. 3.33c. We see that the total stiffness comprises that of the elastic contribution (related to the spring stiffness K) and that of the pretension contribution (related to F¯o /Lo ). This decomposition is fundamental in nonlinear analyses of structures; as discussed in Sect. 3.1, we refer to the first as the elastic stiffness, because it is associated with the elastic stretching of the springs, and the second as the initial stress stiffness. This latter stiffness is also called the geometric stiffness because it is associated with the change of orientation of the springs which is manifested by the second-order displacement (vo /Lo )2 . That is, for a single spring UG = 12 (F¯o /Lo )vo2 . We thus see how the model used to describe the linear behavior of a cable through Eq. (1.16) is based on the assumption that the pretension is dominant. This is an elaboration

262

3 Nonlinear Elastic Shapes

of what we have already encountered; namely, that stiffness is related to axial force and hence membrane stresses. To summarize, an applied load generates two stiffness changing mechanisms: first it changes the shape, and second it generates axial (membrane) stresses. Depending on the circumstance, either one could be dominant, but next we demonstrate a useful approximation when the change of shape is small. Case IV: Approximations When Changes of Shape Are Small The approximation of Case II makes clear that the current total stiffness comprises the two distinct contributions KE = 2K sin2 αo ,

KG = 6K

v u¯ L sin αo cos2 αo = 6K cos2 αo Lo Lo

where u¯ L is the component of displacement resolved along the original orientation of the spring. But this displacement is related to the axial force by K u¯ L = F¯ . We therefore see that KG is related to the axial force; this in turn is related to the applied force by F¯ = Po /(2 sin α) where α is the current orientation of the spring. We now do a linearized version of the analysis. First consider the initially unloaded case; in local coordinates oriented with respect to a spring, the displacements at the end of the spring are u¯ L = v sin αo ,

v¯L = v cos αo

The spring is then both stretched and given a transverse displacement similar to the situation covered with respect to the cable in Fig. 1.14c. Equilibrium of the loaded joint gives   2 Kv sin αo sin αo = Po

or

    Po = K sin2 αo 2v = KLo sin3 αo 2v˜

This gives the axial force as F¯o = Po /(2 sin αo ) = KLo sin2 αo v; ¯ an axial force is generated by the local transverse displacement v¯L (or equivalently, the local axial displacement u¯ L ). Now consider when the springs are already stressed but assume the geometry has not changed by much; the total strain energy for an additional displacement ξ is estimated as U ∗ = UE + UG = 2 12 K v¯L2 + 2 12 [F¯o /Lo ]v¯L2 = Kξ 2 sin2 αo + F¯o ξ 2 cos2 αo /Lo where we have included the prestressed energy due to F¯o . This is an estimate of the energy in the loaded structure based on the unloaded geometry. The total stiffness is then KT∗ =

∂2 U∗ F¯o = K2 sin2 αo + 2 cos2 αo 2 Lo ∂ξ

3.4 Monitoring Changes of Shape and Stiffness

263

where it is seen to have a contribution from both the axial stiffness and the axial force. The stiffness due to the axial force depends on the load Po and hence displacement v (and not ξ ), that is,     KT∗ = K sin2 αo 2 + 2v˜ cos2 αo = Ko 2 + 2v˜ cos2 αo ,

Ko ≡ K sin2 αo

This is similar to the earlier result but the slopes are slightly different. Chapter 4 looks more closely at the cases where a small displacement is superposed on an already nonlinearly deformed structure. A primary conclusion is that the spectral stiffness is a direct monitor of the consequences of this.

Explorations 3.1 An initially deflected plate has a curvature and as such could be considered a curved plate. Let the initial (mid-surface) deformed shape be given by ξ(x, y) so that the initial position of an arbitrary point is described dy rˆ o = x o iˆ + y o jˆ + (ξ + zo )kˆ ,

d rˆ o = dx o iˆ + dy o jˆ + [(ξ + zo ),x + (ξ + zo ),y ]kˆ

and let the additional deformations be given by u(x ¯ o , y o ) = uo − zw,x ,

v(x ¯ o , y o ) = v o − zw,y ,

w(x ¯ o , y o ) = wo

where the unbarred displacements uo , v o , w o are of the mid-plane and are functions of x o and y o . • Show that the differential of the position vector is given by d rˆ = [(1 + uo,x − zw,xx )dx o + (uo,y − zw,xy )dy o ] iˆ o o − zw,xy )dx o + (1 + v,y − zw,yy )dy o ] jˆ +[(v,x

+[(ξ + z + wo ),x dx o + (ξ + z + wo ),y dy o ] kˆ • Show that the nonlinear strains are given by 2 2 + w,x ] + ξ,x w,x − zo w,xx + · · · E¯ xx = u,x + 12 [u2,x + v,x 2 2 + w,y ] + ξ,y w,y − zo w,yy + · · · E¯ yy = v,y + 12 [u2,y + v,y

2E¯ xy = u,y + v,x + [u,x u,y + v,x v,y + w,x w,y ] +ξ,y w,x + ξ,x w,y − 2zo w,xy + · · ·

264

3 Nonlinear Elastic Shapes

• Simplify for plates with only slight initial curvature, — Reference [3], pp. 96 3.2 With reference to Fig. 3.4, with K1 = K2 : • Explore the existence of an equilibrium position either very far to the left (or right) of the lower fixed points. • Explore in what ways the nature of the problem changes when the horizontal springs are not of equal stiffness. 3.3 Cables with single supports used to support gravity generated loads are usually called hangers; the figure shows two hangers supporting the mass 2Mo . • It is obvious that the system has vertical stiffness due to EA, investigate its horizontal stiffness due to the axial stress. Specifically, determine the contributions of pretension due to gravity and change of potential energy due to the pendulum action of the cable.

3.4 Obtain a Ritz approximate solution for an elliptical plate under transverse uniform pressure. Assume a deformed shape of    y2  x2 w(x, y) = 1 − 2 − 2 a00 + a20 x 2 + a02 y 2 a b and that the deflections can be large. • Use an FE analysis to check the accuracy. — Reference [3], pp. 138

References 1. Bathe, K.-J.: Finite Element Procedures. Prentice-Hall, Englewood Cliffs, NJ (1996) 2. Bradford, M.A., Uy, B., Pi, Y.L.: In-plane elastic stability of Arches under a central concentrated load. J. Eng. Mech. 128(7), 710–719 (2002) 3. Chia, C-Y., Nonlinear Analysis of Plates. McGraw-Hill, New York (1980) 4. Dhondt, G.: The Finite Element Method for Three-Dimensional Thermomechanical Applications. Wiley, Chichester (2004) 5. Doyle, J.F.: Wave Propagation in Structures. Springer, New York (1989), 2/E 1997

References

265

6. Doyle, J.F.: Nonlinear Analysis of Thin-Walled Structures: Statics, Dynamics, and Stability. Springer, New York (2001) 7. Doyle, J.F.: Modern Experimental Stress Analysis: Completing the Solution of Partially Specified Problems. Wiley, Chichester (2004) 8. Doyle, J.F.: Mechanics of Structural Materials: From Atoms to Continua. Supplemental Class Notes. Purdue University, West Lafayette, IA (2010) 9. Doyle, J.F.: Nonlinear Structural Dynamics using FE Methods. Cambridge University Press, Cambridge (2015) 10. Elishakoff, I.: Controversy associated with the so-called follower forces: critical overview. Appl. Mech. Rev. ASME 58, 117–142 (2005) 11. Kang, Y.J., Yoo, C.H.: Thin-walled curved beams. I: analytical solutions for buckling of Arches. J. Eng. Mech. 120, 2102–2125 (1994) 12. Kang, Y.J., Yoo, C.H.: Thin-walled curved beams. I: formulation of nonlinear equations. J. Eng. Mech. 120, 2072–2101 (1994) 13. Langhaar, H.L., Boresi, A.P.: Snap-through and postbuckling behavior of cylindrical shells under the action of external pressure. Bull. Eng. Exp. Stat. 54(59) (1957) 14. Novozhilov, V.V.: Foundations of the Nonlinear Theory of Elasticity. Graylock Press, Rochester, NY (1953) 15. Pi, Y.-L., Trahair, N.S.: Nonlinear buckling and postbuckling of elastic Arches. Eng. Struct. 20, 571–579 (1998) 16. Pi, Y.-L., Bradfort, M.A., Francis T-L.: Nonlinear in-plane Buckling of rotationally restrained Shallow Arches under a central concentrated load. Int. J. Non-Linear Mech. 43, 1–17 (2008) 17. Sachkov, Yu.L., Levyakov, S.V.: Stability of inflectional elasticae centered at vertices or inflection points. Proc. Steklov Inst. Math. 271, 177–192 (2010) 18. Timoshenko, S.P., Gere, J.M.: Theory of Elastic Stability. McGraw-Hill, New York (1963) 19. Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity. McGraw-Hill, New York (1970)

Chapter 4

Buckling Shapes

Slender members and thin-walled structures are susceptible to a failure mechanism or instability known as buckling. That is, the current load state cannot be maintained without a significant change of shape. Instability is not to be confused with nonequilibrium; stability is an attribute of the equilibrium of a system. The next chapter delves into the post-instability behavior of a structure, in this chapter we focus on how the structure reaches an unstable state. As presented in Chap. 3, in assessing the stability of a structure, we are interested in what happens to the structure when it is disturbed slightly from its current equilibrium position: does it tend to return to its equilibrium position, or does it tend to depart even further? Buckling analysis of stability is based on two approximations. First, only second-order energy terms are used; second, the load is assumed known from a separate analysis. In this way, the problem can be linearized and reduced to a limited number of unknown variables (e.g., transverse deflection). Although this formulation is approximate, for stiff structures it yields many useful results. Reference [7] gives a good background setting for the study of the buckling of structures, whereas Ref. [11] is an excellent compendium of examples and solutions. The primary source for the buckling instability in structures is the presence of compressive membrane stresses in the structure caused by the loads. Figure 4.1 shows the schematic outline of our approach. The spectral analysis gives a set of shapes ordered with respect to their stiffness. It also shows how the order changes with loads and therefore identifies the critical modes.

4.1 Buckling Shapes of Straight Beams The buckling instability in slender members is caused by the axial compressive stresses. Accounting for the effects of stresses on the behavior of structures fun© Springer Nature Switzerland AG 2020 J. F. Doyle, Spectral Analysis of Nonlinear Elastic Shapes, https://doi.org/10.1007/978-3-030-59494-7_5

267

268

4 Buckling Shapes

Fig. 4.1 Schematic for the spectral analysis of buckling shapes

damentally requires a nonlinear analysis as elaborated on throughout the previous chapter but especially in Sect. 3.2. The buckling problem is easier to tackle if it can be divided into two somewhat unrelated problems. First is the prestress stage where the geometry is assumed to not change by much and linear analysis is used to compute the axial forces; in this section, for simplicity, we mostly consider problems where the stresses in members are straightforward to determine or approximate. Second is the buckling analysis where the axial forces are considered as pre-existing and the only unknown is the out-of-plane deflection. We introduce an important technique known as second-order analysis which allows us to essentially linearize the problem. It is then shown that the buckling problem can be posed in the form of an eigenvalue problem. This allows us to detect the onset of an instability without doing a costly nonlinear analysis. The two aspects explored here is that of distributed elastic constraints and mode interaction.

4.1.1 Second-Order Approximation for the Energy Contributions Figure 4.2a shows a rigid bar attached to a torsional spring at its base. The bar was initially in the upright position before the loads were applied. In the upright position under the action of Po alone, we do not expect any transverse displacement because a free body would show that the forces are balanced and there is no resultant force in the horizontal direction. The key to understanding the effects of nonlinearities is to consider the energies in the slightly disturbed configuration, in the present case a small extra rotation. This discussion has commonalities with that associated with Fig. 3.1 but the types of springs (torsion versus linear) affect the specifics of the results. Therefore, in this analysis we add the new case but the presented results are for both of them. (The cases are made comparable by having Kt = K1 L2 .) Using φ as the single DoF, the strain energy and potential of the loads are, respectively, U = 12 Kt φ 2 ,

V = −Qo u − (−Po )v = −Qo L sin φ + Po L[cos φ − 1]

Total potential, equilibrium, and stiffness are then given by, respectively,

269

♦

♦

4.1 Buckling Shapes of Straight Beams

♦

♦

0.

30.

60.

90. 0.

30.

60.

90.

Fig. 4.2 Nonlinear deflections of a simple system. (a) SDoF system with rigid bar and torsional spring. (b) Load-deflection behavior. Heavy line is for χ = 0.02, Pc = Kt /L = K1 L. (c) Stiffness behavior. Heavy line is for χ = 0.02 and the horizontal arrow is the zero reference

= 12 Kt φ 2 − Qo L sin φ + Po L[cos φ − 1] ∂

= Kt φ − Qo L cos φ − Po L sin φ = 0 ∂φ KT =

∂ 2

= Kt + Qo L sin φ − Po L cos φ ∂φ 2

Consider the case of proportional loading where Qo = χ Po and we think of χ as being small. The equilibrium equation then gives Po as a function of φ according to Po L = Kt φ/[χ cos φ + sin φ] This is shown plotted in Fig. 4.2b as the heavy line for χ = 0.02. It shows a monotonic increase in deflection in contrast to the earlier case (dashed line) which exhibits a limit point. The stiffness plot of Fig. 4.2c shows a significant decrease of stiffness but it does not go through zero, and in fact, on further deflection it increases its stiffness. When the proportional loading is changed to χ = 0.001 (the light lines in the figure) the same behaviors are exhibited but the limit point and zero stiffness are confined closer to the vertical orientation. One can imagine that if χ = 0, these would occur along the vertical axis. But, as stated earlier, if Qo = 0, we do not expect any rotation in which case the plot ceases to exist. This situation needs to be explained. Our test for stability of a deformed configuration is to disturb the structure slightly and ask if the energy increases or decreases. The specific situation we consider is that the bar is in the nonlinear state with deformation φ, under the actions of loads Po and Qo . As we did in Sect. 3.2, consider a small change given to the deformation according to φ¯ = φ + φ b

270

4 Buckling Shapes

but with the loads left intact. We use the small parameter  to track the different orders of approximation and the superscript b (for buckling) designates the additional rotation. Expand the total potential to ¯ = 1 Kt φ¯ 2 − Qo L sin φ¯ + Po L[cos φ¯ − 1]

2 1  = 2 Kt φ 2 − Qo L sin φ + Po L[cos φ − 1]   + Kt φ − Qo L cos φ − Po L sin φ φ b   + Kt + Qo L sin φ − Po L cos φ 12  2 φ b2 + · · · = 0 + 1 φ b + 2 21  2 φ b2 + · · · We recognize the bracketed terms as being, respectively, the potential, equilibrium, and stiffness. In our conception, only the superscripted b terms can change; therefore, at equilibrium with 1 = 0, it is the sign of 2 that informs us of the stability of the system. In one sense this is just a restatement of what we already know in terms of the stiffness; what we are emphasizing here is the distinction between the nonlinear deformation and the buckling deformation. In our current SDoF case, they are in the same direction but nonetheless distinct; in the general case, the directions can also be different. Thus, even if we set Qo = 0 such that φ = 0, we get a nontrivial expansion for the potential as ¯ ≈ [ 0 ] + [ 0 ] φ b + [Kt − Po L] 1  2 φ b2 + · · ·

2 If Po > Kt /L, we get a negative stiffness. This does not contradict our discussions of Fig. 4.2c because that was for a different equilibrium path that involved some rotation. The remainder of our discussion of Fig. 4.2 focusses on the situation where the bar is predominantly vertical and we need not distinguish the deformed state and the buckling state. In general, however, when there is noticeable change of shape before buckling, it is important to make the distinction between the deformed state and the buckling state. When Qo is absent the load point Po = Kt /L is a bifurcation point because it is the intersection of two equilibrium paths. To elaborate on this, consider the equilibrium equation and expand the trigonometric terms for small angle to get    Kt φ − Po L φ − 16 φ 3 + · · · = 0

⇒

   Kt − Po L 1 − 16 φ 2 φ = 0

Implicit in this relation is that if the structure does deflect off the vertical, then it is only by a small amount. The equilibrium has two solutions (the equilibrium paths) I: φ = 0 ,

  II: Kt − Po L 1 − 16 φ 2 = 0

4.1 Buckling Shapes of Straight Beams

271

The first does indeed show that the upright position is a solution. But, as is common in nonlinear problems, there is also a second possible solution. Therefore, in a given context, which of the two solutions prevails? The answer to this question is given by the stiffness KT =

  ∂ 2

1 2 = K − P L cos φ ≈ K − P L 1 − φ t o t o 2 ∂φ 2

Separately, consider the two equilibrium paths so that for path I get I:

KT = Kt − Po L

For Po > Kt /L, the stiffness is negative and therefore the equilibrium state is unstable. Thus the presence of even a very small Qo would easily shift the bar significantly off the vertical. The second path is analyzed similarly so that II:

φ 2 = 6[Po L − Kt ]/Po L ,

KT = −2Kt + 2Po L

Because this path exists only for Po > Kt /L (φ 2 must be positive), then the stiffness is positive and the new found equilibrium state is stable. A similar analysis for the case of Fig. 3.1 shows that the stiffness is negative and there are no nearby stable equilibrium states. Being able to determine the post-instability state is an important part of stability analyses but that is not addressed in this chapter for reasons discussed next; it is taken up in Chap. 5. Thus far, the analysis has been fully nonlinear made tractable by the choice of mechanical model. For more general deformable structures tractable solutions are usually not in the offing and therefore we seek an approximate solution. Specifically, we settle for determining the critical point (where multiple solutions can exist) but at the expense of not knowing the stability state of the alternate equilibrium paths. The practical justification for this is that a structure is designed so that the loading does not get close to the critical value and therefore the post-instability states are not relevant. This, of course, is wishful thinking which is why Chap. 5 considers these post stability states. Here, we develop the approximation because it is very useful in many situations. In any event, it is a necessary analysis before the full nonlinear analysis can be embarked upon. This type of analysis that seeks just the critical values of load we shall refer to as a buckling analysis. The key approximation made is in the treatment of the change of shape during deformation as represented by the off-vertical angle φ caused primarily by the transverse load Qo . The angle affects the potential of the applied loads differently. To see this, consider an expansion of the total potential for small angle; we get the sequence when truncated after quadratic terms

= 12 Kt φ 2 − Qo L sin φ + Po L[cos φ − 1] ≈ 12 Kt φ 2 − Qo Lφ − Po L 12 φ 2

4 Buckling Shapes

♦

Fig. 4.3 Nonlinear deflections of a simple system. (a) Two-DoF system. (b) Buckling shapes

♦

272

♦

♦

∂

= Kt φ − Qo L − Po Lφ = 0 ∂φ ∂ 2

= Kt − Po L ∂φ 2 Observe that the potential of the load Po looks similar to the strain energy term for Kt (i.e., quadratic); in fact, we refer to it as the geometric strain energy term. The final equation shows that Po contributes to the structural stiffness but Qo does not. Therefore, in its most elemental form, we have for the total stiffness KT = Kt −Po L implying that a change of shape due to load does not make a contribution. In this manner, the equilibrium equation is linearized to [Kt /L − Po ] φ = Qo Obviously φ can be determined given any value of Po and Qo ; but suppose Qo = 0 then [Kt /L − Po ]φ = 0 This is known as a [1×1] eigenvalue problem (EVP); that is, nontrivial solutions are obtained only when Kt /L − Po = 0. Thereby, the critical value for Po is obtained through a linearized analysis. As stated earlier, we get this useful information at the expense of not knowing the final value of φ. The following example problem shows that nonetheless significant additional information can be obtained besides the critical value. It also elaborates on more aspects of EVPs as pertains to buckling problems. Example 4.1 If the model in Fig. 4.2a is to represent a continuous beam structure, then it must have additional DoFs. Investigate the consequences of adding one additional DoF. With reference to Fig. 4.3a, each link is of length 12 L and each spring has stiffness Kt . The potential associated with Po is

4.1 Buckling Shapes of Straight Beams

V = −(−Po )v = Po



1 2 L cos φ1

273

 + 12 L cos φ2 − L ≈ − 12 Po [φ12 + φ22 ] 12 L

This result readily generalizes to a system of N links each of length dx = L/N. Note that on replacing the summation with an integral and using φ = ∂v/∂x, we get the geometric stiffness for a cable as given by Eq. (1.16); in fact, this is the elastica modeling discussed in the Introduction. The total potential is

= 12 Kt φ12 + 12 Kt (φ2 − φ1 )2 − 12 Po [φ12 + φ22 ] 12 L − Qo [φ1 + φ2 ] 12 L Differentiation with respect to the DoF then gives the equilibrium equations     # $ # $  2 −1 10 1 φ1 Kt = Qo 21 L − Po 21 L φ2 −1 1 01 1 This can be solved as a regular equilibrium problem given particular values of Po and Qo . But suppose Qo = 0, does this mean that {φ } = 0? The answer is no because we get an eigenvalue problem (EVP). Rewrite the equilibrium equation as  # $ 2 − λ −1 φ1 = 0, −1 1 − λ φ2

λ = 12 Po L/Kt

(4.1)

where λ (and hence Po ) is the eigenvalue. Set the determinant to zero to get the eigenvalues det = λ2 − 3λ + 1 = 0 ⇒ λ = [3 ∓

√ 9 − 4]/2 = 0.382, 2.618 ≈ 3/8, 21/8

Because there are two DoF, there are two critical load values given by Pc1 =

3 Kt , 4 L

Pc2 =

21 Kt 4 L

We also get information about the buckling shape. Using the first equilibrium equation then gives that the amplitude ratios are related by φ2 = (2 − λ) φ1 . In turn, substitute for the critical values to get λ1 :

φ2 =

13 φ1 , 8

λ2 :

5 φ2 = − φ1 8

These are shown plotted in Fig. 4.3b; we later establish through Fig. 4.6b that these shapes do indeed begin to resemble those of a continuous beam. We refer to this eigenanalysis of the critical loads as an eigenbuckling analysis. At present, we are more interested in the fact that extra DoFs give extra modes generally resulting in coupled stiffnesses and hence coupled modes as discussed in connection with Fig. 3.6. We use a spectral analysis to put these results in a bigger

274

4 Buckling Shapes

context. The mass matrix can be established using the approach demonstrated in Sect. 1.4 plus some physical assumptions about the mass distributions. A reasonable set of generalized inertia forces is #

$  # $ P1 3 1 φ¨1 = −M¯ , P2 1 1 φ¨2

M¯ = ML2 /2

so that the EoM are  # $     # $ # $  3 1 φ¨1 2 −1 10 1 φ1 1 1 M¯ + K = Q L L − P t o2 o2 φ2 1 1 φ¨2 −1 1 01 1 Consider Qo as short-lived so that the response thereafter is a harmonic motion. The corresponding spectral EVP is  # $ 2Kt − Po 21 L − ω2 3M¯ φˆ 1 −Kt − ω2 M¯ =0 1 2 2 Kt − Po 2 L − ω M¯ −Kt − ω M¯ φˆ 2

(4.2)

In spectral analysis, it is the frequency ω2 that is the unknown eigenvalue and (in this case) the load Po the control parameter. The characteristic equation is quadratic in ω2 and Fig. 4.4a shows how the spectral stiffness values change with load. Both modes go through a zero stiffness value and these occur at precisely the critical values of load just obtained. This must be so because if we put ω = 0 in Eq. (4.2), we recover the EVP of Eq. (4.1). The critical values of load are, therefore, said to occur at static instability points. The spectral analysis also helps us understand what happens when ω2 becomes negative. In that case, ω becomes imaginary-only as ±iωI , and for a particular mode we have that

Fig. 4.4 Spectral analysis of a two-DoF model of a beam. (a) Spectral plot of the stiffness. (b) Amplitude ratios for the two modes

4.1 Buckling Shapes of Straight Beams

275

φ(t) = φˆ eiωt = φˆ ei(0±iωI )t = φˆ e∓ωI t From this it is clear that an exponential increase in displacement occurs—this is the instability. Where it goes is not determinable from this analysis. Figure 4.4b shows how the amplitude ratios change with load and at the critical load values they give (in this case) precisely the same values as the eigenbuckling analysis. That the ratio φˆ 2 /φˆ 1 changes with load is indicative that there is mode interaction. It is not strong in this case and the first mode goes to zero directly because it is dominated by the 1D model shown in Fig. 4.2a. Because the eigenbuckling analysis and the spectral analysis give the same results in the vicinity of critical points, then what is to choose between the two? Here is the difference: the eigenbuckling analysis is just one point along a whole spectrum of points provided by the spectral analysis. Therefore, the spectral analysis gives a much richer and deeper appreciation of what is occurring leading up to the critical load points, and not just the critical values themselves. In fact, the situation identified in Fig. 3.32 as case I coming close to being unstable is not registered in a buckling eigenanalysis. Also, what is not obvious here but made explicit later, is that the spectral analysis is applicable irrespective of the nonlinearities—it is part of the nonlinear analyses—whereas eigenbuckling analyses are limited to the second-order approximation introduced in this section. Consequently, among its other attributes, spectral analysis is a good tool to assess the performance of the eigenbuckling analyses.

4.1.2 Straight Beam Under Axial Loads Earlier we considered the expansion of the total potential about a deformed configuration and showed that the second-order term is related to the stiffness. Let us now do the same for the continuous beam. The situation we have in mind is that the beam is in the nonlinear state with deformations u and v under the actions of distributed loads qu (x) and qv (x). A small change is given to the deformations according to u¯ = u + ub ,

v¯ = v + v b

but the loads are left intact. The shape (ub , v b ) must satisfy the geometric constraints and therefore the spectral shapes are suitable candidate shapes. The nonlinear total strain becomes E¯ xx = u, ¯ x + 12 v, ¯ 2x = [u,x + 12 v,2x ] + [u,bx +v,x v,bx ] + 12  2 v,b2 x We see our first interaction term in v,x v,bx . As shorthand, we write the strain and the corresponding curvature as, respectively,

276

4 Buckling Shapes b + 12  2 v,b2 E¯ xx = Exx + Exx x ,

κ¯ xx = v,xx +v,bxx

The total potential under constant load becomes ¯ =

≈



2 dx + EAE¯ xx

1 2

1

 +

EI v, ¯ 2xx



dx −

2 dx + EAExx

1 2



[qu u¯ + qv v] ¯ dx 



2

 +



 1 2

EI v,2xx dx −

[qu u + qv v] dx





b EAExx Exx dx +

EI v,xx v,bxx dx −



 b2 EA[Exx + Exx v,b2 x ] dx +

b2 EI vxx dx



 [qu ub + qv v b ] dx 

1

2

2

+ ···

= 0 + 1  + 2 12  2 + · · · The approximation is because we are truncating after  2 terms. This has identified for us (through 2 ) the energy contributions to the stiffness. It consists of  UE =

1 2

 b2 dx UG1 = EI vxx

1 2

UG2 =

1 2

 EAExx v,b2 x dx =



1 2

F¯o v,b2 x dx

 b2 dx = EAExx

1 2

EA[u,bx +v,x v,bx ]2 dx

We see a second level of interaction in the F¯o v,b2 x term. There are two identifiable interactions between the current deformation state (v,x ) and the buckling deformation state (v,bx ). One occurs in UG1 through Exx v,b2 x , the other in UG2 through v,x v,bx . In a good number of beam buckling problems, the transverse deflections are initially negligible (i.e., qv = 0 and v ≈ 0) so that the second interaction term can be neglected. This is generally not true for soft structures whose geometry changes during the deformation up to the point of buckling. A good example of this is the buckling of arches and we save our further discussions until then. Furthermore, because the buckling is flexurally dominated we can also neglect u,b2 x . The relevant energies are then just the flexural and first geometric energies. In our conception, only the superscripted b terms can change during buckling ¯ achieves (individually) an extremum. We use and each of the bracketed terms in

this idea to convert each potential into a strong formulation. For example, we have for 1    b b b δ 1 = EAExx δ[u,x +v,x v,x ] dx+ EI v,xx δv,xx dx− [qu δub +qv δv b ] dx Integration by parts then gives (for piece-wise uniform properties) −EA Exx ,x −qu = 0 ,

−EA (Exx v,x ),x +EI v,xxxx −qv = 0

4.1 Buckling Shapes of Straight Beams

277

Noting that F¯o = EA Exx , the first equation says that F¯o (x) = constant when qu = 0 (only point axial loads are applied) and the second equation becomes EI v,xxxx −F¯o v,xx = qv This is the equilibrium equation for a beam under axial load subjected to a transverse load; it is not the buckling equation. Now consider 2 .  δ 2 =

  b EA Exx δ[u,bx +v,x v,bx ] + v,x δv,bx dx +

 b EI v,bxx δvxx dx

Integration by parts then gives (again for piece-wise uniform properties) b −EA Exx ,x = 0 ,

b −EA (Exx v,x ),x −EA (Exx v,bx ),x +EI v,xxxx = 0

b is uniform along We conclude that the axial force during buckling F b = EAExx the length. Consequently, the second equation becomes b v,xx EI v,bxxxx −F¯o v,bxx = EA Exx

This is a very interesting equation. The left-hand side has the same form as our equilibrium equation except that the superscript is “b”; this is the buckling shape equation. More specifically, this is the differential equation for the buckling shape that gives an extremum for the stiffness. The RHS is inhomogeneous related to the current deformation state (represented by v,xx ) and a contribution from the axial force generated during buckling. This contribution can be neglected for beams so that the equilibrium and stiffness differential equations are the same. This is the same as what happens in linear analyses where the stiffness relation can be interpreted as an equilibrium equation. To summarize, our final energy expression comprises the contributions from the rod-like axial deformation, the beam-like flexural deformation (these two comprise the elastic energy UE ), and what we call the cable-like transverse deformation which contributes the geometric strain energy UG . Dropping the superscript b, we have 

 UE =

1 2

EIzz L

∂ 2v ∂x

2

 dx ,

UG =

1 2

 2 ∂v dx F¯o ∂x L

(4.3)

The geometric term plays the key role in causing buckling instabilities because it is the stiffness term that is load dependent. In buckling analyses, we assume that the relation between F¯o (x) and the applied loads is known from a separate (typically linear and based on the initial geometry) prestress analysis. Using our variational principle (on the single variable v) leads to the governing equation [5]

278

4 Buckling Shapes

∂  ¯ ∂v  ∂2  ∂ 2v  Fo = qv (x) EI − zz ∂x ∂x ∂x 2 ∂x 2 The associated boundary conditions are specified using the terms   ∂  ∂ 3v  ∂v v, − EIzz 3 + F¯o = Vy , ∂x ∂x ∂x

  ∂ 2v φz , EIzz 2 = Mz ∂x

All the relationships for the transverse deflections may be summarized as Displacement : Slope : Moment/Stress : Shear : Loading :

v = v(x) φz =

∂v ∂x

Mz = +EIzz

∂ 2v , ∂x 2

σxx = Ey

∂ 2v ∂x 2

∂ 2v  ∂v ∂  EIzz 2 + F¯o ∂x ∂x ∂x   2 2 ∂  ¯ ∂v  ∂ v ∂ Fo qv = + 2 EIzz 2 − ∂x ∂x ∂x ∂x

Vy = −

(4.4)

These equations are often referred to as the coupled-beam equations (although they are sometimes also called the beam-column equations) and are to be compared to those of Eq. (1.19). The only difference in comparison with the Bernoulli–Euler beam equations is the addition of the F¯o related terms in the expressions for the loading and shear. The BCs are specified identically as those itemized earlier in Fig. 1.17. Note, however, that a zero applied shear BC does not imply that the third derivative of deflection is zero, but is moderated by the presence of the axial force. We reiterate that for the purpose of integrating these governing equations, F¯o is considered known from the prestress analysis. As an example, to integrate the loading relation, we assume that the material properties EIzz , the axial force F¯o , and the loading qv are all at least piece-wise constant. Integrating twice gives EIzz

d 2v − F¯o v = c1∗ x + c2∗ + 12 qo x 2 dx 2

(4.5)

This is an inhomogeneous second-order differential equation. The complete solution comprises a solution to the homogeneous equation plus a particular solution. The character of the solution differs depending on whether F¯o is tensile or compressive. The general deflection solution for compressive loading is v(x) = c1 cos βx + c2 sin βx + c3 x + c4 +

qo x2 2EIzz β 2

β2 ≡

−F¯o EIzz

(4.6)

4.1 Buckling Shapes of Straight Beams

279

Fig. 4.5 Pinned–pinned beams. (a) Geometry. Top beam has an eccentric (or imperfect axial) load. (b) First two buckling mode shapes

and for tensile loading v(x) = c1 cosh βx+c2 sinh βx+c3 x+c4 +

qo x2 2EIzz β 2

β2 ≡

F¯o EIzz

(4.7)

The constants of integration cn are evaluated by imposing particular BCs according to Eq. (4.4). Example 4.2 Use an eigenanalysis to determine the buckling loads for the pinned– pinned beam shown in Fig. 4.5. We begin with the imperfect loading ( = 0) case. Except for Po , the problem does not state any other applied loads, therefore, by inspection for small L get that F¯o = −Po and the deflection solution of Eq. (4.6) is appropriate. The BCs are at x = 0 :

v = 0 ⇒ 0 = c1 + c4 Mz = EIzz

at x = L : Mz = EIzz

d 2v = 0 ⇒ 0 = −c1 β 2 dx 2 v = 0 ⇒ 0 = c1 cos βL + c2 sin βL + c3 L + c4

d 2v = LPo ⇒ LPo = −c1 β 2 cos βL − c2 β 2 sin βL dx 2

These four equations can be put into matrix form as ⎡

1 0 2 ⎢ 0 −β ⎢ ⎣ cos βL sin βL −β 2 cos βL −β 2 sin βL

⎤⎧ ⎫ ⎧ ⎫ 0⎪ c1 ⎪ ⎪ 01 ⎪ ⎪ ⎨ ⎪ ⎬ ⎨ ⎪ ⎬ ⎪ 0 LPo 0 0⎥ c 2 ⎥ = ⎦ 0 ⎪ EI c ⎪ ⎪ L1 ⎪ ⎪ ⎩ ⎪ ⎭ ⎩ 3⎪ ⎭ ⎪ 1 00 c4

280

4 Buckling Shapes

Po appears on both sides of the equation (it is embedded in β) but with Po known it is an ordinary linear elastic problem. Suppose, however, we set  = 0 we then have an EVP with Po unknown. We now inquire if the determinant of this matrix can be zero so as to obtain nontrivial values for Po . On multiplying out, get det = β 4 L sin βL = 0 There are many values of β that satisfy this equation. The obvious one of β = 0 corresponds to the trivial case of zero axial load and zero transverse displacement. The other possibilities are βL = π , 2π , 3π , · · · , nπ On substituting for β in terms of the force, this gives Pc = n2 π 2

EI L2

There are many critical loads, and corresponding to each there is a different buckling shape. To determine these shapes, let us reconsider the relation among the coefficients. At the special values of βL = nπ , the matrix for determining { c } reduces to ⎡

1 ⎢ −β 2 ⎢ ⎣ 1 −β 2

0 0 0 0

⎤⎧ ⎫ c1 ⎪ 01 ⎪ ⎪ ⎨ ⎪ ⎬ ⎥ 0 0 ⎥ c2 =0 c ⎪ L 1⎦⎪ ⎪ ⎩ 3⎪ ⎭ 00 c4

From this we get c1 = 0 ,

c3 = 0 ,

c4 = 0

But we cannot determine c2 . In other words, the equations are satisfied for any value of c2 . Hence the buckling shape is v(x) = c2 sin(nπ x/L) The first two shapes are shown in Fig. 4.5b. Note that because c2 is unknown we have determined the shape of the deflection but not the actual deflection. For more general BCs, we eventually establish the homogeneous system of equations [A(λ)]{ c } = { 0 }

4.1 Buckling Shapes of Straight Beams

281

Table 4.1 Critical load factor α for the first three buckling modes of beams with different BCs where Pc = απ 2 EI /L2 BC Cantilever Pinned–pinned Fixed–pinned Fixed–fixed

Mode = 1 0.25 1.00 2.05 4.00

2 2.25 4.00 6.05 8.18

3 6.25 9.00 12.05 16.00

Characteristic equation cos βL = 0 sin βL = 0 tan βL − βL = 0 2(cos βL − 1) + βL sin βL = 0

where λ is some parameter (which we identify in our case as the axial load) and { c } represents the vector of unknown coefficients. The special values of λ which cause the determinant of [A(λ)] to be zero are the critical values of our EVP; actually, because [A(λ)] contains trigonometric and hyperbolic functions, det[ A ] = 0 ⇒ (sin βL, cos βL, sinh βL, cosh βL) = 0 , β = Po /EI it is usually referred to as a transcendental eigenvalue problem. We obtain the special values of λ (the eigenvalues) by setting the determinant to zero. Thus the critical load is deeply imbedded in the trigonometric and hyperbolic functions. In general, therefore, some numerical method is required to effect a solution. Corresponding to each eigenvalue, we can find a solution for { c }, this is called the eigenvector and when it is used to reconstruct the distribution v(x), this distribution is called the buckling shape. For future reference, Table 4.1 gives results for some other BCs. The critical load is given by Pc = απ 2

EI L2

The numerical parameter α is given in the table. In each case, the critical load is inversely related to length squared where a very long beam has a very small buckling load. If such a long beam is supported transversely against buckling, then what happens is that a higher buckling mode occurs. Example 4.3 The cantilevered beam shown in Fig. 4.6 has a combination of axial and transverse forces. Use the Ritz method to formulate the equilibrium equations in a second-order analysis context. The load Po causes an axial displacement which results in an axial stress and strain. In our second-order analysis, this stress and strain are treated as being pre-existing and therefore accounted for by the F¯o geometric term in the energy expression. By inspection, F¯o = −Po so that the total potential energy is 

=

1 2

L

EI 0

 ∂ 2 v 2 ∂x 2

 dx +

2 1 2 αvL

−

1 2

L

Po 0

 ∂v 2 ∂x

dx − Qo vL

282

4 Buckling Shapes

♦

♦

Fig. 4.6 Cantilevered beam with end loads. (a) Geometry. (b) Buckling shapes for a very soft spring; FE results are the circles, continuous line is model II

That is, the axial force is accounted for through the energy-like third term and not through the load potential −Po uL as done for the transverse load. Note that, consequently, the total potential can be written in terms of the single deflection distribution v(x). Assume that the deflection can be adequately represented by v(x) = a0 + a1 x + a2 x 2 + a3 x 3 The geometric BCs are v(0) = 0,

dv(0) =0 dx

which gives a0 = 0 and a1 = 0. It remains now to determine the other two coefficients. Substituting for v(x) into the potential function gives 

=

1 2

L

0

EI [2a2 + 6a3 x]2 dx + 12 α[a2 L2 + a3 L3 ]2



−

1 2

L

 2 Po 2a2 x + 3a3 x 2 dx − Qo [a2 L2 + a3 L3 ]

0

Invoking the stationarity of with respect to a2 and a3 gives, respectively, 

L

 EI [2a2 + 6a3 x] 2dx + α[a2 L + a3 L ]L − 2

3

2

0

L

  Po 2a2 x + 3a3 x 2 2xdx

0

−Qo L = 0 2



L



EI [2a2 + 6a3 x] 6dx + α[a2 L2 + a3 L3 ]L3 −

0

0

−Qo L = 0 3

L

  Po 2a2 x + 3a3 x 2 3x 2 dx

4.1 Buckling Shapes of Straight Beams

283

From these, after performing the integrations, we obtain the system of equilibrium equations       # $ # 2$  Po L3 40 45L 4L 6L2 a2 L 4 1 L EI + αL − = Qo 6L2 12L3 a3 L3 L L2 45L 54L2 30 It is clear that Po and Qo play different roles with Po appearing on the left-hand side as a stiffness term similar to that of the spring. However, because Po is compressive its contribution to the total stiffness is to decrease it. To elaborate on our earlier nomenclature, we refer to the combination of bending stiffness and spring stiffness as the elastic stiffness [KE ] and to the contribution from Po as the geometric stiffness [KG ] so that the equilibrium equations can be written as   [KE ] + [KG ] { a } = {Q} where    4L 6L2 4 1 L + αL , [KE ] = EI 6L2 12L3 L L2 

Po L3 [KG ] = − 30



40 45L 45L 54L2



This decomposition is a recurring result in this chapter. Also note that the matrices are symmetric which is always the case when obtained from the total potential . Consider a one-term solution then   Po L3 Q o L2 [40] a2 = Qo L2 or a2 = EI [4L] + αL4 [1] − 30 EI 4L + αL4 − Po L3 4/3 Observe that an increasing Po decreases the denominator and therefore the magnitude of the displacement increases. Indeed, there is a value of Po at which the displacement is infinite—this is an example of beam buckling. The critical load is Pc =

3 EI (4 + αL3 /EI ) 4 L2

It is important to note that this does not depend on the transverse load Qo ; that is, it is a system property represented by the stiffness parameters EI , α, and the load Po . The role of Qo in an actual situation is to manifest the instability. The one-term solution with the spring indicates that as the spring stiffness increases so does the buckling load. For a very stiff spring this would imply that there is no buckling. This is not quite what happens. The determinant of the twoterm solution with the spring is 2 ¯ + (12 + 4α)L ¯ 2 = 0, 135λ2 L2 − (156 + 4α)λL

α¯ = αL3 /EI,

λ=

Po L2 30EIzz

284

4 Buckling Shapes

In the limit as α becomes very large, we get ¯ λ1 −→ (8/270)α,

λ2 −→ 1.0

The first mode is suppressed (i.e., its eigenvalue is very large), but the second mode is evident. The second buckling shape is given by a3 = −

a2 4 + α¯ − 40λ2 a2 −→ −1.0 , 6 + α¯ − 45λ2 L L

v2 (x) = a2 [x 2 − x 3 /L]

This is the Ritz function obtained by imposing zero displacement at the right end.

Example 4.4 Continue the previous example problem but with a focus on determining an accurate critical value for the lowest mode. The spring can be ignored. Rewrite the equilibrium equation for the two-term approximation in the form 

   # $ 4 6L 40 45L a2 − λ = 0, a3 6L 12L2 45L 54L2

λ=

Po L2 30EIzz

Setting the determinant to zero gives 135λ2 L2 − 156λL2 + 12L2 = 0

⇒

λ1 = 0.0829 ,

λ2 = 1.0727

This gives the critical loads Pc1 = 2.49

EI , L2

Pc2 = 32.18

EI L2

These are the values shown as model II results in Table 4.2. We also get information about the buckling shape from a3 = −

4 − 40λ a2 [6 − 45λ]L

In succession, substitute for λ1 and λ2 to get λ1 :

a3 = −0.03 a2 /L

v1 (x) = a2 [x 2 − 0.03x 3 /L]

Table 4.2 Comparison of the critical loads for a cantilevered beam using different models. The table gives the value α where Pc = αEIzz /L2 Mode 1 2

FE 2.467 22.207

Strong 2.467 22.207

I 3.000 –

II/IV 2.486 32.181

III 2.500 –

4.1 Buckling Shapes of Straight Beams

λ2 :

a3 = −0.92 a2 /L

285

v2 (x) = a2 [x 2 − 0.92x 3 /L]

These are shown plotted in Fig. 4.6b as the continuous lines; there is a nice comparison with the FE results using 16 elements. To assess the accuracy of approximate solutions it is necessary to have available good reference solutions. Best is when we have exact (within the limitations of the modeling) solutions. Table 4.2 shows results from a variety of models now to be discussed. Our prime model for comparison is the FE results using many elements to ensure convergence. The strong formulated results are indicated as the “strong” column in the table. We have for the cantilever beam strong:

Pc1 = 0.25π 2

EI EI = 2.47π 2 2 , 2 L L

Pc2 = 2.25π 2

EI EI = 22.2π 2 2 2 L L

This shows that the one-term solution of the previous problem (shown as the model I results in the table) is off by about 20%. The two-term solution gives the critical loads Ritz II:

Pc1 = 2.49

EI , L2

Pc2 = 32.18

EI L2

These are shown as the model II results in Table 4.2. We get a much improved mode 1 result, and we also get an estimate (although crude) for the second mode. One way to get improved results within the Ritz formulation is to increase the number of DoF and this is shown by the two-term approximation. We can continue this process of adding extra DoF to improve the results but our preferred way of doing this is via the finite element method (but, of course, this requires numerical methods for solution). Sixteen elements are used to get the strong-form result although just two elements give the first mode within 0.07% of the strong form. Suppose, however, we want an improved result but still have just a SDoF model, what approach should be taken? The crux of the matter is the choice of Ritz function. The restriction on this function is that it must rigorously satisfy the geometric BCs but not necessarily the natural BCs. Realizing that if by accident (say) our Ritz function also satisfied the natural BCs, then our solution would actually yield the exact strong formulation solution; thus on comparing two functions which satisfy the geometric BCs but one has a better approximation of the natural BCs, then we expect this latter function to give a better result. The procedure suggested here does not generalize very well but it is useful when simplified models not involving simultaneous equations are desired. As in the previous problem, we start with a polynomial but truncated after more terms. Imposing zero deflection and slope at x = 0, and zero second derivative associated with the moment (this is one of the natural BCs) at x = L leads to v(x) = a3 [x 3 − 3Lx 2 ] + a4 [x 4 − 6Lx 2 ]

286

4 Buckling Shapes

It is tempting to impose that the third derivative (shear) is also zero at x = L but this is not true as the strong formulation of Eq. (4.4) makes clear. Using just the first term, the energy integrals evaluate to

= UE + UG ,

UE = 12 EIzz a32 12L2 ,

24 UG = 12 F¯o a32 L5 5

This gives the total stiffness and critical value as, respectively, KT =

24 ∂ 2

= EIzz 12L2 + F¯o L5 , 5 ∂a32

Pc = 2.50

EI L2

This is shown as the model III results in Table 4.2. Clearly, this simple model is quite good especially considering it has only a single DoF. Example 4.5 Determine the geometric stiffness matrix for a general frame element. Consider a typical beam element as shown in Fig. 1.26a where it is assumed that it also has a pre-existing axial force F¯o . The DoFs at each node are { u } = {v, φ}T even though there must be axial displacement associated with the axial force; in this analysis we are concerned only with the transverse behavior of the beam, the axial effect is already accounted for in the geometric strain energy expression of Eq. (4.3) which depends only on the transverse deflection. Assume that the same deflection shape functions as Eq. (1.30) can be used to describe the buckling shape, then the geometric stiffness matrix evaluates to ⎡

[kGI J ] =

 ∂2 U  G ∂uI ∂uJ

⇒

⎤ 36 3L −36 3L 2 2⎥ F¯o ⎢ ⎢ 3L 4L −3L −L ⎥ [ k¯G ] = ⎣ 30L −36 −3L 36 −3L ⎦ 3L −L2 −3L 4L2

(4.8)

This is the consistent geometric stiffness for a beam; it is called consistent because the same interpolation functions are used in deriving the elastic stiffness. The stiffness of a general frame element is obtained through a rotation transformation of the beam stiffness in local coordinates according to [ kE ] = [ T ]T [ k¯E ][ T ] ,

[ kG ] = [ T ]T [ k¯G ][ T ]

where [ T ] is the [6 × 6] rotation matrix for plane frames and [12 × 12] for space frames as described in connection with Eq. (1.33). Explicit forms for the matrices (including the mass matrix) are given in Ref. [4] and coded in module > of the SDframe program. One element solution to the cantilever beam problem uses the lower right quadrant of the matrices. It is interesting this leads to the same critical values as model II. The reason is that model II uses a third-order polynomial which is the

4.1 Buckling Shapes of Straight Beams

287

same as the element formulation. The results are indicated as model IV in Table 4.2.

4.1.3 Effect of Distributed Elastic Supports In general, the effect of additional elastic supports and constraints is to increase the elastic stiffness of a structure. Consequently, we would expect that the buckling loads increase. The situation is slightly more complicated in that we must consider the effect on the different modes. While we only consider beams here, the discussions are also relevant to arches and shells because as shown, for example, in Eq. (2.18), the effect of the curvature is that of an elastic foundation. Consider a uniform beam on an elastic foundation, the governing equation with axial loads present is EI

2 ∂ 4v ¯o ∂ v + Kv v = 0 − F ∂x 4 ∂x 2

(4.9)

where Kv is the foundation stiffness per unit length. Because this has constant coefficients, we seek solutions of the form v(x) = Ae−ikx , which leads to the characteristic equation k 4 + P˜ k 2 + K˜ = 0,

Po −F¯o = , P˜ ≡ EI EI

Kv K˜ ≡ EI

This gives the four load spectrum relations '  ˜ k1,3 (P ) = ± − 12 P˜ + 14 P˜ 2 − K˜ ≡ ±α, '  1 ˜ ˜ k2,4 (P ) = ± − 2 P − 14 P˜ 2 − K˜ ≡ ±α¯ While these expressions are simple enough, the actual behavior of kn (P˜ ) is quite complicated as seen in the 3D plots of Fig. 4.7a. Thebranches occur where the inner ˜ ˜ square root  are given by P = ± 4K; it is only for compressive  changes sign and ˜ that the spectrums are real-only, this means the P˜ < − 4K˜ (i.e., |P˜ | > 4K|) buckling loads are real-only also. For the buckling problem, we look for solutions only in this real-only region. The following example problem considers the complex values because there is an applied transverse load and the solution is sought up to the point of buckling; here we do an EVP in the high compression region. If we plot the real-only k against compressive Po , we get a plot similar to Fig. 2.50 as indicated by the projection in Fig. 4.7a.

288

4 Buckling Shapes 10. 8. 6. 4. 2. 0.

0.

25.

50.

75.

100. 125.

Fig. 4.7 Pinned–pinned beam on an elastic foundation. (a) Load spectrum relations for ki (Po ). (b) Buckling loads as a function of length for fixed foundation and beam parameters

The general solution is represented by ¯ ¯ + Ce+iαx + De+i αx v(x) = Ae−iαx + Be−i αx

(4.10)

or alternatively, ¯ + c4 sin(αx) ¯ v(x) = c1 cos(αx) + c2 sin(αx) + c3 cos(αx) because α and α¯ are real-only for high compressive Po . Consider a problem similar to that of Fig. 1.20 with simply-supported BCs. We have that at x = 0 v = 0 ⇒ v = 0 = c1 + c3 M=0⇒

d 2v = 0 = −α 2 c1 − α¯ 2 c3 = 0 dx 2

Because α 2 − α¯ 2 =



1 ˜2 4P

− K˜

is not zero in the high compressive load region, then we conclude that c1 = 0 and c3 = 0. At x = L, we have ¯ v = 0 ⇒ v = 0 = c2 sin(αL) + c4 sin(αL) Mxx = 0 ⇒

d 2v = 0 = −α 2 c2 sin(αL) − α¯ 2 c4 sin(αL) ¯ =0 dx 2

These two equations lead to the eigensystem

4.1 Buckling Shapes of Straight Beams



289

#

sin αL sin αL ¯ 2 2 ¯ −α sin αL −α¯ sin αL

c2 c4

$ =0

The characteristic equation is obtained by setting the determinant equal to zero and gives (α 2 − α¯ 2 ) sin(αL) sin(αL) ¯ =0 Again, because (α 2 − α¯ 2 ) is not zero, then this equation has the solutions '  sin αL = 0

⇒

αL = nπ

⇒

1 2

nπ P˜ 2 − 4K˜ − 12 P˜ = L

and ' sin αL ¯ =0

⇒

αL ¯ = mπ

⇒

− 12

 mπ P˜ 2 − 4K˜ − 12 P˜ = L

Actually, on expanding and rearranging both give  nπ Kv L 2  ( ) −Pc = EI ( )2 + L EI nπ The corresponding buckling shapes are given by vn (x) = c2 sin(αx) = c2 sin(nπ x/L) These shapes are the same as for the buckling of a simply-supported beam. One effect of the elastic foundation is to set a minimum buckling load regardless of beam length. To elaborate, Fig. 4.7b shows the variation of buckling load with beam length. It is noted that for a beam of given length, the lowest buckling load does not necessarily coincide with the simplest buckling shape (that is, lowest n). As the length increases, it is a higher and higher mode that buckles. For example, at L/ h = 60, it is the n = 2 mode that buckles. The buckling value is not too different than when L/ h = 30. In other words, the beam of L/ h = 60 buckles as if it is two beams each of length L/ h = 30. Actually, for very long beams there is a clustering of the modes as is beginning to be observed in the lower right of Fig. 4.7b. The limiting buckling load is independent of length. We can get this overall minimum buckling by setting  2 nπ ∂Pc Kv 2 L 2  = 0 = EI − ( )2 + ( ) ∂L L L EI L nπ from which (nπ/L)4 = Kv /EI giving

290

4 Buckling Shapes

Pmin =



4Kv EI

Note that the value of Pmin coincides with the branch points in Fig. 4.7a; we consider this further in the next example problem. Generally, the effect of a spring or elastic attachment is to increase the buckling load. However, not all modes are affected equally. As an example, for a single spring attached to the center of a simply-supported beam, the first antisymmetric mode is not affected at all because the beam center does not deflect in this mode. Consequently, as the attachment stiffness is increased, another mode may become the critical mode. Furthermore, it is easy to visualize that the position of the spring also has different effects depending on the different modes. The continuously distributed spring affects all modes in such a way that there is a minimum buckling load irrespective of beam length. Example 4.6 Determine the deflected shape for an infinite beam resting on an elastic foundation under the action of an axial compressive load Po and a concentrated transverse load Qo . The deflection of an infinite beam resting on an elastic foundation due to a concentrated load is covered in Sect. 1.2. With reference to Fig. 4.7a, the roots for that problem correspond to Po = 0 and are of the form ±[1 ± i1]. We now extend that solution by considering the effect of the axial load. As the compressive load is increased, the real part of k increases while the imaginary part decreases so the roots are of the form ±[α ± iβ]. The axial load does not change the symmetry of the problem, so the solution for x ≥ 0 has the form   v(x) = c1 cos αx + c2 sin αx e−βx The zero slope and equilibrium condition at x = 0 leads to, respectively, −c1 β + c2 α = 0 ,

−EI [c1 (3α 2 β − β 3 ) + c1 (3αβ 2 − α 3 )] = − 12 Qo

where Qo is the transverse applied load concentrated at x = 0. Solving for the coefficients gives c1 =

Qo , 4EIβ(α 2 + β 2 )

c2 =

Qo 4EI α(α 2 + β 2 )

The deflection shape is v(x) =

Qo [α cos αx + β sin αx]e−βx 4EI αβ(α 2 + β 2 )

and this shape is symmetric about the origin.

4.1 Buckling Shapes of Straight Beams

291

Fig. 4.8 Normalized deflections of a very long beam (with fixed transverse load Qo ) for increasing axial compressive loads

The remarkable feature about this solution is the presence of β in the denominator; from Fig. 4.7a, as the compressive load is increased, the imaginary part (β) ˜ At this stage, the deflection is ˜ decreases and eventually goes to zero at P = 4K. infinite and buckling has occurred. The progression of deflection shapes is shown plotted in Fig. 4.8 for different values of axial compressive load as a percentage of the critical value (note that each plot has been normalized to unity for plotting purposes). The plot exhibits a definite limit to a sinusoid with wavenumber    1 ˜ kc = αc = − 2 P = 12 4K˜ = [Kv /EI ]1/4 This is the same limit as discussed earlier. It is interesting that even though there is no periodicity in the infinite beam there is a definite periodicity in the buckling shape. The reason is because there is a minimum stiffness and just like Fig. 2.50, a minimum frequency (stiffness) has a minimum wavenumber. Buckling patterns such as Fig. 4.8 where the number of half-waves increase progressively, is referred to as cellular buckling [13]. The very long plate under compression (Fig. 4.18) and shear (Fig. 4.26) loadings also exhibits this form of buckling. In fact, most structures of elongated aspect ratio with constraints exhibit this cellular buckling because it is the smaller dimension that dictates the buckling mode. The example problem for the plate in Fig. 4.21 illustrates an exception to this because of the long free edge. The spectral plots in Fig. 4.9 (coming from a nonlinear analysis of the beam behavior) show how the foundation stiffness affects the buckling loads; for easy reference, the horizontal positions of the plots are proportional to Ko . Reference [6] shows from a linear spectral analysis that the structural stiffness is related to the system parameters by ωn2 =

Kv  Po EI  4 n¯ + n¯ 2 , + ρA EI EI

n¯ =

nπ L

(4.11)

♦

♦

♦

♦

♦

4 Buckling Shapes ♦

292

♦

♦

Fig. 4.9 Spectral plots showing the effect of foundation stiffness on the buckling loads

There seems to be two primary effects. First, an increase of foundation stiffness increases the stiffness of the lower modes proportionally more than for the higher modes; that is, the stiffness ratio of the lower mode to the higher mode approaches unity. Second, because the higher modes are more sensitive to load, they are more likely to intersect the lower modes and become critical first. While there does not seem to be significant mode interactions, a consequence is that more modes tend to become unstable at about the same load level. This cellular buckling effect makes it more likely that the postbuckle state is unstable if there is an interaction with a global mode. Such a case for flange buckling is considered in Sect. 5.3. For all the present cases tested, the beam is postbuckle stable.

4.2 Buckling of Arches As mentioned in connection with straight beams, buckling problems are easier to tackle if they can be divided into two somewhat unrelated problems: first is the prestress stage where the geometry is assumed to not change by much and linear analysis is used to compute the axial force distributions; second is the buckling analysis where the axial forces are considered as pre-existing and the only unknown is the out-of-plane deflection. The linear analysis of arches is more complicated than for beams as witnessed by Sect. 2.1, but because of our desire to focus on buckling, we take as a given that the axial force is available from a separate linear analysis. Section 2.1 developed a quite good approximate strong method and Ref. [14] shows how the principle of complementary virtual energy can be used quite effectively to get exact solutions. Obviously our approach is not true of a genuine nonlinear analysis of the deformation, but we nonetheless get useful practical solutions from the approach so long as we verify the solutions with a nonlinear analysis. We use the arch to highlight the deficiency of our basic approach.

4.2 Buckling of Arches

293

4.2.1 Basic Buckling Method Arches are the archetypal problem for the case where there can be significant deformations before buckling occurs. Therefore, it behooves us to have a derivation of the governing equations through which we can judge the approximations being invoked. We follow the procedure done for the straight beams. Consider the arch to be in a state of deformation (u, v) under the action of the distributed loads qu , qv . A small change is given to the deformed configuration according to v¯ = v + v b

u¯ = u + ub ,

where the superscripted quantities are the buckling displacements. The total nonlinear strain is given by 2 ¯ s −v/R ¯ + 12 [u, ¯ s −v/R] ¯ + 12 [u/R ¯ + v, ¯ s ]2 ≈ u, ¯ s −v/R ¯ + 12 v, ¯ 2s E¯ ss = u,

Substitute for the displacements and regroup as   E¯ ss = u,s −v/R + 12 v,2s + [u,bs −v b /R + v,s v,bs ] + 12  2 v,b2 s b = Ess + Ess + 12  2 v,b2 s

As presented, the total strain has a simple decomposition but note that there is coupling between v and v b in the buckling strain. We take the total bending curvature as ¯ s ≈ u, κ¯ ss = φ, ¯ s /R + v, ¯ ss ≈ v, ¯ ss = v,ss + v,bss Our total potential can then be written as ¯ =

≈



2 ds + EAE¯ ss

1 2

1  +  +

2 EI κ¯ ss



ds −

1 2



[qu u¯ + qv v] ¯ ds 

 2 ds + EAEss

2



 1 2

2 ds − EI κss

[qu u + qv v] ds

 b EAEss Ess ds +

 EI v,ss v,bss ds −



 b2 EA[Ess v,b2 s +Ess ] ds +

EI v,b2 ss ds

= 0 + 1  + 2 21  2 We can parse the energy contributions to the stiffness as



 [qu ub + qv v b ] ds  1

2

2

(4.12)

294

4 Buckling Shapes

 UE =

1 2

UG2 =

1 2

 EI v,b2 ss ds, UG1 =



1 2

 EAEss v,b2 s ds =

1 2

F¯o v,b2 s ds

 b2 ds EAEss

=

1 2

EA[u,bs −v b /R + v,s v,bs ]2 ds

The second geometric contribution is quite interesting because in its expanded form we see that it has an interaction term (v,s v,bs ) involving the current deformation state and the buckling deformation state. In contrast to the straight beam, we generally cannot neglect this contribution for arches. Using our variational principle on the stiffness potential 2 gives 

  b EA Ess v,bs δv,bs +Ess [δu,bs −δv b /R + v,s δv,bs ] ds

δ 2 =

 +

EI v,ss [δv,ss +δv,bss ] ds

where the current deformed state (u, v) is not subjected to variation. Doing the integrations by parts leads to b δu : −(EAEss ),s = 0 b b δv : −(EAEss v,bs ),s −EAEss /R − (EAEss v,s ),s +EI v b ,ssss = 0 b during buckling is The first equation shows that the membrane force F b = EAEss constant. The other equation becomes

EI v b ,ssss −F¯o v b ,ss = F b [1/R + v,ss ]

(4.13)

This is our buckling equation. The RHS shows the dependence on the current deformed state. Our basic buckling method for arches ignores the effect of the deformed geometry; later, we assess this approximation. To further simplify matters, assume that the axial force is known from a separate analysis and that the buckling shape (as opposed to the deformation state leading to the buckling) is predominantly flexural so that u/R v,s . The flexural energy (which is now the total elastic energy) is then approximated by  UE = UF =

1 2

 EI [u,s /R + v,ss ]2 ds ≈

 =

1 2

(EI /R 3 )[v,θθ ]2 dθ

The geometric strain energy is expressed as

1 2

EI [v,ss ]2 ds

4.2 Buckling of Arches

295

 UG =

1 2

 2 ∂v F¯o ds = ∂s

 1 2

 2 (F¯o /R) v,θ dθ

Both energies are dependent only of the transverse deflection. The strain energies are approximate and therefore the BCs to be imposed must be consistent with the approximation. The simplest way to make the BCs unambiguous is to derive them from our variational principle (or PoVW). To that end, we add the virtual work of the boundary loads (similar to Fig. 2.3b) to have L  δ We = − Mδv,s +V δv] o This leads to the governing equation (as a modified form of the second of Eq. (2.6)) EI

2 ∂ 4v ¯o ∂ v = qv − F ∂s 4 ∂s 2

or

v,θθθθ +β 2 v,θθ = qv R 2 /EI,

β2 = −

F¯o R 2 EI

The BCs come from  L (EI v,ss −M)δv,s +(−EI v,sss +F¯o v,s −V )δv o All relations are similar to those for a straight beam given by Eq. (4.4). We can immediately write the general solution for compressive axial force as v(θ ) = c1 cos βθ + c2 sin βθ + c3 θ + c4 + Qv

(4.14)

where Qv is the particular solution for the distributed load. Inherent in this solution is the approximation that we can replace F¯o (θ ) with its piece-wise constant average value. Example 4.7 The consequences of the changes of shape can be quite subtle and therefore intuition is not always reliable when it comes to modeling nonlinear problems. Use the case of a point loaded arch to illustrate some of these subtleties. Figure 3.17 shows the deformed shape (not exaggerated) of a point loaded arch. The geometry is symmetric and loading is symmetric; therefore, it would seem reasonable to model the loading problem as being symmetric. That is, only one half of the arch need to be meshed with rollers placed at the line of symmetry as shown in Fig. 4.10a. These assumptions can be erroneous when the deformations are large or singular. A linear buckling analysis, itself, cannot clarify the difficulties involved—a fully nonlinear analysis is required. The spectral monitors, as part of the nonlinear analysis, present the information in a convenient form. The zero-load spectral shapes are shown in Fig. 4.10b, and the full spectral plots for different BC implementations are shown in Fig. 4.10c. The case {1, 1} is when no constraints are applied (the full arch is modeled) and case {0, 0} is when the horizontal displacement and rotation

0.0

♦

♦

♦

♦

♦

♦

♦

4 Buckling Shapes ♦

296

0.5

1.0

Fig. 4.10 Effect of modeling the BCs on the nonlinear deflections of an arch under central point loading. (a) Possible implementation for modeling symmetric conditions. (b) Some spectral shapes at zero load. (c) Spectral plots designated by the center BC as {u, φ} where 1 means free and 0 means restrained

are restrained (this is the case shown in Fig. 4.10a). The effect of the constraints is to suppress the antisymmetric deformation mode (shape I in Fig. 4.10b). The critical load is consequently increased to 1.16Pc . Allowing the rotation but restraining the displacement as in case {0, 1} does not restore the antisymmetric mode. Restraining the rotation but allowing the displacement as in case {1, 0} does give a sort of antisymmetric mode (shape II in Fig. 4.10b); it has slightly less sensitivity to load compared to shape II but because it starts higher it is not the shape that becomes critical. Allowing the sway is, therefore, essential to capture the lowest mode. Example 4.8 Estimate the buckling load for the arch shown in Fig. 2.2b. Keep in mind that it is the buckling shape we seek and not the deformation due to the primary loading. The consequence of various assumptions about the BCs is illustrated in the nonlinear analyses of Fig. 4.10. The deformation mode with the lowest critical load corresponds to mode I in the figure and is the antisymmetric mode in spite of the fact that the primary loading and geometry are symmetric. Therefore, with θ measured off the vertical, we impose that the BCs are at θ = 0 :

v = 0, M = 0;

at θ = α/2 :

v = 0, M = 0

Using Eq. (4.14), the first two BCs lead to c1 = 0, c4 = 0, and the second two lead to the EVP  # $ sin β α¯ α¯ c2 =0 −β 2 sin β α¯ 0 c3

4.2 Buckling of Arches

297

with α¯ = α/2. Consequently, we have that ¯ =0 det = αβ ¯ 2 sin(β α)

⇒

β α¯ = nπ ,

v = c2 sin(βθ )

where n = 1, 2, · · · . The shapes are illustrated by the bottom row of Fig. 2.10. But β is related to the axial load so that the critical values of axial load are given by − F¯o =

critical:

EI R2



n2 π 2 α¯ 2

 or

Pc =

1 EI γ R2



n2 π 2 α¯ 2



where F¯o = γ Po . The FE results of Figs. 2.13 and 2.46 show that the axial force is not constant but could reasonably be considered so. The average is computed as Fˆo = −1.01 Po and gives critical values for the first two critical loads of model: Pc = {15.9, 62.7}

EI R2

vs

FE: Pc = {16.0, 64.0}

EI R2

The difference between the model and the FE generated results for the first mode is quite small. For the symmetric shapes, we impose the BCs that at θ = 0 :

v,θ = 0, V ∝ v,θθθ = 0;

at θ = α/2 :

v = 0, M = 0

These lead to the EVP 

# $ cos β α¯ 1 c1 =0 −β 2 cos β α¯ 0 c4

from which we get det = β 2 cos(β α) ¯ = 0 ⇒ βα/2 = (n + 12 )π,

v = c1 [cos(βθ ) − θ cos(β α)] ¯

In this case, n = 0, 1, 2, · · · . The shapes are illustrated by the top row of Fig. 2.10. The first few critical load values are model: Pc ={4., 36., 100.}

EI R2

vs

FE: Pc ={ ∗ , 35.5, 99.3}

EI R2

(4.15)

The model and FE results agree for the second and third symmetric shapes but the first model shape is completely absent in the FE results. Its absence is troubling because it predicts the lowest buckling load; we could reject it on the basis that it is not inextensible [11] but this is not satisfactory and seems too ad hoc. We consider this further later on in this section. A convenience of the solution given by Eq. (4.14) is that it easily handles different BCs. Consider the case where the end BCs are changed to being fixed. The BCs at

298

4 Buckling Shapes

θ = 0 are the same as for the earlier examples, so that they and the new BCs for the antisymmetric shapes are at θ = 0 :

v = 0, M = 0;

These lead to the EVP  # $ sin β α¯ α¯ c2 =0 β cos β α¯ 1 c3

at θ = α/2;

⇒

v = 0, v,θ = 0

det = sin β α¯ − β α¯ cos β α¯

This is a transcendental equation and must be solved numerically. The simplest scheme is to scan det for different values of β α¯ and identify the zero crossings. The first two found this way are β α¯ = 4.49, 7.72. The critical values of axial load are given by critical:

EI  − F¯o = 2 β 2 R

or

Pc =

1 EI 2 β γ R2

where the average force (taken from a separate linear analysis) is Fˆo = γ Po = −1.19Po . The comparison with the FE results is model: Pc = {27.5, 81.2}

EI R2

vs

FE: Pc = {28.7, 80.4}

EI R2

The difference is reasonable considering the multiple assumptions made. The symmetric shapes with the BCs at θ = 0 :

v,θ = 0, V ∝ v,θθθ = 0;

lead to the EVP  # $ cos β α¯ 1 c1 =0 −β sin β α¯ 0 c4

⇒

at θ = α/2;

det = β sin β α¯ ,

v = 0, v,θ = 0

β α¯ = nπ

The comparison with the FE results is model: Pc = {13.5, 53.8, 121.}

EI R2

vs

FE: Pc = { ∗ , 45.4, 112.}

EI R2

Again we see the presence of a lower symmetric mode and the difference for the second mode of 16% is large. The EVPs also give information about the buckling shape; we have anti: v(θ ) = c2 [sin βθ − βθ cos β α] ¯ ,

symm:

v(θ ) = c1 [cos βθ − cos β α] ¯

4.2 Buckling of Arches

-45. -30. -15. 0. .

299

♦

♦

♦

♦

15. 30. 45.

-45. -30. -15. 0. .

15. 30. 45.

Fig. 4.11 Buckling shape for an arch with fixed ends. Circles are FE data, continuous lines are model results. (a) First antisymmetric mode. (b) First symmetric mode

These are shown in Fig. 4.11 compared to the FE results, there is very good agreement for the antisymmetric shape but very poor for the symmetric shape. Because the results are from an eigenanalysis, in making the plots, their amplitudes were matched at θ = ±16o . To get an estimate of the axial displacement distribution, it was assumed that the buckling shape is inextensible, this results in  u(θ ) =

 v/R ds =

v dθ =

1 1 c1 sin βθ − c2 cos βθ + 12 c3 θ 2 + c4 θ + c5 β β

The additional coefficient was chosen so that u(θ = α) ¯ = 0. The distributions are shown in Fig. 4.11 using the same scaling as for v(θ ). The comparison for the antisymmetric shape is very good but completely wrong for the symmetric shape. We cannot help but conclude that there is something fundamentally different about the behaviors of the antisymmetric and symmetric modes. This would not be a (big) concern if we were assured that it is always the antisymmetric mode that buckles first; but even the simple cases of Fig. 4.10 show that the symmetric mode could become dominant. Therefore, the modeling of the symmetric modes is revisited later in this section. It turns out that the symmetric modes have two significant features which are minimally present in the antisymmetric shapes, namely, they are extensible and they are associated with significant changes of shapes. Both features undermine the basic buckling method used here. Example 4.9 Find the buckling loads for the plane frame shown in√Fig. 4.12. Each member has the same material and section properties. The load of 2Po is applied at an angle of 45o to the horizontal. It is simplest to use an FE model of this problem as described by Eq. (4.8). We use two elements to model the problem although an accurate analysis would require many more. Numbering the nodes as shown, the total degrees of freedom are { u } = {u1 , v1 , φ1 ; u2 , v2 , φ2 ; u3 , v3 , φ3 }T

300

4 Buckling Shapes

Fig. 4.12 Frame with fixed–fixed supports. (a) Geometry and loading. (b) First two FE generated buckling mode shapes. (c) First two buckling mode shapes for a semi-circular arch

The fixed boundary conditions require that u1 = v1 = φ1 = 0 and u3 = v3 = φ3 = 0 giving the reduced system as {uu } = {u2 , v2 , φ2 }T ,

{Po } = Po {1, −1, 0}T

The reduced system is of size [3 × 3]. In our previous example problem, we assumed that the axial force is known from a separate analysis, here we find it informative to illustrate this aspect of the problem. Thus, we first solve the problem neglecting the geometric stiffness. The reduced element stiffness matrices for both members are for the nonzero DoFs {u2 , v2 , φ2 }. For Member 1–2 with connectivity 1 to 2, the orientation is θ = 90◦ , giving ⎡ ⎤ ⎡ ⎤ 12 0 6L 000 EA EI ⎣ 0 0 0 ⎦ ⎣0 1 0⎦ + [k (∗12) ] = L L3 6L 0 4L2 000 For Member 2–3 with connectivity 2 to 3, the orientation is θ = 0◦ , giving ⎡ ⎤ ⎡ ⎤ 0 0 0 100 EA EI ⎣ 0 12 6L ⎦ ⎣0 0 0⎦ + [k (∗23) ] = L L3 0 6L 4L2 000 The reduced structural stiffness relation is, therefore, assembled as ⎡

⎡

10 ⎣ EA ⎣ 0 1 L 00 Solving this gives

⎫ ⎡ ⎤⎤ ⎧ ⎫ ⎧ ⎤ 12 0 6L u2 ⎬ ⎨ 1 ⎬ 0 ⎨ EI 0 ⎦ + 3 ⎣ 0 12 6L ⎦⎦ v2 = −1 Po ⎩ ⎭ ⎩ ⎭ L 6L 6L 8L2 0 φ2 0

4.2 Buckling of Arches

301

u2 = −v2 =

Po L , (EA + 12EI /L2 )

φ2 = 0

This gives the axial loads as f¯o(12) = −

Po EA , (EA + 12EI /L2 )

f¯o(23) = −

Po EA (EA + 12EI /L2 )

In this case, both axial loads are the same and compressive and therefore the prebuckle problem is symmetric. However, it would be wrong to impose symmetry on the buckling problem because, as it turns out, the antisymmetric buckling load is smaller than the symmetric buckling load. The denominator for the axial force has 1 + 12EI /EAL2 = 1 + h2 /L2 ; if h/L < 1/10, the flexibility has a negligible (12) (23) contribution to the static problem giving u2 = v2 = Po L/EA and f¯o = f¯o = −Po . It is now possible to form the geometric stiffnesses. These are ∗(12) [kG ]

⎤ ⎡ 36 0 3L λf¯o(12) ⎣ = 0 0 0 ⎦, 30L 3L 0 4L2

∗(23) [kG ]

⎤ ⎡ 0 0 0 λf¯o(23) ⎣ = 0 36 3L ⎦ 30L 0 3L 4L2

The reduced structural stiffness matrix can, therefore, be assembled. Introducing the notations β1 ≡

EA , L

β2 ≡

EI , L3

γ ≡

(12) (23) −f¯o −f¯o β1 = = 30L 30L (β1 + 12β2 )30L

we get the EVP as ⎤ ⎡ ⎤⎤ ⎧ ⎫ β1 + 12β2 36 0 3L 0 β2 6L ⎨ u2 ⎬ ⎣⎣ 0 β1 + 12β2 β2 6L ⎦ − λγ ⎣ 0 36 3L ⎦⎦ v2 = 0 ⎩ ⎭ 3L 3L 8L2 β2 6L β2 8L2 φ2 β2 6L ⎡⎡

The characteristic equation is (β1 + 12β2 − 36λγ )[135λ2 γ 2 − λγ 4(β1 + 39β2 ) + 4β2 (β1 + 3β2 )] = 0 One of the roots is λ=

(β1 + 12β2 )2 30L β1 + 12β2 = 36γ 36β1

Substituting this into the equations of the eigenvalue problem, we get φ2 = 0

and

u2 = −v2

302

4 Buckling Shapes

This is the symmetric mode. In the inextensible case (β1 or EA/L very large) we get Pc = 0.833β1 L which shows that the frame does not buckle in this mode because β1 (directly related to E) → ∞. The other two critical loads occur at    2 2 2 (β1 + 39β2 ) ± β1 − 57β1 β2 + 1116β2 λ= 135γ Again, suppose that the axial stiffness is very large, then Pc = 30 Lβ2 ,

0.777 Lβ1

The second of these critical loads approaches infinity, indicating the unlikelihood of buckling for this mode. The first of the critical loads corresponds to u2 = v2 = 0 ,

φ2 = 0

This deformation is the same as if the structure is pinned at node 2 and corresponds to an antisymmetric buckling shape. Of course, if the actual structure is pinned at the corner, then the applied load would not produce any axial loads and hence no buckling. Using one element per member gives a rather poor result compared to the FE converged value of 19.9 Lβ2 . However, if the number of elements is doubled to two per member, then the predicted value is 20.6 Lβ2 which is much closer. Figure 4.12b shows the first two buckling shapes. The plots are rotated so that the frame resembles an arch albeit with straight sides (it could be called an A-frame). It is thus interesting to compare the shapes to those of an actual arch as shown in Fig. 4.12c. In both cases, the antisymmetric mode with a rotation about the center point is the fundamental mode but the arch also exhibits a relatively large deflection that includes a left-to-right sway. This indicates that the arch would deflect before buckling as seen in Fig. 3.17. This example shows again that even though the geometry and loading is symmetric, that the antisymmetric buckling shape could be the dominant one. Therefore, the use of symmetry (exclusively) in the modeling would lead to erroneous results.

4.2.2 Assessment of the Basic Buckling Method By a series of approximations and assumptions we have reduced the buckling problem to determining a reasonable estimate of the average of the distributed axial

Fig. 4.13 Spectral plots for the buckling of a shallow arch with simply-supported BCs under central point loading. Circles are nonlinear FE data, light dashed lines are second-order FE data, heavy lines are model results. PR = EI /R 2

303

♦

4.2 Buckling of Arches

♦

0.

4.

8. .

12.

16.

force F¯o (θ ) with which we can then determine the buckling shape and load from a single equation. The example problems showed the usefulness of the approach. But sometimes our approach gives poor results and its drawback is that, inherently, it is incapable of explaining why. A good case in point is the symmetric buckling loads of the arch examined earlier. In these circumstances, we must resort to a full nonlinear analysis for clarification. Figure 4.13 shows results from the nonlinear analysis of a shallow arch under increasing central point loading. The spectral plot shows that it is the antisymmetric mode that goes directly to being critical, but it also shows that the first symmetric mode nearly became the critical mode. In other words, both buckling modes are as likely to occur and should be amenable to the same level of analysis. However, this is not true of our basic method. Our basic method assumes solutions of the form v(x, t) = vo (t) sin(nπ θ/α) where n is either 2 or 3, and a separate linear analysis shows that the average axial force is related to the applied load according to F¯o = −1.01 Po . The computed stiffnesses are shown in Fig. 4.13 as the full light lines. The antisymmetric mode is modeled very well but the symmetric mode is off by a significant amount. Furthermore, the discrepancy in slope for this mode begins at low load levels implying that significant changes of shape should not have occurred. We conclude that our modeling for the symmetric mode is missing something significant. The dashed lines in Fig. 4.13 are for a linear FE analysis taking second-order effects into account, that is, the total stiffness is given as [ KT ] = [KE ] + χ [KG ] where both [KE ] and [KG ] are evaluated with respect to the undeformed geometry and χ is a scaling on the loading. This linear FE analysis represents the antisymmetric mode very well and is close to the basic model for the symmetric mode, but there is a serious discrepancy with the nonlinear FE symmetric results. Thus, if the

304

4 Buckling Shapes

♦

0

2

4

.

6

8

10

♦

0

10

20

30 .

40

50

60

Fig. 4.14 Amplitude spectrums for a shallow arch under central point loading. There are five equal-increment load levels. (a) Expanded plot showing dominant contributions to the shapes. (b) Extended plot showing the dominant contributions to the axial load

symmetric mode were to become the critical mode, a linear eigenbuckling analysis would not correctly identify the critical load. As shown earlier in the section, the prediction for symmetric buckling to occur is at Pc = 35.5EI /R 2 , around twice that of the nonlinear analysis. We use a full spectral analysis of the nonlinear deformation to begin to explain what is happening. The arch was modeled using 128 2D frame elements. The data reduction did not use curved coordinates but treated each element as an independent straight segment. The data stored were the global displacements (and rotation) and the local gradients ∂ u/∂ ¯ x, ¯ ∂ 2 v/∂ ¯ x¯ 2 , and ∂ 3 v/∂ ¯ x¯ 3 for each element. The shape information is shown in Fig. 2.45 and the fourth column of Fig. 1.30. The amplitudes of the spectral shapes were decomposed up to a load level close to critical and the results are shown in Fig. 4.14a. The shapes at each load level are referenced to that of mode 2. Modes 1, 3, 5, 9, and 10 have antisymmetric shapes and do not contribute to the total deformed shape even though the load level is near critical. It is interesting that mode 4 changes sign over load (starts plus, goes minus), whereas mode 6 shows the most relative growth with load. The deformed shape is reconstructed well using just shapes 2, 4, 6, and 7 as shown in Fig. 4.15a. Figure 4.14b compares the relative contributions to the axial force referenced to mode 6. As expected, we need many axial modes to get converged results for the member load distributions. Figure 4.15b shows these load reconstructions at the maximum load level with shapes spanning the first 89 and compared to the FE generated distributions. All comparisons are very good including the axial force. Observe that the axial force F¯ does not have the same shape as the local space derivative ¯ = ∂ u/∂ ¯ x. ¯ With a good full representation of the spectral shapes, we are now in a position to compute the various stiffness contributions. Although the antisymmetric shapes do not contribute to the deformed shape, they do contribute to the stiffnesses. The expressions for the complete stiffnesses are developed in Sect. 3.4 and Fig. 4.16a shows that the spectral analysis reconstructs the FE stiffnesses quite well. This allows us to dissect the different stiffness contributions. Because of the nonlinearities and the imprecise shapes, the stiffness matrix is fully populated but a threshold analysis can identify the significant contributors. At

4.2 Buckling of Arches

305

♦ ♦

♦ ♦

♦ ♦

0.0

0.2

0.4

.

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 4.15 Reconstruction for a shallow arch at maximum load level. Circles are the FE nonlinear results, continuous lines are spectral reconstructions, and horizontal arrows are the zero reference. (a) Global displacements and rotations. Thin lines are at load level P /PR = 9, ¯ × 20. (b) Distributions for the natural BCs

♦

♦

♦

♦

♦

♦

0.

4.

8. .

12.

16.

0

5

10 .

15

20

Fig. 4.16 More reconstruction for a shallow arch. (a) Change of stiffness over load. Circles are the FE nonlinear results. Modes 3 and 4 are shifted vertically downward for plotting. (b) Change of the mode amplitude spectrums with load. Load values are greater than zero

load level P /PR = 9, the total stiffness has the partial contributions for the first six rows ⎡ ⎤ 1 2 3 4 5 6 ⎢ 0.08 ⎥ 0 0.03 0 −0.05 0 ⎢ ⎥ ⎢ ⎥ 0 0.37 0 −0.23 0 −0.55 ⎢ ⎥ ⎢ ⎥ [ KT ] ∝ ⎢ 0.03 ··· ⎥ 0 0.58 0 −0.16 0 ⎢ ⎥ ⎢ ⎥ 0 −0.23 0 1.20 0 −0.12 ⎢ ⎥ ⎢ ⎥ 0 2.09 0 ⎢ ⎥ ⎢ symm ⎥ 0 1.88

306

4 Buckling Shapes

Fig. 4.17 Change of symmetric mode shapes for three load levels. Arrows indicate increasing load

It is interesting that the main diagonal terms have increased stiffness relative to the no load case (coefficients KE11 = 0.01, KE22 = 0.05); the geometric contributions are negative as expected but they are dominated by the other nonlinear terms. And yet, Fig. 4.16a shows the principal stiffnesses decreasing with load. This must be due to mode interactions. The off-diagonal contributions are significant and are dominated by the geometric stiffness terms. Consequently, the interactions between the modes are significant which we now further dissect. When the load is increased, the stiffness is no longer diagonal, the modes are coupled, and the eigenvector of the EVP (here called the mode shape) has contributions from the other spectral shapes. Figure 4.17 shows examples of changing mode shapes with load. Observe how the third symmetric shape (mode #6) gets a reduction in its uniform component. At the load level being discussed, the geometric stiffness contributes the interactions. This contribution is detailed in Sect. 3.4 and can be summarized from Eq. (3.34) as KG → ¯ φI φJ + φ[I φJ + J φI ] = KG1 + KG2 with J being the higher mode. The distributions ¯ and φ are shown in Fig. 4.15a as the thin lines. Let us contrast the interactions between the first antisymmetric mode (#1) and the next two higher antisymmetric modes (#3,#5) with the interactions of the first symmetric mode (#2) and the next two higher symmetric modes (#4,#6). The off-diagonal terms are displayed as follows: modes KG1 KG2 KG (1,3) 0.02 –0.00 0.02 (2,4) 0.09 –0.05 0.04

modes KG1 KG2 KG (1,5) 0.02 –0.01 0.01 (2,6) 0.03 –0.51 –0.48

The coefficients can be compared to the earlier ones for [ KT ]. The first point to note is that the most significant interaction is for modes (2,6). Second, and this is the more subtle point, the overwhelming contribution comes from KG2 . Our basic buckling scheme is predicated on KG1 being dominant, that is, UG = 12 F¯o φ 2 . Figure 4.16b shows the results of the decomposition of the mode shapes. When comparing the first symmetric mode to the first antisymmetric mode it is clear from the first load increment that the former has changed (because there are now

4.2 Buckling of Arches

307

contributions from the other spectral shapes). The antisymmetric mode hardly changes shape even up to the maximum load. This reinforces our earlier conclusion that the first symmetric buckling mode has significant coupling to other modes, whereas the antisymmetric mode is not sensitive at all to other modes. What we are highlighting is the interaction between modes and this is significant even when the deformations are relatively small. In modeling this interaction, we cannot simply assume, say, v = v1 sin n¯ 1 θ + v3 sin n¯ 3 θ because our basic buckling analysis is linearized and the functions are orthogonal. In other words, if F (θ ) is pre-assumed, then there is no possibility of including the coupling between F (θ ) and v,θ (θ ). This just points out a deficiency of typical eigenbuckling analyses. We have already produced good nonlinear solutions for this problem in Sect. 3.3; here we want to examine it within a buckling context, and in as simplified a manner as possible expose the interactions. To this end we use Eq. (4.12) and specifically the stiffness contribution    b2 b2

2 = 12 EI v,b2 ds + EAE v, ds + EAEss ds ss s ss For demonstration purposes, and consistent with the buckling view point, we assume that the axial force EAEss = F¯o and current deformation v(θ ) is known from a separate linear analysis. The first two energy terms yield diagonal stiffness matrices when the trigonometric functions are used for the buckling shapes. Therefore, the b . We begin by examining this coupling arises through the buckling strain term Ess term. The constant axial strain (force) condition becomes   1 1 b b [(u,bθ −v b )R + v,θ v,bθ ] dθ Ess = Ess dθ = 2 α R α  1 = 2 [−v b R + v,θ v,bθ ] dθ R α Although the applied loading generates a symmetric deformed shape, the buckling shape can  be either symmetric or antisymmetric. Suppose it is the latter, then the integral v b dθ iszero. In addition the integral of a symmetric and antisymmetric distribution (here v,θ v,bθ dθ ) is also zero. The buckling membrane strain is then zero and the buckling energy is just the flexural and geometric contributions; this is our basic buckling method. It is a different story when we consider a symmetric buckling shape. Let the buckling and currently deformed shapes be represented by, respectively, v b (θ ) = b1 sin(n¯ 1 θ )+b3 sin(n¯ 3 θ ), v(θ ) = c1 sin(n¯ 1 θ )+c3 sin(n¯ 3 θ ), n¯ j = j π/α These satisfy the zero displacement and moment end conditions. To reiterate, the coefficients c1 and c3 are known from a separate linear analysis. The integrals for

308

4 Buckling Shapes

the flexural and geometric energies have been computed before (e.g., in Sect. 3.3); therefore, we just state the corresponding stiffnesses as [KF ] =

  β2 α n¯ 41 0 , 0 n¯ 43 2

[KG ] =

  β3 α n¯ 21 0 0 n¯ 23 2

where β3 = F¯o /R. These are associated with the DoF {b1 , b3 }T . We now focus on the additional buckling strain. The buckling strain evaluates to b Ess =−

   1  2 b1 b3 + c1 b1 n¯ 21 + c1 b3 n¯ 23 + 2 Rα n¯ 1 n¯ 3 2R

b2 α. Note This is constant; therefore, the strain energy integral is simply 12 EAEss that the energy is only quadratic in b1 , b3 which is consistent with our notion of buckling being a second-order energy analysis. On squaring the strain and double differentiating with respect to the DoF we get the membrane stiffnesses

  β1 4 1/n¯ 21 1/n¯ 1 n¯ 3 [KM1 ] = α 1/n¯ 1 n¯ 3 1/n¯ 23   β1 c1 n¯ 21 /n3 ¯ + c3 n¯ 23 /n¯ 1 2c1 n¯ 1 [KM2 ] = − ¯ + c3 n¯ 23 /n¯ 1 2c3 n¯ 3 R c1 n¯ 21 /n3  2 4  β1 α c1 n¯ 1 c1 c3 n¯ 21 n¯ 23 [KM3 ] = 2 4R c1 c3 n¯ 21 n¯ 23 c32 n¯ 43 Not surprisingly, these have a structure similar to the fully nonlinear matrices developed in Sect. 3.3. The combined stiffnesses change nonlinearly with the deformation (even when based on a linear analysis) as seen by the presence of, for example, c12 and c1 c2 in [KM3 ]. When we say the present analysis is linear (in the second-order sense) we mean the behavior with respect to the buckling DoF b1 , b3 is linear. A spectral analysis makes this clearer. To this end, let us estimate the axial displacement through a linear analysis. This gives ub (θ ) = b1 [1 − cos(n¯ 1 θ ) − 2θ/α]/n¯ 1 + b3 [1 − cos(n¯ 3 θ ) − 2θ/α]/n¯ 3 The mass for the I shape is  T =

1 2

ρA[u˙ 2 + v˙ 2 ] ds ,

MI I = ρAR[1 + 3/(2n¯ 2I )]α/2

The second term in the brackets (which comes from the axial inertia) is about 1/3 for the first shape and 1/24 for the second; because an accurate accounting of the mass is not necessary for our purpose here and the second shape dominates, let

4.3 Plate and Shell Buckling

309

us replace the second term with 1/10 so that we get matching frequencies at zero load. Ignoring any mass coupling effects, we can now write the spectral form of the buckling (dynamic) equilibrium equation as  [KF ] + [KG ] + [KM1 ] + [KM2 ] + [KM3 ] −ω

2 ρARα

2



1+γ 0 0 1+γ

  # b$ bˆ1 =0 bˆ b 3

with γ = 1/10. The linearity of the buckling DoF is now obvious. The key point is that in comparison to our basic buckling method, we have three additional stiffnesses each of which adds coupling but of a different type. We itemize each coupling. The stiffness [KM1 ] is not deformation dependent but it couples the two shapes in such a way that it is the b3 dominated shape that has the lower frequency. This explains why the FE results of Eq. (4.15) do not exhibit a buckling shape of n = 1. The behavior over load is shown as the thin full line in Fig. 4.13 and is comparable to the basic buckling model result. Adding [KM2 ] causes a strong softening effect indicated by the heavy dashed line in Fig. 4.13. This arose from an interaction between the buckling axial strain and the current deformed state. Although the consequences are similar, it is not analogous to the geometric stiffness which arose from an interaction between the current axial strain (force) and the buckling deformed state. Adding [KM3 ] restores some of the stiffness and makes the model results close to the nonlinear FE results as shown by the heavy line in Fig. 4.13. Deviations occur for large loads because the referenced deformed shape is based on a linear analysis; if we based it on the nonlinear deformed shape, we would expect results similar to that in Fig. 3.20a.

4.3 Plate and Shell Buckling We reiterate our contention that the best way to solve a shell problem is via the FE method which is a computerized version of our weak formulations. In this context, the role of analysis is to highlight distinctive features of the problem and the relevant parameters involved, and not to generate accurate solutions per se. The cylinder is chosen here (as in the previous chapters) as representing a complex system that is amenable to simplification through decomposition into its primary modes of deformation. Flat plates are relatively simple to analyze for their buckling behavior, by contrast, the analysis of curved plates and shells is immensely complicated as indicated by the nonlinear analyses in Sect. 3.2. We try to present both in as simplified a form as possible.

310

4 Buckling Shapes

4.3.1 Buckling of a Flat Plate The buckling analysis of thin flat plates follows in a similar fashion to that of the straight beam. The situation we have in mind is that the plate is in the nonlinear deformed state with deformations u, v, w under the actions of distributed loads qu , qv , qw . A small change is given to the deformations according to u¯ = u + ub ,

v¯ = v + v b ,

w¯ = w + w b

but the loads are left intact. The nonlinear total strains (using the von Karman approximations) are ¯ x + 12 w, ¯ 2x = [u,x + 12 w,2x ] + [u,bx +w,x w,bx ] + 12  2 w,b2 E¯ xx = u, x ¯ y + 12 w, ¯ 2y = [v,y + 12 w,2y ] + [v,by +w,y w,by ] + 12  2 w,b2 E¯ yy = v, y 2E¯ xy = u, ¯ y +v, ¯ x +w, ¯ x w, ¯ y = u,y +v,x +w,x w,y +[u,by +v,bx +w,x w,by +w,y w,bx ] + 12  2 [w,bx w,bx ] As shorthand, we write these and the corresponding curvatures as, respectively, E¯ ij = Eij + Eijb + 12  2 w,bi w,bj ,

κ¯ ij = w,ij +w,bij

(4.16)

with i, j = 1, 2 corresponding to x, y. The total potential under constant load can then be constructed. The stiffness term is of most current interest and is given by 

2 =

1 2

b2 E ∗ h[(Exx + νEyy )w,b2 x +(Eyy + νExx )w,y ] dxdy

 + 12

Gh[2Exy w,bx w,bx ] dx 

+ 12

b2 b2 b b + Exx + 2νExx Eyy ] dxdy + E ∗ h[Exx

(4.17)  1 2

b2 ] dxdy + UFb Gh[4Exy

where UFb is the linearized flexural energy given by Eq. (2.25). As for the straight beam this has multiple interaction possibilities. A common assumption for flat plates is that they do not deflect out-of-plane up to the point of buckling. Therefore, the interaction terms in Eijb can be neglected so that Eijb reduces to the small strain measure. The remaining interaction terms are recognized as stresses, and by treating them as pre-existing (or known from a separate analysis) then the flexural buckling problem is uncoupled from the membrane behavior. To summarize, and dropping the superscript b, the energy contributions are  UE =

1 2

A

 2 D ∇ 2 w dx dy + Ug

4.3 Plate and Shell Buckling

UG =

1 2

311

  ∂w 2 ∂w ∂w ∂w 2  σ¯ xx h[ ] + 2τ¯xy h + σ¯ yy h[ ] dx dy ∂x ∂x ∂y ∂y A

(4.18)

where σ¯ xx , σ¯ yy , and τ¯xy are the in-plane membrane stresses just before buckling occurs. For completeness, we state the governing equation for the flexural behavior D∇ 2 ∇ 2 w − σ¯ xx h

∂ 2w ∂ 2w ∂ 2w − σ¯ yy h 2 = q(x, y, t) − 2τ¯xy h 2 ∂x∂y ∂x ∂y

(4.19)

The associated BCs are, for example, # w #

 $ ∂ 3w ∂ 3w ∂w + σ ¯ + (2 − ν) h xx ∂x ∂x 3 ∂x∂y 2 $  2 ∂ w ∂ 2w =D +ν 2 2 ∂x ∂y 

or Vxz = −D

∂w or Mxx ∂x

(4.20)

As stated, the plate buckling problem is linear and invariably leads to an EVP. Flat plate buckling is essentially like that of a beam and most buckling shapes can be grasped by thinking of the plate as a grille (i.e., beam shapes in x and beam shapes in y). The exception is for the shear loading case which is an entirely different mode of deformation and quite complicated to solve. Reference [5] shows how a strongform solution can be obtained, an example problem to follow illustrates the simpler Ritz method approach. Example 4.10 Figure 4.18 shows some FE generated results for the first 16 buckling loads for a simply-supported rectangular plate under in-plane compression as the aspect ratio (a/b) is changed. There is no obvious pattern to the data but two aspects do seem striking. First, just like the beam on the elastic foundation, there is a definite minimum buckling load irrespective of the length of the plate. And second, the eigenmodes form clusters, for example, at a/b = 4 there are clusters at λ = 8 and λ = 30. Construct a model to explain these various results. A good way to construct a model is to use the Ritz approach. To do this, we need an idea of the deflected shape and a model to convert the deflection to strain energy. To this end, first look at some of the buckling shapes shown in Fig. 4.19; it is no coincidence they resemble the spectral shapes of Fig. 2.47; These are all for the same aspect ratio of a/b = 2. The bottom row is the lowest three buckling loads and we designate their shapes in the form [n, m] as [2, 1], [3, 1], [1, 1] indicating the number of half-waves in each direction. Note that the progression is not necessarily from simple (i.e., low number) to complex (i.e., high number). This contrasts with the spectral behavior of Fig. 2.47 which has the same shapes but in a different order. The top row is the eighth, ninth, and tenth buckling modes which we designate as [4 : 2], [5 : 2], [3 : 2]. Shapes like these are replicated at the other aspect ratios.

312

4 Buckling Shapes

50. 40.

.

30. 20. 10. .

0.

1.0

2.0

3.0

4.0

Fig. 4.18 Finite element results for the eigenbuckling analysis of a simply-supported flat plate under axial compression. (a) Geometry and properties. (b) Effect of aspect ratio on the buckling behavior load

Fig. 4.19 Some buckling shapes for aspect ratio 2 : 1. The sequence is lower row first left-to-right, then the top row

In the spirit of the Ritz approach, let the shapes be represented as w(x, y) = w1 sin(nπ x/a) sin(mπy/b) = w1 sin(nx) ¯ sin(my) ¯ This satisfies the geometric constraint that the boundaries have zero deflection and nonzero slope. It also satisfies the natural BC of zero moment. This is a single mode representation, but we have parameterized it using the numbers n and m; in this way, we can conveniently estimate multiple mode behaviors. Because there is only a uniform σ¯ xx stress, then from Eq. (4.18), we have UG =

1 2

   ∂w σ¯ xx h ∂x

2

dx dy = 12 w12 σ¯ xx h

 nπ 2 a

ab 14 = 12 w12 σ¯ xx hn¯ 2 ab 14

4.3 Plate and Shell Buckling

313

50. 40.

.

30. 20. 10. .

0.

1.0

2.0

3.0

4.0

Fig. 4.20 Comparison of model (continuous line) and FE (circles) results for a simply-supported plate

It is pointed out that the geometric strain energy depends on the parameter n but not m. The elastic energy is only flexural and given by  UF = 12 Dw12

nπ a



2

+

mπ b

2 2

 2 ab 14 = 12 Dw12 n¯ 2 + m ¯ 2 ab 14

In this case, the Gaussian curvature term is identically zero. We can now estimate the total stiffness as KT =

    2  ∂2  2 2 2 w12 ab 14 U = D n ¯ + U + m ¯ + σ ¯ h n ¯ E G xx ∂w12

The buckling occurs when the total stiffness is zero giving the critical stress as σ¯ xx h = −D

2 1 2 n¯ + m ¯2 2 n¯

(4.21)

Figure 4.20 shows this model superposed on the results in Fig. 4.18. Some of the results match quite closely. The unconnected FE results are associated with the other [n, m] modes. Thus the model gives coherence to the seemingly random results of Fig. 4.18b. The model clearly indicates a minimum buckling load that is independent of the length. We get the minimum critical load by setting m = 1 and differentiating with respect to a (keeping b constant) leading to

314

4 Buckling Shapes

1.25 1.00

.

0.75 0.50 0.25 0.00

1.

2.

.

3.

4.

Fig. 4.21 FE results for the eigenbuckling analysis of a flat plate simply supported on three sides. (a) Geometry (ss=simply supported). (b) Effect of aspect ratio on the buckling behavior. The normalizing stress is given by σo = Gh2 /bo2

∂ σ¯ xx h =0 ∂a

⇒

a =b n

⇒

σ¯ min h = −4D

 π 2 b

The buckling (half) wavelength is the width of the plate. This behavior of a limiting wavelength is quite similar to that for a beam on an elastic foundation as discussed in Sect. 4.1. The difference, however, is that it is the m ¯ = (mπ/b) term that enters the solution in the same way as Kv ; consequently, we can think of the higher modes as being associated with higher orders of elastic constraint [3]. Example 4.11 Figure 4.21 shows some FE generated results (circles) for the lowest buckling load for a simply-supported rectangular plate with one free edge. There is a strong b-dependence but a weak a- dependence. Construct a model to explain these results. As illustrated in the previous example problem, a good way to construct a model is to use the Ritz approach and to do this, we need an estimate of the deflected shapes. To this end, look at the contours for the buckling shapes shown in Fig. 4.22. These are for the aspect ratio of a/b = 2.5 (a = ao , b = 2bo ), the other aspect ratios have similar contours. The deflection w(x, y) has a strong sinusoidal shape as expected. The rotation φx (x, y) is nearly constant with respect to y, this is more pronounced for the larger aspect ratios.

4.3 Plate and Shell Buckling

315

.

.

Fig. 4.22 Contours for the first buckling shape for aspect ratio a/b = 2.5

A reasonable representation for the shapes is w(x, y) = w1 sin(nπ x/a)y/b = φo sin(nπ x/a)y = φx (x)y

(4.22)

in which n = 1 is the fundamental mode where the plate deflects to just one side or the other. Using the energy expressions from Eq. (2.25) substituting for w(x, y) and integrating then gives  nπ 4  nπ 2 1 UF = 12 w12 D ab 16 + 12 w12 4D¯ ab 1 a a b2 2   1 = 12 w12 D n¯ 2 61 + 4D¯ n¯ 2 21 2 ab b In contrast to the previous example, the Gaussian curvature term is nonzero; in fact, as seen presently, it turns out to be the significant term. Because there is only a uniform σxx stress, we have UG =

1 2

   ∂w σxx h ∂x

2

 dx dy = 12 w12 σxx h

nπ a

2

ab 16

We now estimate the total stiffness as KT =

   ∂2  2 ¯ n¯ 2 3 + 2σxx hn¯ 2 ab 1 U = D n ¯ + U + 4 D F G 6 b2 ∂w12

The buckling occurs when the total stiffness is zero giving the critical stress as  − σxx h = D n¯ + Gh 2

3

1 b



2

or

− σxx

 2 n2 π 2 b2 h = +1 G 2 6(1 − ν) a 2 b

(4.23)

where the relationship D = Gh3 /6(1 − ν) was used; a similar result is presented in Ref. [11]. Figure 4.21b shows this model (as the full line) superposed on the FE results. Clearly, the model captures the FE behavior quite closely.

316

4 Buckling Shapes

The coefficient of the first term in square brackets is approximately 2.19 for n = 1; therefore, for a > 5b the contribution of this term is less than 10% and the critical load is dominated by the Gh2 /b2 term. That is, the critical load is given by − σxx = G

h2 b2

(4.24)

and is independent of the length of the plate. The dashed line in Fig. 4.21b is the model without the first term; it is clear that the first term can be neglected. We make use of this to manipulate the simple model to extract more insight into the plate behavior. Additionally, the structure of the solution is different than Eq. (4.21); the former gives parabolas whereas here they monotonically decrease to the Gh2 /b2 solution. Consequently, the sequence of shapes remains the same. Using the third form of Eq. (4.22), we have the kinematic approximations u(x, ¯ y, z) ≈ −z

∂φx ∂w = −zy , ∂x ∂x

v(x, ¯ y, z) ≈ −z

∂w = −zφx , ∂y

w(x, ¯ y, z) ≈ w(x, y) = yφx where only φx is a function of x. This gives the strains ¯xx =

∂ 2 φx ∂ u¯ = −zy , ∂x ∂x 2

γ¯xy =

∂ v¯ ∂φx ∂ u¯ + = −2z ∂y ∂x ∂x

all others being zero. Note that the axial strain has a second derivative in x, but the shear strain has only a single derivative. We make the assumption that the elastic energy is dominated by the shear strain energy, so that UF ≈

1 2

=

1 2

   2 Gγxy dV =  

 3 1 3 Gh b

∂φx ∂x

1 2

   2 ∂φx G4z ∂x 2 dx

2

dx dy dz

The integration with respect to z is between ±h/2. We recognize from Eq. (2.41) that the 13 Gh3 b term is the St. Venant torsional stiffness of a thin rectangular cross section (see Sect. 2.2) and therefore we can write   UF =

1 2



∂φx GJ ∂x

2

dx

and interpret the strain energy as arising solely from the torsional (shearing) behavior of the plate. In a similar vein, the geometric strain energy is

4.3 Plate and Shell Buckling

317

Fig. 4.23 First three buckling shapes for a flat plate flexurally fixed at the bottom







2

∂w UG = dx dy = σxx h ∂x  2  ∂φx = 12 σxx Io dx ∂x 1 2

1 2



∂φx σxx h ∂x

2

y 2 dx dy

(4.25)

 where Io = hy 2 dy = 13 hb3 is the polar moment of inertia of the cross section about the center of rotation which is at y = 0. This result is used in the next section to help explain the effect of axial loads on thin-walled members under torsion. Example 4.12 Figure 4.23 shows the first three buckling mode shapes of an axially compressed plate simply supported on the ends, flexurally fixed on the bottom, and free at the top. Construct a model to explain why the shape is wrinkled and not all to one side like Fig. 4.22. Figure 4.24a shows the deflection distribution for the first buckling shape. The comparison is with the sinusoid sin(2π x/L) and it is very good. Therefore, we take the buckling shape as w(x, y) = w1 sin(nπ x/a)Y (y) = w1 sin(nx)Y ¯ (y) where we select Y (y) later so as at a minimum it gives zero deflection and slope along the fixed edge. Following the earlier examples, the energies are     4 2 2 UF = Dw2 n¯ Y − 2n¯ 2 Y Y  + Y 2 dy 1 2

 +D2(1 − ν)n¯ 2 UG =

1 2

   2 Y + Y Y  dy 12 a

   2 2  1 σxx w22 n¯ Y dy 2 a

318

4 Buckling Shapes

♦

♦

♦

0.0

0.2

0.4

.

0.6

0.8

1.0

0.0

0.2

0.4

.

0.6

0.8

1.0

Fig. 4.24 Transverse displacement distributions. (a) Along free edge. (b) Along x = a/2. Dashed line is model I and full line is model II

where the prime indicates differentiation with respect to y. The Gaussian curvature term has the interesting rearrangement 

 2 Y + Y Y  ] dy =



b  d   Y Y ] dy = Y Y  ] o dy

If either Y or Y  is zero at a boundary, then there is no Gaussian curvature contribution at that boundary. The stiffness is obtained from the second derivative of the total energy and leads to # −σxx = D

 

nπ a

2





Y − 2Y Y + 2

a nπ

2

Y

2

 dy

$  b  + D2(1 − ν)[Y Y ] o / Y 2 dy 

By appropriate substitution for Y (y), we can recover our earlier plate solutions. The simplest choice for the Y (y) function that satisfies the fixed BC is Y (y) = [y 2 /b2 ], call this model I and is shown plotted in Fig. 4.24b as the dashed line. There is not precise correspondence with the FE displacement but it could be argued that “on average” it represents the FE distribution. The integrals evaluate to  Y 2 dy =

b , 5



Y Y  dy =

2b , 3b2



Y 2 dy =

4b , b4

  b 2b Y Y ] o = 2 b

giving the critical load as −σxx =

 2 20 a 2 D  20 2b + (nπ ) − + 20(1 − ν) 3 hb2 a2 (nπ )2 b2

4.3 Plate and Shell Buckling

319

2.0 1.5

♦

♦

.

1.0 0.5 0.0

0.0

2.0

4.0 .

6.0

8.0

0.0

2.0

4.0 .

6.0

8.0

Fig. 4.25 Change of buckling eigenvalues with aspect ratio. The normalizing stress is σ0 = E/1000 and circle are the FE results. (a) Basic model results. (b) Refined model results

For each mode, the critical load is a parabola in a for fixed b as shown in Fig. 4.25. Hence, as the aspect ratio is changed, it is a different sequence of modes that occurs, e.g., at the arrow in Fig. 4.25 we get n = 3, 4, 2, 5; the n = 1 value is very high. In contrast, the simply-supported case with a free edge of Fig. 4.21 is monotonically decreasing in a for a fixed b and hence always has the same sequence. The minimum occurs at a/b = nπ/201/4 = 1.486 n giving −σmin =

 20 D √ 20 π 2D D + 20 − = 1.75 + 20(1 − ν) = 17.27 √ 3 hb2 hb2 hb2 20

Reference [11] used the strong formulation to get a coefficient of 1.328 and a minimum location of a/b = 1.635 n. This is shown as the horizontal dashed line in Fig. 4.25a; the circles are the FE results and it is clear that both agree quite well with each other. The current model captures the essential behaviors but the predicted numbers are off by a large 33% which we need to correct. We have a number of possible routes to explore in order to improve the quality of the predicted numbers; the two most relevant are adding an extra DoF and improving the guessed displacement shape. The ultimate implementation of the former is the finite element method; therefore, we do not pursue that here. Besides, when extra DoF are added to a model, the resulting equations tend to obscure the value and purpose of the model. Getting improved guessed functions requires insight into the problem, but often results can be borrowed from an analogous problem; this is illustrated next. Based on the distributed load solution for a cantilever beam given in Eq. (1.23), we suggest the function   y2   y  Y (y) = y 2 − 4yb + 6b2 Y (y) = 4y 2 − 12yb + 12b2 , 3b4 3b4

320

4 Buckling Shapes

 1  Y  (y) = 12y 2 − 24yb + 12b2 3b4 This is shown plotted in Fig. 4.24b as the full line. There is improved correspondence over model I with the FE distributions; this is especially true in the slope behavior. The integrals evaluate to  Y 2 dy =

104b , 45 × 9



Y Y  dy =

12 , 7 × 9b2



Y 2 dy =

144 , 5 × 9b4

  b 4 Y Y ] o = 3b giving the critical load as −σxx =

2 D  135 163 1 a 2 135  2b (nπ ) − + 2(1 − ν) + 91 13 (nπ )2 b2 26 hb2 a2

The minimum occurs at a/b = nπ [13/163]1/4 = 1.67 n which is closer to the reference value. This gives a minimum critical load of − σmin

D = 2 hb

'

163 135 − + 13 91

'

 135 π 2D 163 + 2(1 − ν) = 1.36 (4.26) 13 26 hb2

This is quite close to the exact value and FE values as shown in Fig. 4.25b. The dominant contribution to the elastic energy comes from the Gaussian curvature term. If we use it alone (it would yield a coefficient of 1.52), we get − σmin =

D 20(1 − ν) = hb2

2 10 h G 3 b2

(4.27)

This result is similar to Eq. (4.24) but is larger by more than a factor of three. Thus, in contrast to Fig. 4.21b, the constraining effect of the fixed boundary is to enhance the contribution of the flexural behavior relative to the torsional behavior. Example 4.13 Figure 4.26 shows some FE results for the shear buckling of a rectangular plate. The plate has simply-supported BCs with respect to the out-ofplane deflections and the shear loading is applied only along the top surface so that the two ends are stress free. The predominant feature of Fig. 4.26a is that all buckling loads converge to a common threshold as the aspect ratio is increased to an infinitely long plate of fixed width. Construct a model to help explain the observed results. Based on the deflection contours shown in Fig. 4.26b, we assume a buckling shape given by w(x, y) = c1 sin(πy/b) sin[π(x − αy)/s]

4.3 Plate and Shell Buckling

321

10. 8. .

.

6. 4.

.

2. .

0.

1.

2.

3.

4.

.

5.

6.

Fig. 4.26 Shear buckling of a plate. (a) Buckling loads as a function of plate aspect ratio. (b) Outof-plane deflection contours for the first three (bottom upwards) FE buckling modes with aspect ratio a/b = 4. Top is analytical result for an infinitely long plate

In this, α affects the slope of the nodal lines and s affects the periodicity of the shape; both are unknown and the parameterization makes it similar to Ref. [11]. This function satisfies the lateral geometric BC of zero displacement but not the natural BC of zero moment. This assumed shape repeats itself over the infinitely long plate; this is not the case for the actual FE examples which tend to show clusters of buckles. Our assumption essentially is that in the limit of large aspect ratio all buckling modes occur simultaneously; this is given further insight in Fig. 4.8 where the idea of cellular buckling [13] is introduced. Because of the zero displacement boundaries, the strain energy is given by  U = UF + UG =

1 2D

 [∇ w] dxdy + 2

2

A

1 2



∂w 2σ¯ xy h ∂x A



∂w ∂y

 dx dy

Substituting for w(x, y) and performing the required integrations leads to σ¯ xy h =

 π 2D  2 s2 b2 2 2 6α + 2 + + (1 + α ) 2αb2 b2 s2

We choose α and s so as to give the lowest value of σ¯ xy h. That is, ∂ σ¯ xy h 2s 2b2 = 0 = 2 − 3 (1 + α 2 )2 ∂s b s ∂ σ¯ xy h 2 s2 b2 (1 + α 2 )2 b2 =0=6− 2 − 2 2 − 2 + 4(1 + α 2 )2 ∂α α α b s α2 s2 This gives

322

4 Buckling Shapes

  s = b 1 + α 2 = b 3/2 ,

α=



1/2

which leads to a critical load of √ π 2D π 2D σ¯ xy h = 4 2 2 = 5.66 2 b b The normalizing factor used in Fig. 4.26a is σo = π 2 D/b2 h so that the vertical axis actually represents the coefficient. The value of 5.66 (shown as the full line) is reasonably close to the exact value of 5.34 [5] (shown as the dashed line). As further elaboration on this problem, consider the effect of bi-directional membrane stresses on buckling because it is known that a state of simple shear can be simulated using equal (but opposite) biaxial stress. The total energy is  U = 12 D

(∇ 2 w)2 dxdy + A

1 2

   ∂w σ¯ xx h ∂x A



2

+ σ¯ yy h

∂w ∂y

2

dxdy

Proceeding as before, we get the total stiffness as #   2 nπ nπ mπ 2 KT = D ( )2 + ( + σ¯ xx h ) a b a



2

+ σ¯ yy h

mπ b

2$

ab 14

The interaction of the two stresses is interesting. Suppose we have proportional loading with σ¯ xx = σ¯ o ,

σ¯ yy = γ σ¯ o

then the critical load occurs when  −σ¯ o h = D

nπ a



2

+

mπ b

2 2

 nπ / a



2

+γ

mπ b

2

If γ is positive, the stress state is biaxially compressive and we get the lowest critical load. On the other hand, if γ is negative, then the critical load is raised. Indeed, if the plate is square and γ = −1 (this is a state of pure shear), then the plate is predicted to never buckle. The panel in the tension beam of Fig. 5.15 is almost in a state of pure shear but they have finite buckling loads. To fully explain the result requires a postbuckling analysis. Also note that γ affects the shape of the parabolas so that in some cases there is no minimum. Reference [11] solves the problem of nearly square plates under shear loading, it then proposes the relation 

 b σ¯ xy h = 5.35 + 4 a

2

π 2 D/b2

4.3 Plate and Shell Buckling

323

200.

.

100.

.

0.

0

4

8

12

16

20

24

Fig. 4.27 Cylinder under gravity loading. (a) Geometry. (b) Buckling eigenvalue against mode number. Circles are thickness ho , squares are thickness 2ho , full symbols are radius Ro , and the dashed line is the model results

as a reasonable approximation for the minimum critical load for sheared plates with arbitrary aspect ratio. This underestimates the exact solution by about 6%.

4.3.2 Buckling of a Cylindrical Shell Under Axial Load Consider the cylindrical shell in Fig. 4.27a. It is completely fixed at the base, open and completely free at the top, and loaded under gravity. The gravity loading means that the membrane stresses are nonuniformly distributed being maximum at the bottom. The cylinder is modeled using the Hex20 element [6]. We use this model to highlight some essential different behaviors of shells (and curved plates) versus those of flat plates, beams, and frames. A very good review of shell buckling problems is given in Ref. [1]. Figure 4.27b shows the FE generated results for buckling eigenvalues for three radii Ro , 12 Ro , 2Ro , and two thicknesses ho , 2ho . The remarkable feature about these results, especially for the h/R = 0.02 cases is that they are almost flat with respect to mode number; that is, all buckling modes have nearly the same critical value. Figure 4.28 shows a sampling of the buckling shapes: some are axisymmetric, some are star-shaped in the hoop direction, some are localized to the ground, and some extend the full length of the cylinder. Irrespective of buckling shape, the buckling loads are all about the same magnitude. This is a significant point. In a situation like that presented in Fig. 4.27b, the sequence of shapes is unimportant because a small change of parameter changes the sequence. We use this feature to construct a simplified model for the buckling behavior and specifically, for convenience, consider one of the axisymmetric modes. In the discussion of the pressurized cylinder of Fig. 2.15, it was shown that the curvature has an effect similar to that of an elastic foundation so that the total elastic energy of a small band of cylinder is

324

4 Buckling Shapes

Fig. 4.28 Sampling of buckling shapes for a cylinder under gravity loading

UE = 12 E ∗ h

 u 2 R

 2π R dz + 12 D

∂ 2u ∂z2

2 2π R dz

where R is the cylinder radius, h is the thickness, and u(z) is the radial displacement. To this we add the geometric strain energy; think of the shell unrolled so that it is like a flat plate, then  UG =

1 2 σzz h

∂u ∂z

2 2π R dz

We have essentially reduced the problem to a 1DoF problem with energies resembling that of a beam. While not true, let us assume that the particular BCs are not that important, and we can consider the problem as a cylindrical shell with fixed end radii and simply-supported BCs. Also assume that the gravity load is replaced by its average and uniformly applied at both ends so that σzz is constant. Motivated by the spectral analysis in Sect. 3.4, we assume the buckling shape is given by u(z) = un sin(nπ z/L) Substitute into the energy expressions and integrate over the length L to get  UE + UG = 12 u2n

 4  2  Eh nπ nπ 2π R 12 L + D + σ h zz 2 L L R

Equilibrium is when  ∂  UE + UG = 0 ∂un

 ⇒

 4  2  Eh nπ nπ un = 0 +D + σzz h 2 L L R

The nontrivial equilibrium solution is when the bracketed term is zero leading to the critical stress

4.3 Plate and Shell Buckling

325

− σc h =

 2  2 nπ Eh L + D L R 2 nπ

(4.28)

This forms a parabola in (σc , L) space for each n. The minimum is obtained by differentiating with respect to nπ/L and setting the result to zero. This gives ' minimum:

− σc = 2

Eh DE =  2 R h R 3(1 − ν 2 )

(4.29)

As it turns out, this is close to the exact (strong form) result [11] discussed in the first example problem to follow. This minimum buckling load is independent of the stressed length L. Let the yield stress be approximated as σY ≈ E/200, then the critical stress is σc ≈ 100σY

h R

Thus, unless h/R < 0.01, then yielding dominates, i.e., σc > σY . Thus, buckling occurs only for cylinders that are very thin walled. This highlights the exceptional stiffness character of shells as regards withstanding buckling. The origin of this is the curvature contribution to strain energy coming from E ∗ h/R 2 . As discussed in Sect. 2.2, this acts like an elastic constraint and therefore sets a minimum critical value. Look more closely at the two contributions to the elastic stiffness  4 Eh nπ +D L R2 The second term is the flexural energy for a bar of length L; as the length increases, this goes to zero. In the absence of the elastic constraint, long bars are very susceptible to buckling because of this reduction in stiffness. By contrast, the first term is independent of length and therefore sets a minimum stiffness level. Another point worth noting is the connection between the present results and those of Figs. 4.7b and 4.20. The latter two exhibit intersecting modes with a minimum threshold as the aspect ratio is changed. It is also worth contrasting the form of Eqs. 4.21, 4.28 and Eqs. 4.23, 4.26; the parabolic form of the former is clear. Thus the lateral dimension of a plate plays a role similar to an elastic foundation or distributed constraint. The higher modes in Fig. 4.20 correspond to an increase in elastic constraint. This effect is further exaggerated in shells because, the effect of the shell curvature is that of an elastic foundation. This helps to explain to some extent, the flat behavior observed in Fig. 4.27—the modes are in the cluster region of Figs. 4.7b and 4.20. The results given in Fig. 4.25 for the flat plate with a free edge also exhibit this behavior and therefore it seem this is intrinsic to plate systems. However, it is Fig. 4.7b that confirms that it is the distributed elastic constraint that is the origin of the phenomenon.

326

4 Buckling Shapes

Returning to the gravity problem, assume that the stress is linearly distributed given by σzz = −ρ2π Rha[1 − z/L] so that its average is σzz = −ρ2π RhaL/2. The results of this model are shown as the dashed lines in Fig. 4.27b. The model says the critical stress depends linearly on the ratio h/R, this is approximately confirmed by the results in Fig. 4.27b. Accepting that the simplified model is reasonable, it has a couple of important implications. First, a cylindrical shell made of steel or aluminum will never buckle under self-weight because λ=

Eh σzz 25 ≈ √ × 103 ≈ ρgL L R 3ρgL

where L is in meters and we used h/R = 0.02. Reference [8] discusses the buckling of silicone cylinders under self-weight loading; these have a low modulus of E = 1.41 MPa but standard density of ρ = 1200 kg/m3 and exhibit buckling patterns similar to Fig. 4.28. Reference [10] considers the FE analysis of some other cylinder cases such as bending; it also presents the eigenanalyses in the context of nonlinear analyses, it is worth consulting for its discussion of the accuracy required for good shell buckling analyses. Example 4.14 Estimate the buckling loads for a cylinder under axial compression. We already have the linear stiffness of the shell given in Sect. 2.4 and are the contributions to Eq. (2.52). To add the effect of axial load, think of a longitudinal strip of cylinder as a stressed cable (oriented along z) that is displaced from its current position in the plane by u and v. The cable-energy contributions are 2 1 ¯ 2 1 ¯ 2 Fo u,z + 2 Fo v,z

Replacing F¯o in terms of the axial stress σzz , this leads to contributions to the governing equations of δu :

o hu,zz ; σzz

δv :

o σzz hv,zz

Let the displacements be represented by ¯ sin(mz), ¯ u = uo sin(ns)

v = vo cos(ns) ¯ sin(mz), ¯

w = wo sin(ns) ¯ cos(mz) ¯

with n¯ = n/(2R), m ¯ = mπ/L, then the contribution to Eq. (2.52) is ⎡

⎤ m ¯2 0 0 o ⎣ −σzz h 0 m ¯2 0⎦ 0 0 0

327 ♦

♦

4.3 Plate and Shell Buckling

♦

0

2

4

6

8

10

♦

0

2

4

6

8

10

Fig. 4.29 Cylindrical shell under axial compression. (a) Comparison of model buckling loads (lines) with the FE generated buckling loads (symbols). Numbers are for n. (b) Approximate model for cylinder using curved plate coordinates. Numbers are for m

o as the unknown eigenvalue. Setting For the buckling EVP, we set ω = 0 and treat σzz the determinant to zero gives a quadratic equation for the stress which is readily solved. Using parameters of the cylinder in Fig. 2.49, Fig. 4.29 shows the variation of buckling load with the number of half-wavelengths in the axial direction. For the infinitely long cylinder, we can treat m as a continuous number. In contrast to flat plates, Fig. 4.20 for example, the cylinder does not exhibit a common minimum among the curves. The fine horizontal line in the figure is the critical value given by Eq. (4.29); an extended plot for large m shows that all curves cluster to this line. The same FE model as shown in Fig. 2.49 was used to generate the data shown as symbols in Fig. 4.29; although there are only a dozen or so data, they were taken from a total of 99 buckling loads. A longer cylinder would generate intermediate m values, but would not give the high stress critical values. The agreement is good which allows us to explore some simplifications of the cylinder equations. We do this in the form of the shallow curved plate.

Example 4.15 Develop an approximate cylindrical shell model useful for buckling problems. Our buckling analysis is similar to that for the flat plate; therefore, we just highlight the differences. To help illustrate these differences, we use the curved plate notation given in Fig. 2.14b and the nonlinear membrane strain-displacement relations given by Eq. (3.28). We begin by neglecting the nonlinear contributions of the in-plane displacements u and v, thus, the total strain during buckling is E¯ xx ≈ u, ¯ x + 12 w, ¯ 2x , E¯ ss ≈ v, ¯ s +w/R ¯ + 12 w, ¯ 2s , 2E¯ xs ≈ u, ¯ s +v, ¯ x +w, ¯ x w, ¯ s which is the same as for the flat plate except for the w/R ¯ contribution in E¯ ss . b Substituting u¯ = u + u and so on, and grouping terms according to powers of ,

328

4 Buckling Shapes

we can write the total strain in the form of Eq. (4.16) with i, j = 1, 2 corresponding to x, s, and where we identify the buckling strains as b = u,bx +w,x w,bx , Exx

b Ess = v,bs +w b /R + w,s w,bs

b 2Exs = u,bs +v,bx +w,x w,bs +w,s w,bx

The second-order total potential is then the same as given in Eq. (4.18). A common assumption for cylinders is that the prestress deformation state is negligible so that we can write the total potential as b

2 = U M + UFb + UG

where the energies are given by  b = UM

1 2

  b b 2 b b b E ∗ h u,b2 x +(v,s +w /R) 2νu,x (v,s +w /R) dsdx 

+ 12  UFb =

1 2

UG =

1 2



2  Gh u,bs +v,bx dsdx

2  D w,xx +w,ss dsdx   b2 b b σxx h w,b2 x +σss h w,s +2τxs h w,x w,s dsdx

(4.30)

and the pre-existing stresses are σxx = E ∗ [u,x +ν(v,s +w/R)], σss = E ∗ [(v,s +w/R) + νu,x ], τxs = G[u,s +v,x ] The buckling strain energies are the linearized versions but for the buckling deformation. Applying our variational principle to 2 , we get two sets of equations. The first is the membrane equilibrium equations similar to Eq. (2.34) but superscripted b. The second is the transverse equilibrium given by b D∇ 2 ∇ 2 w b − σxx hw,bxx −σss hw,bss −2τxs hw,bxs +σss h/R = 0

(4.31)

b /R is analogous to F /R in Eq. (4.13) and unlike the other stresses in The term σss Eq. (4.31) is not known from a separate linear analysis. In general, this stress cannot be neglected but can be determined by using the compatibility equation

∇ 2 ∇ 2 ψ − Ew,bxx /R = 0

(4.32)

4.3 Plate and Shell Buckling

329

Eqs. (4.31) and (4.32) constitute our governing equations for a cylinder written in curved plate coordinates and are the equations developed by Donnell [2]. o . Let Consider a very long cylinder under the action of an applied axial stress σxx the displacement and stress function be represented by ¯ sin(ms) ¯ , w = wo sin(nx)

ψ = ψo sin(nx) ¯ sin(ms) ¯

with n¯ = nπ/L, m ¯ = m/(2R). Both must have the same functional form because of the even powers of derivatives occurring in the governing equations. The buckling stress is related to the stress function by b = ψ,xx = −n¯ 2 ψo sin(nx) ¯ sin(ms) ¯ σss

Substitute for the displacement and stress to get 

o h −n ¯ 2 ]2 + n¯ 2 σxx ¯ 2 h/R D[n¯ 2 + m 2 2 [n¯ + m ¯ 2 ]2 E n¯ /R

# $ wo =0 ψo

Observe that the buckling stress coupled the membrane stiffness E/R to the flexural stiffness D. This is an EVP giving the critical stress as c −σxx h=

n¯ 2 ¯ 2 ]2 Eh [n¯ 2 + m + D R 2 [n¯ 2 + m ¯ 2 ]2 n¯ 2

This is in a form similar to Eq. (4.28), in fact, if we set m ¯ = 0, we recover that solution. Figure 4.29b shows the complete behavior when the parameters of the previous example problem are used. The significant difference is that the longer wavelength buckling loads are overestimated. However, because of the clustering effect, it is expected to give good results when the buckling mode has many half-waves.

4.3.3 Buckling of Curved Plates In contrast to flat plates, curved plates are susceptible to instabilities due to transverse loadings as well as axial loading. The archetypal model for the former is the arch, the archetypal model for the latter is the straight beam but on an elastic foundation. The reason is that the curvature gives extra transverse stiffness and buckling, if it occurs, occurs in a higher mode. The curved plate problem has a total of three unknowns but to simplify matters, the stresses are assumed to be known separate from the buckling analysis. The relevant energies are then just the flexural and geometric energies containing just v and w. Constructing a model thus requires the selection of two displacement functions

4 Buckling Shapes

200.

♦

330

0. .

♦

-200.

.

-400. ♦

0

5

10

15

20

0.

25.

50.

75.

100.

Fig. 4.30 Shallow curved plate loaded uniformly in the transverse direction. (a) Mesh and parameters. (b) FE generated distribution for the mid-length contributions to hoop strain during nonlinear loading. Empty symbols are 90% critical load, full symbols are 100%, continuous lines are during snapthrough. (c) Amplitude spectrum for three load levels

with two independent variables. We, therefore, explore some simplifications and their implications. The generic types of loadings considered are transverse and axial.

Loading I: Transverse Figure 4.30a shows a mesh for a shallow curved plate uniformly laterally loaded. All boundaries allow rotations, the straight sides are displacement fixed, and the curved ends are constrained to move in the plane. The mesh shown uses the MRT/DKT shell element [5] and the parameters of the case are taken from Ref. [12]. Although the loading and geometry are symmetric, the large-load nonlinear behavior shown in Fig. 4.30b exhibits a distinct asymmetry in the deformation which we now explain. We begin by looking at the change of displacement over load. Figure 4.30c shows the spectrums at load levels of: half-critical, critical, and during the snapthrough, ratioed to mode 4. Modes 4, 10, 12, and 13 are symmetric and hardly change (their ratio) during the loading indicating very little change of the symmetric shape, all the activity is only in mode 1 which is antisymmetric. In constructing a model, we would like to have just a single variable (e.g., transverse displacement w(s)) to deal with. In general, there is no simple relation between v(s) and w(s) that can be used to replace v(s) but in some cases of thin-walled structures we can utilize the inextensibility condition to essentially achieve the same condition. To elaborate, Fig. 4.30b shows the contributions to the hoop strain during the nonlinear analysis. What is remarkable is that while the separate contributions of v,s and w/R vary significantly, their combination yields an almost constant strain. The distributions change as the critical load is approached, but the hoop strain still remains almost constant even during the snapthrough. We, therefore, can conclude that a reasonable approximation in some cases is that

♦

300.

331 ♦

4.3 Plate and Shell Buckling

.

200.

100. ♦

0.

♦

0.0

0.5

.

1.0

1.5

0.0

0.5

1.0 .

1.5

2.0

Fig. 4.31 Laterally loaded shallow curved plate. (a) Average hoop strain as a function of load. Circles are FE data, continuous line is model. (b) Comparison of FE generated spectral stiffness values for the lowest mode (triangles use shell elements and squares use Hex20 elements) and model (continuous line)

w ∂v + = constant = o ∂s R and in some cases o ≈ 0 (the inextensibility condition). We utilize this in constructing a model for this problem. Figure 4.31a shows the average hoop strain in response to the load. The reference load corresponds to the FE mesh having unit vertical loads at each node and is qo = 43 kPa (6.2 psi). The relation is predominantly linear over the load range; a slight deviation is observed toward the maximum load. A model for this behavior is based on elementary pressure vessel modeling as introduced in association with Fig. 2.15. On the assumption that the hoop stress is constant everywhere, the equilibrium between the edge forces and the applied load gives m 2σss Lh sin(α/2) = q2R sin(α)L

or

m σss = qR/ h

m = νσ m The end constraints impose that xx = 0 so that, because of plane stress, σxx ss giving

ss =

12 1 − ν2 qR [σss − νσxx ] = σss = ∗ = o E E E h

This is shown plotted as the straight line in Fig. 4.31a, the agreement is quite good. Note that this simple modeling gives wo = Ro and wo is interpreted as the average radial displacement. We conceive of the buckling shape being superposed on the nonlinear deformation. Figure 4.32a shows the first four spectral shapes as candidate superposition shapes; the ratios of stiffness (ω2 ) are 1.0 : 1.9 : 4.3 : 4.4. The spectral plots

332

4 Buckling Shapes

♦

0.

25.

50. .

75.

100.

Fig. 4.32 Spectral analysis of a shallow curved plate. (a) First four spectral shapes. Sequence is lower left to upper right. (b) Deformation distributions for the first mode. The arrow indicates the zero reference

of Fig. 4.31 shows that the stiffness of the first mode goes to zero directly without apparent interference or interaction with the other modes. Adopting the first spectral shape as the buckling shape essentially makes the problem similar to that of the arch. Figure 4.32b shows that the first mode is inextensible and therefore, as a superposition, would not affect o . This is what we see in Fig. 4.30b through the full lines. Let the nonlinear deflected shape be w(s) = wo + wm sin(mπ s/a) = wo + w2 sin(2π s/a) where wo is the uniform deflection associated with the membrane strain o —neither are subjected to variations and instead are treated as parameters. The nonlinear strains are Exx ≈ 0 ,

Ess ≈ o + 12 w,2s −zw,2ss ,

2Exx ≈ 0

The strain energy becomes  UM =

1 2

UF =

1 2

2   ¯ 2 w22 21 + E ∗ h o + 12 w,2s b ds = 12 E ∗ h o2 + o m

3 4 4 ¯ w2 32 m

 ab

 ¯ 4 w22 21 ab Dw,2ss b ds = 12 D m

where m ¯ = mπ/a = 2π/a. The three contributions to the membrane energy are, respectively, linear, second-order nonlinear (geometric), and high-order nonlinear. The kinetic energy is  T =

1 2

  ρhw˙ 2 b ds = 12 ρh w˙ 22 21 ab

4.3 Plate and Shell Buckling

333

The stiffness and mass are obtained as KT =

∂2 U ∂2 T ∗ 21 41 9 4 2 = E h[ m ¯ + m ¯ w ]ab + D m ¯ ab, M = = ρh 12 ab o 2 2 8 2 ∂w22 ∂ w˙ 22

The spectral stiffness is then given as ω2 = KT /M. This is shown plotted in Fig. 4.31b as the continuous line and compared to two sets of FE results. The agreement is very good which means our basic conception of the problem is correct. The second and third spectral shapes are also antisymmetric but they have more complicated distributions in x. Let us apply our method to the second mode. The nonlinear deflected shape is reasonably represented by w(x, s) = wo + w2 sin(2π s/a) cos(π x/L) = wo + w2 sin(m ¯ 2 s) cos(n¯ 1 x) where m ¯ 2 = 2π/a and n¯ 1 = 1π/L. Following our earlier manipulations, we form the stiffness (to second order) and mass as, respectively,    2   ¯ 22 14 ab + D n¯ 21 + m ¯ 22 KT = E ∗ h o ν n¯ 21 + m

1 4

ab ,

M = ρh 14 ab

The spectral stiffness is shown plotted in Fig. 4.31b as the light continuous line. It does not compare well to the FE results. The main reason for the poor result is that the distributions (especially in x) are more complicated than assumed. For example, the transverse deflection is better represented by w(x, s) = wo + w2 sin(2π s/a)[a1 cos(π x/L) + a2 sin(2π x/L)] Both functions of x are antisymmetric and therefore give an interaction effect. This is not pursued here because it rightly belongs in Chap. 2 which deals with initial stiffness effects. Furthermore, a shift of the model results as indicated by the dashed line in Fig. 4.31b follows the trend of the FE data quite well; this indicates that our modeling of the buckling process is correct, we need to only improve the initial stiffness model. To summarize, we think of the problem as having two deformations. The applied loading causes the almost uniform strain o associated with a transverse displacement wo so that o = wo /R. The load changes the structural stiffness softening each of the modes. The stiffness of the first antisymmetric mode goes to zero and this is why it appears in the deformation distributions of Fig. 4.30b. The symmetric buckling problem must be tackled similar to that for the arch. Our conclusion is that the problem is not amenable to the basic buckling method—a fully nonlinear analysis is needed and this is given in Sect. 5.2.

4 Buckling Shapes ♦

2000.

♦

334

1500.

.

1000. 500. ♦

0.

200.

400.

600.

800.

0.

♦

0

2

4

6

8

10

12

Fig. 4.33 Axial compression of a shallow curved plate. (a) Spectral stiffness plots. Circles are FE nonlinear results. The full lines are for the [1, 2] and [6, 1] modes, the dashed lines are for the [2, 2] and [5, 1] modes. (b) Dependence of buckling load on the number of half-wavelengths along the length. Symbols are FE data, lines are model results

Loading II: Axial Compression In addition to transverse loads, the curved plate is expected to withstand axial compression. The shallow curved plate considered is the same as that shown in Fig. 2.52a. The problem has much in common with that of the cylinder under axial compression, in particular, it exhibits a good deal of mode clustering. Figure 4.33a shows the FE generated spectral plots. They are complicated and quite unlike anything we have encountered before. It seems that the most significant change of stiffness occurs in the higher modes; this contrasts with the transversely loaded curved plate. Also, it seems that the higher modes interact with the lower modes also forcing them (the lower modes) to become singular. These are aspects we wish to investigate and we use a fuller spectral analysis to this end. The change of amplitude spectrum during loading is shown in Fig. 4.34b referenced to the first symmetric mode #3. As expected, mode #3 [1, 1] is initially dominant because it is the bulging effect. Over load, however, the most significant activity is observed in mode #10 [5, 1]. It seems that this mode competes with the even higher mode #14 [6, 1] as to which becomes critical first. This, interestingly, is not reflected in the amplitude spectrum, highlighting again the difference between the deformed shape and the buckling shape. A zero-load stiffness model was already constructed in connection with Fig. 2.53b, here we add the geometric strain energy. From a static analysis, we can make the reasonable approximation that the stress state is predominantly axial o = P /Rαh. The geometric and uniform along the length and given simply by σxx o strain energy is then estimated from Eq. (4.30) as  UG =

1 2

o σxx hw,2x

dsdx =

2 2 1 o ¯ aL/4 , 2 σxx wo n

[KG ] =

o σxx h

 2  n¯ 0 aL 0 0 4

4.4 Load Coupling of Deformation Modes

335

Fig. 4.34 Shallow curved plate. Amplitude spectrum for five load levels

This contribution is independent of the shape being symmetric or antisymmetric. Figure 4.33a shows this model superposed on the FE generated results for the first two modes (which are antisymmetric shapes) as the dashed lines. The agreement is good until the intersection with the higher modes. The model correctly predicts that it is not the antisymmetric modes that become critical; this contrasts with the transversely loaded curved plate. Figure 4.33a also shows model results for the [5, 1] and [6, 1] modes as the full lines. The agreement with the FE data is reasonable and show the dramatic way in which the higher modes can become critical. Figure 4.33b shows the collection of buckling loads and how a minimum is achieved in the vicinity of n = 5 ∼ 6. The flexural and geometric stiffnesses for the curved plate and simple straight beam have similar longitudinal dependence on n, ¯ namely, n¯ 4 and n¯ 2 , respectively. The difference is that the curvature raises the stiffness of the lower modes more than the higher ones. Consequently, the steep slope of the higher modes results in intersections with the lower modes and becomes critical at lower relative loads. The analysis of mode interactions near a singular point is very complicated and certainly not part of the “usual” buckling analyses. Section 5.3 attempts a beginning of this analysis exploration of mode interaction but clearly it will require its own volume to elucidate.

4.4 Load Coupling of Deformation Modes When we considered the buckling of a simple beam and determined the buckling shapes, all shapes correspond to the flexural behavior of the beam but they coupled to the axial deformation or load. The arch and shell analyses of Chap. 2 are complex examples having initially coupled deformations, but these also coupled to the axial loads. As we apply our analyses to more complicated structures, we encounter other types of deformations which possibly can be coupled. This section explores what happens when different deformation types interact with each other. Of course, there must be a coupling mechanism, this subsection looks at coupling that can be described in terms of loads. There is a distinction to be made between coupled deformations and coupled modes; we clarify this.

336

4 Buckling Shapes

Fig. 4.35 Lateral buckling of a beam. (a) Buckling shape. (b) Translation and rotation of a typical cross section. (c) Coordinates describing the motion of the cross section

4.4.1 Lateral Buckling of Beams Figure 4.35 shows a beam with applied moments about the z-axis at each end that cause lateral (out-of-plane) buckling. Figure 4.35b shows the buckling position of the cross section at midlength; it is apparent that there is both translation (w) and rotation (φx ) of the cross section. That is, there is coupling of the bending and torsional deformations. These coupling effects are fundamentally nonlinear, but as demonstrated earlier, by retaining just second-order effects we can linearize the analysis and this is what we call load coupling. The key to this analysis is to consider equilibrium in the slightly deformed configuration. Consider the beam cross section shown in Fig. 4.35c undergoing lateral deflections. We introduce two sets of coordinates: (x, y, z) are the global coordinates oriented with respect to the original beam, (ξ, η, ζ ) move with the beam as it deflects. The deflection of the cross section is given in terms of the v, w displacements and the rotation φx . Conceive of the cross section as arriving at its current orientation through a sequence of rotations about the axes: first about x, then y, and then z. That is, ⎡

⎤⎡ ⎤ ⎤⎡ cos φz − sin φz 0 1 0 0 cos φy 0 sin φy [ R ] = ⎣ sin φz cos φz 0 ⎦⎣ 0 1 0 ⎦⎣ 0 cos φx − sin φx ⎦ 0 sin φx cos φx 0 0 1 − sin φy 0 cos φy Assuming the rotation angles are small, then the triple product yields ⎡

⎤ 1 −φz φy [ R ] ≈ ⎣ φz 1 −φx ⎦ −φy φx 1 When the components of an initial vector are multiplied by [ R ], the result {v  } = [ R ]{ v } is the components of the rotated vector referred to the same reference

4.4 Load Coupling of Deformation Modes

337

frame. In our case, however, the vector remains unchanged, it is the reference frame that is rotated. It is straightforward to show that the components referred to the rotated frame are given by {v  } = [ R ]T { v } where the superscript “T ” refers to the matrix transpose. Thus, an arbitrary fixed vector has the resolved components in the rotated coordinate system of ⎧ ⎫ ⎡ ⎤⎧ ⎫ ⎡ ⎤⎧ ⎫ 1 φz −φy ⎨Mx ⎬ 1 v,x w,x ⎨Mx ⎬ ⎨ Mξ ⎬ = ⎣ −φz 1 φx ⎦ My = ⎣ −v,x 1 φx ⎦ My M ⎩ ⎭ ⎩ ⎭ ⎩ η⎭ Mζ φy −φx 1 Mz −w,x −φx 1 Mz where the subscript comma means partial differentiation. The relation of the rotations to the displacement gradients can also be seen in Fig. 1.16b. Based on Eq. (1.19) for bending and Eq. (1.20) for torsion, the governing equations can be written in local coordinates as EIζ ζ

∂ 2v = Mζ , ∂x 2

EIηη

∂ 2w = −Mη , ∂x 2

GJξ ξ

∂φx = Mξ ∂x

What gives rise to the possibility of buckling is that the moments (via the displacement gradients and axial rotation) are deformation dependent. For example, consider the case of a pure bending moment Mo about the z-axis. The resolved local components are M ζ = Mo ,

Mη = φz Mo ,

Mξ =

∂w Mo ∂z

The governing equations now become EIζ ζ

∂ 2v = Mo , ∂x 2

EIηη

∂ 2w + Mo φ x = 0 , ∂x 2

GJξ ξ

∂φx ∂w − Mo =0 ∂x ∂x

We see that the rotation φx and displacement w are coupled as demonstrated in Fig. 4.35b and the coupling comes from the applied load in conjunction with secondorder deformation effects. As another case, consider a cantilevered I-beam with a transverse load applied at its end at the centroid of the cross section. Should the beam rotate (but the load keeps its original orientation), this force no longer acts along the web. The moments at any cross section along the length are then Mx = −Vy [wL − w] ,

My = 0 ,

Mz = Vy [L − x]

where wL is the lateral displacement of the end. Resolving these components to the local coordinate system and dropping nonlinear terms gives

338

4 Buckling Shapes

Mξ = −Vy [wL − w] +

∂w Vy [L − x], ∂x

Mη = φx Vy [L − x],

Mζ = Vy [L − x]

The governing equations for w and φx become ∂ 2w − Vy [L − x]φx = 0 ∂x 2 ∂φx ∂w + Vy [L − x] + Vy [wL − w] = 0 GJξ ξ ∂x ∂x

EIηη

Again, we see the coupling effect. Because of the presence of x in the equations, in general, numerical methods are required to solve these governing equations. Casually, it was just stated that “should the beam rotate . . . ,” the question is why should the beam rotate? A rule of thumb in buckling analyses is that if a soft deformation can occur, it will occur. This is where spectral analysis can be additionally very useful because it identifies possible soft mechanisms independent of any particular type of loading. For example, Fig. 1.29 shows the hierarchy of possible shapes a plate-like beam can have. We see that the out-of-plane shapes (both bending and torsional) are the softest and this would still be true if end constraints (as in the present problem) are imposed. Even if there is no explicit load available to induce the soft mode, nonlinear coupling or (in practice) load imperfections might be sufficient. The FE modelings are general enough to handle the variety of possibilities for causing buckling and therefore it is not essential for us to itemize them here. However, if strong formulations are developed to help understand the FE analysis, it is important to understand the various implementations of the BCs and their influence on the different possible modes. Example 4.16 Figure 4.36 shows the buckling shapes for the beam of Fig. 4.35 with two different BCs. Explain how coupling affects the specification of the BCs (f15.lateral1) The relevant coupled governing equations are EIηη

∂ 2w + Mo φ x = 0 , ∂x 2

GJξ ξ

∂φx ∂w − Mo =0 ∂x ∂x

(4.33)

for the out-of-plane bending w and axial rotation φx . Eliminate φx from the first equation and w from the second to get EIηη

∂ 3w Mo2 ∂w = 0, + GJξ ξ ∂x ∂x 3

GJξ ξ

∂ 2 φx Mo2 + φx = 0 EIηη ∂x 2

The deflection equation has a first integration so that both equations have the same forms, namely,

4.4 Load Coupling of Deformation Modes

339

♦

♦

♦

.

.

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 4.36 Axial rotation during the lateral buckling of a beam; circles are the FE data, continuous lines are model results. (a) Case I: fixed–fixed BC. (b) Case II: fixed–free BC

∂ 2w + α 2 w = constant , ∂x 2

∂ 2 φx + α 2 φz = 0 , ∂x 2

α2 =

Mo2 EIηη GJξ ξ

Consequently, both sets of solutions also have the same forms φx (x) = c1 cos αx + c2 sin αx ,

w(x) = c1∗ cos αx + c2∗ sin αx + c3∗ (4.34)

It appears from the second-order differential equations that w and φx are uncoupled and therefore the problem for both can be posed independently of each other. This is not quite true which we elaborate on presently in the context of the FE implementation. For now, treat the problems as uncoupled and Fig. 4.36 shows the FE generated axial rotations for two possible sets of BCs. Let the rotations at each end be zero, then the characteristic equation gives c2 sin αL = 0

or

αL = nπ

Substituting for α gives the critical moments and shapes as, respectively, Mc =

nπ  EIηη GJξ ξ , L

φx (x) = c2 sin(nπ x/L)

Precisely the same answer is obtained if the zero displacement conditions are imposed. Call these results case I. Noting that for a beam of height H and thickness h (similar to Fig. 1.8a) GJξ ξ = G 13 H h3 , then

1 EIηη = E 12 H h3

340

4 Buckling Shapes

Mc =

√ nπ H h3 √ 2nπ Iηη EG EG = L 6 L

Thus, within the thin plate approximation, it is the small moment of inertia that controls the buckling. That is the reason why thin-walled structures are usually triangulated so as to increase the transverse stiffness. As an alternative BC, let the rotation at x = 0 be zero but at x = L be free. This latter BC implies that ∂φx /∂x = 0. The characteristic equation gives c2 cos αL = 0

or

αL = 12 π, 32 π, . . .

Substituting for α then gives Mc =

n + 12 π  EIηη GJξ ξ , L

φx (x) = c2 sin((n + 12 )π x/L)

Call these results case II. Care must be exercised when implementing the FE boundary conditions to recreate the model situations: while the reduced model deals only with φx (or w), the FE model deals with all DoF. For example, using a shell element with translation and rotation DoF, we have at x = 0 for both cases {u, v, w; φx , φy , φz }T = {0, 0, 0; 0, 1, 0}T Note that φy is unconstrained because φy = −

GJηη ∂φx ∂w = ∂x Mo ∂x

the last relation coming from the governing equation. Because φx is specified as zero, then its derivative must be free. The BCs at x = L for the fixed and free cases are, respectively, {u, v, w; φx , φy , φz }T = {1, 1, 0; 0, 1, 1}T ;

{1, 1, 1; 1, 0, 1}T

The u, v, and φz DoFs are free to allow the applied bending action. For case II, because φx is free, then φy must be fixed. In contrast, the Hex20 element does not have rotations as inherent DoF and therefore rotations must be implemented at the structural level (i.e., across multiple nodes). At x = 0, we specify {u, v, w}T = {0, 0, 0}T along the vertical centerline. This allows u displacement of the other end-face nodes, thus facilitating rotation about the y-axis. At x = L, we specify

4.4 Load Coupling of Deformation Modes

341

Table 4.3 Lateral buckling of a beam. The table gives the value α where Mc = απ 2 EIzz /4L2 for the first two buckling modes Case I II

Model 2.649 1.325

Shell 2.664 1.338

5.299 3.974

case I: {u, v, w}T = {1, 1, 0}T ,

5.373 4.034

Hex20 2.635 1.263

5.307 3.803

case II: {u, v, w}T = {0, 1, 1}T

The case I conditions are for the vertical centerline and the case II conditions are for the horizontal centerline. The case II conditions allow displacement in the y-z plane (and hence rotation about the x-axis) but it restricts rotation about the y-axis. The results in Table 4.3 show that this does not quite faithfully reproduce the conditions of the shell or simplified modelings. Now treat the original differential system of Eq. (4.33) as a coupled system of equations and assume solutions of the form w(x) = wo e−ikx ,

φx (x) = φo e−ikx

Substitute and cancel common terms to get  2 # $ Mo −k EIηη wo =0 ikMo −ikGJξ ξ φo

(4.35)

Setting the determinant to zero gives the characteristic equation and roots as ik 3 EIηη GJξ ξ − ikMo2 = 0

k = 0,

or

 k = ±Mo / EIηη GJξ ξ = ±α

These roots lead to the solutions given in Eq. (4.34) but with the difference that the coefficients ci and ci∗ are related through the amplitude ratio wo /φo obtained from Eq. (4.35). That is, GJξ ξ wo Mo = 2 = φo Mo α EIηη

or

G 1 h3 wo 4D¯ = 3 = , H φo Mo Mo

D¯ = Gh3 /12

These show the coupled nature of the deformation modes. Using the Hex20 data for the fixed–fixed case, we get for the first two modes wo = 4.00 H φo

vs

32 = 4.05; 3λ1

wo = 2.01 H φo

vs

32 = 2.01 3λ2

Note that the appropriate moment to use is the buckling eigenvalue times the applied moment. The results are quite close. For the Hex20 element, H φo is computed as

342

4 Buckling Shapes

the difference of the top and bottom out-of-plane displacements and wo is taken as the displacement of the centerline. Example 4.17 Explain in what sense lateral buckling of a beam is an example of coupled mode interactions. First, the meaning of mode is used in the spectral analysis sense of the shapes we have referred to beginning with Figs. 1.29 and 1.30, that is, shapes that have embedded in them the geometric BCs. In the present case they refer to shapes such as those in Fig. 1.29 but with the ends restrained. These are the zero applied load modes. The interaction effect or question is: does the load cause an interaction of modes? Let us first write the dynamical form of the appropriate equations. Double differentiate the first of Eq. (4.33) so that it is a beam deflection equation with inertia given by qv = −ρAw. ¨ Single differentiate the second of Eq. (4.33) so that it is a ¨ Using the subscript comma shaft rotation equation with inertia given by qφ = ρI φ. notation, the two equations become EI w,zzzz +Mo φ,xx = −ρAw¨ ,

GJ φ,zz −Mo w,xx = ρI φ¨

where A is the cross-sectional area (H h) and I is the rotational moment of inertia 1 hH 3 ). Consider the example case of simply-supported end conditions so that ( 12 ¯ , w = wo sin(nπ x/L) = wo sin nx

φ = φo sin(nπ x/L) = φo sin nx ¯

ˆ o }eiωt , and with both DoF having the harmonic time responses {wo , φ,o } = {wˆ o , φ, we get the EVP  # $ wˆ o −EI n¯ 4 + ρAω2 Mo n¯ 2 =0 Mo n¯ 2 −GJ n¯ 2 + ρI ω2 φˆ o

(4.36)

The characteristic equation is quadratic in ω2 and given by aω4 + bω2 + c = 0 ,

   ω2 = − b ± b2 − 4ac /2a

where a = ρAρI ,

b = −[EIρI n¯ 4 + GJρAn¯ 2 ] ,

c = EI GJ n¯ 6 − Mo2 n¯ 4

Figure 4.37a shows the two computed eigenvalues and their comparison with nonlinear FE results; the comparison is very good. The modes are uncoupled at zero load, that is, the matrix in Eq. (4.36) is diagonal. Figure 4.37b shows that as Mo increases, there is stronger coupling and thus interaction; the mode shape is now a strong mixture of the two uncoupled shapes.

0.6

80.

0.4

60. .

40. .

♦

100.

343

♦

4.4 Load Coupling of Deformation Modes

0.2

20. 0. -20.

.

♦

0.0 1.0 2.0 3.0 4.0 5.0

0.0

.

♦

0.0 1.0 2.0 3.0 4.0 5.0

Fig. 4.37 Lateral buckling of a beam. Circles are nonlinear FE results, continuous lines are model results. (a) Spectral plots of the stiffness. (b) Amplitude ratio for the lowest mode

This is seen in the solution of the characteristic equation by the changing loaddependent term c. In contrast to some of our earlier examples, the lower mode stiffness does not go directly to zero tracing a straight line but has a curve. In either case, the eigenbuckling analysis predicts the correct buckling load. The spectral analysis shows the path to get there. This was also observed in conjunction with Fig. 3.6. A final point to highlight is that the stiffness (frequencies) need not be close to each other in order for there to be an interaction; it is the wavelength of the shapes that is important. The presence of the square root indicates the coupling.

4.4.2 Buckling of Thin-Walled Open Sections Under Axial Loads Consider a long thin-walled member of arbitrary cross section with an axial load acting along the centroid. We make the kinematic assumptions that the motion of any cross section entails a rigid body translation (u, v) and a rotation (φz ) about the shear center. These are sufficient to represent the motion of all points on the cross section and is represented by the rectangle in Fig. 4.38. We identify two significant points in the plane: C is the centroid, and S is the shear center. Because the cross section does not distort, these two points move with the cross section. Assuming small rotations, an arbitrary point (x, y) located in the plane displaces the amount u∗ = u − (y − yS )φz ,

v ∗ = v + (x − xS )φz

where (xS , yS ) is the position of the shear center relative to the centroid. Thus the centroid (originally located at x = 0, y = 0) has the displacements uC = u + yS φz ,

vC = v − xS φz

344

4 Buckling Shapes

Fig. 4.38 Schematic for the translation and rotation of a section

The resultant force moves with the centroid and therefore creates moments (both bending and torsional). We now add these effects to the governing system of Eq. (2.49). We have already considered the energies in Sect. 2.3, here we add the potential of the axial load. Let L be the end shortening, the potential is  V=−

σxx L dA A

The new length of a fiber originally at (x, y) is dl =



1 ∗2 dz2 + du∗2 + dv ∗2 ≈ [1 + 12 u,∗2 z + 2 v,z ] dz

Substitute for the displacements and noting that L = L − l then gives  V = − 12

L

F¯o [u,2z +v,2z +2yS u,z φz ,z +2xS v,z φz ,z +(IS /A)φz2 ,z ] dz

where  IS =

Ixx + Iyy + A[xS2

+ yS2 ] ,

Ixx =

 2

y h ds,

Iyy =

x 2 h ds

 σxx dA. The cross-sectional integrations were performed on the and F¯o = assumption that the axes are principal bending axes. The coupled equations for the distributed load case are then obtained using the variational principle giving  2  2 ∂ 4u ¯o ∂ u + yS ∂ φz = qu − F ∂z4 ∂z2 ∂z2  ∂ 2v ∂ 4v ∂ 2 φz  = qv EIxx 4 − F¯o − x S ∂z ∂z2 ∂z2

EIyy

4.4 Load Coupling of Deformation Modes

345

Table 4.4 Critical loads for an axially compressed I-beam with pinned displacement boundary conditions # 1 2 3 8

Mode Pcy1 Pcx1 Pcy2 Pcφ1

ECw

Shell 252.4 783.9 915.7 1194.

Hex20 256.8 747.1 930.7 1082.

Model I 264.5 923.8 1058. 517.5

% +5 +18 +16 −57

Model II 259.5 807.3 978.7 1304.

 2 2 2  ∂ 4 φz ¯o IS /A] ∂ φz + F¯o xS ∂ v − yS ∂ u = qφ − [GJ + F ∂z4 ∂z2 ∂z2 ∂z2

% +3 +3 +7 +9

(4.37)

The BCs that apply to Eq. (2.49) also apply here. The following example problems show applications of these equations. Example 4.18 Investigate the buckling of an I-beam due to axial loading. Assume the beam has simply-supported BCs. Table 4.4 shows the buckling load factors Pc /Po for an I-beam in axial compression. The I-beam is modeled using the MRT/DKT shell element [5] and has relatively stiff end plates. This model used 80 modules through the length and eight modules in each flange and web. The global FE BCs are set for the centroid of both end plates as {u, v, w; φx , φy , φz }0 = {0, 0, 0; 1, 1, 0}, {u, v, w; φx , φy , φz }L = {0, 0, 1; 1, 1, 0} These correspond to simple supports for the bending action. It is always a good idea to confirm results for shell modeling of open sections by also running a 3D solid analysis in case shear effects are significant. The column labeled “Hex20” shows the results. This model used 20 elements through the length and four elements is each flange and web. Only one element through the thickness was used. The comparisons with the shell element results are acceptable and give us a basis for judging the model results. The shear center is located at the centroid for an I-beam (xS = 0, yS = 0), consequently, the governing equations uncouple to EIyy

∂ 4u ∂ 2u + Po 2 = 0 4 ∂z ∂z

∂ 4v ∂ 2v + P =0 o ∂z4 ∂z2   ∂ 4 φz IS ∂ 2 φz =0 ECw 4 − GJ − Po A ∂z2 ∂z EIxx

346

4 Buckling Shapes

where F¯o = −Po . Also, for an I-beam 1 hH 3 , Iyy ≈ 16 hW 3 , J ≈ 13 [2W + H ]h3 , A ≈ [2W + H ]h, Ixx ≈ 12 W hH 2 + 12 1 and Cw = 24 hH 2 W 3 . The notations are defined in Fig. 2.31. To get some model results, begin by assuming solutions of the form

u(z) = uo sin(nπ z/L) ,

v(z) = vo sin(nπ z/L) ,

φz (z) = φo sin(nπ z/L)

which satisfy the simply-supported BCs. Substitute to get EIyy (nπ/L)2 − Po = 0,   IS ECw (nπ/L)2 + GJ − Po =0 A

EIxx (nπ/L)2 − Po = 0,

The three sets of critical loads are  Pcx = EIxx

nπ L



2

,

Pcy = EIyy

nπ L

2

,

Pcφ =

  nπ A GJ + ECw IS L

2

The first two are, of course, the Euler buckling loads. We call these the model I results in Table 4.4. The comparison for Pcy1 is acceptable, but there is a significant difference for the others with the torsion buckling being especially poor. Because the comparisons between the two FE results are acceptable, the discrepancies for the model I results must lie with the model itself. When dealing with shells, it is not always an obvious matter as to how to impose particular sets of BCs; this was an important consideration in the lateral buckling example. To elaborate, a simply-supported BC for a skeletal beam requires v = 0 and M = 0. For a 3D I-beam modeled with shell elements, is this condition imposed at a single point, the centroid say, or does it apply to the whole cross section? A 3D I-beam modeled with solid elements does not have moments or rotations as part of its formulation making the BC specification even more vague. Additionally, the use of stiff end plates is essential to avoid severe stress concentrations and local deformations at applied load and support locations but this further complicates the nature of the actual BCs. We take the attitude that the FE model is the “real” model, be it implemented with skeletal elements, shell elements, or solid elements and choose the model that best reflects its behavior. That is, more or less, we do not try to force the 3D model to conform to the elementary assumptions but try to force the elementary model to conform to the 3D model. Figure 4.39 shows the behavior of the centroidal line for the four buckling modes given in Table 4.4. The bending mode shapes have a definite sinusoidal aspect to them, thus justifying the initial assumption of model I. The twist distribution, however, is closer to the form

4.4 Load Coupling of Deformation Modes

0.0

0.2

0.4

0.6

0.8

347

1.0

Fig. 4.39 FE generated data for the deformation distributions for the centerline of an axially compressed I-beam with pinned displacement boundary conditions

φz (z) = φo sin2 (nπ z/L) = 12 φo [1 − cos(2π z/L)] When substituted into the torsion buckling equation, this gives   IS ECw (2π/L)4 + GJ −Po (2π/L)2 = 0 or A

Pcφ =

  2π A GJ +ECw IS L

2

The warping contribution is enhanced. This is shown in the table labeled as model II. There is a significant improvement in the estimated critical load. More discussion of the restrained warping effect at boundaries is given in connection with Figs. 2.39 and 2.40. We still would prefer to have improved results for the bending buckling modes. Because the buckling shapes, as described by the distributions shown in Fig. 4.39, are as assumed, we must conclude that the reduced model is missing some aspect that is present in both FE analyses. The model overestimates the critical load in each case suggesting either the elastic stiffness is overestimated or the geometric stiffness is underestimated. As discussed in Sect. 2.3 and elaborated on in Fig. 2.33, deep beams exhibit extra deformation (are less stiff) due to the effect of the shear stress. That is, the Bernoulli–Euler beam model overestimates the stiffness. We argue that the bending in effect in the buckling problem is essentially the same as a distributed load, and thus governed by Eq. (2.38), but using the shear dominant areas—Aw = Ayy = 2W h for Pcy1 and Pcy2 , Aw = Axx = H h for Pcx2 . The derivation of the appropriate deep-beam governing equations is as already done in Sect. 2.3 but we add

348

4 Buckling Shapes

 UG =

1 2

 2 ∂v dz F¯o ∂z

to the strain energy. The resulting equations for bending about the x-axis are      ∂v ∂ ¯ ∂v ∂ GAxx − φx + =0 Fo ∂z ∂z ∂z ∂z     ∂ ∂v ∂φx EIxx + GAxx − φx = 0 ∂z ∂z ∂z

(4.38)

and only the shear BC is affected. The equations for bending about the y-axis are similar, just permute the subscripts. Based on the distributions of Fig. 4.39, it is reasonable to assume a buckling shape of the form v(z) = vo sin(nπ z/L) ,

φx (z) = φo sin(nπ z/L)

which satisfy the simply-supported BCs. Substituting into the governing equations leads to the EVP # $  −GAxx vo GAxx (nπ/L) + F¯o (nπ/L) =0 −GAxx (nπ/L) EIxx (nπ/L)2 + GAxx φo Let Pcx = EIxx (nπ/L)2 and Ps = GAxx , and setting the determinant to zero leads to Ps Pcx + Ps2 + F¯o Pcx + F¯o Ps − Ps2 = 0

or

− F¯o =

Pcx 1 + Pcx /Ps

This gives a very simple correction to the original estimate of buckling load. The corrected estimates are shown as the model II results in Table 4.4. There is a definite improvement. What is worth noting is that because Ps does not change with bending mode number, then the correction Pc /Ps becomes more significant at the higher modes. This is reflected in the improvement of Pcy2 . The significant improvement in Pcx1 is because the shearing area is just the vertical web. A final point is that the torsional constant contributes to increasing the torsional buckling load because ECw contributes a positive stiffness. Example 4.19 Figure 4.40 shows the FE generated results for the first and ninth buckling shapes for a C-channel with nominal simply-supported end conditions and an axial load acting through the centroid. The intermediate buckling shapes correspond to local wrinkling of the flanges. The channel has rigid end plates (not shown). It is not obvious from Fig. 4.40 (but made explicit in Fig. 4.42a), that the first deformed shape has both a vertical deflection as well as a rotation of the middle section. That is, it has coupled bending–twisting behavior. The second deformed

4.4 Load Coupling of Deformation Modes

349

♦

♦

Fig. 4.40 Exploded view of the buckling shapes for an axially compressed C-channel of length L = Lo /2

♦

0.

100. 200. 300. 400. 500. .

♦

0.

200.

.

400.

600.

Fig. 4.41 Spectral plots for an axially compressed C-channel. Circles are nonlinear FE data, light lines are extensions of the low load data. (a) Length L = Lo /2. (b) Length L = Lo /4

shape is a more classic Euler buckling shape for a beam with pinned end conditions. The less stiff axis of the C-channel is about the y-axis, yet that is not the global mode that buckles first; construct a model to explain these results. Let us begin by doing a spectral survey. Figure 4.41a shows the behavior of the first 11 modes for L = Lo /2. The continuous lines are obtained from a linear vibration eigenanalysis at zero and a nonzero load and then extended as required. The initially two lowest modes are the global modes indicated with the full circles. They neither intersect nor interact. The other modes are mostly the wrinkling (or local plate) modes. Some of the initially high modes have very steep slopes and therefore intersect the lower modes. As a consequence, their critical load value is less than that of the second global mode. The first mode goes directly to zero and therefore keeps its initial spectral properties. As shown at the end of Sect. 2.4, the spectral shapes for a C-channel has coupled bending–torsion deformations as illustrated in Fig. 2.58. Thus this is a problem with coupled deformations (bending and twisting) but not coupled modes. The coupling with the torsion reduces the overall structural stiffness and for that

350

4 Buckling Shapes

reason the lowest buckling mode has bending about the stiffer axis. We develop a model from a buckling perspective that also illustrates this. A utility of Fig. 4.41a is that it shows we can develop a lowest mode buckling model without considering the other modes. Figure 4.41b, on the other hand, shows a case where one of the higher modes becomes critical with likely interactions with the lower modes. The next chapter takes up the very difficult problem associated with the wrinkling modes intersecting the global modes. First we establish the geometric parameters of the problem. The centroid and shear center locations, based on simple modelings, are given by xC =

W2 , 2W + H

e=−

3W 2 6W + H

respectively, measured from the web along the line of symmetry. As discussed in relation to the FE results of Fig. 2.32, the actual shear center is expected to be slightly different than this if Hex20 modeling is used, here the MRT/DKT shell element [5] is used. We also have Ixx ≈ 12 W H 2 h +

Iyy ≈ 13 W 3 h ,

3 1 12 H h ,

Cw =

H 2 W 3 [3W + 2H ]h 12[6W + H ]

The area moments are given by A = [2W + H ]h,

J = 13 [2W + H ]h3 ,

IS = Ixx + Iyy + A[xS2 + yS2 ]

with xS = |e| + |xC |, yS = 0. For the C-channel with yS = 0, the governing system of Eqs. (4.37) (with F¯o = −Po ) simplify to ∂ 4u ∂ 2u + P =0 o ∂z4 ∂z2  ∂ 2v ∂ 4v ∂ 2 φz  − xS 2 = 0 EIxx 4 + Po 2 ∂z ∂z ∂z EIyy

ECw

∂ 4 φz ∂ 2v ∂ 2 φz − [GJ − P I /A] − P x =0 o S o S ∂z4 ∂z2 ∂z2

The first is uncoupled and behaves similar to the I-beam of the previous example. That is, it has a buckling shape given by u(z) = uo sin(nπ z/L) which leads to

4.4 Load Coupling of Deformation Modes

351

 EIyy (nπ/L)4 − Po (nπ/L)2 ]uo = 0

 or

Pcy = EIyy

nπ L

2

This gives an eigenvalue of λ = 526.4 for L = Lo /2 (Fig. 4.41) which is high relative to the FE result for the reasons discussed for the I-beam. We now concentrate on the coupled mode. The buckling position of the cross section located at z = L/2 is shown in Fig. 4.42a as the full circles; it is clear that there is both a translation and a rotation. The average vertical deflection and rotation are, respectively, va = −0.0822 ,

φa = −0.0187 ,

ADev(φ) = 0.00248

(Remember that buckling shapes do not have an absolute scale.) That the average deviation (ADev) [9] is nonzero means there is a distortion of the cross section. We assume, nonetheless, that the average displacement coincides with the displacement of the centroid and therefore the displacement of the shear center can be computed as v = va − φa [|e| + xC ] = −0.0250 This, in turn, allows us to compute the position of each node according to ui = −φz [yio − yS ],

vi = v + φa [|e| + xio ];

xi = xio + χ ui ,

yi = yio + χ vi

♦

♦

where χ is the same scale factor used for the FE results. These model results are shown on the plot as empty squares. There is good agreement which reinforces our conclusion that this buckling mode couples the vertical deflection and the axial rotation.

♦

♦

0.0 0.2 0.4 0.6 0.8 1.0 .

0.0 0.2 0.4 0.6 0.8 1.0 .

Fig. 4.42 First buckling mode of an axially compressed C-channel. Symbols are FE data, continuous lines are model results. (a) Rotation of center cross section. (b) Axial distribution of displacements along the web centerline. (c) Axial distribution of rotations along the web centerline. Circles are L = Lo /2, squares are L = 2Lo

352

4 Buckling Shapes

Figure 4.42b and c shows sampled vertical deflection and rotation distributions along the web centerline for two lengths. The distributions are bracketed by the functions model I:

sin(π z/L);

model II:

1 2 [1 − cos(2π z/L)]

Neither model matches the data well, but the displacements are close to model I while the rotations are close to model II. The reason the rotations are not of the sine form is explained in the discussions associated with Figs. 2.38 and 2.44. The different models means we cannot use them directly to satisfy the governing equations. Instead, we must use them as Ritz functions in conjunction with energy expressions. The relevant energy from Sect. 2.3 and potential from earlier are, respectively,  2 2  2 2  2 2    ∂ u ∂ v ∂ φz U= EIyy dz + EIxx + ECw 2 2 ∂z ∂z ∂z2 L  1 V = −2 F¯o [v,2z +2xS v,z φz ,z +(IS /A)φz2 ,z ] dz 1 2

L

Let the deformation be given by v(z) = vo sin(π z/L) = vo sin(n¯ 1 z);

φz (z) = φo [1 − cos(n¯ 2 z)]/2

where n¯ 1 = π/L and n¯ 2 = 2π/L. Computing the integrals is straightforward but use must be made of   4L 2L cos(1π z/L) sin(2π z/L) dz = + , sin(1π z/L) cos(2π z/L) dz = − 3π 3π L L Introduce the critical loads Pcx = EIxx n¯ 21 , Pcφ =

A [GJ + ECw n¯ 22 ] IS

which correspond to the uncoupled buckling cases. Then after some simplification, the coupled equilibrium equations reduce to 

Po − Pcx −xS Po α −xS Po A/Is α Po − Pcφ

#

$ vo = 0, φo

α=

8 3π

This makes it clear how the coupling is due to the moment arm associated with the shear center relative to the centroid. The critical load and amplitude ratio are obtained by setting the determinant to zero

4.4 Load Coupling of Deformation Modes

353

φo Pc − Pcx = vo xS Pc α

[1 − xS2 A/IS ]Pc2 − [Pcx + Pcφ ]Pc + Pcx Pcφ = 0 , Solving as a quadratic equation gives for the case of L = Lo Pcφ = 178, Pcx = 230,

λ1 = 110, λ2 = 1074;

FE:

λ1 = 115, λ50 = 1241

The most significant result here is that the critical load is much smaller than for either of the uncoupled modes. To reiterate, coupling tends to reduce the lowest principal value of stiffness. It is possible to explore in more detail the exact nature of the effective BCs in the FE modeling and construct a model that reflects these. Such a model would entail using more Ritz functions. This is not done here because the model in its present form achieved its main purpose which is to show that when coupling occurs, the consequent buckling load can be less than that predicted by the separated modes. It is reiterated, however, that the same insight is obtained by taking a spectral perspective on the stiffness behaviors.

Explorations 4.1 Consider the buckling of a structure with a cruciform cross section. • Use the results of Fig. 4.21 to estimate the buckling load. • Compare the results with an FE analysis. —Reference [11, pp. 225] 4.2 Obtain a strong-form solution for the problem in Fig. 4.6. • Compare with the Ritz solution. • Do an FE analysis and compare with both solutions. 4.3 The C-channel of Sect. 2.4 can be adapted to buckling problems by adding the geometric energy contribution. • Use the same interpolation functions to derive the geometric stiffness matrix. • Compare the performance to a standard FE analysis using either shell elements or solid elements. • Devise difference situations to investigate the different types of BCs.

354

4 Buckling Shapes

References 1. Bushnell, D.: Buckling of shells – pitfall for designers. AIAA J. 19(9), 1183–1226 (1981) 2. Donnell, L.H.: A new theory for the buckling of thin cylinders under axial compression and bending. ASME Aeronautical Eng. 56, 795–806 (1934) 3. Doyle, J.F.: Wave Propagation in Structures. Springer, New York (1989). 2/E 1997 4. Doyle, J.F.: Static and Dynamic Analysis of Structures. Kluwer, The Netherlands (1991) 5. Doyle, J.F.: Nonlinear Analysis of Thin-walled Structures: Statics, Dynamics, and Stability. Springer, New York (2001) 6. Doyle, J.F.: Nonlinear Structural Dynamics using FE Methods. Cambridge University Press, Cambridge (2015) 7. Hoff, N.J.: The Analysis of Structures. Wiley, New York (1956) 8. Mandal, P., Calladine, C.R.: Buckling of thin cylindrical shells under axial compression. Int. J. Solids Struct. 37, 4509–4525 (2000) 9. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes. Cambridge University Press, Cambridge (1986). 2/E 1992 10. Schmidt, H.: Stability of steel shell structures: general report. Comput. Constructional Steel Res. 55, 159–181 (2000) 11. Timoshenko, S.P., Gere, J.M.: Theory of Elastic Stability. McGraw-Hill, New York (1963) 12. Ventzel, E., Krauthammer, T.: Plates and Shells. CDC Press, New York (2001) 13. Wadee, M.A., Gardner, L.: Cellular buckling from mode interaction in i-beams under uniform bending. Proc. R. Soc. A 455 (2011). https://doi.org/10.1098/rspa.2011.0400 14. Yang, T.Y.: Finite Element Structural Analysis. Prentice-Hall, Englewood Cliffs (1986)

Chapter 5

Studies of Postbuckled Shapes

It can reasonably be said that an analysis identifying the Euler critical load of a perfect structure but does not identify the postbuckle behavior as stable or unstable could lead to catastrophic consequences due to imperfections. Therefore, for an actual complex structure, stability analyses should always require a fully nonlinear analysis in addition to the linear eigenbuckling analyses. These considerations have serious implication for the postbuckle behavior of shells compared to plates. We show, for example, that plates are most often postbuckle stable, while shells (and curved plates) are most often postbuckle unstable. We conclude this book with an introduction to postbuckled shapes. Postbuckling behavior is a large deformations problem initiated by a singularity and this singularity helps delineate between the pre- and post-phases. A buckling analysis can identify the region of this singular point; but the buckling analyses simplified the problem by assuming the membrane stress (axial force) is known a priori from a separate linear analysis. Here, we must determine the stress as an integral part of the nonlinear transition to the postbuckled state. This complicates the analyses considerably and we try to emphasize the distinction between this analysis and the buckling analyses. For background material, mention should be made of Refs. [21, 28] which place stability in a broad context and Ref. [8] that considers many dynamic effects.

5.1 Postbuckling of Beams and Arches The essential deformations for the buckling of a beam is an axial compression and an out-of-plane flexure. Because the buckling process is inherently nonlinear, these actions are coupled. The coupling for arches is more complicated in that the deformations are inherently coupled and that the loading deformation may or may

© Springer Nature Switzerland AG 2020 J. F. Doyle, Spectral Analysis of Nonlinear Elastic Shapes, https://doi.org/10.1007/978-3-030-59494-7_6

355

356

5 Studies of Postbuckled Shapes

not be congruent with the buckling shape. We explore some relatively simple cases so as to get a sense of the transition to the postbuckle state.

5.1.1 Beam Under Axial Loading Figure 5.1 shows some FE generated results for the postbuckling behavior of a thin beam with pinned BCs. Observe that the end shortening is negligible before buckling and almost linear with load after buckling. The transverse deflection is also negligible before buckling and appears quadratic or higher order after, that is, it has zero initial slope. The plot of u against v has the important implication that these two DoF are not independent of each other and it should be possible to write one as a function of the other, thus reducing the problem to a single unknown. We construct a model that achieves this. An elastica is a slender member that supports both axial and bending loads; however, it does not experience any axial stretching. Some exact solutions can be found in Refs. [6, 14] which are based on Euler’s original analysis as discussed in the Introduction. We adopt the elastica as our first beam model because it lends itself to convenient approximation. Although there are no axial strains in the elastica model, there are axial forces, and these are obtained through equilibrium equations and not through elastic relations. We choose loading situations that make this aspect of the problem relatively simple. Consider the plane deflection of the elastica shown in Fig. 5.2a. Some of the geometrical descriptions were already covered in Sect. 3.2 but are repeated here for completeness. Let s be the distance of a material point along the

1.0

.

0.5

0.0 0.0

0.2 .

0.4

Fig. 5.1 Postbuckling behavior of a beam with pinned BCs. (a) FE generated deformed shapes at different stages of the loading after buckling. (b) Transverse deflections (v) and end shortening (u) against load. Circles are the FE results, lines are the model results, and the plots are shifted for clarity

5.1 Postbuckling of Beams and Arches

357

Fig. 5.2 An elastica with tip loads. (a) Notations. (b) Shape and second derivative (plotted with respect to original position). Circles are FE data, continuous lines are model results

elastica; the point always has the same value of s because the elastica is assumed to be inextensible. That is, a point originally at position x o = s, y o = 0, moves to a location (x, y) where its distance along the curve is s. Hence, the two displacements are given by u = x − xo = x − s ,

v = y − yo = y − 0

In addition, we have that dx = cos φ , ds

dy = sin φ ds

where φ is the slope. We can put these in differential form as u,s =

dx du = − 1 = cos φ − 1 , ds ds

v,s =

dy dv = = sin φ ds ds

where the subscript comma indicates differentiation. It is clear that these equations satisfy the inextensibility constraint [1+u,s ]2 +v,2s = 1. Furthermore, this says that the transverse and axial deflections are related to each other. We use this connection to develop some approximate forms for the elastica equations. The total potential for a beam in uniaxial stress with tip loads Px and Py is 

=

1 2

  2 EEss dA ds − Px u(s=L) − Py v (s=L)

The large deflection strain in the beam is   Ess = u,s + 12 u,2s +v,2s − yφ,s = 0 − yφ,s ,

2 Ess = y 2 φ,2s

Substitute into the total potential and integrate on the cross section to get

358

5 Studies of Postbuckled Shapes



=

1 2

 EI φ,2s ds − Px

u,s ds − Py vL

Note that the end shortening needs to be calculated as the integral of u,s over the length of the beam. Some additional elastic details are presented in Sect. 3.2 dealing with flexure in beams. Rather than work with the exact equations, it is instructive to consider some approximate versions. For example, we have the important geometric approximation that  1 u,s = 1 − v,2s − 1 ≈ 0 − 12 v,2s − 18 v,4s − 16 v,6s − · · · When truncated after the first nonzero term, this is then the second-order approximation that relates axial displacements (end shortening) to transverse displacements that is fundamental in the beam buckling analyses of Chap. 4. The presence of the zero is a reminder that the dominant term of the approximation is a second-order term. So as to introduce the familiar v,ss term associated with beam bending, we also have the approximation φ,s =

1 d 2v dφ 1 = = v,ss , 2 ds cos φ ds 1 − v,2s

  φ,2s ≈ v,2ss 1 + v,2s +v,4s + · · ·

Substitute this into the potential and replace u,s with its approximation to get 

=

1 2

−Px

  EI v,2ss 1 + v,2s +v,4s + · · · ds    1 v,6s − · · · ds − Py vL 0 − 12 v,2s − 18 v,4s − 16

A similar expression for the total potential can be found in Ref. [19]. Of interest to us is that the flexural behavior has a second- order contribution and so too the axial load as expected from the buckling analyses. In this way it is seen that the large deflection of a beam can be reduced to determining a single function v(s). Furthermore, it depends only on the flexural stiffness EI and does not depend on the axial stiffness EA. Figure 5.1 shows the FE generated results for the postbuckling behavior of a thin beam with pinned BCs. Observe that the end shortening (after buckling) is almost linear with load while the transverse deflection appears quadratic or higher order, that is, it has zero initial slope. The plot of u against v has the important implication that these two DoF are not independent of each other as utilized in our analysis. For the pinned–pinned beam it is reasonable to assume that the deflected (postbuckled) shape is described by v(s) = v1 sin(nπ s/L) with n = 1; this satisfies the kinematical end conditions of zero transverse displacement. This function and its second derivative are compared to the FE results in Fig. 5.2b; the comparison is

5.1 Postbuckling of Beams and Arches

359

quite good. Substitute into the total potential, set Px = −P , vL = 0, and integrate over the length L to get   1 6 v1 + · · · L

= 12 EI n¯ 4 12 v12 + n¯ 21 81 v14 + n¯ 41 16   1 4 5 4 −P n¯ 2 0 + 14 n¯ 1 v12 + 18 n¯ 21 83 v14 + 16 n¯ 1 16 v1 + · · · L where n¯ 1 = 1π/L. Again, the presence of the zero in the load potential is a reminder of the order of the terms retained, thus second order in the load potential is matched with first order in the strain potential. Equilibrium is given by    ∂

= EI n¯ 41 1 + n¯ 21 21 v12 + n¯ 41 83 v14 + · · · ∂v1   15 4 −P n¯ 21 1 + n¯ 21 83 v12 + n¯ 41 64 v1 + · · · 12 Lv1 = 0 This has two equilibrium paths: the first corresponds to v1 = 0 where there is no transverse displacement, the second is when the braced term is zero and gives a nonzero transverse displacement. To the lowest order, this second path gives EI n¯ 41 − P n¯ 21 = 0

or

P = EI

 nπ 2 L

= Pc

which is the Euler buckling load. That is, the lowest order approximation corresponds to a buckling analysis which identifies the critical load but gives no information about the postbuckle state. It is indicated by the fine horizontal lines in Fig. 5.1b. To the next order terms, we have     EI n¯ 41 1 + n¯ 21 21 v12 − P n¯ 21 1 + n¯ 21 83 v12 = 0

or

P = Pc

1 + n¯ 21 21 v12 1 + n¯ 21 83 v12

This is indicated by the dashed lines in Fig. 5.1b. Some information is given about the postbuckle state (such that it is stable) but the accuracy is not very good. However, it does have the interesting further approximation that   P ≈ Pc 1 + n¯ 2 81 v12 giving the displacements . v1 =

√ P − Pc 2 2L , Pc nπ

uL = − 14

 nπ 2 L

v12 L =

P − Pc 1 L Pc 2

(5.1)

360

5 Studies of Postbuckled Shapes

Note that the end shortening is linear with load. As a heuristic model, these indicate that the deformations in the postbuckle state depend on the excess of applied force over that necessary to cause the buckling. The full heavy line in Fig. 5.1b is our highest approximation given by P = Pc

1 + n¯ 21 21 v12 + 38 n¯ 41 v14 1 + n¯ 21 83 v12 +

15 4 4 ¯ 1 v1 64 n

uL = −

,



1 2 2 ¯ 1 v1 4n

+

3 4 4 ¯ 1 v1 64 n

+

5 ¯ 41 v14 4096 n

 L

This model seems to catch the main postbuckle features. The constraint relationship between u and v is also nicely captured as seen by the plot in the lower right portion of the figure. While the elastica solution gives good modeling of the postbuckle deformations, it does not give any insights into the singular behavior during the buckling. The next example problem delves into this aspect. Example 5.1 The beam in Fig. 3.26 is given a large compressive Po and a small Qo , and Fig. 5.3 shows the displacement results. In contrast to the elastica results, there is definite end shortening before the onset of gross postbuckling deformation. Construct a model to explore this stage of the deformation. We begin by simplifying the nonlinear beam equations to conditions of moderately large deflections and rotations. Assume that the dominant nonlinear contribution comes from the transverse deflection v so that Eqs. (3.19) and (3.21) reduce to   2  2 1 1 UM = 2 EAExx dx = 2 EA u,x + 12 v,2x dx  =

1 2

UF =

1 2

  EA u,2x +u,x v,2x + 14 v,4x dx



 1 2

EI v,2xx dx

♦

♦

2 dx = EI κxx

♦

♦

-0.5

0.0 .

0.5

-40. -20. 0. .

20.

40.

Fig. 5.3 Transverse deflection and end shortening for an axially loaded pinned–pinned beam. Circles are FE generated data. (a) Expanded scale. Lines are model results. (b) Full scale

5.1 Postbuckling of Beams and Arches

361

In contrast to the elastica formulation, we allow axial extension/contraction as represented by EA. The nonlinearity is only in the membrane term but it couples both actions. To illustrate the mechanics involved, we use a Ritz approach. Based on the separate load cases, a reasonable assumption for the deflected shape is that u(x) =

u1 x, L

v(x) = v1 sin n¯ 1 x ,

n¯ 1 = π/L

This satisfies the geometric BCs for a simply-supported beam and for a rod fixed at one end. Substitute into the energy expressions and integrate to get 3 UM =

1 2 EA

u21 u1 + v12 n¯ 21 21 + 14 v14 n¯ 41 83 L L2

4 L,

& % UF = 12 EI v12 n¯ 41 21 L

The potential of the applied loads is V = −Po u(L) − Qo v(L/2) = −Po u1 − Qo v1 There are two equilibrium equations given by $ # u1 1 2 21 ∂

= EA + v1 n¯ 1 4 L − Po = 0 ∂u1 L L2     u1 ∂

v1 n¯ 21 21 + 12 v13 n¯ 41 83 L + EI v1 n¯ 41 21 L − Qo = 0 = EA ∂v1 L It is clear that equilibrium is governed by a set of coupled nonlinear equations. The structural stiffness matrix is ) [KT I J ] =

∂ 2

∂uI ∂uJ

* =

    EI 0 0 EA 1 v1 n¯ 21 21 L + 9 2 L L v1 n¯ 21 21 L u1 n¯ 21 21 L + v12 n¯ 41 16 L3 0 n¯ 41 21 L4

The flexural term is the linear transverse stiffness of a simply-supported beam and the K11 term is the linear stiffness of a rod. The other terms depend on the deformation and are therefore nonlinear contributors. Of particular note is the K22 membrane term: because it contains u1 then this term can decrease causing buckling. The off-diagonal terms are the coupling of the deformations which increases with increasing transverse deformation. Consider Qo = 0 in the equilibrium equations, then  1 2 21 u1 + v1 n¯ 1 2 = Po , EA L L 

One equilibrium path is

 EA

u

1 2 n¯ 1

L

  + 38 v12 n¯ 41 + EI [n¯ 41 ] v1 = 0

362

5 Studies of Postbuckled Shapes

I:

v1 = 0 ,

u1 = Po L/EA

This corresponds to an indefinite vertical line emanating from the origin in Fig. 5.3c. Note that for this path KT 22 = [EAu1 /L + EI n¯ 21 ] 12 Ln¯ 21 which can go to zero if u1 (or Fo = EAu1 /L) is sufficiently negative. If Po > EI n¯ 21 = Pc , transverse displacements are more likely to occur and this is what the plot shows. For the equilibrium path with v1 = 0, the two equilibrium equations can be rewritten as u1 /L + 14 v12 n¯ 21 = −Po /EA ,

u1 /L + 38 v12 n¯ 21 + Pc /EA = 0

Subtract these to get ' II:

v1 =

√ Po − Pc 8 , EA n¯ 1

u1 + Po L/EA = (Po − Pc )2L/EA

This path exists only for Po > Pc , again as shown in the plot. The v1 relation resembles that of Eq. (5.1) except that the denominator has EA instead of Pc . Because these two quantities are different by orders of magnitude it means our result is overly stiff and this is reflected by the fine dashed line in Fig. 5.3a. There must be an additional softening mechanism generated during the transition to the postbuckled state which we now explore. The FE data were generated using a load imperfection of Qo = Fo /1000. This is small but sufficient to give a noticeable transverse deflection before buckling as seen in Fig. 5.3a; it is not noticeable in the full scale plot of Fig. 5.3b. Figure 5.4a shows some FE generated data for the shapes during the buckling transition. The shape for v(x) hardly changes even though its magnitude does. A significant change of shape for u(x) occurs even at the critical load and Fig. 5.3a shows that this is accompanied by a large end shortening. A precursor for this was given by the spectral analysis in Fig. 3.27a. The axial strain distribution while initially uniform becomes sinusoidal during and after buckling. These data suggest that softening mechanism is generated during buckling. A first guess at adding softening to the transverse deflection would be to add an extra v3 sin 3π x/L shape. This does increase the displacement but only as a small fraction of v1 as is evident by its absence in Fig. 5.4a for v(x). We conclude that an altogether different softening mechanism must be in play. Based on the FE data for u(x), a reasonable assumption for the deflected shape is that u(x) = u1 x/L + u2 sin n¯ 2 x ,

v(x) = v1 sin n¯ 1 x ,

n¯ 1 = π/L, n¯ 2 = 2π/L

5.1 Postbuckling of Beams and Arches

363

♦ ♦

♦

♦

0.0

0.2

0.4

.

0.6

0.8

1.0

0.

1.

2.

Fig. 5.4 FE generated data for the large deflections of an axially loaded pinned–pinned beam. (a) Displacement and strain distributions. Circles, squares, and triangles are at 80%, 100%, and 120% of the the critical load, respectively. The scales on the displacements are arbitrary, chosen for easier comparison of shapes. (b) Spectral plots, scaled and shifted. Full lines are model III results, dashed line is model II results

This satisfies the geometric BCs for a simply-supported beam and for a rod fixed at one end. The axial strain is xx (x) = u1 /L + u2 n¯ 2 cos n¯ 2 x which seems to reflect the distribution in the FE data. Substitute into the full energy expressions and integrate to get 3

u21 u1 UM = + u22 n¯ 22 21 + 14 v14 n¯ 41 83 + v12 n¯ 21 21 + u2 v12 n¯ 2 n¯ 21 41 2 L L   UF = 12 EI v12 n¯ 41 21 L 1 2 EA

where use was made of  cos n¯ 2 cos2 n¯ 1 x dx = L

1 4

L

The potential of the applied loads is V = −Fo u(L) − Qo v(L/2) = −Fo u1 − Qo v1 ⇒ Pu1 = Fo ,

Pu2 = 0,

Pv1 = Qo

4 L,

364

5 Studies of Postbuckled Shapes

There is no contribution associated with u2 ; we could add an imperfection load associated with this DoF, but as shown presently, it is tightly coupled to the v1 DoF which in turn is manifested through the imperfection Qo . There are three equilibrium equations given by $ # u1 1 2 21 ∂

L − Fo = 0 v = EA + n ¯ ∂u1 L 1 14 L2   ∂

= EA u2 n¯ 22 21 + 12 v12 n¯ 2 n¯ 21 41 L = 0 ∂u2    u ∂

1 v1 n¯ 21 21 + u2 v1 n¯ 2 n¯ 21 41 + 12 v13 n¯ 41 83 L + EI v1 n¯ 41 21 L − Qo = 0 = EA ∂v1 L The structural stiffness matrix is obtained as ⎤ ⎡ 1 0 v1 n¯ 21 21 L EA ⎣ ⎦ [KT I J ] = v1 n¯ 2 n¯ 21 41 L2 0 n¯ 22 21 L2 L 1 1 2 1 1 2 9 2 2 2 2 2 2 4 v1 n¯ 1 2 L v1 n¯ 2 n¯ 1 4 L u1 n¯ 1 2 L + u2 n¯ 2 n¯ 1 4 L + v1 n¯ 1 16 L ⎡ ⎤ 00 0 EI + 3 ⎣0 0 0 ⎦ L 0 0 n¯ 41 21 L4 with the DoF arranged as {u1 , u2 , v1 }T . The new membrane term shows a growing coupling between u2 and v1 as v1 increases. We get from the second equilibrium equation that u2 = − 14 v1 n¯ 21 /n¯ 2 which modifies the third equilibrium equation (with Qo = 0 and v1 = 0) to EA

u

1 21 n¯ 1 2

L

−

1 2 4 ¯1 16 v1 n

+

3 2 4 ¯1 16 v1 n



  + EI n¯ 41 21 = 0

The negative term is the softening. The results generated by this model are shown as the continuous line in Fig. 5.3a; the agreement with the FE results is quite good. It should be said, however, that the postbuckle slope of the P versus v curve is so shallow that small changes in the model could affect the predictions significantly. We use a spectral analysis to illustrate this. The kinetic energy is given by  T =

1 2

% & ρA[u˙ 2 + v˙ 2 ] dx = 12 ρA u˙ 21 31 + u˙ 22 21 + v˙12 21

5.1 Postbuckling of Beams and Arches

365

where, as an approximation, we neglect the mass coupling contributions. The diagonal masses are, therefore, clear so that we have the initial (unloaded) stiffnesses ωv2 =

EI 4 n¯ , ρA 1

2 ωu1 =

EA 3 , ρA L2

2 ωu2 =

EA 2 n¯ ρA 2

These are ordered as ωv < ωu1 < ωu2 with ωv being substantially smaller than the other two. Figure 5.4b shows some results for the current model labeled as model III. The most striking difference with the previous model is the reduced stiffness for loads greater than Pc .

5.1.2 Arches and Effects of Boundary Constraints An accurate model for the large deflection of an arch under point loading was developed in connection with Fig. 3.19. For the most part it is implicit and therefore not of good heuristic value. We now develop an approximate solution more amenable to interpretation. Let the mechanical model discussed in connection with Fig. 3.33b be our initial conception of the postbuckling behavior of an arch under distributed radial loading. That is, the arch is “squished” in compression through to the other side in a symmetric manner. Take the second of Eqs. (2.6), modified for geometric stiffness contributions, as our basic model. We write this as EI v,ssss −F¯o v,ss −F¯o /R = qv The u,s flexural contribution is ignored, and based on data such as Fig. 3.18, the axial force being sensibly constant with respect to s allows replacing the EA term with F¯o /R. This is the same as Eq. (3.31). At this stage the governing equation looks like the strong formulation for the buckling of arches covered in Sect. 4.1. However, because of the later nonlinear manipulations, we do not want to deal with differential equations and to that end we assume solutions of the form v(s) = vn sin(nπ s/a) where n = 1, 3, . . . are the symmetric shapes. On substituting this we conclude that it is not a solution (not all terms have a common sine term), but we can invoke the Galerkin approximate approach [2] of working with a weighted integral of the differential equation; it has commonalities with the Ritz method but is most useful when the governing DEs have already been established. That is, substitute the assumed solution and weight with respect to sin ns ¯ to impose 

a

%

& EI n¯ 4 vn sin ns ¯ ds = 0 ¯ + F¯o n¯ 2 vn sin ns ¯ − F¯o /R − qv sin ns

0

where n¯ = nπ/a. Performing the integrations and dividing by 2/n, ¯ we get

366

5 Studies of Postbuckled Shapes

EI n¯ 5 41 avn + 14 F¯o n¯ 3 avn − F¯o /R = qv If we set vn = 0, then the remaining equation is the same relation between hoop stress and pressure established for cylinders in Sect. 2.2. On the other hand, if those two terms are set equal (and cancel), then the remaining equation is an EVP, the simplest of solutions of which are F¯o = −EI n¯ 2 . These are the Euler buckling loads and correspond with an FE analysis as Model FE

0.65 –

2.62 2.57

5.89 5.72

10.5 10.3

16.3 16.0

None of this good agreement should be too surprising because we began with the linear equilibrium equation modified for geometric stiffness effects and we already know that the buckling analysis gives reasonable results. The conception we have is that the deformed shape v(s) = vn sin(nπ s/a) is being imposed, then the nonlinearities are associated with the axial load-deflection relation. Based on an approximation of Eq. (3.25), we can write the hoop strain and averaged axial force as, respectively, 1 F¯ = EA a

Ess ≈ u,s −v/R + 12 v,2s ,

 [u,s −v/R + 12 v,2s ] ds

As utilized in Sect. 3.3, because the u(s) displacement is zero at each end, the first term in the force expression (when integrated) is zero and we have the approximation after integration of   2 1 2 2 ¯ F = EA − vn + 4 n¯ vn R na ¯ This force has a peak in the vicinity of where the arch becomes flat. Substitute the force into the equilibrium equation to get 

1 ¯ 5a 16 EAn



vn3 −



3 ¯ 2 /R 4 EAn



  vn2 + EI [n¯ 5 /4]a + EA2/(R 2 na) ¯ vn = qv

This is a cubic equation and is shown plotted in Fig. 5.5a where qv is computed for specified real values of vn . The plot shows that a maximum load point is reached (the limit point), under displacement control the load then decreases to a minimum after which it can then increase. The descending part of the curve is unstable. This solution has the same general form as the middle curve in Fig. 3.19b. But this is not what actually occurs with the arch as shown by the FE generated data in Fig. 5.5a. The dashed arrows indicate that bifurcations occur at a much lower load than the limit load. This low buckling load has been used to illustrate that curved plates and shells are susceptible to imperfections, however, that does not

367

♦

5.1 Postbuckling of Beams and Arches

5.

♦

10. ♦ ♦

0. .

♦

-5. -10. ♦

0.

2.

4.

.

6.

8.

10.

0.

2.

4. .

6.

8.

Fig. 5.5 Postbuckling behavior of a shallow arch under radial loading. Circles and triangles are FE results, continuous line is the model result. (a) Load-deflection behavior showing snapthrough. (b) Spectral behavior showing when the antisymmetric and symmetric modes become unstable

quite reflect the situation. Figure 5.5b shows the spectral stiffness plots; the circle data are the perfect structure and show that while the first mode (the antisymmetric mode) becomes unstable it is not triggered or manifested because there is no agent. It is only after the second mode (the symmetric mode) becomes unstable is the instability manifested. It is manifested because the new increment in load is congruent to the instability. In both cases, the instability is initiated at a load significantly lower than the limit load. A small initial asymmetric imperfection in the form of a small moment causes the antisymmetric mode to manifest its instability, these data are indicated by the triangle symbols in Fig. 5.5b. What these results make clear is the distinction between the deformation state at the current loading and the buckling state. Think of a perfect beam under axial compression: the primary equilibrium path is indefinite compression, it is the occurrence of a secondary equilibrium path that leads to the buckling. Equations (4.12) are our formal way of testing for this second equilibrium path. Two separate questions arise, one is in regard to the significance of the instability, the second is in regard to the agents to manifest the instability. The latter seems minor because there are so many possibilities available in a real situation that it can always be assumed that at least one of them manifest the instability. The other question, the significance or otherwise of an instability is a hugely complex question. An example problem in the next section of a tension beam intentionally allowing instabilities to occur beautifully illustrates that instabilities can be allowed to promote some other positive design objective such as enhanced stiffness or maximum load. Reference [16] discusses structures intentionally with zero stiffness and gives desk lamps as an example. The arch behavior appears to be sensitive to the precise values of its geometry and BC parameters; we, therefore, take a closer look at the effect of the BCs. Figure 5.6 shows the FE generated member distributions for a curved and straight beam subjected to gravity loading. The properties are the same as in Fig. 2.2a. Two

368

5 Studies of Postbuckled Shapes

Fig. 5.6 Comparison of member distributions in a simple arch and straight beam subjected to gravity loading. Both ends have immovable pins. (a) Arched frame. (b) Straight beam. (c) Freebody diagram for the arch

points are striking. First the bending stresses are orders of magnitude different between the curved and straight beam, and two, there is a significant axial stress in the curved beam but none in the straight beam. We explain the relevance of this to the postbuckling behavior. The free body of Fig. 5.6c explains the equilibrium situation; the gravity loading is resisted primarily by the axial force in the arch but entirely by the bending moment in the beam. That is, F¯o H + Mo

vs

Mo

or

17 × 29 − 9 vs

486

Reference [10] refers to the effect of H as the structural depth and claims that a good part of efficient structural design consists of selecting and utilizing the geometry so as to maximize the structural depth. The downside is that because the arch generates compressive axial forces, there is the possibility of buckling; whereas the straight beam does not generate axial forces and there is no possibility of buckling. Figure 5.7 shows the member distributions for a gravity loaded arch similar to the one in Fig. 5.6a; the only difference is that the right boundary is a pin on rollers so that it can move horizontally. Designate Figs. 5.6 and 5.7 as case I and case II, respectively; the two significant differences between the cases are that the axial force is close to zero in the middle portion, and the moments are nearly two orders of magnitude larger. That is, support for the transverse load has shifted to being mostly flexural, just like a straight beam. Because the roller case has smaller axial forces, it would be expected that it has a larger computed buckling load (less likely to buckle). This turns out not to be so, the first two buckling loads for the two cases are I pin–fixed:

λ = 539, 1259 ,

II pin–roller:

λ = 437, 2024

Figure 5.7b makes a comparison of the buckling shapes. For case I, the n = 1 symmetric shape is suppressed but it appears as the first shape for case II. The n =

5.1 Postbuckling of Beams and Arches

369

♦

Fig. 5.7 Buckling of a simple arch subjected to gravity loading. Right end has pin on rollers. (a) Member distributions for case II. (b) Buckling mode shapes. Top three are case II

1.0

0.5

♦

0.0

-2.

-1. .

0.

Fig. 5.8 Large deflection analysis of the arch. (a) Deflections against load. (b) Deformed shapes during the snapthrough of case I. (c) Deformed shapes during the push-through of case II

2 antisymmetric shape looks about the same in each case (observe that there is no noticeable horizontal displacement for II) but the eigenvalues are very different reflecting the fact that case II has the smaller axial force. More or less, we can say that the third case II shape is like an n = 3 symmetric shape although distorted by the horizontal displacement. When anomalous or unexpected behavior is encountered in a buckling analysis, it is usually a good idea to do some fully nonlinear large deflection analyses. Figure 5.8a shows the displacement of the arch apex against load. Case I exhibits a snap-through buckling in the vicinity of λ1 , and very little deformation up to the critical load. Case II, however, simply exhibits a large deflection as shown in Fig. 5.8c; that is, it first flattens out and then shows a bowed-down shape, and as the load is further increased, the bow just gets deeper with the boundaries closer together. The dashed line in (a) shows the movement of the roller—first left-to-right and then right-to-left.

370

5 Studies of Postbuckled Shapes

Fig. 5.9 Spectral stiffness behavior of an arch with an elastic boundary. Circles are the first mode, squares are the second mode. Horizontal arrows indicate zero stiffness and all plots have the same scales

Figure 5.8b shows the deformed shapes during the snapthrough buckling of case I; the snapshots are at equal time intervals. A static nonlinear analysis could identify the onset of the instability, but to navigate the snapthrough under load control required a fully nonlinear dynamic analysis. The apparent buckling load is slightly larger than predicted because it takes time to set the arch in motion during which the load continued to increase. A small imperfection of an applied torque was applied at the apex in order to initiate the motion in a reasonable (computational) time. It is notable that the deformed shape at the onset of the instability is similar to the eigenbuckling shape (i.e., antisymmetric). The spectral plots are similar to those of Fig. 5.5b. There is no buckling as such (no severe change of shape) for case II. The spectral plot shows a gentle increase of stiffness during all stages of the loading. The deformed shapes in Fig. 5.8c are at equal load increments and indicate this stiffening effect. It can be imagined that if a spring is added to the roller, this adds stiffness and the response is, therefore, somewhere in between our fixed–pinned and roller–pinned states. The situation is slightly more complex because of the modes involved as discussed next. Figure 5.9 shows the spectral stiffness behavior when a roller spring is changed by four orders of magnitude; lower left is essentially unconstrained, upper right is essentially fixed. Keep in mind that in each case, the arch passes through a horizontal position and goes through to the other side. We first observe (lower right) that the spring causes an increase in the initial stiffness The numbers on the left of the vertical axis indicates the shapes essentially as given in Fig. 5.7b. n = #. The big change that occurs (upper left) is that adding sufficient spring stiffness shifts the n = 1 symmetric mode to a higher frequency where it does not participate in the deformation process. The larger intermediate spring stiffness case is quite interesting because both the antisymmetric and symmetric modes vie for causing the instability. As a consequence, their mode interaction causes a critical load that is smaller than

5.2 Plates and Cylinders

371

expected based on their initial slope behavior. Additionally, the postbuckle behavior is more energetic (as seen by the fluctuations in the postbuckle stiffness). This type of complex mode interaction behavior is the subject of the final section of this chapter. A spectral analysis of the stiffness matrix correctly identifies the onset of the instability and shows that for an elastically restrained arch it can be either the n = 1, 2, or 3 mode that buckles. It does not give much insight about the actual snapthrough process itself because that is a truly dynamic event. (Dynamic results are discussed in Ref. [8].) Once the vibrations die down, the spectral stiffnesses are observed to be larger than at the beginning of the loading and this is because the gravity loading puts the deflected arch in tension. To summarize, the eigenbuckling analysis accurately predicted the critical load for case I because the arch is very stiff and there is not much change in shape up to the critical load level. The eigenbuckling analysis is irrelevant for case II because there is a significant change in shape up to the predicted critical load level. Thus buckling analyses of thin-walled structures that exhibit significant deflections (plates and shells with free edges, for example) should always be treated with circumspection unless accompanied by a nonlinear analysis.

5.2 Plates and Cylinders As is apparent from the studies of the last three chapters, the significant factor when discussing general plates is that of initial plate curvature. Therefore, we divide the following into separate analyses connected to this. First we look at flat plates, then shallow curved plates, and finish with circular cylinders.

5.2.1 Flat Plates One aspect that distinguishes a plate from a beam is that the plate typically has more edge constraints. Consequently, the displacements and rotations are not as large and the full nonlinear energy expressions need not be used. In these circumstances, the von Karman plate equations are generally adequate. To recap, the strain energies for a flat plate using the von Karman plate assumptions are summarized as UM =

1 2

   2 2 2 dxdy E ∗ h[Exx + Eyy + 2νExx Eyy ] + Gh4Exy

UF =

1 2

   2 2 2 ¯ + w,yy + 2νw,xx w,yy ] + D[4w ] D[w,xx ,xy dx dy

(a)

372

5 Studies of Postbuckled Shapes

where the membrane strains are approximated as 2 , Exx = u,x + 12 w,x

2 Eyy = v,y + 12 w,y ,

2Exy = u,y + v,x + w,x w,y

(b)

The flexural energy is just the linear contribution and the membrane nonlinearity is simply related to the out-of-plane displacement. The variation of membrane and linear flexural strain energies are, respectively,  δ UM =

  E ∗ h [Exx + νEss ] δExx + [Ess + νExx ] δEss dxds 

+  δ UF =

  D [w,xx +νw,ss ] δw,xx +[w,ss +νw,xx ] δw,ss dxds 

+

  Gh 4Exs δExs dxds

  D¯ 2w,xs 2δw,xs dxds

Similar to our flat plate analysis in Sect. 2.2, we recognize the terms in small brackets as related to stresses such as σxx = E∗ [Exx + νEss ]. Add to these energies the virtual work of the applied loads δ We = qw δw dxds; for simplicity, we take that u and v loadings are applied along the edges. Then through our PoVW, after integration by parts, the variations with respect to u and v lead to the equilibrium equations in terms of stresses identical to those of Eq. (2.26) without the body loads. The variation with respect to w has contributions such as −(σxx w,x ), x. On expanding and using the membrane equilibrium equations to replace the derivatives of stress, we finally get for transverse equilibrium equation that is identical to Eq. (4.19) for buckling. The big difference here is that the stresses are unknown and obtained as part of the nonlinear solution. Because of this, we modify the stress equilibrium equations some more. As done in Sect. 2.4, we can satisfy the membrane equilibrium equations exactly by representing the stresses in terms of an Airy stress function φ(x, y). The stresses (and hence strains), however, are not necessarily compatible. We form a collection of derivatives similar to what was done in the linear case (but using nonlinear Eij instead of the linear ij ), the leading terms disappear leaving Exx ,yy +Eyy ,xx −2Exy ,xy = w,xx w,yy −w,2xy This is the compatibility condition for the nonlinear strains. Substitute for the strains in terms of the stresses, and the stresses in terms of the stress function, to get the inhomogeneous biharmonic equation ∇ 2 ∇ 2 ψ − E[w,2xy −w,xx w,yy ] = 0

(5.2)

♦

2.

373

♦

5.2 Plates and Cylinders

.

1.

0.

♦

0.0

0.5

♦

0.0

0.5

1.0

1.5

2.0

Fig. 5.10 FE results for the postbuckling behavior of a simply-supported flat plate. (a) End shortening. (b) Quarter point deflection

where all the inhomogeneous contributions are associated with the transverse displacement. This relation elegantly connects the membrane stresses to the transverse deflection. Equation (5.2) in combination with D∇ 2 ∇ 2 w − σxx hw,xx −σyy hw,yy −2τxy hw,xy = qw

(5.3)

constitutes the two nonlinear equations for the strong formulation of large deflections of flat plates. The former is a compatibility condition (or geometric constraint) which we should satisfy exactly, but the latter is an equilibrium equation which, where necessary, we can approximate in a Ritz method sense. It is useful to realize that the LHS operators of these equations are the same and linear. Thus, if w(x, s) is known (or assumed), then these are linear differential equations to be solved, but the displacement coefficients will be nonlinear. The example problems to follow focus on the effect of BCs. Example 5.2 Figure 5.10 shows some FE generated results for the axial compression of the simply-supported flat plate shown in Fig. 4.18. In comparison to Figs. 5.1 and 5.3 for beams, the transverse deflections are an order of magnitude smaller. Construct a model to explain the postbuckling decrease in transverse displacement. Unlike the modeling of the initial buckling of the simply-supported plate done in Sect. 4.3, here, in order to control the BCs the plate of size [ao × b × h] was actually modeled with the size [2ao × b × h] as shown in Fig. 5.11 where simplysupported BCs are also imposed at x = ao . This modeling essentially uses periodic BCs (pBCs). If this is not done, there is excessive and nonuniform end shortening. The same result could be achieved by imposing displacement controlled BCs; the choice taken better illustrates how deformed shapes naturally follow from the applied loadings. In the following, only the plate segment of length ao is analyzed.

374

5 Studies of Postbuckled Shapes

.

.

.

Fig. 5.11 FE results for the postbuckling behavior of a simply-supported flat plate. Results are at a load level of twice the buckling load. (a) Contours of w(x, y) displacement. (b) Contours of σxx (x, y) and σyy (x, y) stress

Note that a segment of length 12 ao or 32 ao could equally well have been used in the analysis. The plate was modeled with [80 × 20] modules using the MRT/DKT [6] shell element. Loading was done quasi-statically to a level approximately 2.0 times the buckling load. Very small end moments were used as agents to initiate the postbuckling deformations. Figure 5.11a shows contours of the out-of-plane deflections in the postbuckling region. These have the familiar sinusoidal buckle pattern which we can take as w(x, y) = w2 sin(nπ x/a) sin(mπy/b) ,

a = ao ,

n = 2,

m=1

The stress contours shown in Fig. 5.11b do not resemble the uniform uniaxial state that was present at buckling. However, a study of the average stress against average strain shows there is very strong linearity into the postbuckling range. Furthermore, the average σyy stress is essentially zero, and the average membrane Exx and Eyy strains essentially follow Poisson’s ratio. We thus conclude that the plate, on average, maintains the uniaxial state of stress into the postbuckling range; more discussions of the stresses are given later. With these results in mind, assume the in-plane displacements are given by u(x, y) = −u1 [x/a] ,

v(x, y) = νu1 [y/a]

where u1 is the uniform displacement of the edge of the plate. From Eq. (a) we have the nonlinear membrane strain

5.2 Plates and Cylinders

375

2 Exx = u,x + 12 w,x =−

 nπ 2 u1 + 12 w22 cos2 (nπ x/a) sin2 (mπy/b) a a

The membrane strain energy is then given by  UM =

1 2

2 dxdy = 12 Eh Eh Exx

 u2 1 a2

−

u1 1 2  nπ 2 1 4  nπ 4 9  ab w + 4 w2 a 4 2 a a 64

where integral relations from Sect. 2.1 was used. The flexural strain energy is the same as done for the initial buckling problem (but labeled UE there) and is 3 UF =

1 2D

1 2 4 w2

  nπ 2 a

+

 mπ 2 2 b

4 ab

The traction distribution along x = ao changes shape after buckling occurs and therefore the loading is not proportional. However, thinking in terms of a resultant load (or average stress) we can write the potential of the applied load as  V=−

σxx (a, y)u(a, y) dyh = −σa h

u1 ab a

The same result is obtained using virtual work. There are two equilibrium equations given by  ∂  UM + UF + V = 0 , ∂u1

 ∂  UM + UF + V = 0 ∂w2

The first leads to   u1 1 1 2  nπ 2 1 − σa h = 0 or Eh 2 − 8 w2 a a a a

 nπ 2 u1 σa = + 18 w22 (5.4) a E a

The leading term on the right is the uniaxial compression, the last term is the extra shortening because of the arching of the plate. The second equilibrium equation yields 3

   4  u1 1  nπ 2 1 2  nπ 4 9 nπ 2  mπ 2 2 1 Eh − 4 + D4 + 2 4w2 + w2 = 0 a a a 64 a b 

This has two solutions. The first is when w2 = 0 and then we simply have that u1 = (σa /E)a from the first equilibrium equation. When w2 = 0 the term in braces must be zero. First consider just at buckling when σa = σc , w2 = 0 then σc h = D

  nπ 2 a

+

 mπ 2 2  a 2 b nπ

376

5 Studies of Postbuckled Shapes

These are the buckling loads already found in Sect. 4.3. We can now solve for the displacements in terms of the loading '

σa − σc 4 σa u1 = + , E E 5

w2 =

σa − σc a E nπ

'

32 5

(5.5)

Reference [27] solves the same problem by a different method and yields a similarly structured solution but with a slightly different coefficient '

√ 32 versus 8 = 5

'

32 4

Note that in the solution, u1 changes linearly with load but with a reduced modulus. Both displacements are shown plotted in Fig. 5.10 as the continuous lines; the model captures the main features of the behavior. It is interesting to compare the results obtained here with those of the buckling analysis done earlier. We already used the same elastic energy but in the buckling problem it was associated entirely with the flexural action. The geometric strain energy was   ) ∂w 1 UG = 2 σ¯ xx h ∂x

2

* dx dy = 18 σ¯ xx h w22

 nπ 2 a

ab

The load potential of the present analysis in combination with u1 obtained from the membrane straining gives  V=−

σxx (a, y)u(a, y) dyh = −σa h

 nπ 2  σ u1 a ab = −σa h + 18 w22 ab a E a

For plates that do not plastically yield, σa /E is very small, and thus V and UG are essentially the same. This example gives us an opportunity to discuss stress within the context of an approximate Ritz solution. The first point to make clear is that the applied stress along a boundary (the tractions) is associated with the natural BCs, and because these are not specified, then we do not expect them to be satisfied exactly. In other words, the stresses may violate what we know to be the natural BCs and yet the formulation can still be legitimate. The model gives the axial stress as    nπ 2 u1 σxx = E Exx = E − + 12 w22 cos2 (nπ x/a) sin2 (mπy/b) a a This indicates that the stress has similar range of variation in the x and y directions. But this is not true of the contours in Fig. 5.11b which show an almost uniform variation in x. This is further emphasized in Fig. 5.12a which shows a plot against

377

♦

1.0

♦

5.2 Plates and Cylinders

0.8

.

0.6 0.4 0.2 ♦

♦

0.0

-4.

-3.

-2. .

-1.

-4.

0.

-3.

-2. .

-1.

0.

Fig. 5.12 Stress distributions at σa = 2σc . Circles are the FE data. (a) Axial stress values for all nodes. (b) Traction distributions at x = ao

y of the stress for every node in length ao of the plate; it is clear that there is not a significant variation along the length. Figure 5.12b shows a plot of the FE axial stress at x = ao as the circles; this constitutes an applied traction BC. Also shown is the simple model distribution as the dashed line (note that the cosine term is unity); there is a significant difference. There is a slight asymmetry in the FE distribution, but a reasonable model of the distribution is given by  nπ 2



2πy a b  nπ 2  nπ 2  mπy  = −σa − 18 Ew22 + 14 Ew22 sin2 a a b

σxx = −σa −

2 1 8 Ew2

cos

where σa is the applied average stress. This is shown as the full line in the figure. In comparison, the original model has an off-set of 18 Ew22 (nπ/a)2 and a coefficient change of 12 → 14 . The virtual work of the generalized force must equal the virtual work of this traction. Because u1 is the generalized DoF, we find P1 from  δ We = P1 δu1 =

σxx (a, y) δu(a, y) dyh 

=

  −σa − 18 Ew22 (nπ /a)2 cos(2πy/b) δu1 dyh= − σa bh δu1

The generalized force is the average stress times the cross-sectional area. We conclude that the model captures the main features of the postbuckling of plates but the detailed stress distributions are off. Let us, therefore, look closer at this aspect of the problem. With our assumed transverse deflection, the compatibility of Eq. (5.2) becomes

378

5 Studies of Postbuckled Shapes

2.

.

1.

0. 0.0

0.1

0.0

1.0

2.0

3.0

4.0

Fig. 5.13 FE results (circles) for the postbuckling behavior of a long flat plate with one free edge. (a) End shortening. (b) Edge deflection

∇ 2 ∇ 2 ψ = Ew22 n¯ 2 m ¯ 2 [cos2 nx ¯ cos2 my ¯ − sin2 nx ¯ sin2 my] ¯ = Ew22 n¯ 2 m ¯ 2 12 [cos(2nx) ¯ + cos(2my)] ¯ Observe the doubling of the wavenumber for both functions. The particular solutions are simply ψ = A cos(2nx) ¯ + B cos(2my) ¯ ,

A=E

¯2 1 m , 32 n¯ 2

B=E

1 n¯ 2 32 m ¯2

This then leads to the stress system ¯ 2 cos(2my) ¯ , σxx = −B4m

σyy = −A4n¯ 2 cos(2nx) ¯ ,

τxy = 0

The full stress distributions have superposed the uniaxial stress system σxx = −σa , σyy = 0. Based on equilibrium and compatibility, we have a stress system where σxx is independent of x and has a superposed cos(2my) ¯ dependence, and σyy is independent of y and has a strong cos(2nx) ¯ dependence. The contours of Fig. 5.11b and distributions of Fig. 5.12 for the most part reflect these conclusions. We could incorporate these modified stresses as an improved model, the main effect would be that of adding a softening mechanism giving better agreements in Fig. 5.10. We are content with the results from the simplified uniaxial stress model because it better highlights the main contributors to the postbuckle state. In fact we use the same basic notion in the next example problem. Example 5.3 Figure 5.13 shows some FE generated postbuckling results for the plate in Fig. 4.21. In comparison to Fig. 5.10, the ratio of transverse deflection to end shortening is about seven times different. Construct a model to help explain this almost order of magnitude difference.

5.2 Plates and Cylinders

379

Fig. 5.14 Finite element generated contours for the postbuckling deformed shape of a long flat plate. The load level is twice the buckling load

As done in the previous example problem, the actual plate model was a multiple of ao as shown in Fig. 5.14. Symmetry is imposed at the left side with simplysupported BCs imposed at x = 12 ao , 32 ao . The plate was modeled with [60 × 8] modules using the MRT/DKT [6] shell element. Loading was done quasi-statically to a level approximately 2.0 times the buckling load. Very small end moments were used as agents to initiate the postbuckling deformations. The data for Fig. 5.13 were taken from the line of symmetry for w2 and at x = 12 ao for u1 . The deformation pattern is similar to that of Fig. 4.22 except that there is also some end shortening; the contours of u(x, y) show that the end shortening at x = 1 2 ao is essentially uniform with respect to y in contrast to what happens at x = 3 2 ao . As discussed in connection with Figs. 3.16 and 4.22, the membrane behavior is mostly uniaxial stress so that we can write the displacements as u(x, y) = −u1 [x/a] ,

w(x, y) = w2 sin(nπ x/a)[y/b] ,

n=1

and ignore the contributions from the v(x, y) displacements. The energies can then be computed from  UM =

1 2

UF =

1 2

2 Eh Exx

dxdy =

1 2

 

 Eh[u,2x +u,x w,2x + 14 w,4x ] dxdy

   2 2 2 1 3 + w,yy + 2νw,xx w,yy ] + G 12 h [4w,xy ] dx dy D[w,xx

which lead to  nπ 4 3  u1 2  nπ 2 1 w2 + w24 ab a a 6 a 160   nπ 4 1  2 2  1 3 2 nπ + G 12 UF = 12 Dw22 ab h w2 a 6 a b2

UM = 12 Eh

 u2 1 a2

−

The flexural energy is the same as used for the plate in Fig. 4.21. As discussed in the previous example, we can write the potential of the applied load as

380

5 Studies of Postbuckled Shapes

 V=−

σxx (a, y)u(a, y) dyh = −σa h

u1 ab a

There are two equilibrium equations, the first leads to Eh

u

1 2 a

−

1 2  nπ 2  1 w − σa h = 0 12a 2 a a

or

 nπ 2 1 u1 σa = + w22 a E a 12

The second equilibrium equation gives #   4 3  u1  nπ 2 1 2 nπ Eh − + w2 a a 6 a 80  nπ 4 1  nπ 2 2 $ 1 3 1 3 w2 = 0 + G 12 +E ∗ 12 h h a 6 a b2 This has two solutions. The first is when w2 = 0 and then we simply have that u1 = (σa /E)a from the first equilibrium equation. When w2 = 0, the term in braces must be zero and this simplifies to −

 nπ 2 9 E ∗ h3  nπ 2 Gh3 1  2 9 σc u1 u1 2 nπ +w22 + = +w + = 0 or 2 a a 40 Eh 12 a Eh b2 a E a 40

where σc is the buckling load for the plate in Fig. 4.21. It is interesting that both equilibrium conditions lead to similarly structured equations with the former using σa and the latter σc . These equations are readily solved to get the displacements in terms of the loading as σa − σc 10 σa u1 = + , E E 17

' w2 =

σa − σc a E nπ

'

120 17

(5.6)

Note that u1 changes linearly with load. Both displacements are shown plotted in Fig. 5.13 as the continuous lines; the model captures the main features of the behavior. The results in Eq. (5.6) are remarkably similar to those in Equation (5.5), the respective coefficients compare as 0.8 vs 0.6, 2.5 vs 2.6. What accounts for the difference in the w2 displacement is simply the length a: as seen in Fig. 5.11 it is the shorter dimension b for n = 1, but is the full length (from support to support) in Fig. 5.14. Also, because of the similarity, the discussions of the stress apply here too. Example 5.4 Investigate the reinforcing effect of shear panels on the postbuckling behavior of frame structures. Figure 5.15 shows a frame with a distributed end load. A frame loaded like this would suffer out-of-plane torsional or lateral buckling as discussed in connection

5.2 Plates and Cylinders

381

♦

4.0 3.0 2.0 1.0 .

0.0

♦

0.0

1.0

2.0

3.0

4.0

5.0

Fig. 5.15 Deformation of a reinforced structure. (a) Geometry and properties. (b) Load-deflection response

Fig. 5.16 Large-load nonlinearly deformed shapes of the frame structures. (a) No bracing showing just large deflections. (b) With bracing showing less overall deflections and local buckling

with Fig. 4.35; therefore, the top and bottom edges are constrained to move in the plane only; the constraints can be thought as arising from a box-beam structure. We consider three cases: case I is the frame with no additional members, case II adds northeast members, and case III replaces the diagonal members with a shear panel of the same amount of material. The shear loading puts the top members in compression, being largest close to the fixed end. An eigenbuckling analysis predicts that there is local buckling of the top two left members. The buckling loads for case I are λ1 = 3.37 and λ2 = 5.24. Figure 5.16a shows the nonlinearly deformed shape when the load is only λ = 1.0. The structure is very flexible and therefore no buckling actually occurs. The tip deflection against load is shown in Fig. 5.15b as case I. It is clear from Fig. 5.16a that the shear action causes the northeast diagonals to increase substantially. Therefore, a weight-efficient way to increase the stiffness is to add diagonal bracing. Figure 5.15b shows this as case II and indeed there is a significant change in stiffness. Interestingly, the predicted buckling loads hardly change being λ1 = 3.22 and λ2 = 4.41. The reason there is little change is that

382

5 Studies of Postbuckled Shapes

the predicted underlying buckling mechanism (i.e., top left members buckle) is unchanged by the bracing. The maximum load achieved is close to the predicted buckling load and the postbuckled shape (obtained from the nonlinear FE analysis) shown in Fig. 5.16b is close to the eigenbuckling shape. A few additional comments are worth making in regard to the nonlinear analysis. Small transverse loads (Qy = Vo /1000) were applied to the top left members to act as initiators (agents) for the buckling. After initiation of the buckling, there is a very large, nearly instantaneous, displacement to the configuration shown in Fig. 5.16b. This suggests that the immediate postbuckle state is unstable. The load increments were adjusted (smaller steps taken) in the vicinity of buckling so as to better capture the behavior shown in Fig. 5.15b. For case III, the diagonal members are replaced with panels of approximately the same amount of material. This resulted in a panel thickness of h = 1.27 mm (0.05 in) which is quite thin. In some ways, this is then like a membrane plate problem but the frame makes a big difference. A linear static stress analysis revealed the superposition of two stress systems in the panel: a beam bending-like distribution σxx increasing toward the fixed end, and a constant τxy distribution. The top part of the first panel, therefore, has a significant compressive principal stress. In looking at Fig. 5.16b, there is a large rotation of the first top joint resulting in a large transverse displacement of the three attached members. It is expected that the panel should minimize this. An eigenbuckling analysis gives buckling loads of λ1 = 0.57 and λ2 = 0.63 which are significantly lower than for the braced case. The situation we have is that the thin panel has very little flexural stiffness and is therefore susceptible to out-of-plane buckling due to compression. The eigenbuckling analysis identifies that this is the buckling mechanism. Thus, do we conclude that the substitution of the panel is a bad design? Figure 5.15b shows this not to be true; indeed, the panel design shows increased stiffness with no hint of buckling over a load range that is nearly an order of magnitude larger than the predicted buckling load (for the frame without reinforcements). This is worth investigating some more. The nonlinear analysis used small transverse loads (Qz = Vo /10,000) in the upper part of each panel. As part of the analysis, the spectral stiffnesses were monitored and Fig. 5.17a shows how these change with applied load. The lowest value achieves four minimums (zero is not achieved because of the load imperfections) and these minimums coincide with the separate buckling of the four panels. This is confirmed in Fig. 5.17b which shows the deflection contours a few load increments after the minimums are achieved. The important point is that each panel buckling is postbuckle stable; therefore, the stiffness of that particular buckling mode is recovered (more or less) and the overall structural stiffness is hardly affected. To finish this discussion, Fig. 5.18 shows the nonlinear out-of-plane displacement contours at the largest load level. There is a definite pattern to the buckled shapes with the folds being parallel to the tension direction. This means that after buckling of the panels, there is still a small northwest compression resistance, but the northeast tension direction is unaffected by the buckling. That is, on further loading,

5.2 Plates and Cylinders

383

.

0.3 .

0.2

.

.

0.1 .

0.0

0.0

0.5

1.0

1.5

.

Fig. 5.17 Nonlinear analysis of frame and panel structure. (a) Spectral stiffness plots. (b) Out-ofplane deflection contours after each panel buckling

.

Fig. 5.18 Contours of postbuckled shape at large load of λ = 4.0

it continues to function similar to the brace in tension which is why this design is often referred to as a tension beam. Reference [3] has a nice photograph (fig. 7-4.3) of experimentally induced folds in a diagonal tension beam. Example 5.5 The flat plate problems exhibited postbuckled deformed shapes very similar to the buckling shape. Explain why. Consider the vibration EVP 2 [ KT ]{φ }m = ωm [ M ]{φ }m

and let the spectral shapes be normalized such that [  ]T [ M ][  ] =  I ,

[  ]T [ KT ][  ] =  ω2 ,

[  ] = [{φ }1 , {φ }2 , · · · {φ }N ]

384

5 Studies of Postbuckled Shapes

This is also a statement of their orthogonal properties. Because the shapes are linearly independent of each other, then any vector can be written as a linear combination of them (this is called the expansion theorem). Consider a load increment close to a singular point, and let the increment of displacement be represented by {u} = η1 {φ }1 + η2 {φ }2 + · · · =

 m

ηm {φ }m

where ηm are the associated amplitudes. Now substitute this into Eq. (3.7) representing the loading path, premultiply the resulting equation by {φ }Tm , then making use of the orthogonality properties leads to 2 ηm ωm − λ{φ }Tm { P} = 0

or

ηm =

1 χ {φ }Tm { P} 2 ωm

where χ { P} is the increment of proportional loading. We, therefore, have the general spectral representation for the displacement increment {u} = λ

 1   {φ }Tm { P} {φ }m 2 ωm

The eigenvalues (stiffnesses) are ordered so that ω12 is the smallest and therefore, typically, this is a series with decreasing higher terms. At a singular point, the lowest eigenvalue is zero, consequently just past the singular point, ω12 is very small so that the expansion is dominated by the first term. That is, {u} ≈ η1 {φ }1 The displacement increment resembles the first spectral shape. Furthermore, at the singular point, the EVP for the first mode reduces to [ KT ]{φ }1 = 0

or

[KE + λKG ]{φ }1 = 0

which in fact is the EVP for the first buckling mode. Thus, we conclude that the displacement increment has the shape of the first buckling mode shape. This does not imply that the final postbuckled state has the same shape; however, for relatively simple postbuckle stable situations such as for beams and plates it does imply this.

5.2 Plates and Cylinders

385

Fig. 5.19 Stress contours at maximum load for the axial compression of a shallow curved plate. o The shear stress contours vary between ±0.09 σxx

5.2.2 Shallow Curved Plates In this subsection we concentrate on the effects of the membrane stresses on the postbuckling behaviors. To this end, we choose the simplest of all shell models, that of Donnell [5]. This model treats the flexural behavior as essentially the linear behavior of a flat plate with the membrane stresses incorporated via second-order energy terms as done in the previous chapter. Consider the curved plate shown in Fig. 2.52. All boundaries are flexurally simply supported and a uniform axial compressive load is applied equally at both ends. The stress contours at maximum load are shown in Fig. 5.19. They are quite complex even though the applied stress is uniform along the edge and contrast with the flat plate contours shown in Fig. 5.11. That is, both sets of contours show periodicities in x. Additionally, no FE data are available beyond buckling indicating that this shallow curved plate, in contrast to the flat plate, is postbuckle unstable. We look closer at both the stress and the stability. Using the curved plate coordinates of Fig. 2.14, the strains are approximated from Eq. (3.28) as Exx ≈ u,x + 12 w,2x ,

Ess ≈ v,s +w/R+ 12 w,2s ,

2Exs ≈ u,s +v,x +w,x w,s

The only difference in comparison to the flat plate is the presence of w/R in Ess . Following the procedure for the flat plate, we get the compatibility and transverse equilibrium equations to be, respectively, ∇ 2 ∇ 2 ψ − E[w,2xy −w,xx w,yy −w,xx /R] = 0

(5.7)

D∇ 2 ∇ 2 w − σxx hw,xx −σss h[w,ss +1/R] − 2τxs hw,xs = qw

(5.8)

Donnell [5] is credited with being the first to present compatibility for nonlinear curved plates and shells in this form. The contribution σss h/R is a carry-over from the w/R hoop strain contribution to the membrane energy. Equations (5.7) and (5.8) constitute the two nonlinear equations for the Donnell formulation for the nonlinear deformation of shallow curved plates and shells.

386

5 Studies of Postbuckled Shapes

Based on the spectral analysis of Sect. 2.4, assume the deflected shape is given by ¯ sin my ¯ w(x, y) = wo sin(nπ x/L) sin(mπy/a) = wo sin nπ Substitute into Eq. (5.7) and replace powers of the trigonometric function in terms of double angles to get ∇ 2 ∇ 2 ψ = Eh





1 2 2 2 ¯ m ¯ [cos(2nx) ¯ + cos(2my)] ¯ + (wo n¯ 2 /R) sin(nx) ¯ sin(my) ¯ 2 wo n

The leading terms are the same as for the flat plate problem. The particular solution is given by ψ(x, y) = Awo2 cos(2nx) ¯ + Bwo2 cos(2my) ¯ + (C/R)wo sin(nx ¯ sin(my) ¯ where A=E

¯2 1 m , 32 n¯ 2

B=E

1 n¯ 2 , 32 m ¯2

C=E

n¯ 2 (n¯ 2 + m ¯ 2 )2

o y2 + To this, we add the homogeneous solution ψ = − 12 σxx corresponds to a biaxial stress state. The stresses are

1 o 2 2 σyy x

which

o σxx = ψ,yy = −Bwo2 4m ¯ 2 cos(2my) ¯ − (C/R)wo m ¯ 2 sin(nx) ¯ sin(my) ¯ − σxx o σyy = ψ,xx = −Awo2 4n¯ 2 cos(2nx) ¯ − (C/R)wo n¯ 2 sin(nx) ¯ sin(my) ¯ − σyy

τyy = −ψ,xy = −(C/R)wo n¯ m ¯ cos(bnx) cos(my) ¯ The simple assumed deflection shape has given rise to a rather complicated stress state. But keep in mind that the stresses are in equilibrium (even at a local level) and they (and the strains) are compatible. Furthermore, the stress coefficients are written in terms of the single unknown wo . The stress contours at maximum load are shown in Fig. 5.19. Aside from the uniform stress distributions, the dominant contributions come from the C related terms which accounts for the x-periodicity for σxx . When we substitute the stresses and displacements into Eq. (5.8). we get terms such as cos(2my) ¯ sin(bnx) sin(my), ¯ cos(2nx) ¯ sin(bnx) sin(my); ¯ in other words, the differential equation is not satisfied. However, we can get an approximate solution in a weighted-average sense [2] by setting  [ D ] g(x, y) dxdy = 0 ,

g(x, y) = sin(bnx) sin(my) ¯

5.2 Plates and Cylinders

387

where [ D ] is the differential equation. In the present context, this is usually called the Galerkin method [27]. The integrals are readily evaluated using trigonometric relations such as cos 2nx ¯ sin nx ¯ =

1 2

sin 3nx ¯ −

1 2

sin nx ¯ ,

cos2 nx ¯ =

1 2

+

1 2

sin nx ¯

Grouping terms according to powers of the deflection, we get   n¯ 4 Eh o 2 1 D(n¯ 2 + m ¯ 2 )2 + 2 2 − σ h n ¯ xx 4 aLwo R (n¯ + m ¯ 2 )2    n¯ 3 m ¯ Eh 1 8 Eh  4 wo2 + + n¯ + m − ¯ 4 aLwo3 = 0 2 2 2 R 6 3 (n¯ + m 64 ¯ ) The load is related to the transverse deflection by   ¯ 2 )2 n¯ 2 (n¯ 2 + m Eh o σxx h= D + n¯ 2 R 2 (n¯ 2 + m ¯ 2 )2    1 8 n¯ 3 m ¯ Eh 4 Eh  4 wo + + n¯ + m − ¯ 4 wo2 2 2 2 2 R aLn¯ 6 3 (n¯ + m 16 ¯ ) The leading bracketed term on its own is the linear buckling EVP; interestingly, these are the buckling loads for a complete cylinder of radius R [22] and coincide o = with Eq. (4.28) when m = 0. This came about because of our assumption that σyy 0. The third bracketed term would suggest that the postbuckle state is stable and symmetric as for a flat plate. The significant aspect of the solution is the second bracketed term which shows that the load has a linear in wo contribution giving an asymmetric buckling as illustrated in Fig. 3.3. This conforms that the plate is postbuckle unstable. The solution succinctly shows the effect of the initial curvature: the buckling load is increased due to the stiffening but at the expense of making the plate initially postbuckle unstable. References [23, 27] present results for a square plate with initial imperfections; apparently the results are based on Ref. [24].

5.2.3 Circular Cylinders Figure 5.20 shows the spectral plots for the axial compression of a circular cylinder. The BCs are that the ends are radially fixed but allow rotations and axial displacements (warping). The compression is applied at both ends. The sequence of zero-load spectral and buckling shapes in the form [hoop, length] are, respectively:

5 Studies of Postbuckled Shapes ♦

388

♦

0.

1.

2.

3.

4.

5.

Fig. 5.20 FE generated spectral stiffness plots for the axial compression of a circular cylinder. Only every other mode (odd number) is plotted, and the circles are for two initially consecutive even-number modes

spectral: [n, m] = [6, 1], [8, 1], [4, 1], [10, 1], [8, 2], [10, 2], [12, 1], [6, 2] buckling:

[n, m] = [6, 1], [8, 2], [10, 3], [ ?, ? ], [ ?, ? ], [10, 4], [12, 6], [ ?, ? ]

The question marks are for ambiguous shapes. It turns out that the spectral plots are fairly easy to interpret: the stiffness of the original lowest zero-load mode goes directly to zero. There are no intersections or interactions. And yet, there are severe FE convergence difficulties once ω12 → 0. This suggests that the postbuckle state is unstable. If so, this is something worth investigating. It should be pointed out that the nonlinear deformation of the cylinder does not exhibit any hints of the impending instability. It is only because of the sensitivity of the spectral stiffness that we actually get to monitor or observe the approaching instability. The solution we present is based on the classic paper of von Karman and Tsien [25]; it definitely makes clear how the shape changes during the postbuckling stage. It turns out that the crux of the postbuckling behavior of cylinders lies in establishing a good estimate of the stresses after the buckling. In other words, we need an accurate estimate of the membrane stiffness and our previous scheme of assuming a deflected shape to get the strains (and then the stresses) is simply not adequate enough and we directly use the equilibrium equations (in conjunction with assumed shapes) to establish the membrane stresses and strains. This proved effective in connection with arches. Because we leave out some details, we primarily use the notation of the cited reference, and this most closely resembles that of the flat plate in Fig. 2.14 but (w, z) is directed toward the center of curvature, (y, v) is in the hoop direction, and the origin is at the center of the cylinder. The membrane and flexural strains are

5.2 Plates and Cylinders

389

approximated by the von Karman relations of Eqs. (3.23) with the effect of curvature (−w/R) added to the Eyy strain. The elastic relation is assumed the same as plane stress so that the stresses and displacements are related by σxx = E ∗ [u,x + 12 w,2x +ν(v,y + 12 w,2y −w/R)] σyy = E ∗ [v,y + 12 w,2y −w/R + ν(u,x + 12 w,2x )] τxy = G [u,y +v,x +w,x w,y ]

(5.9)

The stress state clearly depends on all the displacements and is the same as for a flat plate except for the presence of w/R in the normal stresses. Compatibility is the same as for the shallow curved plate but with a sign change for the w/R term, giving ∇ 2 ∇ 2 ψ = E[w,2xy −w,xx w,yy −w,xx /R] ,

∇2 =

∂2 ∂2 + ∂x 2 ∂y 2

What it means for us here is that once we assume a reasonable distribution for the transverse displacement, this gives us the means to compute a stress distribution that is guaranteed to be in equilibrium. A subsequent integration of (a re-arranged) Eq. (5.9) then gives the associated membrane displacements. The membrane strain energy in terms of stress is given by Eq. (1.13), the flexural strain energy in terms of curvatures is given by the linear expression of Eq. (3.22), and the potential of the applied loads is 

L/2

V=−



L/2

σxx |a h dy

0

u,x dx , 0

u,x =

1 [σxx − νσyy ] − 12 w,2x E

These equations comprise the totality of the formulation. All that is now required is an assumed transverse deflection shape. The postbuckled shape is assumed to be a combination of the shapes I:

cos(mx/R) cos(ny/R);

II:

cos2 [(mx + ny)/2R] cos2 [(mx − ny)/2R]

The cylinder is assumed to have periodic BCs in x and therefore both wavenumbers can be referenced to R. Shape I is the linear buckling shape similar to that of a simply-supported flat rectangular plate. Shape II is the so-called diamond shape experimentally observed in the large postbuckled state. The combination of the two shapes is taken in the particular form w/R = [f0 + 14 f1 ] + 12 f1 [cos(mx/R) cos(ny/R) +

1 4

cos(2mx/R)

+ 14 cos(2ny/R)] + 14 f2 [cos(2mx/R) + cos(2ny/R)]

390

5 Studies of Postbuckled Shapes

where f0 , f1 , f2 are the nondimensional amplitudes (generalized DoFs wi /R) and f0 is introduced to account for the radial expansion of the cylinder. Observe that the cos2 terms result in double wavenumber shapes. On substituting this shape into the nonhomogeneous part of the compatibility equation, we get the collection of particular solutions R2  1 1 Ac01 cos m ¯ 2 x + 16 B cos n¯ 2 y + Cc11 cos m ¯ 1 x cos n¯ 1 y n2 16 +Dc13 cos m ¯ 3 x cos n¯ 1 y + Gc31 cos m ¯ 1 x cos n¯ 3 y  1 H c12 cos m ¯ 2 x cos n¯ 2 y + 12 αx 2 − 12 σo y 2 + 16

ψ = −Eμ2

where m ¯ i = im/R, n¯ i = in/R, μ = m/n, and c01 = 1/(0+μ2 )2 ,

c11 = 1/(1+μ2 )2 ,

c13 = 1/(1+9μ2 )2 ,

c31 = 1/(9+μ2 )2

The amplitudes are given by A = −Z0 + 18 f12 n2 ,

B = 18 f12 n2 ,

C = 12 f12 n2 Z0 − 12 f1

D = G = 14 f12 n2 Z0 ,

H = n2 Z02 ,

Z0 = 12 f1 + f2

Note that none of these coefficients contain the radial expansion f0 . Also note that triple wavenumbers have resulted but because μ is of order unity the associated coefficients c13 and c31 are quite small. The separate stresses are obtained by differentiation and result in the membrane energy  2  3 2 1 1 2 U / Uo = 4 (1 − ν 2 )(σo /E)2 + n4 64 f1 + 16 f1 f2 + 16 f2 + (f0 + 14 f1 )2    3 2 1 1 2 −2n2 64 f1 + 16 f1 f2 + 16 f2 (f0 + 14 f1 )   1 + 18 A2 + μ4 18 B 2 + c11 C 2 + c13 D 2 + c31 G2 + 16 c11 H 2 where Uo = 12 Ehab. The potential of the applied load becomes V/ Uo = 2(1 − ν 2 )(σo /E)2 + n2 (σo /E)(ν + μ2 )



3 2 32 f1

+ 18 f1 f2 + 18 f22



−2ν(σo /E)(f0 + 14 f2 ) Only UM and V contain f0 , therefore the equilibrium equation associated with this DoF becomes  ∂

3 2 = 2(f0 + 14 f2 ) − 2n2 64 f1 + ∂f0

1 16 f1 f2

+

1 2 16 f2



− 2ν(σo /E) = 0

5.2 Plates and Cylinders

391

This determines f0 in terms of f1 , f2 , and σo . It also shows that the average hoop stress is zero. The two other equilibrium equations can be combined to eliminate σo resulting in the cubic equation for the amplitude ratio ρ = f2 /f1 A0 + A1 ρ + A2 ρ 2 + A3 ρ 3 = 0 where on introducing η = n2 h/R, ξ = f1 R/ h = w/ h the coefficients are expressed as   1 1 + (ηξ )2 32 − 58 c11 − 18 c13 − 18 c31 μ4 + 2c11 μ4 − 12 A0 = (ηξ )2 32   − 23 η2 (1 + μ4 ) − 14 (1 + μ2 )2 /(1 − ν 2 )     1 1 A1 = (ηξ )2 16 + (ηξ )2 16 + 14 c11 + 14 c13 + 14 c31 μ4 − ηξ 12 + 8c11 μ4   +4c11 μ4 − 1 − 23 η2 2(1 + μ4 ) − 12 (1 + μ2 )2 /(1 − ν 2 )     A2 = (ηξ )2 92 c11 + 32 c13 + 32 c31 μ4 − (ηξ ) 12 + 8c11 μ4 A3 = (ηξ )2 [3c11 + c13 + c31 ]μ4 Figure 5.21a shows the amplitude ratio as a function of the deflection ξ for the given aspect ratio μ = m/n = 1 and R/ h = 100. The plot is antisymmetric about ξ = 0, and the common point has the value ρ = −0.5 giving f2 = − 12 f1 . This corresponds to shape I as introduced earlier. What is striking about the plot is the very rapid change of shape with deflection—the predominant change occurs within one plate thickness. With the value of ρ known, we can then determine the axial stress from the equilibrium equations. A useful form is σo R 1 μ2 (1 + ν 2 )2 η = + Eh η (1 + μ2 )2 12(1 − ν 2 ) μ2    1  2μ4  ρ + 2 (ηξ )2 c11 + 14 c13 + 14 c31 μ4 (ρ 2 + ρ) − (ηξ ) 18 + ημ (1 + μ2 )2    2μ4  1 1 1 1 1 −(ηξ 2 ) 64 + ( 64 + 14 c11 + 16 c13 + 16 c31 − (ηξ ) 16 + (1 + μ2 )2 This is shown plotted in Fig. 5.21b for different values of n (η) and μ = 1. The plots are symmetric with respect to the deflection but they have the significant feature that the immediate postbuckle behavior is unstable. That is, the buckling is of the symmetric unstable type. A second notable feature is the clustering of the modes along the ξ = 0 axis; Fig. 5.22a shows how, for large R/ h, there is a shallow

5 Studies of Postbuckled Shapes

0.0

1.0

♦

0.8

-0.4

0.6

.

.

-0.2

♦

♦

392

-0.6

0.4

-0.8

0.2

-1.0

0.0 -6.

-4.

-2.

0.

2.

.

4.

♦

-6.

6.

-4.

-2.

0. .

2.

4.

6.

1.0

3.

♦

♦

Fig. 5.21 Axial compression of a circular cylinder with R/ h = 1000. (a) Amplitude ratios. (b) Reduced compression stress against wave amplitude. Numbers correspond to n

0.8 2. .

.

0.6 0.4

1. 0.

0.2 ♦

2

6

10

14

.

18

22

26

30

0.0

♦

0.0

0.5

1.0 .

1.5

2.0

Fig. 5.22 Axial compression of a cylinder. (a) Stress at buckling as a function of n. Numbers are the ratio R/ h. (b) Compression stress against end shortening. Numbers are n

minimum stress behavior indicating that many modes buckle almost simultaneously. Because there are mode intersections for ξ > 0, there is a lower envelope on the curves; when μ = 1 this reaches a minimum and then increases similar to Fig. 3.1, but when μ = 0.5, the envelope exhibits a continuous decrease. The end shortening is obtained from the re-arranged stress-displacement relation in the form u,x =

1 (σxx − νσyy ) − 12 w,2x E

Substituting for the stresses and averaging on the cross section leads to σo R o R = + h Eh

1 16 ξ(ηξ )

  ρ 2 + ρ + 34

5.3 Mode Interactions with Softening

393

This is shown plotted in Fig. 5.22b. The interpretation of the plot is that once the critical load is reached, there is a substantial horizontal jump to one of the rising curves. However, because the curves are initially asymptotic to the primary path, then small disturbances can easily cause jumps to the unstable paths with consequent early buckling. Reference [12] presents a clear study of the effects of imperfections on the postbuckling behavior. While the solution confirms that the cylinder is postbuckle unstable, it does not identify the origin of this. The main reason for this difficulty is that a cubic equation is solved to get the amplitude ratio ρ. In this connection, mention should be made of the report by Langhaar and Boresi [13]; in a systematic way, they lay out the different orders of contributions to the postbuckle state. They show, at least for radial pressure loading, that the unstable behavior is due, not so much to the presence of shear, but to the nonlinear shear interaction arising from −w,x (v/R − w,s ).

5.3 Mode Interactions with Softening The buckling of thin-walled open sections is quite complicated because of the occurrence of both local and global buckling modes as discussed in connection with Fig. 4.41. The tension beam illustrates a situation where local buckling does not adversely affect the overall structural stiffness but there are a number of situations, however, where the structure experiences a softening mechanism due to the interaction of the local and global modes and results in catastrophic consequences. For example, Ref. [7] explores the case of elastic-plastic yielding during buckling, the case considered in Ref. [26] is that of wrinkling of flanges in a thinwalled structure. References [1, 6, 9, 17] examine the allied phenomenon called mode jumping which involves the interaction between two flexural modes in a plate. What these cases have in common is that there is a nonlinear coupling mechanism between the modes leading to an initially unstable postbuckled state. These problems of mode interactions with softening for continuous structures are not amenable to simple analytical exposition. Therefore, we first present some spectral plots data for flange buckling of an I-beam to motivate the analyses, this is followed by some mechanical modeling to help expound on the flange data.

5.3.1 FE Results for Flange Buckling Figure 5.23 shows some load-deflection FE data for an I-beam. In contrast to results such as Fig. 5.1b for a rectangular cross section, a nearby equilibrium state cannot be found, we emphasize nearby. This seems to indicate that this I-beam is postbuckle unstable. It must be that the flange makes the difference by creating a softening mechanism.

5 Studies of Postbuckled Shapes ♦

80.

♦

394

60.

.

40. 20. 0.

♦

0.00

0.01

.

0.02

0.03

♦

0.00

.

0.01

♦

Fig. 5.23 Load-deflection curves for two I-beams. (a) hf = 1.04ho , hw = 2hf . (b) hf = 1.00ho , hw = 2hf

♦

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 5.24 Postbuckling behavior of a flange. (a) Geometry indicating flexural DoFs and parameters. (b) Spectral plot behaviors, heavy lines are for simple beam modeling (scale σo = E/10)

Figure 5.24b shows the spectral behavior for the buckling of a plate with BCs chosen to mimic those of the flange of an I-beam. The question we want to explore is why the buckling predictions, using beam modeling, are different compared to the actual FE generated results. To help focus on the mechanisms involved, the FE results are for an isolated single (half) flange of an I-beam. Figure 5.24a indicates the flexural BCs, the membrane DoFs allow the axial compression and allow the plate to flex in the plane (this corresponds to the I-beam bending about the x-axis in Fig. 2.1a). The small extender to the right is like a periodic BC and is used to prevent excessive twisting in the plate. The bottom part of the extender cannot flex in the plane. We first establish the global mode behaviors. These correspond to two flexural modes and an end-shortening mode, all of which are in the plane of the plate. Figure 5.25 shows some spectral shapes, the first and third shapes on the left are

5.3 Mode Interactions with Softening

395

Fig. 5.26 Expanded spectral view of postbuckling behavior showing mode interaction and subsequent softening

♦

Fig. 5.25 Orthogonal views of the first three spectral shapes (bottom is lowest) for different levels of loading. Left is zero load and right is load level II in Fig. 5.26

♦

0.2

0.3

0.4

0.5

the first and second global bending shapes at zero loads. The middle shape is an example of a local mode which is described as a [3 × 1] mode (the extender is not counted). To get the global picture, the global stiffnesses are plotted on the vertical axis as indicated by the heavy lines in Fig. 5.24b. Then, after doing a buckling eigenanalysis, identify the shapes most similar to the global shapes and plot the load values on the horizontal axis. Finally, connect the data points for similar shapes; the lines B1 and B2 in Fig. 5.24b are for the first and second bending modes, respectively. Parenthetically, in this particular example, the stress levels are so high that in reality plastic yielding would occur but this is ignored here. The results indicate that the two flexing global modes do not interact; therefore, we just focus on the lowest modes. Figure 5.26 shows an expanded plot of the lowest four modes; the vertical dashed lines correspond to the load levels for the shapes shown in Figs. 5.25 and 5.27.

396

5 Studies of Postbuckled Shapes

Fig. 5.27 More spectral shapes for different levels of loading. Left is load level III and right is load level IV in Fig. 5.26

The right shapes in Fig. 5.25 show that it is the [5 × 1] local mode that first becomes critical. The postbuckled shape is dominated by this shape. Observe that the third mode is an interaction between the global and a local mode. On further load increase, there is a mode interaction between the global mode and the [4 × 1] local mode. This is confirmed at load level IV, Fig. 5.27. It is the global bending mode that becomes critical. Numerically, a nearby postbuckled configuration could not be found so that we can surmise that the postbuckled state is unstable. Returning to Fig. 5.24b, we see that two effects have occurred. First, the global buckling mode is at a lower load level than predicted by beam modeling. Second, and perhaps more importantly, when buckling does occur, the postbuckle state is unstable and therefore can lead to a catastrophic consequence. The next analyses construct simple (mechanical) models to help explain why this happens.

5.3.2 Mechanical Models for Mode Interactions The two aspects we focus on are the origin of a possible source of softening and the meaning and consequences of mode interactions. The objective is to develop a sense of how significant this can be.

Aspect I: Softening Mechanisms To make the general discussions more concrete, consider the spectral behavior of the stiffness matrix for the mechanical model in Fig. 5.28. The model in a simple way

397 ♦

5.3 Mode Interactions with Softening

♦

Fig. 5.28 Simple mechanical model for postbuckling of plates and beams. (a) Two-DoF mechanical model. (b) Spectral behavior for the mechanical model with K1 = K2 = 1, Kt = 1, and L = 1. Dashed lines indicate unstable paths

represents the behavior of beams and plates; it has axial compression and transverse flexure [20]. The model is made of rigid links each of length L with a torsional spring Kt representing the flexural stiffness, the axial springs are associated with the end shortening u1 . The two linear springs have different effects, as demonstrated presently; the spring K1 controls the postbuckling behavior, while the spring K2 introduces a nonlinear coupling into the system even for small displacements w2 and u1 . With the small-deflection assumption, the angles are given by φ1 ≈

w2 , L

φ2 ≈

w2 L

and the twist of the spring is φ = φ1 + φ2 = 2w2 /L. The end shortening of the links is computed as       = L[2 − cos φ1 − cos φ2 ] ≈ L 2 − 1 − 12 φ12 + · · · − 1 − 12 φ12 + · · · = Lφ12 ≈ w22 /L The strain energy for the membrane (axial spring) and flexure (torsion spring) actions become, respectively, UM = 12 K1 u21 + 12 K2 [u1 − ]2 = 12 [K1 + K2 ]u21 − 12 K2 [2u1 w22 /L − w24 /L2 ]  w 2 2 UF = 12 Kt [φ]2 = 12 Kt [2φ1 ]2 = 12 Kt 2 L We see the nonlinear contribution of the K2 spring in the last term of UM . What is also interesting to observe is that this spring couples the in-plane and out-of-plane

398

5 Studies of Postbuckled Shapes

Fig. 5.29 Postbuckling behavior of a model with a softening mechanism. (a) Nonlinear torsional spring behaviors. (b) End shortening with different slopes. (c) Transverse deflection

deflections because of the u1 w22 term. With V = −χ Po u1 , the total potential for the problem can be written as  w 2 2 + 12 [K1 + K2 ]u21

= U + V = 12 Kt 2 L − 12 K2 [2u1 w22 /L − w24 /L2 ] − χ Po u1 We detail the equilibrium paths for a modified system presently, here we show that the stiffness is     2 ∂

−2K2 w2 /L [K1 + K2 ] = [ KT ] = −2K2 w2 /L 4Kt − 2K2 [u1 L − 3w22 ]/L2 ∂u1 ∂w2 The spectral in Fig. 5.28b shows that the second mode, the flexural mode, goes to zero directly and although there is an intersection, there is no interaction. This is simple buckling behavior. As an illustration of softening/hardening, we introduce an additional nonlinear torsion spring stiffness. Specifically, we modify the total potential by replacing UF with UF = 12 Kt [φ 2 + 12 αφ 4 + · · · ] ,

T =

∂ UF = Kt [1 + αφ 2 + · · · ]φ ∂φ

Note that α can be either positive (hardening) or negative (softening) as shown in Fig. 5.29a. When α is negative, the torque-twist behavior resembles that of a beam in elastic-plastic bending and represents our softening situation. The total potential for the problem can now be written as

= U + V = 12 Kt [4φ 2 + 12 α32φ 4 ]+ 12 (K1 +K2 )u21 − 12 K2 L2 [2φ 2 u1 /L−φ 4 ]−λPo u1 where it is useful to use φ = 2w2 /L. Note that α has a contribution φ 4 similar to the noncoupling nonlinear term arising from K2 . The equilibrium equations are

5.3 Mode Interactions with Softening

399

∂

= (K1 + K2 )u1 − K2 Lφ 2 − λPo = 0 ∂x1  ∂  = 4Kt [1 + α8φ 2 ] − 2K2 L2 [u1 /L − φ 2 ] φ = 0 ∂φ There are two equilibrium paths. We obtain the first by setting φ = 0; we obtain the second by setting the bracketed terms to zero. This results in λPo w2 = 0 (K1 + K2 ) ' λPo − 2K1 Lγ λPo [1 + α16Kt /K2 L2 ] − 2Kt /L w2 = L II: u1 = K1 [1 + α16γ ] K1 L[1 + α16γ ] I: u1 =

where γ is the stiffness ratio introduced earlier and φ = 2w2 /L. The first path is simply the linear end shortening of the springs before buckling occurs. For the second path, buckling occurs when λPo = 2K1 Lγ and is independent of the torsional nonlinearity α. Where α plays a role is in the postbuckling behavior as shown in Fig. 5.29b, c. The end-shortening slope is given by ∂P K1 [1 + α16γ ] = ∂u1 1 + α16KT /K2 L2 When α is positive, the postbuckle slope is steeper; when it is negative, the postbuckle slope is shallower. Indeed, when α = −1/(16γ ), the postbuckle slope is flat, indicating a neutral stability. Of interest to us here is that, if α is sufficiently negative, that is, α < −1/(16γ ), then the postbuckling is unstable. The crucial ingredient here is the nonlinear contribution to the torsional stiffness represented by α. In the simple mechanical modeling, it is just a nonlinear spring but bear in mind that in actual situations such as the beam on an elastic foundation or a plate with constrained edges, this is a fairly complex nonlinear interaction associated with the actual large deflections behavior of the structure.

Aspect II: Secondary Buckling and Mode Jumping As pointed out earlier in reference to Fig. 5.28, the essential deformations for the buckling of plates is an in-plane compression with nonlinear coupling to the outof-plane flexure. Secondary buckling and mode jumping are associated with the interaction of two modes, hence we need at least two flexural DoFs. The mechanical model shown in Fig. 5.30 (which is a modified version of that introduced by Stein [18] and the following analysis is an extension of that in Ref. [6]) is made

400

5 Studies of Postbuckled Shapes

Fig. 5.30 Three-DoF mechanical model for a plate

of rigid links each of length L with torsional springs Kt representing the flexural stiffness. It also has axial springs associated with the shortening u1 . The spring K2 introduces a nonlinearity into the system even for small deflections w1 and w2 ; it is, therefore, sufficient for our purpose to work with small deflections. Consider the torsional springs first. With the small-deflection assumption, the angles are given by φ1 ≈ w1 /L ,

φ2 ≈ w2 /L ,

α ≈ (w1 − w2 )/L

The twists of the springs are then φ1 + ψ ≈ (2w1 − w2 )/L ,

φ2 − ψ ≈ (w1 − 2w2 )/L

The flexural strain energy (embedded in the torsional springs) is  UF =

1 2 Kt

2w1 − w2 L



2 +

1 2 Kt

2w2 − w1 L

2 =

 Kt  2 2 5w − 8w w + 5w 1 2 1 2 2L2

Because we do not intend to compare our results directly to some FE results, we find it useful to introduce new generalized DoFs defined as x1 ≡ 12 [w1 + w2 ],

x2 ≡ 12 [w1 − w2 ]

or

w1 = 12 [x1 + x2 ],

w2 = 12 [x1 − x2 ]

where x1 and x2 are the amplitudes of the symmetric and antisymmetric deformation modes, respectively. These play a role in helping us distinguish between the two flexural modes. After substitution, we get UF =

 Kt  2 2 x + 9x 2 4L2 1

This result is interesting for two reasons. First, the energies of the two modes are uncoupled and we can utilize this to add springs without affecting the coupling. Second, the antisymmetric mode has a larger coefficient and hence we expect the symmetric mode to buckle first.

5.3 Mode Interactions with Softening

401

The end shortening of the links is computed as  = L[3−cos φ1 −cos α −cos φ2 ] ≈

1 1 2 [w12 +8w22 +(w1 −w2 )2 ] = [x +3x22 ] 2L 4L 1

The membrane strain energy (embedded in the axial springs) is, therefore, UM = 12 K1 u21 + 12 K2 [u1 − ]2 = 12 [K1 + K0 ]u21 + K2 [x12 + 3x22 ]2 /32L2 − K2 [x12 + 3x22 ]u1 /4L The nonlinear contribution of the K2 spring is observed in the second and third terms. This spring also couples the in-plane and out-of-plane deflections. The total potential for the problem is

= UF + UM + V =

  Kt  2 K2  2 2 2 2 1 x x + + 9x [K + K ]u − + 3x 1 2 2 1 2 u1 2 4L 1 4L2 1 2 K2  2 2 x + + 3x − χ Po u1 (5.10) 1 2 32L2

This is the general structure of the potential function. This model can exhibit a bifurcation, but it cannot exhibit a mode jump because additional nonlinearities are required. We can utilize the uncoupling of the modes in the strain energy expression to modify the potential in various ways and, in particular, to add the extra nonlinearities. Consider a nonlinear axial and a nonlinear torsional spring attached at the center of the middle link, then we just add to the potential UA = 12 α1 x12 + 14 α2 x14 + · · · ,

UT = 12 β1 x22 + 14 β2 x24 + · · ·

(5.11)

respectively. Note that α2 and β2 can be either positive or negative. This gives us a mechanism to change the parameters of the system without affecting the mechanics of the problem. Rather than perform a parametric study using the general form of the potential— an exhaustive example of such a study for a plate on a nonlinear foundation is given in Ref. [4]; as done in Ref. [6], we find it more to the point to compare the above to a system known to exhibit mode jumping. Allman [1], as part of a discussion of mode jumping in plates, introduced an idealized problem with chosen parameters that make the manipulations less cumbersome. The potential energy (modified slightly) of the system is

=K

1

2 2 [x1

 + 4x22 + x32 ] − 12 [x12 + 3x22 ]x3 + 14 [x14 + 5x12 x22 + 6x24 ] − χ P u1

A combination of Eqs. (5.10) and (5.11) can be made to coincide with Allman’s by making the following associations:

402

5 Studies of Postbuckled Shapes

Kt 1 + α1 = K, α2 = K 6 2L2 3 9Kt 3 K2 = 2K, L = , + β1 = 4K, β2 = − K L 10 2L2 2

K1 + K2 = K, Po = K,

The main point of this comparison is that, in terms of the mechanical model with nonlinear springs, in order to have mode jumping we need a softening mechanism associated with the antisymmetric mode in the large postbuckling region; that is, β2 must be negative. The softening mechanism in Fig. 5.29 is associated with plasticity; it seems the softening mechanism in Fig. 5.26 is associated with the wrinkling. Having identified β2 as the key nonlinear effect for mode jumping, let us make the association of our system with that of Allman but parameterize the x24 contribution. That is, write the total potential as

= 12 [x12 + 4x22 + u21 ] − 12 [x12 + 3x22 ]u1 + 14 [x14 + 5x12 x22 + 6γ x24 ] − χ u1 The equilibrium equations are   ∂

= x1 1 + x12 + 52 x22 − u1 = 0 ∂x1   ∂

= x2 4 + 52 x12 + 6γ x22 − 3u1 = 0 ∂x2   1 2 ∂

= − 2 (x1 + 3x22 ) + u1 − χ = 0 ∂u1 There are four equilibrium paths even though there are only three equilibrium equations. The first three are obtained by setting: primary, x1 = 0, x2 = 0; symmetric, x2 = 0; and antisymmetric, x1 = 0; respectively, giving I: x1 = 0 x2 = 0 u1 = χ √ u1 = 2χ − 1 II: x1 = 2χ − 2 x2 = 0 √ x2 = (6χ − 8)/γ3 u1 = (12γ χ − 12)/γ3 III: x1 = 0 with γ3 = 12γ − 9. These paths are shown plotted in Fig. 5.31. Note that they do not exist for all positive loads; for example, the second path exists only for χ ≥ 1 and the third path for χ ≥ 43 . The third path is affected by γ ; if, for example, γ = 34 , then an infinite u1 occurs and the system is unstable. When γ < 34 , real solutions exist only for χ < 43 and these are unstable. The fourth path has nonzero x1 , x2 and is obtained by setting the first two bracketed terms to zero. This results in IV:

x1 =



[(30 − 24γ )χ − (34 − 24γ )]/γ4 ,

u1 = [(25 − 24γ )χ + (12γ − 11)]/γ4

x2 =

 [4 − 2χ ]/γ4

403

♦

♦

♦

5.3 Mode Interactions with Softening

2.0

.

1.0

0.0 .

♦

♦

♦

0.

2. 0.

.

2. 0.

1.

2. .

3.

4.

Fig. 5.31 Equilibrium paths. Solid lines γ = 1.0, dashed lines γ = 1.2

with γ4 = 13 − 12γ . This path is also shown plotted in Fig. 5.31 and exists only for a narrow load range, for example, 53 < χ < 2 when γ = 1.0. The deformed shape is a combination of the symmetric and antisymmetric shapes. We are also interested in distinguishing the stable and unstable portions of the paths. We get this stability information by looking at the spectral properties of the tangent stiffness matrix. This matrix is given by ) [ KT ] ≡

∂ 2

∂uI ∂uJ

*

⎡

⎤ 1 + 3x12 + 52 x22 − u1 5x1 x2 −x1 = ⎣ 5x1 x2 4 + 52 x12 + 18γ x22 − 3u1 −3x2 ⎦ −x1 −3x2 1

Of course, this matrix is symmetric but it is also nonlinear because it changes with the deformation. For consistency with Ref. [1], let the “mass” matrix be specified as [ M ] = [KT (uI = 0)] or ⎡

⎤ 100 [ M ] = ⎣0 4 0⎦ 001 The EVP to be solved for each equilibrium path is [ KT ]{φ } − ω2 [ M ]{φ } = 0 For path II, for example, with x2 = 0 we get ⎤⎧ ⎫ 1 + 3x12 − u1 − ω2 0 −x1 ⎨ xˆ1 ⎬ ⎣0 ⎦ xˆ2 = 0 4 + 52 x12 − 3u1 − 4ω2 0 ⎩ ⎭ 0 1 − ω2 uˆ 1 −x1 ⎡

Substituting for x1 , u1 and setting the determinant to zero leads to

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5 Studies of Postbuckled Shapes

   2 − χ − 4ω2 ω4 + (3 − 4χ )ω2 + 2χ − 2 = 0 giving the three eigenvalues path II:

ω12 = [2 − χ ]/4 ,

2 ω2,3 = [4χ − 3 ∓



16χ 2 − 32χ + 17]

Checking all three eigenvalues, we find the system is stable for 1 ≤ χ ≤ 2 and unstable for loads χ > 2. A similar analysis of the first path gives path I:

ω12 = 1 − χ ,

ω22 = 1 − 34 χ ,

ω32 = 1

which shows the path is unstable for loads χ > 1. We, therefore, have the following loading scenario: as the load increases from zero, the deformation is just the end shortening, at χ = 1 there is a bifurcation onto the stable path II and the load can continue to increase up to χ = 2 when the path becomes unstable. What happens next depends on the stability of paths III and IV which are affected by the parameter γ . Path IV is the nearby path and is stable if γ > 13 12 , this is shown as the dashed line in the figure. In this case for increasing load, there is a gradual decrease in x1 and an increase in x2 . When this path intersects III, there is another bifurcation onto path III, but the load can increase indefinitely because the path is stable. When γ < 13 12 , path IV is unstable and shown by the full line for γ = 1.0. Therefore, for χ > 2 there is no nearby stable state. However, for γ = 1.0, the eigenvalues for path III are path III: ω12 = χ −

5 3

2 = 1 (6χ − 7) ∓ , ω2,3 2

1 2



3(12χ 2 − 30χ + 19)

All are positive for χ > 2 and therefore there is a jump onto this path analogous to a snap-through buckling. This occurs in a violent manner. In this model only two flexural modes interact, but it can be imagined that in more complex systems such as shells or plates with distributed elastic constraint that give mode clustering (as illustrated in Figs. 4.33 and 5.24), that a complicated sequence of mode jumps could occur with outcomes quite different depending on small changes of parameters. This, once again, highlights the need for a full nonlinear analysis when dealing with stability problems. In this regard, Ref. [9] and its cited references are a good starting for learning which computational tools are required for exploring FE based mode jumping analyses.

5.3.3 Comments Related to Flange Buckling Returning to the flange problem, looking at Fig. 5.24b, if local buckling (or wrinkling) did not occur, buckling would have occurred at 0.8 and the postbuckle

5.3 Mode Interactions with Softening

405

state would be stable. But because the postbuckle behavior of the wrinkling (beginning around 0.25) is stable, the subsequent instability (beginning around 0.45) is postbuckle unstable. While this was not navigated, the fact that there were numerical difficulties is indicative that equilibrium is very far away and not at what would have been the unwrinkled equilibrium path. For the I-beam, we see that the global buckling modes (beam bending) are affected by the wrinkling. While the system remains stable after the wrinkling, it is reasonable to conjecture that the subsequent behavior is affected by the softening. That is, think of the flange as like a buckled strut, it can continue to support the current load but its ability to support additional load is diminished. It seems, however, that the situation is worse than that, because once the second instability occurs, there are no nearby equilibrium states and the system is postbuckle unstable.

Explorations 5.1 Consider an “extensible elastica” where the axial force is related to the axial straining by F = EAE1 , E1 = s/so . • Show that the potential energy for the case shown in Fig. 5.1 is 



=

1 2

=

1 2

EI 

 EI

dφ ds dφ ds

2

 ds + 12 EA

2 ds −

P2 2EA

E12 ds + P u(L) 

 cos2 φ ds + P

[cos φ − 1] ds

• Show that the governing equation is EI

d 2φ P2 cos φ sin φ = 0 + P sin φ − EA ds 2

• Obtain an approximate solution and compare to the inextensible elastica. —Reference [15, pp. 8446] 5.2 The potential energy of a particular system is given by

= 12 K1 l 2 [sin2 φ1 + γ (1 − cos φ1 )2 ] + 12 K2 [sin φ2 − l(1 − cos φ1 )]2 −P [l(1 − cos φ1 ) + L(1 − cos φ2 )] where K1 , K2 are stiffnesses, L, l are lengths, φ1 , φ2 are rotation DoFs, γ is a positive parameter, and P is the load. • Determine the equilibrium paths and their stability. —Reference [11, pp. 104]

406

5 Studies of Postbuckled Shapes

References 1. Allman, D.J.: On the general theory of the stability of equilibrium of discrete conservative systems. Aeronaut. J. 27, 29–35 (1989) 2. Bathe, K.-J.: Finite Element Procedures. Prentice-Hall, Englewood Cliffs, NJ (1996) 3. Boresi, A.P., Sidebottom, O.M., Seely, F.B., Smith, J.O.: Advanced Mechanics of Materials, 3rd edn. Wiley, New York (1978) 4. Chen, Y-C.: Stability and bifurcation of finite deformations of elastic cylindrical membranes— Part I: stability analysis. Naturwissenchaften, 34(14), 1735–1749 (1997) 5. Donnell, L.H.: A new theory for the buckling of thin cylinders under axial compression and bending. ASME Aeronaut. Eng. 56, 795–806 (1934) 6. Doyle, J.F.: Nonlinear Analysis of Thin-walled Structures: Statics, Dynamics, and Stability. Springer, New York (2001) 7. Doyle, J.F.: Guided Explorations of the Mechanics of Solids and Structures: Strategies for Solving Unfamiliar Problems. Cambridge University Press, Cambridge (2009) 8. Doyle, J.F.: Nonlinear Structural Dynamics Using FE Methods. Cambridge University Press, Cambridge (2015) 9. Fritsche, L., Haughk, M.: Static path jumping to attain postbuckling equilibria of a compressed circular cylinder. Comput. Mech. 26, 259–266 (2000) 10. Hanaor, A.: Principles of Structures, Blackwell Science, Oxford (1998) 11. Hunt, G.W., Williams. K.A.J.: Closed-form and asymptotic solutions for an interactive buckling model. J. Mech. Phys. Solids 32(4), 101–118 (1984) 12. Hutchinson, J.: Axial buckling of pressurized imperfect cylindrical shells. AIAA J. 3(8), 1461– 1466 (1965) 13. Langhaar, H.L., Boresi, A.P.: Snap-through and postbuckling behavior of cylindrical shells under the action of external pressure. Bull. Eng. Exp. Station 54(59), 33 (1957) 14. Lau, J.H.: Large deflections of beams with combined loads. Eng. Mech. ASCE 108, 180–185 (1982) 15. Magnusson, A., Ristinmaa, M., Ljung, C.: Behaviour of the extensible elastica solution. Int. J. Solids Struct. 38, 8441–8457 (2001) 16. Schenk, M., Guest, S.D.: On zero stiffness. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 228, 1701–1714 (2013) 17. Stein, M.L.: Loads and deformations of buckled rectangular plates. NASA Technical Report R-40 (1959) 18. Stein, M.L.: The phenomenon of change in buckle pattern in elastic structures. NASA Technical Report R-39 (1959) 19. Thompson, R.B.: Application of elastic wave scattering theory to the detection and characterization of flaws in structural materials. In: Johnson, G.C. (ed.) Wave Propagation in Homogeneous Media and Ultrasonic Non-destructive Evaluation. Applied Mechanics Division (AMD), vol. 62, pp. 81–73. American Society of Mechanics Engineer, New York (1984) 20. Thompson, J.M.T., Hunt, G.W.: A General Theory for Elastic Stability. Wiley, London (1973) 21. Thompson, J.M.T., Hunt, G.W.: Elastic Stability, Wiley, London (1984) 22. Timoshenko, S.P., Gere, J.M.: Theory of Elastic Stability. McGraw-Hill, New York (1963) 23. Ventzel, E., Krauthammer, T.: Thin Plates and Shells, Marcel Dekker (2001) 24. Volmir, A.S.: Flexible plates and shells. Technical Repost No. 66–216 (translated from Russian), Wright-Patterson Air Force Base, Ohio (1967) 25. von Kármán, Th., Tsien, H.-S.: The buckling of thin cylindrical shells under axial compression. J. Aeronaut. Sci. 8(8), 303–312 (1941) 26. Wadee, M.A., Gardner, L.: Cellular buckling from mode interaction in I-beams under uniform bending. Proc. R. Soc. A 468, 245–268 (2012) 27. Yoo, C.H., Lee, S.C.: Stability of Structures: Principles and Applications. Elsevier, Amsterdam (2011) 28. Ziegler, H.: Principles of Structural Stability. Ginn and Company, Lexington, MA (1968)

Index

A Amplitude ratio, 54, 120, 201, 341 Anticlastic bending, 122 Arch, 79, 160, 230, 302, 368 Asymmetric bifurcation, 192 B Beam, Bernoulli-Euler model, 31 Beam-column equation, 278 Beam, Timoshenko model, 142 Bifurcation point, 270, 401 Biharmonic equation, 124 Boundary condition (BC), 30, 131, 278, 311 Buckling analysis, 271 Buckling instability, 1, 267, 289, 355, 399 C Cable, 18, 26, 209 Cell, 116 Cellular buckling, 291, 321 Centroid, 139 Closed section, 15, 16 Compatibility, 124 Complementary shape, 180 Computer experiment, 4 Conservative system, 186 Consistent geometric stiffness, 286 Continuous system, 7 Coupled-beam equation, 278 Curved beam, 230 D Deformation gradient, 204, 216 Deformation mode, 163

Degree of freedom, 11, 51, 286 Donnell model, 136, 169, 329, 373, 385 Dynamic stiffness, 169

E Eigenbuckling, 273 Eigenvalue/vector, 281 Eigenvalue problem (EVP), 54, 273 Elastica, 215, 273, 356 Elastic constraint, 37 Elastic force, 41, 187, 194, 199 Elastic foundation, 37, 38, 287 Elasticity, 20 Elastic stiffness, 195, 261, 283 Equilibrium path, 187, 260, 270 Expansion theorem, 384

F Flange, 9, 393 Flexural behavior, 17, 76 Flexural stiffness, 31 Folded plate structure, 16, 79 Follower load, 186 Frame, 79 Frame element, 52 Funicular shape, 18

G Galerkin method, 365 Gaussian curvature, 131 Generalized DoF, 41 Geometric BC, 32

© Springer Nature Switzerland AG 2020 J. F. Doyle, Spectral Analysis of Nonlinear Elastic Shapes, https://doi.org/10.1007/978-3-030-59494-7

407

408

Index

Geometric stiffness, 196, 261, 283 Global mode, 176

Moment of inertia, 23 Multiply-connected cross section, 116

H Hanger, 264 Hooke’s law, 20

N Natural BC, 32 Neutral axis, 17 Neutral equilibrium, 190 Normal component, 207

I Imperfection, 200 Indicial notation, 19 Inertia, 76 Inextensible behavior, 60, 87, 97, 98, 119, 215, 253, 330, 357 Inextensible member, 102 Initial stress stiffness, 261 Integration by parts, 29 Interaction energy, 218 Interpolation function, 51

J Jacobian, 205

K Kirchhoff shear, 108 Kirchhoff stress, 205

L Lagrangian strain, 204 Lateral buckling, 336, 380 Limit point instability, 188, 192, 269 Loading equation, 49, 197, 238 Loading path, 195 Load potential, 186 Lumped mass, 76

M Mass matrix, 75 Maxwell’s reciprocity, 167 Mechanical model, 396 Member load, 23, 89, 161, 243 Membrane behavior, 9, 17, 104 Mixed Ritz method, 93, 103, 237 Modal matrix, 56 Mode cluster, 289, 321, 334 Mode interaction, 178, 218, 242, 252, 275, 306, 333, 343, 393 Mode jump, 393, 401 Mode shape, 280

O Open section, 16, 79 Orthogonality, 54, 69, 384

P Periodic BC (pBC), 94, 164, 168, 227, 373, 389 Ping load, 196 Plane stress, 20, 106, 141, 206, 214 Plate, 104 Pluck load, 59 Poisson’s ratio, 15 Poke load, 195 Prestress, 268 Principle of virtual work (PoVW), 26, 186, 222, 295, 372 Proportional loading, 269, 322, 375, 384

R Reduced model, 4 Ritz direct/semi-direct method, 7, 125 Ritz function, 40 Ritz method, 21, 44, 281 Rod element, 52

S Sag, 18, 186 Secondary torsion, 155 Second order effect, 27, 193, 199, 200, 303, 336, 358 Sectorial moment, 155 Shape function, 51 Shear, 208 Shear center, 136, 140 Shear flow, 116, 139 Shear modulus, 15 Shell, 79 Snap-through buckling, 366 Softening mechanism, 192, 223, 238, 333, 362, 364, 378, 393 Solid cross section, 15

Index Spectral analysis, 3, 59, 287 Spectral decomposition, 56, 160, 162, 247 Spectral shape, 59, 165, 254, 275, 383 Spectrum relation, 62, 64, 163 Statically indeterminate, 32 Static instability, 274 Stationary potential energy, 187 Stiffness, 185 Strain, 204 Strain-displacement relation, 19 Strain energy, 22, 83, 106, 400 Stress function, 124, 372 Stretch, 207 Strong formulation, 7, 21 Structural depth, 17, 18, 368 St. Venant’s principle, 131 St. Venant torsion, 117, 147, 156, 316 T Tangent stiffness, 194, 403 Tension beam, 383 Tensor, 204 Thin plate, 16 Thin-wall cross section, 15 Total potential energy, 186, 188

409 Total stiffness, 196 Traction, 10, 20, 130, 144 Tube, 116

U Uniaxial stress state, 14

V Variational principle, 21, 26, 86, 125 Virtual displacement, 25 Virtual work, 25, 26 von Karman plate, 218, 250

W Warping, 13, 112, 153, 168 Warping constant, 148, 155 Wavenumber, 62, 118, 291, 378, 390 Wrinkling, 393

Y Young’s modulus, 15