Integrodifferential Relations in Linear Elasticity 9783110271003, 9783110270303

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De Gruyter Studies in Mathematical Physics 10 Editors Michael Efroimsky, Bethesda, USA Leonard Gamberg, Reading, USA Dmitry Gitman, São Paulo, Brasil Alexander Lazarian, Madison, USA Boris Smirnov, Moscow, Russia

Georgy V. Kostin Vasily V. Saurin

Integrodifferential Relations in Linear Elasticity

De Gruyter

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ISBN 978-3-11-027030-3 e-ISBN 978-3-11-027100-3 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek 7KH'HXWVFKH1DWLRQDOELEOLRWKHNOLVWVWKLVSXEOLFDWLRQLQWKH'HXWVFKH1DWLRQDOELEOLRJUD¿H detailed bibliographic data are available in the internet at http://dnb.dnb.de. © 2012 Walter de Gruyter GmbH & Co. KG, Berlin/Boston Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ˆ Printed on acid-free paper Printed in Germany www.degruyter.com

Preface

The book is the result of the authors’ activity in the linear theory of elasticity and finite element analysis for the last ten years at the Institute for Problems in Mechanics of the Russian Academy of Sciences. In spite of the fact that the basis statements of linear of elasticity were formulated in the 19th century, it still remains very attractive for many scientists. In the last decades, various numerical approaches, in which finite element formulations based on conventional variational principles as well as Galerkin algorithms play a key role, have been actively developed. In these methods, some equations of elasticity (equilibrium equations, boundary conditions in terms of stresses and etc.) are given in generalized forms, whereas the stress-strain relation of Hooke’s law is considered, often implicitly, as a strict local equality. The governing relations in the linear theory of elasticity describe the deformed state of an elastic body at any internal point, and the stresses and displacements in interior should tend to their boundary values. It is implied that the components of the elastic modulus tensor are also continuous functions of spatial coordinates including the boundary. But it should be taken into account that boundary conditions are generated by specific physical and geometrical factors, for example, some part of the boundary can be an interface between two or more media. In this case, any such point belongs simultaneously to the body under consideration and the bodies producing these boundary conditions, that is, to parts of continuum with different mechanical properties. Strictly speaking, the elastic modulus tensor on such surfaces is not rigorously defined. To introduce this uncertainty into linear elasticity problems explicitly, an integral statement of the stress-strain relation is proposed by the authors in the method of integrodifferential relations. The boundary value problem modified in accordance with this approach can be reduced to the minimization of a nonnegative functional over all admissible displacements and equilibrium stresses. Such reformulation became a starting point to develop an advanced numerical technique to stress-strain analysis and solution quality estimation in linear elasticity. The book can be tentatively divided into three essential parts. The first part deals with basic concepts of the linear theory of elasticity that provide the foundation for the other two parts. In Chapter 2, conventional notations for the stress, strain, and displacement fields, governing equations, as well as the formulations of boundary value problems are introduced and discussed. The relations of simplified models, such as elastic beams and membranes, are considered. The classical and generalized

vi

Preface

variational principles in linear elasticity are described in the next chapter. Emphasis is placed on their applications to numerical solution of boundary value problems. The middle part of the book (Chapters 4–7) is devoted to the method of integrodifferential relations. Various ways to weaken the constitutive relation between the equilibrium stresses and kinematically admissible displacements, given usually in the local form, are discussed in Chapter 4. A parametric family of quadratic nonnegative functionals, Hooke’s law in the integral form, is introduced, and corresponding minimization problems for analysis of stress-strain states as well as natural vibrations of elastic bodies and structures are formulated. Numerical algorithms based on polynomial approximations of unknown stress and displacement functions and suitable minimization technique are described, and their efficiency is illustrated on the example of two-dimensional static and dynamic problems. In Chapter 5, variational properties of the quadratic error functionals introduced are discussed. It is shown that the stationary conditions of these functionals together with the equilibrium equation and boundary conditions are equivalent to the complete system of equations in the linear theory of elasticity. In addition, the integrodifferential formulations have a direct relation to conventional variational principles and make it possible to decompose the problem into two independent subproblems of displacements and of stresses, respectively. The bilateral estimates of the elastic energy stored are derived for different types of boundary conditions. Chapter 6 focuses on numerical approaches grounded on the method of integrodifferential relations and advanced finite element technique. An adaptive algorithm using high order piecewise polynomial approximations of kinematically admissible displacements and equilibrium stresses for an arbitrary triangulation is presented. Various mesh refinement and mesh adaptation strategies based on explicit local and integral energy estimates are discussed. Chapter 7 is devoted to the approach in which the original problem in partial differential equations is approximated by a system of ordinary differential equations. The variational technique developed is spread on the case of semi-decrete approximations for the displacement vector and stress tensor including, on the one hand, a polynomial expansion of finite dimension over some coordinate components and, on the other hand, unknown functions over one remaining component. The approach is illustrated on plane static and dynamic beam problems. The last part of the book deals with an alternative to the approaches discussed in the previous chapters. In the asymptotic approach considered in Chapter 8, the consistent boundary value problem is composed of the appropriately selected coefficients in the semi-discrete polynomial expansion of the stress-strain relation. The attractiveness of this technique is that the differential order of the approximate system of equations is half the one following the variational approach. The corresponding algorithms for two- and three-dimensional static modeling as well as natural vibration analysis of elastic beams are presented.

Preface

vii

In what follows, a modification of the Petrov–Galerkin method based on integral projection of the stress-strain equation on a special set of test functions and semidiscretization of admissible displacement and stress fields is described. The relations among the projection, asymptotic, and variational approaches are discussed in Chapter 9. Numerical projection algorithms are developed to design least-dimensional systems that guarantee given solution quality. Chapter 10 focuses on applications of the projection approach to spatial static problems for elastic beams with an asymmetric cross section. Special attention is paid to effective computation of such mechanical characteristics as bending and torsion stiffnesses, flexural center coordinates, etc. The influence of cross-section shape and boundary conditions (Saint–Venant’s principle) on beam deformation is numerically quantified. In Chapter 11, the approach on the basis of the projection technique is spread on the three-dimensional dynamic case. The frequency-wave analysis of natural and forced beam vibrations is performed. General features of eigenfrequencies and eigenforms are illustrated on the example of combined torsional, longitudinal, and lateral vibrations for beams with triangular cross-sections. Spectrum characteristics of the beams and their specific resonance properties caused by the lack of symmetry are discussed. The appendix contains the most important definitions and necessary information on the vector and tensor algebra as well as functional analysis. Potential readers of this book will be mathematicians, engineers, as well as graduate and postgraduate students who are interested in learning the mathematical basis, modeling, and numerical technique in solid mechanics.

Contents

Preface

v

1

Introduction

1

2

Basic concepts of the linear theory of elasticity

6

2.1 Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.2 Linear strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.1 Static statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.2 Dynamic problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3

2.5 Simplified models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Elastic rods and strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Beam models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Plane stress and strain states . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 35 38 40

Conventional variational principles

43

3.1 Classical variational approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Energy relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Direct principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Complementary principles . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43 44 47

3.2 Variational principles in dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3 Generalized variational principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3.1 Relations among variational principles . . . . . . . . . . . . . . . . . . . 55 3.3.2 Semi-inverse approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.4 Finite dimensional discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Ritz method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Galerkin method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Boundary element method . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 62 64 66 72

x 4

Contents

The method of integrodifferential relations

73

4.1 Basic ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.1.1 Analytical solutions in linear elasticity . . . . . . . . . . . . . . . . . . . 73 4.1.2 Integral formulation of Hooke’s law . . . . . . . . . . . . . . . . . . . . . 78 4.2 Family of quadratic functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.3 Ritz method in the MIDR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.3.1 Algorithm of polynomial approximations . . . . . . . . . . . . . . . . . 83 4.3.2 2D clamped plate – static case . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4 2D natural vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.4.1 Eigenvalue problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.4.2 Free vibrations of circular and elliptic membranes . . . . . . . . . . 93 5

Variational properties of the integrodifferential statements

101

5.1 Variational principles for quadratic functionals . . . . . . . . . . . . . . . . . . . 101 5.2 Relations with the conventional principles . . . . . . . . . . . . . . . . . . . . . . 103 5.3 Bilateral energy estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.4 Body on an elastic foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.4.1 Variational principle for the energy error functional . . . . . . . . 114 5.4.2 Bilateral estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6

Advance finite element technique

124

6.1 Piecewise polynomial approximations . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.2 Smooth polynomial splains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Argyris triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Stiffness matrix for the Argyris triangle . . . . . . . . . . . . . . . . . . 6.2.3 C 2 approximations for a triangle element . . . . . . . . . . . . . . . . .

127 127 132 133

6.3 Finite element technique in linear elasticity problems . . . . . . . . . . . . . 136 6.4 Mesh adaptation and mesh refinement . . . . . . . . . . . . . . . . . . . . . . . . . 145 7

Semi-discretization and variational technique

158

7.1 Reduction of PDE system to ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Beam-oriented notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Semi-discretization in the displacements . . . . . . . . . . . . . . . . . 7.1.3 Semi-discretization in the stresses . . . . . . . . . . . . . . . . . . . . . .

158 158 160 162

7.2 Analysis of beam stress-strain state . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 7.3 2D elastic beam vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Contents

8

An asymptotic approach

xi 177

8.1 Classical variational approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 8.2 Integrodifferential approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 8.2.1 Basic ideas of asymptotic approximations . . . . . . . . . . . . . . . . 182 8.2.2 Beam equations – general case of loading . . . . . . . . . . . . . . . . 187 8.3 Elastic beam vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Statement of an eigenvalue problem . . . . . . . . . . . . . . . . . . . . . 8.3.2 Longitudinal vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Lateral vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

190 190 194 199

8.4 3D static problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 9

A projection approach

218

9.1 Projection formulation of linear elasticity problems . . . . . . . . . . . . . . . 218 9.2 Projections vs. variations and asymptotics . . . . . . . . . . . . . . . . . . . . . . 222 10 3D static beam modeling

227

10.1 Projection algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 10.2 Cantilever beam with the triangular cross section . . . . . . . . . . . . . . . . . 241 10.3 Projection beam model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 10.4 Characteristics of a beam with the triangular cross section . . . . . . . . . . 248 11 3D beam vibrations

252

11.1 Integral projections in eigenvalue problems . . . . . . . . . . . . . . . . . . . . . 252 11.2 Natural vibrations of a beam with the triangular cross section . . . . . . . 255 11.3 Forced vibrations of a beam with the triangular cross section . . . . . . . . 266 A

Vectors and tensors

269

B

Sobolev spaces

271

Bibliography

274

Index

278

Chapter 1

Introduction

One of the aims of mathematical physics is development of specific models to study various natural phenomena. The theory and methods of this scientific discipline are widespread because its models are based on fundamental laws of nature such as the laws of conservation of energy, momentum, mass, charge, and so on. In particular, this leads to the fact that the same model can describe different physical processes. Moreover, the methods of mathematical physics can be applied to other areas of science. These approaches are currently used in chemistry, geology, biology, ecology, economy, etc. They also have application in engineering for modeling of various technical systems and devices. Modeling is considered as a reliable replacement of an investigated object with its mathematical description and following analysis, as a rule, by methods of computational mathematics. The objectives of mathematical physics are closely related to investigation of processes in a system with distributed parameters typically occupying a spatial region (so-called continuous medium). The quantities describing the state of the medium and its behavior depend usually on the spatial coordinates and time. The progress of mathematical modeling for such distributed processes is historically stimulated by the advances in continuous mechanics and electrodynamics. Models reflecting the behavior of such systems can be divided into three levels: relation of the whole media with the environment, interaction between the elementary system volumes, and properties of a single volume element. The relation at the first level is defined by external conditions including, in the general case, the boundary and initial constraints. The second level corresponds to the interaction of elementary volumes in accordance with the constitutive laws and takes into account the transfer of material particles in space that gives the possibility to derive equations of exchange processes. Finally, the third level characterizes the properties of the medium in the representative elementary volume. Systems with distributed parameters are typically described by partial differential equations (PDEs) and, in some cases, by integral or integrodifferential relations. These models may also involve functionals of unknown variables. On an admissible set of functions, such a functional attains its stationary value, which corresponds to a stationary point, i.e., desired solution of the problem. This is usually associated with the problem statement based on a variational principle, which has a definite physical meaning. In certain cases, the solution can correspond to an extremum of the functional. The classification of PDE systems and their relations to the calculus

2

Chapter 1 Introduction

of variations can be found in classical books of Courant and Hilbert [16] as well as Morse and Feshbach [51]. A variety of natural phenomena generates a wide spectrum of approaches to solving problems of mathematical physics. Detailed description of developed methods and approaches in computational mechanics are presented in the encyclopedia [67]. Among these methods, it is worth highlighting three majors which have received a significant development, especially in recent years, namely, variational, projection, and least squares (LSQ) methods. All these approaches have obvious advantages as well as some disadvantages, on which we would like to focus our attention. Variational principles and their application in many fields of physics, including the theory of elasticity, have a long history. However, the importance of these principles has been clearly understood only thanks to the advances in the finite element method (FEM), which traces back to Courant [15] and Turner [74]. Since then, it has been repeatedly proved that the variational technique is a powerful tool in the mathematical formulation of FEM problems. Vice versa, the rapid development of this method has stimulated the improvement of variational approaches. The basic ideas of the FEM can be found, for example, in the books [7, 58]. An important feature of variational principles is that the governing equations directly follow as the stationary conditions of the corresponding functional. In addition, variational formulations have several advantages as compared with the statement in PDEs. Firstly, the variational technique is suitable to transform the problem initially given in PDEs to an equivalent one, which is solved simpler than the original. In variational formulations under additional constraints, this conversion is carried out by using the method of Lagrange multipliers, which is a very effective and regular procedure. In this way, it is possible to get a family of variational principles that are equivalent to each other. Secondly, if the exact solution of the problem cannot be found, then the variational method often provides a variety of finite dimensional formulations to obtain an approximate solution. Thirdly, the implementation of variational principles guarantees the stability of numerical algorithms and the optimality of approximate solutions. At that, the resulting system of equations is usually symmetric and positive definite. Among the shortcomings of variational approaches, it may be noted that not all problems of mathematical physics allow one to formulate any variational principles. It is also rather sophisticated to obtain reliable estimates of solution quality. When finding the approximation to a variational problem formulated with Lagrange multipliers, for example, on the basis of the Hu–Washizu principle in linear elasticity [77], the problem loses the property of positive definiteness and symmetry. Projection methods such as Galerkin [9], Petrov–Galerkin method [6], etc., are deprived of some shortcomings inherent in variation approaches. First, these methods are applicable to problems for which variational principles have not been formulated

Chapter 1 Introduction

3

yet. Second, the projection methods are more flexible in assembling the governing system of equations. A recent survey and review concerning the discontinuous Galerkin methods can be found in [28]. Projection approaches certainly have their drawbacks. In particular, the choice of test and trial functions is a procedure that is not always unambiguous and simple. Sometimes, it is difficult to provide the stability of numerical algorithms and to ensure their convergence, especially in nonlinear problems. Just as for variational approaches, it is often expensive to construct reliable estimates of the quality of approximate solutions. The third approach, which can also be attributed to the general methods of mathematical physics, is the LSQ method. The state-of-the-art in the LSQ approach with application to the FEM is presented in [11]. It looks natural enough to compose a non-negative functional in the following way. All the equations describing the phenomenon under study are squared, summarized, and integrated over space and time. Moreover, it is known in advance that the global minimum of this integral is equal to zero. Conventional strategies of the FEM can be applied in the LSQ method to finding approximate solutions. At that, implicit bilateral estimates of solution quality can be accordingly constructed. The lower bound of the functional is known, and the value of the functional on an approximate solution can be always chosen as the upper bound. However, it should be noted that the Euler equations (stationary conditions) for this minimization problem, in general, differ from the PDE system that generates this functional. In other words, the problem resulting from the LSQ method is a variational principle for another boundary value problem. So, the questions of existence and uniqueness of the solution to this system require further studies. One common characteristic feature inherent in all of the above-mentioned methods is some ambiguity in the formulation of a finite approximation problem. It is not clear what relations are best to be weakened. As an example, let us consider the equations of linear elasticity. In the original statement, there are 15 variables, namely, 12 components of stress and strain tensors as well as three components of displacement vector, which correspond to 9 PDEs (equilibrium and kinematic equations) and 6 algebraic constitutive relations (Hooke’s law). If all the relations including the boundary conditions are taken in integral (weak) form, then this statement coincides with the Hu–Washizu principle, which contains 18 variables (three Lagrange multipliers are added) and no constraints are imposed on them. The physical meaning of the Lagrange multipliers follows from the stationary conditions of the corresponding functional. By requiring the implementation of certain governing equations, the number of independent variables in the variational formulation can be reduced. For instance, it is possible to derive the Hellinger–Reissner principle in which there are 12 unknown functions. After successive elimination of variables, the classical principle of minimum total potential energy is obtained, in

4

Chapter 1 Introduction

which the only three variables, components of the displacement vector, remain. The equivalence of these principles was theoretically justified, e.g., in [77], but from a practical point of view, it is a considerable difference whether the problem is solved with respect to either 3 or 15 functions. Similar ambiguity characterizes projection approaches. In composing a projection system, the choice of appropriate test and trial function spaces is of great importance. In addition to limitations pointed above, the LSQ method is rather sensitive to the choice of weighting coefficients. The presence of such factors is stipulated by the fact that the governing relations are of different dimensions. The equilibrium equations have the physical dimension of force per unit volume, the relations of Hooke’s law can be, e.g., dimensionless like kinematic conditions, boundary conditions can be given in units of either length or force per unit area. Note that this is not a simple task to determine the appropriate weighting coefficients for a given set of equations. An approach which benefits of the advantages and takes into account the abovementioned shortcomings of the variational and projection methods, as well as the LSQ technique, is discussed in this book. The authors refer to it as the method of integrodifferential relations (MIDR) [34]. The essence of this approach lies in the fact that some part of governing equations must be exactly satisfied, but the other relations are considered in integral form. It is defined a priori, often from the physical point of view, which of these relations should be weakened. For example, in heat transfer problems, only Fourier’s law is taken into account integrally whereas the first law of thermodynamics, initial and boundary conditions are satisfied exactly [63]. In numerical simulations of linear elasticity problems, approximate stress and displacement fields strictly obey the equilibrium equations, kinematic relations, and boundary conditions. Whereas the relations of Hooke’s law are weakened, i.e., satisfied in some integral sense [61] or projected onto a finite dimensional function subspace [44]. It looks rather reasonable in numerical realization to present Hooke’s law as an integral over a function which is a quadratic form of the stressstrain relations. An approximate solution of the integrodifferential problem is found by minimizing its functional under the differential constraints in the form of equilibrium equations, kinematic relations and boundary conditions. This formulation is fully consistent with the ideas of the LSQ method, but it is simultaneously a variational principle. So, the variational and LSQ methods coincide in this case. Further studies have shown that there exist other positive definite quadratic forms representing Hooke’s law, which are not obligatory perfect squares, but, at the same time, the basis of variational principles. Thus, it was considered the functional of energy error, which gives one the possibility to divide the problem originally formulated in terms of stresses and displacements into two independent subproblems: one in the displacements (the principle of minimum total potential energy), the other in the stresses (the principle of minimum total complementary energy).

Chapter 1 Introduction

5

For various variational formulations following the MIDR, the bilateral energy estimates of approximate solution quality were proposed [35]. Finite element algorithms were developed not only to check for model errors but also to refine adaptively FEM meshes in order to improve the solution quality [39]. In accordance with the ideas of this method, a projection approach was developed as a modification of the Petrov–Galerkin method. By using semi-discrete polynomial approximations and the projection technique, three-dimensional static and dynamic problems of linear elasticity can be solved with high accuracy [44]. Note that the local relations of Hooke’s law can be weakened not only in an integral way but also by applying an asymptotic approach [41]. This approach allows us to work out various beam models, which may serve as a reliable tool to the analysis of new types of buildings and structures [40]. The MIDR, above all, aims at development of more efficient numerical strategies, which are based on the ideas of the FEM and semi-discrete approximations of desired functions. The approaches discussed in this book were applied not only to static and spectral problems of linear elasticity but also to direct and inverse initial-boundary value problems [43] of solid mechanics, hydro- and thermodynamics [62].

Chapter 2

Basic concepts of the linear theory of elasticity

2.1 Stresses There are two types of forces in the linear theory of elasticity, namely, ones applied to the boundary of an elastic body or surface loads and others distributed within the body or volume forces. Surface loads arise if outside objects influence the body. Such forces can be either continuously distributed on the external boundary of the body, for example, as hydrostatic pressure and wind load, or applied pointwise to the body, i.e., locally. Any concentrated force can be considered as a limiting case of surface force under assumption that the boundary fragment subject to this loading is negligible with respect to the whole body surface. Volume forces are continuously distributed in the domain occupied by the body. Gravitation is an example of such loads, which is especially often met in applications. The stress principle of Euler and Cauchy is one of the basic axioms that are used to describe mechanical phenomena in a deformed elastic body. It is read as follows: the interaction of forces taking place on any imaginary surface drawn inside the body is analogous to that appearing on the boundary [20, 21, 12, 13]. Some short historical comments are provided by Truesdell and Toupin in [73]. Firstly, this principle states that elementary surface forces exist on the boundaries of each subdomain arbitrarily drawn in the deformed body. Secondly, it claims that the variability of such loads at any internal point is only determined by the direction of the surface normal vector. It could be also assumed that this elementary force at the given point depends on other geometrical properties of the surface, e.g. its curvature. Nevertheless, as it has been shown by Noll in [52], it is possible to formulate a general theory of surface forces that allows us to leave out any dependencies on such supplementary geometrical characteristics (see also [24, 80]). Thirdly, the stress principle declares that every part of a deformed elastic body is in a static equilibrium. This statement can be interpreted as equality of the corresponding resulting force to zero (the axiom of force balance) and also as absence of the total moment with respect to any given point (the axiom of moment balance). Thus, by means of the mathematical axioms, this principle expresses an intuitive idea that any piece of an elastic body can be balanced by specific forces applied to the internal surface of a subdomain subject to given volume forces and, maybe, given loads on a part of the surface which belongs to the boundary of the body.

7

Section 2.1 Stresses

Suppose that a self-equilibrated system of surface and volume forces effects some elastic body (see Figure 2.1). The body is deformed under the influence of these forces. Let us assume that the deformation process has finished and all body particles are in equilibrium. Such a body state is named stress-strain state. Cut mentally the body into two parts (A and B) by some smooth surface and consider the equilibrium condition for one of these parts, for example A. Generally speaking, those external forces P4 , P5 , P6 , which are directly applied to part A, are not in balance. Nevertheless, this new body A as a part of the whole body must be in equilibrium. To guarantee the total equilibrium and balance with the forces P4 , P5 , P6 , some loads are assumed to be added to the surface between A and B. These forces are nothing else but the influence of body B on body A and are named internal elastic forces. It is considered that they are distributed on the inner surface continuously but, generally speaking, non-homogeneously. Their intensity at some surface point O is usually defined in the following way. Select a small surface of area S located at the point O on the surface generated by the cut in the solid. Internal elastic forces and moments on this area are equipollent to a force R and a couple (moment) M , respectively. Note that these resultants are, in general, different, in both magnitude and orientation, from the corresponding resultants acting on the entire surface of the cut, as shown in Figure 2.1. Contract the contour of the area S in such a way that the point O is always in the interior of the element S . Let the surface decrease until it becomes an element of infinitesimal area S ! 0. Then, the values of R and M are also tending to zero. The limit of ratio R=S , which characterizes the intensity of internal elastic forces, determines stresses at the point O with respect to the area element S R D  .n/ : S!0 S lim

(2.1)

Both value and direction are attributed to the stresses. The stress direction is the same as for the vector R. In accordance with the Euler–Cauchy principle for stresses, the absolute value of ratio M=S tends to zero lim M=S D 0

S!0

if the area S vanishes. The case when the limiting value of this ratio is not equal to zero is considered in the so-called micromorphic theory of elasticity [19]. During the limiting process described above, the surface orientation, as defined by the normal to the surface, is kept constant in space. If a different normal n is chosen, then another stress vector  .n/ must be obtained. So, the vector  .n/ is characterized not only by its value and direction, but also by the surface orientation. Thus, when one speaks about the stress at some point, it is necessary to indicate which area element is considered. By definition, the area direction coincides with its unit normal vector pointing outwards (outward normal) with respect to the part of the body which equilibrium is studied.

8

Chapter 2 Basic concepts of the linear theory of elasticity

Figure 2.1. An elastic body loaded by external forces.

Two stress vectors measured at some point but on areas directed oppositely to each other obey the equation  .n/ D  .n/ : (2.2) As it is seen from Figure 2.1, this relation describes the interaction between two parts A and B of the elastic body and can be treated as Newton’s third law (the forces of action and reaction between two bodies are equal in magnitude and opposite in direction). In general, the direction of a stress vector does not coincide with the direction of the surface outward normal. The stress vector can be decomposed into two independent vectors: one, p, is collinear with the normal to the area S and the other,  , is orthogonal to the normal. These vectors are named normal and tangential stresses, respectively. While the scalar product of the normal stress vector p and outward normal n is positive, then this stress seeks to stretch material, otherwise it tends to compress the body. The tangential stress  seeks to shear or cut material along the area S and so is called shear stress (shearing). The stress vector  .n/ , defined at a point on the surface with the outward normal n, is often presented using the values of the projections of the vector on the Cartesian coordinate axes x1 , x2 , x3 . Then, by denoting these values as n1 , n2 , n3 , the expression for the stress has the form p D  .n/  n D n1 n1 C n2 n2 C n3 n3 ;

(2.3)

Section 2.1 Stresses

9

where ni , i D 1; 2; 3, are the components of the normal n in the Cartesian coordinates. It is considered that the vector n has unit length and, hence, n  n D 1. Here, dot denotes a scalar product of two vectors in the Euclidean space (see Appendix A). The components of a stress vector defined at a point on the surface parallel to one of the coordinate planes, for example x2 x3 , are usually denoted by symbols 11 , 12 , 13 . Here the first index 1 shows that the direction of the outward normal coincides with the axis x1 . Thus, 11 is a component of the normal stress vector p, 12 and 13 are two components of the shear stress vector  . The sign of the normal stress has been defined above. The following rule usually applies to the tangential stress. If the outward normal coincides with positive direction of one of the coordinate orts, then the positive directions of the tangential stress components are said to be the same as the directions of two other orts. If the normal n has the opposite direction, it is also necessary to change the positive directions of the tangential stresses. It has been shown (see, e.g., [25]) that the stress vectors  .n/ defined at some given point O on the different faces S are not independent of one another. All the stresses can be obtained uniquely if three stress vectors for three different areas are known at this point. For instance, these faces can be mutually orthogonal. This fact points out that the values defining the stress state of an elastic body can be characterized as components of some tensor. Nine components of the stress vectors related to the three faces parallel to the coordinate planes can be presented as a stress matrix 8 9

ˆ > < = 1 1 " D 2 12 "22 2 23 : ˆ > ˆ > :1 1  " ; 2 13

2 23

33

The angular distortions 12 , 13 , and 23 are also called the engineering shear strain components. The Cauchy strain tensor is a symmetric second rank tensor. Its properties are similar to the Cauchy stress tensor already introduced in Section 2.1. It is easy to see from eq. (2.44) and eq. (2.45) that six components "11 , "22 , "33 , "12 , "13 , "23 of the stain tensor " are not independent because they can be obtained from the three independent components u1 , u2 , u3 of the displacement vector u, as it has been shown above. This statement can be expressed in different ways. Let us dissect an undeformed body into infinitely many rectilinear parallelepipeds. After that, it is considered that each of such elements is under deformation of an arbitrary intensity. Next, try to connect continuously all these deformed parallelepipeds. In general, this attempt can fail, because some special relations between deformations have to be satisfied to organize such “assembling”. So, if the six components of the strain field are derived from the three functions of the displacement vector u as in eqs. (2.44) and (2.45), they are not independent and must satisfy Saint–Venant’s strain compatibility equations. As it was shown, for example, in [49], these conditions have the matrix form 9 8

ˆ ˆ > > ˆ ˆ >   C 2  0 0 0 ˆ > ˆ > < =    C 2 0 0 0 : 0 0 0  0 0> ˆ ˆ > ˆ > ˆ 0 0 0 0  0> ˆ > ˆ > : ; 0 0 0 0 0  For any tensors  and " given at a body point, it is always possible to find such coordinate systems in which all the nondiagonal components of these tensors are equal to zero. The axes of these coordinate systems are called the stress and strain principal axes. It follows from Hooke’s law for the isotropy that the principal axes of the stresses and strains coincide, that is the stress and strain tensors are coaxial. It is obvious since the only tensile and compressive stresses act along the principle directions. So, they are not able to cause any shear strains as it is seen in eq. (2.64).

25

Section 2.3 Constitutive relations

Instead of the Lame moduli, other coefficients are often used. Let us consider uniaxial stretching of an isotropic body along the axis x1 . In this case, the component 11 is the only non-zero stress, and Young’s modulus is defined by the ratio E D 11 ="11 . Taking into account eqs. (2.64), we have 2"11 D 11  or ED

 11 3 C 2

.2 C 3/ : C

(2.65)

Besides, as it has been experimentally established, there is lateral deformation so that "22 D "33 D  "11 , where the constant is Poisson’s ratio which can be presented as  D : (2.66) 2. C / Vice versa, the Lame moduli are given by E ; .1 C /.1  2 / E D DG: 2.1 C / D

(2.67)

The constant G is known as the shear modulus of the material and defined as the ratio of a shear stress to the corresponding engineering strain, e.g., G D 12 =12 . The elastic modulus tensor C in matrix form can be expressed via the elastic moduli E and as 8 9 1 0 0 0 > ˆ ˆ > ˆ > ˆ 1 0 0 0 > ˆ > ˆ > < = E 1 0 0 0 : (2.68) 0 0 0 1  2 0 0 > .1  2 /.1 C / ˆ ˆ > ˆ > ˆ 0 0 0 0 1  2 0 > ˆ > ˆ > : ; 0 0 0 0 0 1  2 In turn, the compliance tensor C 1 in the constitutive relation (2.61) can be also expressed via the values E and . Then, eq. (2.61) can be rewritten in the matrix form 8 8 9 98 9 "11 > 1   0 0 0 > 11 > ˆ ˆ ˆ > > ˆ ˆ > ˆ > > ˆ ˆ > ˆ > ˆ ˆ > ˆ "   1  0 0 0 > ˆ ˆ > ˆ 22 > 22 > > > ˆ ˆ > ˆ = < < = = < 1   1 "33 33 0 0 0 D : (2.69) "12 > 12 > 0 0 0 1C 0 0 > ˆ ˆ Eˆ > > ˆ ˆ > ˆ > > ˆ ˆ > ˆ ˆ ˆ ˆ " > 0 0 0 0 1C 0 > > > ˆ ˆ > ˆ13 > ˆ ˆ > ; ; : 13 > : ;ˆ : > "23 23 0 0 0 0 0 1C

26

Chapter 2 Basic concepts of the linear theory of elasticity

2.4 Boundary value problems 2.4.1

Static statements

The equations of linear elasticity can be divided into three basic groups. First of all, it is the differential equilibrium equation derived for internal stresses in Section 2.1 as r   C f .x/ D 0 :

(2.70)

Secondly, it is the kinematic relation between the strain tensor " and the displacement vector u D ¹u1 ; u2 ; u3 ºT in the form introduced in Section 2.2 "D

 1 ru C ruT : 2

(2.71)

The third group discussed in Section 2.3 is presented by the strain-stress relation written in tensor notation as  D C.x/ W " : (2.72) Thus, there exist 15 linear equations to define stress-strain state of an elastic body. Sometimes, the compatibility conditions are added to the second group. It is also considered that the stress-strain state is known if all the components of the stress and strain tensors,  and ", are defined at each point of the elastic body. To complete the formulation of a boundary value problem in linear elasticity, suitable boundary conditions must be appended to the system. In what follows, let us restrict ourselves to considering only the linear boundary conditions that can be written componentwise as .1/

I

(2.73)

.2/

I

(2.74)

.3/

:

(2.75)

eN .i/ .x/  u D uN i .x/;

x 2 i

eN .i/ .x/    n D qN i .x/;   eN .i/ .x/    n C .i/ .x/ eN .i/ .x/  u D qNi .x/;

x 2 i x 2 i

Here, uN i and qNi , i D 1; 2; 3, are given functions of boundary displacements and external surface loads, respectively; eN .i/ are the orts of a Cartesian coordinate systems;

.i/ are the coefficients of an elastic (Winkler) foundation. Note that the orts eN .i/ can change depending on boundary orientation, e.g., the ort eN .1/ coincides with the outward normal n. The boundary parts i.k/ , for all types of boundary conditions k D 1; 2; 3, obey the following rule: .1/

i

.2/

[ i

.3/

[ i

DI

.j /

i

.k/

\ i

D ;;

j ¤ k;

where  D @ is the boundary of the body domain .

i; j; k D 1; 2; 3 ;

Section 2.4 Boundary value problems

27

Conventionally, three types of boundary value problems are of interest and importance in solid mechanics. The first boundary value problem, named Dirichlet problem, is to solving the partial differential equations (PDEs) (2.70), (2.71) as well as the algebraic equation (2.72) in the interior of a given domain  with respect to the functions u, ",  so that dis.1/ .1/ .1/ placements take prescribed values on the boundary    1 [  2 [  3 as in eq. (2.73). The second boundary value problem, named Neumann problem, is to solving the system (2.70) – (2.72) so that stresses take prescribed values on the external surface .2/ .2/ .2/    1 [  2 [  3 as in eq. (2.74). The third boundary value problem, or mixed problem, indicates that different types of boundary conditions both in displacements and stresses are imposed on different parts of the boundary . All the boundary conditions in the problems mentioned above are enclosed in the formulation given by eqs. (2.73) and (2.74). Besides, conditions (2.75) enable one to consider the linear elasticity problems with a part of the boundary where neither displacements, nor surface forces are explicitly fixed, e.g., an elastic body on a Winkler foundation. In all these cases, the volume (or mass) forces f are assumed to be known. Furthermore, these forces combined with the surface load q cannot be chosen arbitrary because they have to satisfy all integral equilibrium conditions (2.12) and (2.13). Boundary value problems in the classical linear theory of elasticity are usually formulated either only in displacements or in stresses. It seems rather natural to reduce the number of independent variables by means of elimination of the vector function u or tensor function  from governing equations (2.70) – (2.72). There are two ways to do that. In the first one, three equations (Navier’s equations) remain as a result of eliminating the stresses  and strains " from eqs. (2.70) – (2.72) The obvious advantage of this approach is that the compatibility conditions are not required. On the other hand, the elimination of displacements u and strains " throughout the compatibility relations leads to six differential equations with respect to stresses  (Beltrami–Michell equations of compatibility). These two systems of PDEs are often called the basic equations of the linear theory of elasticity. Both approaches are widely used in the linear theory of elasticity. Which approach to apply depends on the problem statement. It is worth noting that the general solution, which is applicable to all cases, has not yet been obtained for any of the main formulations. The basic equations in displacements are derived consequently if the expression (2.72) is substituted for  in eq. (2.70) and, then, the strain tensor " is eliminated by using eq. (2.71).

28

Chapter 2 Basic concepts of the linear theory of elasticity

After that, Navier’s equation can be written componentwise as # " @2 u2 @2 u3 1  2 @2 f1 D 0 ; C C .1  2 /  C 2 u1 C @x1 @x2 @x1 @x3 G @x1 " # @2 @2 u3 1  2 @2 u1 f2 D 0 ; C .1  2 /  C 2 u2 C C @x2 @x1 @x2 @x3 G @x2 " # @2 u2 @2 1  2 @2 u1 f3 D 0 ; C C .1  2 /  C 2 u3 C @x1 @x3 @x2 @x3 G @x3

(2.76)

where  D r r D

@2 @2 @2 C C @x12 @x22 @x32

is the Laplace operator. Note that it is a difficult mathematical problem to solve Navier’s equations. Indeed, if the second boundary value problem is considered, then conditions (2.74) are rather inconvenient. After substituting expressions (2.71) and (2.72) for the tensor  in eq. (2.74), these conditions take the form   C W ru C ruT  n D 2q.x/ N ; (2.77) where qN is the given boundary stress vector. Nevertheless, Navier’s linear equations (2.76) are the starting point for a set of numerical methods, since they do not use the compatibility conditions. Instead of using the displacements u as an unknown vector function, it is possible to solve first the equilibrium equations in stresses. Then, the strain tensor " is defined explicitly by the inverted form of Hooke’s law (2.61). The compatibility equation   (2.78) r  r  C 1 W  D 0 has to be solved exactly taking into account the equilibrium condition (2.70). As a result, the six Beltrami–Michell equations are obtained for the components of the stress tensor  . These equations can be written in a Cartesian coordinates for homogeneous isotropic material as   @f1 1 @2 @f2 @f3 @f1 2 D C C ; 11 C 2 1 C @x1 1  @x1 @x2 @x3 @x1   @f1 1 @2 @f2 @f3 @f2 D C C ; 22 C 2 2 1 C @x2 1  @x1 @x2 @x3 @x2

Section 2.4 Boundary value problems

  @f1 1 @2 @f2 @f3 @f3 ; 33 C D C C 2 1 C @x32 1  @x1 @x2 @x3 @z   @2 1 @f1 @f2 12 C D C ; 1 C @x1 @x2 @x2 @x1   1 @f3 @2 @f1 ; D C 13 C 1 C @x1 @x3 @x3 @x1   1 @f2 @f3 @2 23 C D C ; 1 C @x2 @x3 @x3 @x2 D tr  D 11 C 22 C 33 :

29

(2.79)

While all the relations in the linear theory of elasticity are fixed, two fundamental questions are raised. Does their solution exist, and if the solution exists, is it unique? As to the former question, it is a purely mathematical problem. But this problem is still not completely solved, and this question is usually avoided. From physical point of view, it is obvious that an elastic body under loading can be in an equilibrium state. Moreover, as the mathematical statement of elasticity problems is based on the fundamental physical principles, it might be supposed that this basis cannot lead to absurd results. The proof of the uniqueness theorem for boundary value problems of elastostatics can be carried out by various methods. In one way, this theorem is proved ex adverso based on the fact that the assumption of uniqueness does not lead to a contradiction. The existence and the positive definiteness of the elastic strain energy play a central role in proving the uniqueness. At that, the starting point is the statement which is known as Clapeyron’s theorem in linear elasticity. Theorem 2.1. Under assumption that external forces remain constant from the initial state to the final state, the potential energy of deformation of a body, which is in equilibrium under the given load, is equal to half the work done by these forces. The proof that the equations of the linear theory of elasticity have a unique solution (if it exists at all) was first given by Kirchhoff [32]. This conclusion is based on the principle of force superposition and the assumption that the displacements do not affect on the external forces. In cases where this principle is not applicable, several different equilibrium forms may correspond to the same loading system. These questions relate to the stability of different equilibrium shapes for elastic bodies. If a body boundary is a multiply connected surface, then this proof is no longer valid since it can be possible to obtain multi-valued solutions. Above all, this is stipulated by the possibility of elastic bodies to be in a prestressed state. Such stresses may arise in elastic bodies during their manufacture and are called prestressing. Prestressing is of great importance in practice, but it is rarely taken into account because there are usually no reliable measurements and exact data on prestressed states.

30

Chapter 2 Basic concepts of the linear theory of elasticity

In conclusion of this subsection, it is possible to say based on the above discussion that if the solution of the boundary value problems (2.70) – (2.75) exists, then it is unique.

2.4.2 Dynamic problems If the external forces acting on an elastic body are changed during a loading process, then the time history of the stress and strain fields has to be analyzed in addition to their spatial distribution. In this case, the stresses  .t; x/, strains ".t; x/, and displacements u.t; x/ depend not only on the spatial coordinates x, but also on the time coordinate t . The functions of the volume force density f .t; x/ and surface load q.t; x/ as well as the boundary displacement vector u.t; x/ can change in a dynamical process. Even in the absence of external loading, natural vibrations can be excited in the body which initially was in a non-equilibrium state. In the dynamic formulation of linear elasticity problems, the kinematic equation (2.71), constitutive relations (2.72), and boundary conditions (2.73) – (2.75) remain in the same form as in a static case, whereas an additional term (inertial force), which is defined by the volume density of an elastic material and the acceleration of body points, must be appended to the equilibrium equation (2.70). The dynamic equilibrium equation has the form r    .x/

@2 u C f .t; x/ D 0 : @t 2

(2.80)

The initial conditions on displacements and velocities of the body points ˇ @u ˇˇ 0 D v 0 .x/ ; u.0; x/ D u .x/; @t ˇ tDt0

(2.81)

must be given to complete this formulation. Here, u0 and v 0 are given vector functions of initial displacements and velocities. Similar to Navier’s equation in elastostatics, the dynamic equation of motion in terms of displacements u can also be written. By using eqs. (2.71), (2.72), and (2.80), it is possible to present it as r  C W ".u/  .x/

@2 u C f .t; x/ D 0 : @t 2

(2.82)

In what follows in the book, harmonic vibrations of elastic bodies will be considered. The method of separation of spatial and time variables is applicable for this type of motions. It means that the unknown functions as well as external loads and boundary displacements can be presented, for instance, as u D u.x/ Q sin !t;  D Q .x/ sin !t; " D ".x/ Q sin !t ; f D fQ.x/ sin !t; qN i D qNQ i .x/ sin !t; uN i D uNQ i .x/ sin !t ;

i D 1; 2; 3 ; (2.83)

31

Section 2.5 Simplified models

Figure 2.2. A rectilinear elastic rod (beam) with a variable cross section.

where ! is the frequency of vibrations. If forced vibrations are analyzed, all unknown variables are defined by harmonic external load and boundary displacements with the prescribed frequency. Otherwise, an eigenvalue problem for natural vibrations is under study, where the value ! is an unknown eigenfrequency of the free vibrations and f D qN D uN D 0. After substituting the expressions (2.83) in eqs. (2.71) – (2.75), (2.82) and eliminating time-dependent term sin !t , the resulted dynamic equilibrium equation for harmonic motions has the form r   C .x/! 2 u C f .x/ D 0 :

(2.84)

The sign “tilde” has been omitted for simplicity. Then, the kinematic and constitutive relations (2.71), (2.72) with the boundary conditions (2.73) – (2.75) do not change their forms in this statement.

2.5 Simplified models 2.5.1 Elastic rods and strings As a primary example of mechanical systems with distributed parameters, let us consider a rectilinear elastic rod with a variable cross section shown in Figure 2.2. Introduce a coordinate axis x directed along the midline of the rod with the origin point O. The characteristic size of the rod cross section is assumed to be much smaller than the rod length. All lateral displacements of this elastic body are ignored, and a deformed state of the rod is described by the displacement function u.X/. The coordinate X of a rod point in the undeformed state and the coordinate x.X/ of this point in a deformed state are related by x.X/ D X C u.X/ : If a stress-strain state of the rod is described by the coordinate x then it is the Euler description, otherwise the system is defined by the Lagrange coordinate X. In what

32

Chapter 2 Basic concepts of the linear theory of elasticity

follows, the displacement function u and its derivatives with respect to these spatial coordinates are assumed to be very small. Then, these both descriptions, as it has been shown in Section 2.2, are equivalent. So, the coordinate x can be chosen as the only variable. Let E.x/ > E0 > 0 be Young’s modulus, A.x/ > A0 > 0 be the cross-sectional area, f .x/ be the linear density of external forces, .x/ be the modulus of elastic foundation, 0;1 be the coefficients of elastic supports at the rod ends. In the absence of deformation, the rod occupies an open segment  D ¹x W x 2 .x 0 ; x 1 /º : Without loss of generality, the coordinate system can be always chosen so that x 0 D 0. In this case, the value x 1 means the length of the rod. The unknown functions which should be found are the longitudinal displacements u.x/, the internal normal stresses  .x/ acting in cross sections of the rod, and the normal strains ".x/. Similarly to the 3D linear theory of elasticity, the one-dimensional relations describing the stress-strain state of an elastic rod can be divided into three groups: a) the kinematical relation "D

du ; dx

x 2 I

(2.85)

b) the constitutive relation (Hooke’s law)  D E.x/" ;

x 2 I

(2.86)

c) the equilibrium equation d .A.x/ /  .x/u C f .x/ D 0 ; dx

x 2 :

(2.87)

By expressing the stress and stain functions through the displacements u from eqs. (2.85) and (2.86), the ordinary differential equation of the second order is obtained   du d A.x/  .x/u C f .x/ D 0 ; x 2  : (2.88) dx dx In analogy with the previous section, the boundary conditions at the rod ends x D x0 and x D x1 can be sorted into three types: a) Dirichlet conditions u.x 0 / D u0 ;

u.x 1 / D u1 I

(2.89)

b) Neumann conditions A.x 0 / .x 0 / D P0 ;

A.x 1 / .x 1 / D P1 I

(2.90)

33

Section 2.5 Simplified models

c) elastic supports A.x 0 / .x 0 /  0 u.x 0 / D P0 ;

A.x 1 / .x 1 / C 1 u.x 1 / D P1 :

(2.91)

Here, u0 and u1 are some fixed end displacements, P0 and P1 are given forces. Note that it is always possible to set u0 D 0 and the boundary loads P0 , P1 have to be compatible and relate to each other through the integral force balance Z P0 C P1 C

x1

x0

.f  u/ dx D 0 :

(2.92)

Only one boundary condition of the three types presented in eqs. (2.89) – (2.91) is possible to be given at each rod end. As a result, nine different boundary value problems can be formulated. Consider here also a dynamical model of an elastic rod under given external load. In this case, the stresses  .t; x/, strains ".t; x/, and displacements u.t; x/ are functions of the time coordinate t and the spatial coordinate x. The linear density of external force f .t; x/, the boundary displacements u0;1 .t /, and stresses P0;1 .t / can also change during the process. By taking into account the inertial forces, the static equilibrium equation (2.87) becomes a PDE A.x/ .x/

@2 u @ .A.x/ / C .x/u D f .x; t / ;  2 @t @x

x 2 ;

t > t0 :

(2.93)

Here, is the volume density of the rod material. The kinematic equation (2.85) is also transformed into a PDE @u "D : (2.94) @x Constitutive relation (2.86) as well as boundary conditions (2.89) – (2.91) remain in the same form. To complete the formulation of an initial-boundary value problem, the initial conditions on displacements and velocities of the rod points have to be added ˇ @u ˇˇ 0 D v 0 .x/ ; (2.95) u.t0 ; x/ D u .x/ ; @t ˇ tDt0 where u0 and v 0 are some initial displacement and velocity functions, respectively. In this linear system, the initial time t0 can be chosen equal to zero without loss of generality. The final equation of elastic rod motions in displacement u.t; x/ has the following form:   @ @2 u @u A.x/ .x/ 2  A.x/E.x/ C .x/u D f .x; t / : (2.96) @t @x @x

34

Chapter 2 Basic concepts of the linear theory of elasticity

If harmonic vibrations of an elastic rod are under consideration, then all unknown functions as well as given external load and boundary displacements can be presented as u D u.x/ Q sin !t ; f D fQ.x/ sin !t ;

 D Q .x/ sin !t ; P0;1 D PQ0;1 sin !t ;

" D ".x/ Q sin !t ; u0;1 D uQ 0;1 sin !t ;

(2.97)

where ! is either the given frequency of forced vibrations or the unknown eigenfrequency of natural vibrations in the case when f D P0;1 D u0;1 D 0. After substituting expressions (2.97) in eqs. (2.86), (2.93), (2.94) and omitting the sign “tilde”, the governing equations of harmonic rod motions are "D

du ; dx

x 2 I

(2.98)

 D E.x/" ;

x 2 I

(2.99)

 dA.x/  C A .x/! 2  .x/ u D f .x/ ; dx

x2

(2.100)

under some of nine types of the boundary conditions (2.89) – (2.91). The constants u0;1 and P0;1 are known amplitude of end displacements and forces, respectively. No initial condition is necessary for this boundary value problem. It is worth noting that the similar system of equations describes the linear behavior of an elastic string. In the string model, the function u is the lateral displacements, " is the angle of the string inclination, and A is the projection of the string tension on the axis normal to the undeformed line of the string. Another useful simplification of 3D linear theory of elasticity, which has a wide application in mechanical engineering, is the model of elastic rod torsion. Let the x-axis be again directed along the reference line of a thin rectilinear rod with a uniform cross section. In this model [70], the lateral displacements are supposed to have a specific shape (2.101) uy D z.x/ ; uz D y.x/ ; where  is the rotation angle of the cross section around the x-axis, y and z are the lateral coordinates. The longitudinal displacement ux D

.y; z/

d.x/ dx

(2.102)

defines the deplanation of the rod cross section and, as a result, the shear stresses 12 and 13 . The function is obtained as the solution of a PDE with respect to the coordinates y and z under special boundary conditions following from the kinematic

35

Section 2.5 Simplified models

and equilibrium relations of the linear elasticity. Under assumption that the deplanation function is small enough and the angle  is insignificantly changed along the x-axis, the value of this angle can be obtained from the following system of equations: "D

d ; dx

M D kGJr " ;

dM D 0 ; x 2 .x 0 ; x 1 / : dx

(2.103)

In this model, the angle .x/ is an unknown function, M.x/ is the torque in the cross section, the torsion constant Jr is specific for the chosen shape of the rod cross section, the product kGJr is called torsional stiffness, k is a correcting coefficient. The different boundary conditions analogous to those defined by eqs. (2.89) – (2.91) can be introduced here as .x 0 / D 0 ; M.x 0 / D M0 ; M.x 0 / C 0 .x 0 / D M0 ;

.x 1 / D 1 I

(2.104)

M.x 1 / D M1 I

(2.105)

M.x 1 / C 1 .x 1 / D M1

(2.106)

with given displacements 0;1 , torques M0;1 , and angular spring coefficients 0;1 at the rod ends x 0 and x 1 .

2.5.2

Beam models

The Bernoulli beam theory is based on the intuitive hypotheses which were proposed by J. Bernoulli in the end of the 17th century. He supposed that a cross-sectional plane which has been normal to the reference beam line prior to deformation remains plane and normal to this line. Moreover, it is implied that the cross section is not noticeably deformed during the beam loading. This theory occupies an important place among the simplified theories in the solid mechanics [18]. In spite of the fact that the beam theory is applicable for a wide class of engineering problems, it does not take into account such important mechanical characteristics of elastic structures like shear and anisotropic properties of material. The qualifying formulae taking into account the influence of Poisson’s ratio have been proposed for static (Timoshenko beam, see [70]) and dynamic problems (Rayleigh correction, see [55]). In the model developed by Reissner in [59], a variational approach is used to derive the bending of think plate (beam) under a priori fixed basis functions for the approximation of displacement fields. The advance approximation have been applied in elastic beam theory to obtain compatible equations of higher orders taking into account the spatial character of displacement and stress distributions in elongated elastic bodies [48]. Consider the plane bending of a rectilinear beam with a cross section symmetrical with respect to two orthogonal straight lines. One of these lines is in the bending plane. The axis x is directed along the beam center line.

36

Chapter 2 Basic concepts of the linear theory of elasticity

If the function w.x/ defines small lateral displacements of the center line, then, according to the Bernoulli hypotheses mentioned above, the longitudinal displacement u.x; z/ of the beam cross-sectional points is a linear function of a lateral coordinate z (axis z is parallel to the bending plane) and has the form u.x; z/ D 

dw.x/ z: dx

(2.107)

The only variable w.x/ (displacement) is presented in this model. This function is related to the normal strain component in accordance with ".x; z/ D 

d 2 w.x/ z: dx 2

(2.108)

The bending moment .x/ is found by integrating the linear moment of this strain function over the beam cross section “ d 2 w.x/ ".x; z/z 2 dydz D E.x/J.x/ ; (2.109) .x/ D E.x/ dx 2 A where J is the moment of cross-sectional area. The axis y (see eq. (2.109)) is orthogonal to the bending plane. After introduction of the shearing force F .x/ D 

d.x/ ; dx

(2.110)

the balance of internal elastic forces and external load in the lateral direction z can be expressed through the bending moment .x/ and the lateral displacement w.x/ in the form d 2 C .x/w D f .x/ : (2.111) dx 2 Here, the coefficient of an elastic foundation and the density f of distributed lateral loads are introduced. After substituting the moment  of eq. (2.109) in eq. (2.111), the beam equation is obtained as   d2 d 2 w.x/ EJ.x/ C .x/w D f .x/ : (2.112) dx 2 dx 2 To complete the statement of the boundary value problem, it is necessary that two boundary conditions are set at each beam ends x D x 0;1 . Likewise for the longitudinal loading of an elastic rod in the previous subsection, the boundary conditions either in displacements w.x 0 / D w0 ;

w.x 1 / D w1

(2.113)

F .x 0 / D F0 ;

F .x 1 / D F1

(2.114)

or in shear forces

37

Section 2.5 Simplified models

can be given (see Figure 2.2). Additionally, the conditions on the incline angle ˛D

dw.x/ dx

(2.115)

of the center line at these points ˛.x 0 / D ˛0 ;

˛.x 1 / D ˛1

(2.116)

.x 0 / D M0 ;

.x 1 / D M1

(2.117)

or on the bending moment

can be imposed. A useful generalization of these boundary conditions is the lateral and angular elastic supports 0 F .x 0 / D w w.x 0 / C F0 ;

1 F .x 1 / D  w w.x 1 / C F1

(2.118)

.x 0 / D ˛0 w.x 0 / C M0 ;

.x 1 / D  ˛1 w.x 1 / C M1 ;

(2.119)

and 0;1 and ˛0;1 are the given elastic coefficients of these supports. respectively. Here, w If bending motions of an elastic beam are considered in the frame of the Bernoulli model, the lateral inertial forces have to be taken into account to derive the partial differential equation of beam motion as it follows:  @2 w @2 w @2 A.x/ .x/ 2 C 2 E.x/J.x/ 2 C .x/w D f .t; x/ : (2.120) @t @x @x

Here, the rotational inertia of the beam cross section is not considered. The initial state must be given to complete the description of beam motions, like for the elastic rod model in the above subsection, via the lateral displacement distribution and initial velocity of the beam points as @w.t; x/ : (2.121) @t In the case of harmonic vibrations of an elastic beam, the unknown functions of displacements, bending moments, and strains can be given in the form w.0; x/ D w 0 .x/ ;

w D w.x/ Q sin !t ;

v.0; x/ D v 0 .x/ ;

 D .x/ Q sin !t ;

v.t; x/ D

" D "Q.x/ sin !t ;

(2.122)

respectively, where ! is the frequency of beam vibrations. By omitting the sign “tilde”, the result equations for harmonic motions are the kinetic and constitutive relations (2.108), (2.109) as well as a new dynamic equilibrium equation  d 2  2  A .x/! 

.x/ w D f .x/ (2.123) dx 2 under four of the twelve boundary conditions (2.113), (2.114), (2.116) – (2.119).

38

Chapter 2 Basic concepts of the linear theory of elasticity

2.5.3 Membranes A two-dimensional analogue of the linear string model introduced in this section is the model of an elastic membrane. From the theoretical point of view, the membrane represents a flexible, infinitely thin plate stretched homogeneously in all directions by such large stresses that this tension does not change appreciably during a lateral loading or motion of the membrane. In the linear approximation, the elastic potential energy is assumed to be proportional to relative change of membrane surface area. Let a Cartesian coordinate system Ox1 x2 x3 be fixed in the 3D space and an undeformed membrane occupy some plane domain   ¹x1 ; x2 ; 0º with the boundary  (see Figure 2.3). This means that the membrane mid-surface belongs to the plane x3 D 0. In applications, the domain  can be considered as a curvilinear polygon with the finite number n of vertices Aj , j D 0; : : : ; n, A0 D Am . The polygon  is assumed to be an open, bounded, and connected domain, so that  D @ D

n [

Nj ;

j D1

where j is an open analytical plane curve j D ¹x1 D 'j ./ ;

x2 D

j ./ ;

 2 I D .1; 1/º

with some analytical functions 'j and j defined on the closed segment IN. These functions obey the inequality ˇ ˇ ˇ ˇ ˇ d'j ˇ2 ˇ d j ˇ2 ˇ Cˇ ˇ ˇ ˇ d ˇ ˇ d ˇ c > 0 : If 'j and j are both linear functions of  then j is an open linear segment. Wherever this linearity is fulfilled for every j D 1; : : : ; n, the domain  is a conventional polygon. The inner angle ˛j at the vertex Aj is defined by the curves j and j 1 . It is supposed that this angle is oriented as it is shown in Figure 2.3 and the following inequality is correct: 0 < ˛j < 2. This case corresponds to the domains with the Lipschitz boundary. Only (rather small) lateral displacements w.x1 ; x2 / of the membrane are admissible in the linear model. Note that in the static case, the external force with density f distributed on the membrane surface can be compensated only by inner elastic stresses. Then the static equation can be written as ! @2 w @2 w C D f .x/ : (2.124)  @x12 @x22 The constant  for the isotopic membrane is called a membrane tension coefficient.

Section 2.5 Simplified models

39

Figure 2.3. A curvilinear polygonal domain .

The constitutive relation between physical and geometrical parameters can be obtained by introducing a vector function  D ¹1 .x1 ; x2 /; 2 .x1 ; x2 /ºT according to  @w  @w 1 D ; 2 D ; (2.125) h @x1 h @x2 where h.x1 ; x2 / is the thickness of the membrane. The functions 1 and 2 are the projections of the corresponding stresses along the axes x1 and x2 on the lateral axis x3 . Similarly to the elastic spring model in Subsection 2.5.1, the strain vector can be composed as ³ ² @w @w T ; : (2.126) "D @x1 @x2 After that, the two scalar equations in eq. (2.125) can be rewritten in a vector form  (2.127)  D ": h By using eqs. (2.125) and (2.127), the static equation (2.124) is expressed via the stress vector as r  .h.x1 ; x2 / / C f .x1 ; x2 / D 0 : (2.128) In order to define the boundary conditions, we restrict ourselves to the case where the boundary of the domain  is a union of three different parts  D N .1/ [ N .2/ [ N .3/ . Each of these parts consists of a unique set of the curves j and does not intersect with the other ones (N .j / [ N .k/ D ;, j ¤ k, j; k D 1; 2; 3). On the boundary segments N .1/ , N .2/ , and N .3/ the following Dirichlet, Neumann, and Winkler conditions can be given, respectively:

40

Chapter 2 Basic concepts of the linear theory of elasticity

a) in displacements w D w0 .x1 ; x2 / ;

¹x1 ; x2 º 2  .1/ I

(2.129)

b) in stresses n.x1 ; x2 /   D q0 .x1 ; x2 / ;

¹x1 ; x2 º 2  .2/ I

(2.130)

c) as an elastic support n.x1 ; x2 /    .x1 ; x2 /w D q0 .x1 ; x2 / ;

¹x1 ; x2 º 2  .3/ :

(2.131)

Here, w0 and q0 are some given functions of boundary displacements and lateral stresses, n is the unique outward normal to the boundary , > 0 is the coefficient of the elastic support. In the dynamical case the inertial forces have to be added to the eq. (2.124) ! @2 w @2 w @2 w C (2.132) h .x1 ; x2 / 2   D f .t; x1 ; x2 / @t @x12 @x22 with the volume density depending on the coordinates x1 and x2 . Just as in all previous examples of this section, the initial conditions on the displacements and velocities w.0; x1 ; x2 / D w 0 .x1 ; x2 / ; v.0; x1 ; x2 / D v 0 .x1 ; x2 / ; @w.t; x1 ; x2 / v.t; x1 ; x2 / D @t

(2.133)

must be imposed to formulate the initial-boundary value problem correctly. Accordingly, the equation of harmonic vibrations for the elastic membrane can be derived based on the method of separation of variables in the form ! @2 w @2 w 2 h .x1 ; x2 /! w C  C (2.134) D f .x1 ; x2 / : @x12 @x22 Here, w.x1 ; x2 / is the amplitude of membrane vibrations, ! is either the given frequency of forced motions or an eigenfrequency of natural vibrations. The PDE (2.134) and boundary conditions (2.129) – (2.131) are compatible with each other in eigenvalue problems only if f D w0 D q0 D 0.

2.5.4

Plane stress and strain states

A particular case of great practical importance is the so called plane state of stress. It is considered that, e.g., all stress components i3 .x/, i D 1; 2; 3, acting along axis x3 vanish.

Section 2.5 Simplified models

41

In applications, this state can naturally be realized, for example, when one size of the body is much smaller than the other two. This model is also applicable if i3 .x/ are small, compared with the other stresses. The only non-vanishing components are 11 , 12 , and 22 , and the original spatial problem of linear elasticity is reduced to a 2D problem. Then, the stress tensor can be presented in a matrix form as 9 8

"11 > ˆ ˆ ˆ > > ˆ < = > ˆ : 22 ; ˆ E 12 ˆ > ˆ ; : ; 12 : > "33   0 In this case, the component "33 is not equal to zero due to Poisson’s ratio. This function can be eliminated from the governing equations because it does not influence the total elastic energy. Then, Hooke’s inverse law can be written in the matrix form as 98 9 8 8 9 1 0 = 0 is the mesh parameter. After that, every rectangle is divided into two triangles. The polynomial degree Nu as an approximation parameter is defined identically for all triangular elements and connected to N through the relation N D Nu  1. The following non-dimensional geometrical and material parameters have been chosen:

Section 6.4 Mesh adaptation and mesh refinement

149

Figure 6.17. An adapted triangular mesh in Example 6.6.

the plate length and width is l D 1 and h D 1, respectively; Young’s modulus is E D 1; Poison’s ratio equals to D 0:3; the density of the external load is q D 1. The computations have been performed with different approximation degrees Nu D 3; 4; 5; 6 and different numbers of the mesh parameter M 18. The best found value of the elastic energy is W   5:633. First, let us comment on the numerical results obtained by using uniform meshes (Figure 6.16) for different numbers of the parameter M and polynomial degree Nu . The coordinates of uniform mesh nodes in this case are given as in Example 6.4. The uniform monotonic vanishing of normalized function .N / D ˆW1 is demonstrated in Figure 6.18 for different degrees Nu . Note that the value of the functional ˆ is decreasing (rather slowly) while the numbers M (h-convergence) and/or Nu (p-convergence) are increasing. The bilateral estimates of the elastic energy W stored by the body versus the number of degrees of freedom N at Nu D 6 are shown in Figure 6.19. The strain energy W" .au / D W .au / is monotonically increasing and can be regarded as a lower bound for the elastic energy in the exact solution, whereas the stress energy W .a / is monotonically decreasing and can serve as an upper bound for W . The relative energy error  reaches the value which is approximately equal to 0:46% at N  40; 000. The distribution of the energy error density '.x1 ; x2 / is presented in Figure 6.20 at M D 12 and Nu D 6. One can see that the function ' is close to zero everywhere except for a small vicinity of the crack tip with coordinates x D ¹0; 0ºT . The maximal value of the energy error distribution ' realized at this point approximately equals to '  674. This maximum is not shown in Figure 6.20 because of its rather high value. In order to increase the accuracy of approximate solutions, an adapted mesh has been generated (see Figure 6.17). This mesh results from the uniform one with the same number M by changing the distances between horizontal and vertical interele-

150

Chapter 6 Advance finite element technique

Figure 6.18. Relative energy error  vs. the number of degrees of freedom N for different degrees Nu (uniform meshes).

Figure 6.19. Bilateral estimates of elastic energy for different numbers of degrees of freedom N at Nu D 6 (uniform meshes).

151

Section 6.4 Mesh adaptation and mesh refinement

Figure 6.20. Distribution of the energy error density '.x1 ; x2 / at M D 12, Nu D 6 (uniform mesh).

ment lines. The new line coordinates x1.i;1/ for i D 1; : : : ; 2M  1 and x2.1;j / for j D 1; : : : ; M  1 are chosen so that the values of the following integrals: Z

h

ˆ1;i Z 2;j

ˆ

0

Z

x .i;1/

x .i 1;1/ x .1;j / Z l

x .1;j 1/

l

'.x1 ; x2 /dx1 dx2 ;

i D 1; : : : ; 2M

'.x1 ; x2 /dx1 dx2 ; j D 1; : : : ; M

were approximately equal to each other for all i and j . The new distribution of the local error '.x1 ; x2 / is presented in Figure 6.21 at M D 12 and Nu D 6. The shape of the function ' in this figure is similar to that shown in Fig 6.20. In contrast to the case of the uniform mesh, the domain where the value of ' is notably high is narrowed down considerably . At the same time, the maximal value of ' increases and gets equal approximately to '  1; 406. The stress fields are shown in Figures 6.22 – 6.24 for the adapted mesh. These figures demonstrate that the components 11 , 22 , and 12 of the stress tensor  obey the boundary conditions given for this problem. It is worth noting the complicated behavior of the stress functions in the crack tip vicinity. All components have discontinuities at the singular point x D ¹0; 0ºT . For example, if x1 ! C0, x2 D 0, then the boundary condition 22 .x1 ; 0/ D 0 must hold, whereas 22 tends to infinity along the other directions. At the numerical modeling, the maximal value of 22 is increasing

152

Chapter 6 Advance finite element technique

when the number of degrees of freedom N grows and attains 22  26 000 for the given parameters. The notable fluctuations of functions 11 and 12 are observed and additionally 12 changes its sign near the crack tip. The important characteristics of numerical solution quality are the component values of the stress error tensor & . The functions &11 , &22 , and &12 are shown in Figures 6.25 – 6.27. Like the local error ' (see Figure 6.21), these functions are close to zero everywhere except for the crack tip vicinity. It is also possible to observe small oscillations of the stresses &ij in the interior of the domain . An essential feature of the FEM realization proposed in this section is the convergence rate with respect to degree of piecewise polynomial approximations of unknown functions (p-convergence). In Figure 6.28, the monotonic decreasing of the relative error  versus the number of polynomial degree Nu are presented for the adapted mesh under consideration. In Figure 6.29, h-convergence rate for uniform meshes at Nu D 6 (upper curve) and different M is compared with the p-convergence rate for the adapted meshes at M D 12 and Nu D 3; 4; 5; 6. It is seen from this plot that the mesh adaptation technique gives one the possibility to increase significantly the accuracy of numerical results on the same dimensions of the FEM model. The bilateral convergence of the elastic energy W .N / is demonstrated in Figure 6.30 for the uniform mesh at the fixed polynomial degree Nu D 6 (dashed lines, see also Figure 6.19) and for the adapted mesh at the fixed parameter M D 12 (solid lines). The presented dependences are in good agreement with the theoretical bilateral estimates given in eq. (5.32). The relative energy error  for the adapted meshes reaches the value of about 0:033% at N  40; 000. This relative integral error is 14 times less than the one obtained by using the uniform mesh with the same approximation parameters M and Nu .

Section 6.4 Mesh adaptation and mesh refinement

153

Figure 6.21. Distribution of the energy error density '.x1 ; x2 / at M D 12, Nu D 6 (adapted mesh).

Figure 6.22. Distribution of the stress component 11 .

154

Chapter 6 Advance finite element technique

Figure 6.23. Distribution of the stress component 22 .

Figure 6.24. Distribution of the stress component 12 .

Section 6.4 Mesh adaptation and mesh refinement

Figure 6.25. Distribution of the stress error component &11 .

Figure 6.26. Distribution of the stress error component &22 .

155

156

Chapter 6 Advance finite element technique

Figure 6.27. Distribution of the stress error component &12 .

Figure 6.28. Function .Nu / at M D 12 (p-convergence, adapted mesh) .

Section 6.4 Mesh adaptation and mesh refinement

157

Figure 6.29. Function .N / for uniform (dashed) and adapted (solid) meshes.

Figure 6.30. Bilateral estimates of elastic energy for different numbers of degrees of freedom N for the uniform (dashed) and adapted (solid) meshes.

Chapter 7

Semi-discretization and variational technique

To construct approximate solutions for boundary value problems in solid mechanics, one uses various simplified models, on which basis the stress and displacement fields can be obtained effectively with an acceptable accuracy. For various mechanical structures, their elements called beams or rods have a special geometrical feature that one of the characteristic sizes is much larger than the other two. Various beam theories occupy a special place among approximate approaches in mechanics. As it has been mentioned in Section 2.5, the simplest model called Euler–Bernoulli beam is based on the hypotheses that a cross-sectional plane which has been normal to the reference line prior to a deformation remains plane and normal to this line after the deformation. Up to now, different refined theories based on variational and asymptotic methods have been developed to increase the accuracy of modeling for thick, thin-walled, and composite beams [59, 70, 79]. A complete literature review is beyond the scope of this book but it is worth citing some recent papers in this field [57]. The rest of the book, Chapters 7 – 11, focuses on different approaches to reliable beam modeling and analysis of resulted solution quality. The current chapter is devoted to an approach in which the original two-dimensional elasticity problem in PDEs is approximated by a system of ordinary differential equations (ODEs). The variational technique discussed in the previous chapters is extended on the case of semi-discrete approximations for the displacement vector and stress tensor including, on the one hand, polynomial expansion of finite dimension over one coordinate component and, on the other hand, unknown functions over the other component. The approach is illustrated with static and dynamic beam problems.

7.1 Reduction of PDE system to ODEs 7.1.1 Beam-oriented notation In what follows, the statement of a linear elasticity problem is given in a specific notation, which is used to describe the behavior of beams. Let a Cartesian coordinate system Oxy be introduced. The plane stress state of an elastic body (beam) occupying an elastic rectangular domain  D ¹x; y W x 2 .l; l/ ; y 2 .h; h/º

(7.1)

Section 7.1 Reduction of PDE system to ODEs

159

Figure 7.1. Rectangular elastic beam.

with the sizes 2h  2l is considered (see Figure 7.1). In the beam-oriented notation, 2l is the length of the beam, whereas 2h is its height. The origin of the Cartesian system is placed at the central point of the rectangle. The x-axis is parallel to the sides with lengths 2l y˙ D ¹x; y W x 2 .l; l/ ; y D ˙hº : (7.2) The y-axis is collinear with the beam end sides x˙ D ¹x; y W x D ˙l; y 2 .h; h/º :

(7.3)

It is assumed that the stress-strain state of the beam is described by the 2D equations of linear elasticity introduced in Chapter 2 but presented in a new notation as @xy @x C C fx .x; y/ D 0 ; @x @y @y @xy C C fy .x; y/ D 0 I @x @y   @v @u @v 1 @u ; "y0 D ; "0xy D C I "0x D @x @y 2 @y @x  D "0 .u; v/  C 1 .x; y/ W  D 0 :

(7.4) (7.5) (7.6)

Here, u.x; y/ and v.x; y/ are unknown displacements along longitudinal, x, and lateral, y, directions, respectively; "0x , "y0 , and "0xy are the components of the strain tensor "0 .x; y/; x , y , and xy are the components of the stress tensor  .x; y/; fx and fy are the given components of the volume force vector f .x; y/; C.x; y/ is the elastic modulus tensor.

160

Chapter 7 Semi-discretization and variational technique

Suppose that the beam edges y˙ are loaded by distributed forces according to xy D ˙px˙ .x/

and y D ˙py˙ .x/

for

y D ˙h :

(7.7)

On the end edges of the beam x˙ , the boundary conditions can be given either in the stresses, (7.8) x D ˙qx˙ .y/ and xy D ˙qy˙ .y/ for x D ˙l ; or in the displacements, u D 0 and v D 0 for

x D ˙l :

(7.9)

Here, px˙ .x/, py˙ .x/, qx˙ .y/, qy˙ .y/ are the components of the boundary load.

7.1.2 Semi-discretization in the displacements To construct a system of approximating ODEs, the principle of minimum potential energy discussed in Section 3.1 is first applied. The potential energy …Œu; v defined in eq. (3.10) is expressed in the proposed notation as Z …D

Z

h h

l l

 U  fx .x; y/u  fy .x; y/v dxdy C

Z

1 U D "0 .u; v/ W C.x; y/ W "0 .u; v/; 2 ´ px˙ .x/u  py˙ .x/v x 2 y˙  ˙  : Jb D ˙ ˙ ˛ qx .y/u C qy .y/v x 2 x˙

Jb d  ; 

(7.10)

where U is the strain energy density and Jb is the potential of the boundary forces. The coefficient ˛ ˙ is introduced so that ˛ ˙ D 1 if the stress conditions (7.8) are given on the corresponding end cross sections x˙ or ˛ ˙ D 0 if the zero displacements (7.9) are fixed there. Let us define the approximate displacement fields in  as polynomial functions with respect to the coordinate y uQ D

Nu X kD0

u.k/ .x/

yk ; hk

vQ D

Nv X kD0

v .k/ .x/

yk : hk

(7.11)

Here, Nu and Nv are the approximation degrees of the displacements, u.k/ and v .k/ are unknown functions of the coordinate x. In accordance with Theorem 3.2, the admissible displacements (7.11) have to obey the homogeneous boundary conditions (7.9) if they are present in the formulation of the problem. The other conditions (7.7) and (7.8) are natural and used in the integral form.

161

Section 7.1 Reduction of PDE system to ODEs

After substituting the approximations (7.11) for u and v in eq. (7.10) and integrating over the coordinate y, the potential energy … can be rewritten as Z

l

Z

l h

…D AD

B˙

h

  A x; a.x/; a0 .x/ dx C B  .a.l// C B C .a.l// ;

 U.u; Q v/ Q  fx uQ  fy vQ dy

 pxC .x/u.x; Q h/  pyC .x/v.x; Q h/  px .x/u.x; Q h/  py .x/v.x; Q h/ ; Z h  da : (7.12) qx˙ .y/uQ C qy˙ .y/vQ dy ; a0 D D ˛ ˙ dx h

Here, ° ±T a D u.0/ ; u.1/ ; : : : ; u.Nu / ; v .0/ ; v .1/ ; : : : ; v .Nv / ;

a.x/ 2 RMu ;

is the vector function of design parameters constituted of the coefficients u.i/ .x/ and v .j / .x/ of approximation (7.10) and defined with the dimension Mu D Nu C Nv C 2 on the interval x 2 .l; l/. The functional … in eq. (7.12) depends quadratically on the vector a and its first derivative a0 . The minimization problem (3.16) formulated in Section 3.1 is reduced to a system of second-order Lagrange–Euler equations d @A.x; a; a0 / @A.x; a; a0 / D 0:  dx @a0 @a

(7.13)

The total differential order of the ODEs (7.13) is equal to 2Mu . Either the essential boundary constraints (7.9) or the following natural conditions # " @A @B ˙ D 0 for ˛ ˙ D 1 (7.14) ˙ 0C @a @a xD˙l

must be added to system (7.13) depending on the type of boundary constraints at x D ˙l. If the beam is clamped at least on one end, i.e., ˛ C ˛  D 0, then the differential order of the system coincides with the full number of boundary conditions (7.9)

162

Chapter 7 Semi-discretization and variational technique

or (7.14). Otherwise, the equations (7.14) are degenerated due to the global force and moment balance Z hZ l Z l  C  C  P P C px C px dx D 0; fx dxdy C FC F C

h h

Z

l l

h



l

MC  M C l FC C F Z Ch

l



l

Z

Z

 

fy dxdy C Z C  C

h

Z

h



l l

px  px dx C

Z

l l l



 pyC C py dx D 0;

 fy x  fx y dxdy

l

l

  x py C pyC dx D 0 :

(7.15)

Here, the tensile forces P˙ , shear forces F˙ , and bending moments M˙ given at the beam ends x D ˙l, respectively, are introduced in eq. (7.15) according to P˙ D ˙ M˙ D

Z

h

h Z h h

qx˙ .y/dy ;

F˙ D ˙

Z

h h

qx˙ .y/ydy :

qy˙ .y/dy ; (7.16)

The signs of the forces and moments are chosen as it is shown in Figure 2.2 in analogy with the Bernoulli beam model. These quantities have to be given if the corresponding boundary conditions (7.8) are imposed. The conditions (7.15) show that there exist three independent constants which are used in the 2D case to define the position and orientation of the body as a whole in the plane Oxy if ˛ C ˛  D 1.

7.1.3 Semi-discretization in the stresses The principle of minimum complementary energy intorduced in Section 3.1 for the stresses can be used to derive another ODE system solving approximately the boundary value problem under consideration. The complementary energy …c Œx ; y ; xy  introduced in eq. (3.27) is given for the elastic body under the homogeneous conditions (7.9) by Z hZ l …c D Uc .x ; y ; xy /dxdy (7.17) h

l

with the stress energy density 1 Uc D  W C 1 .x; y/ W : 2

(7.18)

163

Section 7.1 Reduction of PDE system to ODEs

The approximating stress field has a specific structure Q x D

N X

x.k/ .x/

kD0

Qxy D

px0 .x/

C

yk ; hk   N 1 y2 X yk .k/ C 1 2 xy .x/ k ; h h h

y px1 .x/

kD0

  N y2 X y yk y.k/ .x/ k : Qy D py0 .x/ C py1 .x/ C 1  2 h h h

(7.19)

kD0

.k/

.k/

.k/

Here, x , xy , and y are the unknown stress functions of the coordinate x; N is the approximation degree. The stresses Qxy .x; y/ and Qy .x; y/ identically obey the boundary conditions (7.7) if  1 C px  px ; 2  C  1 p C px ; px1 D 2 x px0 D

 1 C py  py ; 2  1 C p C py : py1 D 2 y

py0 D

(7.20)

We restrict ourselves to the case when the boundary loads qx˙ .y/ and qy˙ .y/ on the edges y˙ are polynomials according to qx˙ D

N X

qx.˙k/

kD0

qy˙

D

yk ; hk

˙px0 .˙l/ ˙



y2 C 1 2 h h

y px1 .˙l/

 NX  1

qy.˙k/

kD0

yk ; hk

(7.21)

where the compatibility conditions qyC .h/ D pxC .l/ ;

qyC .h/ D px .l/ ;

qy .h/ D pxC .l/ ;

qy .h/ D px .l/

(7.22)

for the boundary stresses at the vertices of the rectangle  are taken into account. For such boundary loads, the stress components Q x .x; y/ and Qxy .x; y/ are able to satisfy exactly the boundary conditions (7.8). It is additionally assumed that the volume forces are defined by the polynomials with respect to y fx D

N X kD0

fx.k/ .x/

yk ; hk

fy D

where fx.k/ .x/ and fy.k/ .x/ are given functions.

NX  C1 kD0

fy.k/ .x/

yk ; hk

(7.23)

164

Chapter 7 Semi-discretization and variational technique

Now, we turn our attention to the equilibrium equations (7.4), which are essential constraints on the stresses in the principle of minimum total complementary energy. After substituting the approximations Q x , Qxy , and Qy of eq. (7.19) into eq. (7.4) and taking care that the equilibrium is valid for any y 2 .h; h/, the system of ODEs for .k/ N 2 with respect to the stress functions x.k/ , xy , y.k/ is assembled from all coefficients at monomials y k hk  j C 1  .j C1/ px dx .x/ .j 1/ C xy  fx.j / .x/ ; D  xy dx h h .j C1/

.j /

j D 0; : : : ; N I (7.24)

 k C 1  .kC1/ y  y.k1/ D h dpxk .x/ pykC1 .x/    fy.k/ .x/ ; k D 0; : : : ; N C 1: (7.25) dx h Here, for the sake of uniformity, the following void functions are introduced .k/

.k2/

d xy d xy  dx dx

C

.j / .x/  0 for xy

j < 0 and j N ;

y.j / .x/ pxj .x/

j < 0 and j > N ;

 0 for

D pyj .x/  0

for j < 0 and j > 1 :

It can be shown that all equations (7.25) except, e.g., for the first, that is, .k1/ k D 1; : : : ; N C 1, can be resolved with respect to the function y .x/. In turn, .j 1/ eq. (7.24) for j D 2; : : : N is solved algebraically with respect to xy .x/. After that, only three differential equations remain in eqs. (7.24) and (7.25), which depend .0/ .j / on coefficients xy , x , j D 0; : : : ; N . These equations are equivalent to the equilibrium conditions at any beam cross section of the body which can be presented as equations of force balance Z h Z xZ h Z x Q x .x; y/dy C fx .x1 ; y/dydx1 C px1 .x/dx1 D P  ; Z

h h

h

Z

Qxy .x; y/dy C

l h xZ h

l

and moment balance Z Z h y Q x .x; y/dy C  h

Z C

x l

Z

h

fy .x1 ; y/dydx1 C

Z

l x

l

py1 .x1 /dx1 D F 

(7.26)

h h

.l C x/Qxy .x; y/dy

 .l C x1 /fy .x1 ; y/  yfx .x1 ; y/ dydx1 h Z x

 .l C x1 /py1 .x1 /  2hpx0 .x1 / dx1 D M  (7.27) C h

l

165

Section 7.1 Reduction of PDE system to ODEs

calculated with respect to the point ¹l; 0º. The forces P  , F  , and moment M  have been introduced in eq. (7.16). These constants are found form the corresponding boundary conditions (7.8). The global equilibrium relations (7.26) and (7.27) can be resolved via three stress .0/ functions, for example, x.0/ , x.1/ , and xy . As a result, the approximations Q x , Qxy , and Qy in eq. (7.19) depend on the vector function of design parameters ° ±T b D x.2/ ; x.3/ ; : : : ; x.N / ;

b.x/ 2 RN 1 ;

x 2 .l; l/ ;

(7.28)

as well as its first and second derivatives b 0 , b 00 . In accordance with Theorem 3.4, the equilibrium stresses Q x , Qxy , and Qy also have to obey the stress conditions (7.8) if they are imposed. The displacement conditions (7.9) in the complementary variational problem are taken into account integrally in the corresponding functional. Let us substitute these approximations for x , xy , and y in eq. (7.17) and integrate over the coordinate y. Then, the total complementary energy …c can be rewritten as Z

l

Z

l h

…c D Ac D

h

  Ac x; b.x/; b 0 .x/; b 00 .x/ dx ; Uc .Q x ; Qxy ; Qy /dy ;

(7.29)

where Ac is the linear density of the beam stress energy. Thus, the minimization problem (3.26) can be reduced to the Lagrange–Euler system of fourth-order ODEs d @Ac @Ac d 2 @Ac D0  C 2 00 0 dx @b dx @b @b

(7.30)

under the essential boundary conditions (7.8) for ˛ ˙ D 1 which can be rewritten taking into account the solutions of the system (7.24) and (7.25) as follows Q x .b/ D ˙qx˙ .y/

and Qxy .b 0 / D ˙qy˙ .y/

for

x D ˙l :

If ˛ ˙ D 0, then the following natural conditions: ˇ  @Ac ˇˇ d @Ac @Ac  D D0 dx @b 00 @b 0 xD˙l @b 00 ˇxD˙l

(7.31)

(7.32)

have to be satisfied. The total differential order M D 4.N  1/ of the ODEs (7.30) plus the number of the constants following from the equilibrium balance (7.26) and (7.27) is always equal to the number of the boundary constraints (7.31) or natural conditions (7.32) imposed on the vector of design parameters b at x D ˙l.

166

7.2

Chapter 7 Semi-discretization and variational technique

Analysis of beam stress-strain state

The integrodifferential approach discussed in Chapters 4–6 allows one to estimate the quality of numerical solutions a .x/ and b  .x/ for the approximating ODE systems (7.13) and (7.30) with the corresponding essential constraints, (7.9) or (7.31), and natural conditions, (7.14) or (7.32). As it has been stated in Section 5.2 (see eq. (5.19)), the nonnegative energy error functional ˆ can be presented for the considered boundary conditions as the sum of the total potential and complementary energies in eqs. (7.12), (7.29) according to Z ˆŒa; b D

l l

'x .x; a; a0 ; b; b 0 ; b 00 /dx D …Œa C …c Œb 0 :

(7.33)

Here, the energy error density 'x is introduced as function of the coordinate x Z 'x D

h h

® ¯ U.u; Q v/ Q C Uc .Q x ; Qxy ; Qy /  2Ue .u; Q v; Q Q x ; Qxy ; Qy / dy

(7.34)

with the elastic energy density 1 Ue D  W "0 : 2 The relative integral error can be given by D

ˆŒa; b ; W Œb

W Œb D …c Œb ;

(7.35)

where the stress energy W coincides with the total complementary energy …c in eq. (7.29) for this static problem. Moreover, the boundary conditions (7.7), (7.8), (7.9) fall within the scope of Theorem 5.2 and the bilateral energy estimates (5.30) can be given as follows W" Œa  W Œu ; v  ;    W Œb  

(7.36)

with the strain energy Z W" Œa D

l l

Z

h h

U.u; Q v/dydx Q :

(7.37)

Here, u , v  , and   are the actual state of the body. Up to sign, integral (7.37) equals to the potential energy W" Œa  D …Œa  (7.38) on the approximate solution.

167

Section 7.2 Analysis of beam stress-strain state

Example 7.1. Consider a cantilever rectilinear beam clamped at x D l and loaded at x D l. Two other sides of the beam (y D ˙h) are free of loading. The boundary conditions can be presented as xy D 0 and y D 0

for y D ˙h ;

x D 0 and xy D q0 .y/ for x D l ; u D 0 and v D 0 for x D l

(7.39)

(7.40)

with the boundary load   y2 6FC 1 2 ; q0 .y/ D 8h h

Z

h h

q0 .y/dy D FC ;

where FC introduced in (7.16) is the shear force at x D l. The volume forces are absent (fx .x; y/ D fy .x; y/  0). The elastic beam material is assumed homogeneous and isotropic with Young’s modulus E and Poisson’s ratio . The plane stress state of the beam is considered. In this case, the constitutive relations expressed via the strain error tensor  are given componentswise as  1  x  y D 0 ; E  1  0 y D "y .u; v/  y  x D 0; E 1C 0 xy D 0 : xy D "xy .u; v/  E x D "0x .u; v/ 

(7.41)

Here, "0x , "y0 , "0xy are the components of the Cauchy strain tensor, which have been presented in eq. (7.5). The stress-strain state of the beam is described by the approximate displacements u, Q vQ defined in eq. (7.10) with Nv D Nu  1 and the stresses (7.19) in the form Q x D

Nu X

x.k/ .x/

kD0



Qy D 1 

yk ; hk

 Nu y2 X h2

kD0

 Nu 1  y2 X yk .k/ Qxy D 1  2 xy .x/ k ; h h

y.k/ .x/

kD0

yk hk

:

The parameter Nu defines the total number N D Mu C M D 3Nu of the system variables collected in two independent vectors a and b.

(7.42)

168

Chapter 7 Semi-discretization and variational technique

The optimal approximation of the displacements u, Q vQ and stress tensor Q for this elasticity problem is found via the solution a .x/ and b  .x/ of the Euler equations (7.13) and (7.30). This solution obeys the essential conditions (7.40) as well as the natural conditions (7.14) at x D l and (7.32) at x D l. These conditions can be rewritten for this example as ˇ  @Ac ˇˇ d @Ac @Ac  D D 0 for x D l ; dx @b 00 @b 0 xDl @b 00 ˇxDl ˇ @A ˇˇ @B C for x D l ; (7.43) D  @a0 ˇxDl @a where Z AD

Z

h h

BC D 

Z

U.u; Q v/dy Q ;

Ac D

h h

Uc .Q /dy ;

h h

q0 .y/v.l; Q y/dy ;

 G 0 2 ."x / C ."y0 /2 C 2 "0x "y0 C 2.1  /."0xy /2 ; 1  1 2 2 : x C y2  2 x y C 2.1 C /xy Uc D 2E U D

(7.44)

The full differential order of the ODE system (7.13), (7.30) is equal to Nd D 2Mu C 4M D 8Nu  2 ; while the number of edge conditions in eqs. (7.40) and (7.43) is Nb D 8Nu C 1 : With account of those three undefined constants F , P , and M which arise in eq. (7.27), the two independent systems of ODEs (7.13) and (7.30) can be composed with the compatible number of boundary conditions. The following dimensionless geometrical and material parameters have been chosen in this example: the beam length 2l D 10 and height 2h D 1, Young’s modulus E D 104 and Poison’s ratio D 0:3, the external force FC D 1. The computations have been performed with different polynomial degree Nu . To raise the effectiveness of the semi-discretization algorithm described in this section, take into account the symmetry properties of this boundary value problem. Like in Examples 4.3 and 5.6, the boundary conditions are symmetric with respect to the x-axis and, thus, the solution has specific parity characteristics analogous to those given in Table 5.1. More particularly, this problem relates to the beam bending or lateral deformations which can also be described by the Bernoulli model. The suitable parities of the displacement and stress functions are presented in Table 7.1.

169

Section 7.2 Analysis of beam stress-strain state

Figure 7.2. The strain energy W" and stress energy W vs. the lateral coordinate degree Nu in Example 7.1.

As it can be seen from Figure 7.2, the approximate stress energy W .Nu / (solid curve) strictly decreases when the stress energy value W" .Nu / D W .Nu / (dashed curve) increases. Such behavior is in accordance with Theorem 5.4. Some integral characteristics of the approximation at Nu D 9 are shown in Table 7.2. The solution accuracy obtained on the basis of the dimensionless ratio 1 D 1 

W" Œa  W Œb  

introduced in Section 5.3 is approximately equal to one hundredth percent. The displacement of this 2D rectilinear elastic body v.5; 0/ can be compared with the end lateral deflection vB found on the basis of the beam equation introduced by eq. (2.112) with the same parameters E, FC , h, l and the moment of cross-sectional area JB D 16 . The fact that the elastic body displacement is greater than the corresponding beam one can be due to a violation of the Bernoulli hypothesis of plane cross sections.

Table 7.1. The parity of displacement and stress components with respect to the coordinate y for lateral beam deformation.

Parity to y

u

v

x

xy

y

y 2j C1

y 2j

y 2j C1

y 2j

y 2j C1

170

Chapter 7 Semi-discretization and variational technique

Table 7.2. The solution parameters: differential order Nd , beam displacement vB , displacement v.5; 0/, stress energy W , energy error ˆ, and relative error 1 in Example 7.1.

Nd 70

vB

v

0.4000

0.4024

W

ˆ

0.2012

1 ; % 5

2:19  10

0.0109

7.3 2D elastic beam vibrations For dynamic problems, the conventional beam theory considers only reference line motions (longitudinal and lateral beam displacements) [71]. The next step to extend this model was made by Rayleigh [55]. He proposed to include in dynamical beam equations a correcting term defined by the rotary inertia of the cross section. Refined relations taking into account the influence of shear deformations on static and dynamic beam states were introduced by Timoshenko [70]. In Timoshenko’s theory, correction factors are used to modify the shear and torsion beam stiffness and to consider the deplanation of a cross-section shape. A few analytical solutions of dynamic problems for elastic beam-like structures are also known. See, for example, the solution of free vibrations for cylindrical shaft obtained by Love [49]. To derive the dynamic equations for composite plates based on the principle of virtual displacements, the polynomial functions determining the shear strains and stresses along transverse direction were applied by Reddy [57]. The variational asymptotic method proposed by Berdichevsky [10] was used by Yu and Hodges [78] to develop a finite element approach including cross-sectional analysis for composite beam-like structures. Based on the kinematic assumption that each cross section is infinitely rigid in-plane but free to warp out, the beam model for anisotropic materials was proposed by Bauchau [8]. Now, apply the integrodifferential approach and semi-discretization technique described in the above section to eigenvalue problems of elastic vibrations formulated in Section 2.4. Let a homogeneous and isotropic 2D body (beam) shown in Figure 7.1 occupy the domain (7.1) with the boundary (7.2), (7.3) and be free of loads xy D y D 0 for x D xy D 0

y D ˙h ;

(7.45)

for x D ˙l :

(7.46)

The equations of motion (2.84) for f  0 are given by @xy @x C C ! 2 u D 0 ; @x @y @y @xy C C ! 2 v D 0 : @x @y

(7.47)

Here, is the volume density and ! is the unknown eigenfrequency, for which a non-trivial solution of the problem exists. The displacements u.x; y/, v.x; y/ and

171

Section 7.3 2D elastic beam vibrations

components x .u; v/, xy .u; v/, y .u; v/ of the stress tensor  are eigenforms of natural vibrations corresponding to this frequency. Approximate eigenvalues of beam vibrations can be obtained by exploiting the semi-discretization of the displacements uQ and vQ of eq. (7.10), for example, on the basis of the Hamilton principle formulated in Section 3.2 (Theorem 3.8). To derive the corresponding system of ODEs, the linear density of kinetic energy Z 

! 2 h  2 uQ C vQ 2 dy (7.48) Kx D 2 h minus the potential energy A.u; Q v/ Q given in eq. (7.44) should be varied with respect to all the unknown functions u.j / and v .k/ of eq. (7.10) for j D 0; : : : ; Nu and k D 0; : : : ; Nv . The equations similar to (7.13) and (7.14) can be obtained in the same way as discussed in Section 7.1. An alternative approach, proposed in Chapter 4 on the example of elastic membranes, is based on the integrodiffential statement. Let us formulate the following minimization problem: find such non-trivial displacements u .x; y/, v  .x; y/ and stress tensor   .x; y/ satisfying the equilibrium equations (7.47) and the boundary conditions (7.45), (7.46) and minimizing the functional ˆŒu ; v  ;    D min ˆŒu; v;   D 0 ; u;v; Z Z l 'x dx ; 'x D ˆD l

h h

' dy ;

1 'D WC W  2

subject to the isoperimetric condition Z Z 1 l h W D  W C 1 W  dydx D 1 : 2 l h

(7.49)

(7.50)

Resolve the equilibrium equations (7.47) with respect to the displacement components as follows:     1 @xy 1 @y @x @xy uD 2 C ; vD 2 C : (7.51)

! @x @y

! @x @y Then, the strain tensor "0 introduced by eq. (4.13) can be expressed via the stress tensor  in accordance with eq. (7.51) as ± 1 ° T r .r   / C .r .r   // : (7.52) "0 .; !/ D  2 ! 2 After that, the strain tensor "0 is substituted in Hooke’s tensor  of eq. (7.41) as follows: ± 1 ° T r .r   / C .r .r   //  C 1 W  : (7.53) .; !/ D  2 ! 2

172

Chapter 7 Semi-discretization and variational technique

The first variation of the functional ˆ must be equal to zero on the solution Z ıˆ D

Z

h

h

l l

   W C W ı"0  ı dxdy D 0 :

After integrating by parts and taking into consideration the equilibrium displacements (7.51) and the stress boundary conditions (7.45), (7.46), this functional is rewritten according to 1 ıˆ D .r  & /  rı   W ı dxdy 2 h l ! Z l

0  yDh xy .u; v/ıu C y0 .u; v/ıv yDh dx C Z

h

Z C

Z

l h

h

l



0  xDl 0 x .u; v/ıu C xy .u; v/ıv xDl dy :

(7.54)

Here, the variations of the displacements ıu and ıv are related to the stress variations ı in accordance with eq. (7.51) as     @ıxy @ıy @ıx @ıxy 1 1 C C ; ıv D  2 I (7.55) ıu D  2

! @x @y

! @x @y 0 are the components of the geometrical stress tensor x0 , y0 , xy

 0 .; !/ D C W "0 I

(7.56)

& is the stress error tensor expressed via the strain error tensor  of eq. (7.53) as follows: &.; !/ D C W  : (7.57) Integrating by parts the first term of the double integral in eq. (7.54) and taking into account the boundary conditions (7.45) and (7.46) give the expression Z ıˆ D

h

h

Z

C

Z

Z

l

l h

h

 W ı dxdy C

0 x0 ıu C xy ıv

l



l

xDl

xDl

0 xy ıu C y0 ıv

yDh yDh

dy :

dx (7.58)

Here, an auxiliary tensor of the second rank is introduced as .; !/ D

± 1 ° T r .r  & / C .r .r  & // : 2 ! 2

(7.59)

173

Section 7.3 2D elastic beam vibrations

By using eq. (7.55), the first boundary integral of eq. (7.58) can be transformed according to 1  2

!

Z

l



l

1 D

! 2

 0 xy

Z

l l

"

@ıxy @ıx C @x @y

0 @xy

@x

ıx C

@y0 @x



 C

y0

@ıy @ıxy C @x @y

0 ıxy  xy

@ıxy  @y

 yDh dx

yDh #yDh @ı y y0 dx : @y yDh

(7.60)

In accordance with eq. (7.42), take the stresses Q x , Qxy , and Qy satisfying the boundary conditions on the beam sides at y D ˙h and introduce into consideration the vector a of design parameters as ±T ° .0/ .N 1/ ; : : : ; xy ; y.0/ ; : : : ; y.N / (7.61) a D x.0/ ; : : : ; x.N / ; xy with a.x/ 2 RM . The dimension of this vector is M D 3N C 2:

(7.62)

After substituting approximation (7.42) in eqs. (7.58) – (7.60), the Euler equation of the functional ˆ with respect to the vector a is obtained Z

h h

.y; Q a/ W 1 C 2

!

"

@Q dy @a 0 2 @Qy0 @Qxy @Qxy @2 Qy @Q x 0 @ Qxy C  Qxy  Qy0 @x @a @x @a @a@y @a@y

#yDh D 0 : (7.63) yDh

0 ,  Here, Qxy Qy0 are components of the approximated geometrical stress tensor Q a/ D .Q / is the tensor defined in eq. (7.59). Q 0 D  0 .Q ; !/ and .y; The corresponding natural boundary conditions are derived from the last integral in eq. (7.58) by taking into account eqs. (7.42), (7.55), and (7.56) as Z h @ @Q x dy D 0; Q x0 0 @a @x xD˙l h Z h 0 @ @Qxy dy D 0; Qxy @a0 @x xD˙l h Z h 2 Qy 0 @  dy D 0: (7.64) Qxy @a@y xD˙l h

The equations (7.64) are linearly dependent due to the finite dimension of the poly0 .˙l; y/ passing on the boundary. The number of indepennomials Q x0 .˙l; y/ and Qxy dent natural boundary conditions in eq. (7.64) is equal to the sum of the polynomial

174

Chapter 7 Semi-discretization and variational technique

0 degrees of the functions Q x0 and Qxy

Mn D 4N C 6 :

(7.65)

To constitute a consistent system of boundary conditions, the projection technique can be applied. Then, these conditions have the form Z

h

Q x0 .y; a.x/; !/y j dy D 0 at x D ˙l

for

j D 0; : : : ; N ;

0 Qxy .y; a.x/; !/y k dy D 0 at x D ˙l

for

k D 0; : : : ; N C 1 : (7.66)

h h

Z

h

Note that the number of equations is equal to the dimension M of the design parameter vector a given in eq. (7.62). The full differential order of the Euler equations (7.63) coincides with the total number of the natural conditions defined by eq. (7.64) and Me D 4N C 2

(7.67)

essential boundary conditions expressed through their projections as Z h Q x .y; a.x/; !/y j dy D 0 at x D ˙l for j D 0; : : : ; N ; Z

h h

h

Qxy .y; a.x/; !/y k dy D 0 at x D ˙l

for k D 0; : : : ; N  1:

(7.68)

These constraints follow directly from eq. (7.46) and polynomial representation of the stress functions with respect to the variable y. The full differential order of the resulted system of ODEs (7.63) is Nd D 8N C 8

(7.69)

that coincides with the total number of boundary conditions (7.66) and (7.68). Example 7.2. Consider the rectilinear beam, which is shown in Figure 7.1, under the homogeneous boundary conditions (7.45), (7.46), and analyze its first eigenmode of natural vibrations. The dimensionless parameters of the system are E D D 1;

D 0:3;

2h D 1;

2l D 10:

Due to the beam symmetry, the body motions are decomposed into two types, namely, lateral and longitudinal vibrations. The former has the properties which correspond to the beam bending discussed in Example 7.1. The parity of the stress and displacement approximations is the same as indicated in Table 7.1. The longitudinal motions are described by taking into account the symmetry properties of the corresponding components shown in Table 7.3.

175

Section 7.3 2D elastic beam vibrations

This decomposition allows one to reduce the number of ODE system in eq. (7.63) and, consequently, its full differential order. For the lateral vibrations these numbers are (7.70) M D 3N C 3; Md D 8N C 10 for N D 2N C 1: whereas these parameters for longitudinal motions are equal to M D 3N C 2;

Md D 8N C 6 for

N D 2N C 1:

(7.71)

Only the first vibration mode, which is longitudinal, is considered in this example. The approximation parameter N D 1 has been fixed. As it has been demonstrated for the purely polynomial approximation in Section 4.4, the integrodifferential approach gives complex eigenvalues and eigenforms. The resulting approximations are taken as the real part of Re !1 and Re Q . The stress x computed under the isoperimetric condition (7.50) (W D 1) at the midpoint of the beam and the approximate frequency Re !1 are given in Table 7.4. The difference in the first eigenvalues !B1  Re !1 shows a rather good correspondence between this 2D model and the thin rod model described in Section 2.5. The imaginary part of the eigenfrequency can serve as an implicit estimate of the solution quality. As it is shown in this table, the energy values W" > 1, W D 1, W < 1 are close to each other and the integral error ˆ are relatively small. The distribution of local error function '.x; y/ given by (7.49) is depicted in Figure 7.3. As it can be seen from the picture, this function is rather small and reaches its maximum magnitudes at the corner points of the beam.

Table 7.3. The parity of displacement and stress components with respect to the coordinate y for longitudinal beam deformation.

Parity to y

u

v

x

xy

y

y 2j C1

y 2j

y 2j C1

y 2j

y 2j C1

Table 7.4. The solution parameters: maximum stress x .0; 0/, approximate eigenfrequency Re !1 , its difference from the Bernoulli beam frequency !B1 D = l, imaginary part of eigenfrequency !1 , residual energy W"  W , and energy error ˆ in Example 7.2.

x

Re !1

!B1  Re !1

Im !1

W"  W

ˆ

0.6325

0.3140

1:17  104

5:75  105

1:07  109

1:34  109

176

Chapter 7 Semi-discretization and variational technique

Figure 7.3. The distribution of the energy error ' over the domain  for the first eigenvalue of beam vibrations in Example 7.1.

Chapter 8

An asymptotic approach

In the asymptotic approach considered in this chapter, the governing equations for elastic beams are derived by using a technique that involves the expansion of unknown displacement and stress functions in terms of a small parameter (the ratio of the crosssectional size of the beam to its length) [37]. A consistent ODE boundary value problem is composed from the appropriately selected coefficients in the semi-discrete polynomial expansion of the stress-strain relation (asymptotic approximations). The attractiveness of this technique is that the differential order of the ODE system is two times less than the corresponding order following the variational approach. Moreover, it is possible, within the framework of the plane section hypothesis, to derive explicit expressions for the stress field as well as a system of first-order differential equations for the displacement field in the cases of both isotropic and anisotropic beams. The classical variational principles [77] and the method of integrodifferential relations [39, 41] lead to different approximate equations for the stress-strain fields of an elastic beam. The corresponding algorithms for 2D and 3D static modeling as well as natural vibration analysis of elastic beams are discussed in this chapter.

8.1 Classical variational approach Consider a plane homogeneous elastic body (beam) occupying a rectangular domain  with boundary  (see Figure 8.1). Introduce the Cartesian coordinate system Oxy with the origin O located at the middle point of the left beam edge and the x-axis perpendicular to this edge and pointing toward the opposite side as shown in Figure 8.1. Let h and l be the height and length of the plate, respectively. The thickness of the plate is constant and equal to unit (b D 1). It is assumed that the plane stress-strain state of the body is governed by the system of 2D elasticity equations defined in eqs. (7.4) – (7.6). If the beam length l is much greater than its height h, then the stress-strain state of this body is approximately described by classical beam equations. For example, the lateral bending equation for the beam has been given in eq. (2.112) with the beam bending rigidity EJ (J D bh3 =12). The system of equations (7.4) – (7.6) and boundary constraints are necessary conditions for the principle of minimum total potential energy discussed in Section 7.1. It is also considered that the displacement functions are described by the semi-discrete approximations (7.11).

178

Chapter 8 An asymptotic approach

Figure 8.1. 2D rectilinear elastic beam.

Based on the semi-discretization technique, let us derive several simplest solutions for system (7.4) – (7.6) with the following boundary conditions. The displacements on the left edge of the beam are zero u.0; y/ D v.0; y/ D 0 : The edges parallel to the x-axis are free of loading, i.e.,     h h D y x; ˙ D 0: xy x; ˙ 2 2

(8.1)

(8.2)

The distributed forces are applied to the right beam cross section x .l; y/ D qx .y/

and

xy .l; y/ D qy .y/ :

(8.3)

There are no volume forces (fx .x; y/ D fy .x; y/  0). Example 8.1. Assume that the shear stress is only applied to the beam according to F F D ; (8.4) hb h where F is the magnitude of shearing force (see Figure 2.2). For the zero-order approximation in eq. (7.11) (Nu D Nv D 0) the displacement functions u and v depend only on the coordinate x, i.e., x .l; y/ D 0 and xy .l; y/ D

uQ D u.0/ .x/;

vQ D v .0/ .x/ :

(8.5)

The system of equations (7.13) in this case has the form d 2 vQ d 2 uQ D 0 and D 0: (8.6) dx 2 dx 2 By taking into account the boundary conditions, the solution of system (8.6) is obtained with u D 0 and v D

F x: Gh

(8.7)

179

Section 8.1 Classical variational approach

Solution (8.7) corresponds to the pure shear under the action of the tangential stress xy D F= h, but the boundary conditions (8.2) on the beam edges at y D ˙h=2 do not hold. Example 8.2. For Nu D Nv D 1, the displacement components uQ and vQ are linear functions of y, which can be presented as uQ D u.0/ .x/ C

y .1/ u .x/; h

vQ D v .0/ .x/ C

y .1/ v .x/ : h

(8.8)

In this case, equations (7.13) can be divided into two independent subsystems d 2 u.0/ dv .1/ D 0; C dx 2 dx du.0/  24v .1/ D 0 I  24 h dx h

.1  /h2

d 2 v .1/ dx 2

h h2

(8.9)

du.1/ d 2 v .0/ D 0; C dx 2 dx

dv .0/ d 2 u.1/  6.1  /u.1/ D 0 :  6.1  /h dx 2 dx

(8.10)

System (8.9) governs the compression and tension of the beam, while eq. (8.10) is for bending. Note that ODEs (8.10) can be reduced to one fourth-order differential equation for the unknown function v0 . Thus, this system and the Bernoulli beam equation have the same differential order. Consider eq. (8.10) in more detail. With reference to the boundary conditions (8.1), (8.2), and (8.4), the lateral displacement v .0/ .x/ has the form  3 x lx 2 h2 x .1  2 /F  C C : (8.11) v .0/ .x/ D EJ 6 2 6.1  / The solution of the bending equation (2.112) for a consol loaded by the same shearing force gives the following distribution of the lateral deflection along the beam length:  3 x lx 2 Py vc .x/ D  C : (8.12) EJ 6 2 Unlike the classical Bernoulli solution, the function in eq. (8.11) explicitly depends on Poisson’s ratio . It is important to note that the derivative of v .0/ .x/ on the left edge of the plate is not equal to zero, ˇ F dv .0/ ˇˇ : (8.13) D ˇ ˇ dx Gh xD0

180

Chapter 8 An asymptotic approach

Figure 8.2. The displacement vl vs. Poisson’s ratio for linear (dashed curve) and quadratic (solid curve) approximations.

For the following dimensionless parameters: l D 10, h D 1, E D 106 , and F D 1, D 0:3, the stiffness characteristics of the beam for the linear approximation (8.8) and the classical beam solution are given by ˇ ˇ .0/ ˇ ˇ ˇ .0/ 3 dv 6 3 dvc ˇ v .l/ D 3:67  10 ; D 2:6  10 ; v .l/ D 4  10 ; D 0: ˇ c dx ˇ dx ˇxD0 xD0

Figure 8.2 plots the lateral deflection of the middle point of the beam right edge vl D v .0/ .l/ versus Poisson’s ratio (dashed curve). This displacement is the quadratic function   2.1  /l 2 C h2 lF vl D h3 G. / of and monotonically decreases as Poisson’s ratio increases. Example 8.3. For the quadratic approximation, Nu D Nv D 2, the displacement functions u and v have the form y .1/ y2 u .x/ C 2 u.2/ .x/ ; h h 2 y y v D v .0/ .x/ C v .1/ .x/ C 2 v .2/ .x/ : h h

u D u.0/ .x/ C

(8.14)

181

Section 8.1 Classical variational approach

Similarly to the linear approximation, the Euler equations (7.13) are divided into two independent subsystems, one of which describes tension and the other is for bending. As it has been shown in [38], such a division is characteristic of any approximation degree of the displacements u and v. The functions u.0/ ; v .1/ ; u.2/ ; v .3/ ; : : : define tension, while v .0/ ; u.1/ ; v .2/ ; u.3/ ; : : : describe bending of the beam. In these sequences, the coefficients u.i/ with even indices correspond only to variables v .j / with odd indices, and vice versa. In the case of bending, the Euler equations can be represented in the form d 2 v .0/ d 2 v .2/ du.1/ D 0; Ch C 12 2 2 dx dx dx  .0/  dv dv .2/  12.1  / C u.1/ D 0 ; C .5  1/h dx dx 12h

2h2 20.1  /h2

d 2 u.1/ dx 2

2 .2/ d 2 v .0/ du.1/ 2d v  160v .2/ D 0 : C 3.1  /h  20.5  1/h dx 2 dx 2 dx (8.15)

System (8.15) can be reduced to one sixth-order differential equation with respect to an unknown function, e.g., v .0/ . The order of the ODE system (7.13) increases while the degree of approximation of the displacements grows. For example, for the cubic approximation, the bending of the beam is governed by an eighth-order equation. The solution of system (8.15) subject to the boundary conditions (8.1), (8.4) is cumbersome, and therefore it is not presented. In addition, the exponential terms in this solution have rather large values, which can lead to certain computational difficulties. For example, for lh1 D 10 and the quadratic approximation, one has to operate with terms which have the order of magnitude about 10120 . For the beam parameters from Example 8.2 and D 0:3, the stiffness characteristics for the quadratic approximation (8.15) as compared with those of the classical beam solution are given by ˇ .0/ ˇ dv ˇ vl D 4:02  103 ; D 6:03  104 ; vc .l/ D 4  103 : ˇ dx ˇ xD0

Figure 8.2 (solid curve) shows the vertical deflection of the middle point of the beam right edge, vl , versus Poisson’s ratio . As compared with the linear approximation, the value vl for fixed increases. In addition, the dependence vl . / is no longer monotonic: the function vl . / has a maximum at  0:1.

182

Chapter 8 An asymptotic approach

8.2 Integrodifferential approach 8.2.1 Basic ideas of asymptotic approximations An approach to linear elasticity problems on the basis of integrodifferential relations has been presented in Chapters 4 and 5. In this method, the governing equations of elasticity have the form (see Problem 2. in Section 4.2) Z  W d  D 0;  D "0  C 1 W ; (8.16) ˆ" D 

r   C f D 0; "0x D

@u ; @x

"y0 D

@v ; @y

"0xy D

1 2

(8.17) 

@u @v C @y @x

 :

(8.18)

Unlike the classical variational formulation, the unknown functions are the components of both stress tensor and displacement vector. As an example, these components can be approximated by finite series in powers of the ratio y (8.19) Y D l in the form x D uD

2 X nD0 3 X

x.n/ .x/Y n ; u

.n/

n

.x/Y ;

xy D vD

nD0

3 X nD0 4 X

.n/ xy .x/Y n ;

y D

4 X

y.n/ .x/Y n I

nD0

v

.n/

(8.20)

n

.x/Y :

nD0

Such a representation has been chosen for two reasons. Firstly, the coefficients in eq. (8.20) have the dimension of either stress or displacement. Secondly, the expansion parameter is small for a narrow beam since h l. In expansion (8.20), x is a linear function of Y . This choice of approximations for xy and y is justified by the fact that the derivatives of the stresses in the equilibrium equations (8.17) should have the same order of expansion in terms of the small parameter Y . Substitute the stresses of eq. (8.20) into the equilibrium equations (8.17) and collect the coefficients of the same powers of Y to obtain the system of five linear differential equations (there are no volume forces, i.e., f D 0) .0/

dx .1/ C xy D 0; dx .0/ d xy C y.1/ D 0; dx

.1/

dx .2/ C 2xy D 0; dx .1/ d xy C 2y.2/ D 0; dx

.2/ d xy C 3y.3/ D 0 : dx

(8.21)

183

Section 8.2 Integrodifferential approach

In this section, we confine ourselves to the boundary conditions of the form ˇ ˇ (8.22) xy ˇyD˙ h D y ˇyD˙ h D 0I 2l

2l

h

h

Z2l

Z2l x .l; Y /ld Y D P;

h  2l

xy .l; Y /ld Y D F ; h  2l

h

Z2l x .l; Y /l 2 Yd Y D M I

(8.23)

h  2l

u.0; Y / D v.0; Y / D 0 :

(8.24)

The relations (8.22) – (8.24) correspond to tension and bending of a cantilever beam, the free end of which is loaded by the forces P , F and the moment M . After the boundary conditions (8.22) have been satisfied, the stresses in expansion (8.20) are simplified and take the form   4l 2 Y 2 .0/ .1/ .0/ xy ; x D x C x Y; xy D 1  h2    4l 2 Y 2  .0/ .1/ y D 1  .x/ C  .x/Y : (8.25)  y y h2 By solving the system of equations (8.21) with reference to the relations (8.25) and the boundary conditions (8.23) on the right beam edge, the equilibrium stress field is obtained as   12y P 6F 1 y 2 x D C 3 ŒF .x  l/ C M  ; xy D  2 ; y D 0: (8.26) h h h 4 h Then, the functional in eq. (8.16) takes the form Z  2  2 ˆ" D x C y2 C 2xy d

(8.27)



with du.3/ 3 du.2/ 2 x D Y C Y C dx dx C

du.0/ P  ; dx Eh

F .x  l/ C M du.1/ l dx EJ

! Y

184

Chapter 8 An asymptotic approach

4v .4/ 3 3v .3/ 2 Y C Y C y D l l

F .x  l/ C M 2v .2/ C l l EJ

! Y

v .1/ P  ; l Eh ! dv .2/ .1 C /F l 2 dv .4/ 4 dv .3/ 3 3u.3/ Y C Y C C C D Y2 dx dx l dx EJ ! dv .1/ dv .0/ 3.1 C /F 2u.2/ u.1/ C C  : (8.28) C Y C l dx l dx Eh C

2xy

The minimization of the functional (8.27) is reduced to the solution of a system of second-order differential equations (Euler equations) for nine unknown displacement functions. Rewrite the strain error tensor  defined in eq. (8.28) as follows: x D 1 Y 3 C 2 Y 2 C 3 Y C 4 ; y D 5 Y 3 C 6 Y 2 C 7 Y C 8 ; xy D 9 Y 4 C 10 Y 3 C 11 Y 2 C 12 Y C 13 :

(8.29)

If all coefficients i , i D 1; : : : ; 13, were equal to zero, the functional ˆ would attain its absolute minimum ˆ D 0. This condition cannot be satisfied in general case because the number of the coefficients i exceeds the number of unknown variables. To estimate residuals of the equations i D 0, a new quadratic functional is introduced J D

13 Z X

l

iD1 0

i2 dx :

(8.30)

It can be shown that the minimization of the functional J is reduced to independent minimization of the following integrals: Z J1 Œu.0/  D

l 0

Z J3 Œv .1/ ; u.2/  D Z

0

Z

0

J4 Œv .2/ ; u.3/  D J5 Œv .3/  D

0

Z 42 dx;

J2 Œv .0/ ; u.1/  D

l



 2 22 C 82 C 12 dx;

l



 2 12 C 72 C 11 dx;

l



 2 62 C 10 dx;

l 0



 2 32 C 13 dx;

Z J6 Œv .4/  D

l 0



 52 C 92 dx:

(8.31)

(8.32)

185

Section 8.2 Integrodifferential approach

Note that the functionals J1 and J2 are able to reach their absolute minima. To that end, it is necessary to solve the system of three first-order ODEs P du.0/  D 0; dx Eh du.1/ F .x  l/ C M 3 D  l D 0; dx EJ u.1/ 3.1 C /F dv .0/ 213 D C  D0 dx l Eh 4 D

(8.33)

for three unknown displacement functions u.0/ , u.1/ , and v .0/ . The boundary conditions for system (8.33) follow from (8.24) and have the form u.0/ .0/ D u.1/ .0/ D v .0/ .0/ D 0 :

(8.34)

The function u.0/ defines the tension or compression of the beam midline, while and u.1/ describe the bending of the beam. The system of equations (8.33) can be referred to as the equations of an elastic beam. In accordance with the hypotheses stated by J. Bernoulli [18], straight lines that have been normal to the middle surface prior to deformation remain straight and normal to the middle surface and do not change their length after the deformation. System (8.33) meets all Bernoulli’s hypotheses, except for the statement that the straight lines that have been normal to the middle surface prior to deformation will remain normal to that surface after the deformation. The terms i , which have not been taken into account in the functionals J1 and J2 , are responsible for the distortion of beam cross sections. These terms do not influence the solution of system (8.33) for the prescribed stress field (8.26). The solution of system (8.33) that satisfies the boundary conditions of (8.34) defines the displacement field of the body as v .0/

Px M xy F xy.2l  x/  C ; EJ 2EJ Eh M x2 F x 2 .3l  x/ 3F x v.x/ D C C : (8.35) 2EJ 6EJ 2Gh Unlike the classical beam solution (8.12), the lateral displacements v of eq. (8.35) in the case of M D 0 depends linearly on Poisson’s ratio . The first derivative of the function v on the left edge of the beam is ˇ dv ˇˇ 3F : D ˇ dx 2Gh u.x; y/ D 

xD0

For the beam parameters and the shearing force F specified in Section 8.1 as well as Poisson’s ratio D 0:3, the stiffness characteristics of the plate for the approximation (8.35) and the classical beam solution are given by ˇ dv ˇˇ 3 D 6:04  104 ; vc .l/ D 4  103 : v.l/ D 4:039  10 ; dx ˇ xD0

186

Chapter 8 An asymptotic approach

Figure 8.3. Lateral displacement vl . / vs. Poison’s ratio .

To compare different approaches to the problem of elastic beam bending, the dependence of the lateral deflection v.l; 0/ on Poisson’s ratio is considered in more detail. Figure 8.3 presents the plots of v.l; 0/ versus . These curves have been obtained by three different methods. The dashed curve corresponds to the quadratic approximation of the displacement functions that have been obtained in Section 8.1 on the basis of the principle of minimum potential energy. This principle has been also utilized to solve the problem of linear elasticity. The corresponding algorithm for constructing a numerical solution, in which u and v are represented by complete polynomials of a fixed degree Nu with respect to the variables x and y, has been described in Chapter 4. The dash-dot curve in Figure 8.3 is obtained by this method for Nu D 16. The solution corresponding to eq. (8.35) is shown in Figure 8.3 by a solid curve. The curves of Figure 8.3 indicate the following specific features. The relative error of calculation of the displacement vl obtained by these methods for arbitrary 2 Œ0; 0:5 does not exceed 1%. The maximum spread in the values of vl is observed for D 0:5 (an incompressible material). The dashed curve can be regarded as a lower estimate and the solid curve as an upper estimate for the exact value of vl . The curve corresponding to the asymptotic approach is a strictly increasing function of the argument , whereas the dash-dot and dashed curves (variational approach) have a maximum at certain values of Poisson’s ratio .

Section 8.2 Integrodifferential approach

187

8.2.2 Beam equations – general case of loading Let a beam of a rectangular cross section, which stress-strain state is described by the equations of (7.4) – (7.6), be clamped along the left edge (Figure 8.1), i.e., u.0; y/ D v.0; y/ D 0 :

(8.36)

The upper and lower edges of the beam are loaded by the forces according to     h h ˙ (8.37) xy x; ˙ D ˙px .x/; y x; ˙ D ˙py˙ .x/; 2 2 where px˙ .x/ and py˙ .x/ are prescribed functions. The linear combinations of px˙ .x/ and py˙ .x/ can be regarded as various types of loads distributed along the beam. For example, ps D pyC C py is the shearing load, p t D pyC  py is the lateral compression or tension, pb D pxC  px defines the distributed bending moment, and pl D pxC C px is the longitudinal distributed forces. There are two types of boundary conditions considered on the right edge of the beam. If the forces and the moment of eq. (8.23) are prescribed on the right edge then the problem is statically determinate and the stress fields can be obtained on the basis of the equilibrium equations (7.4). For the boundary conditions specified in the form u.l; y/ D v.l; y/ D 0;

(8.38)

the problem is three times statically indeterminate and the stress fields can be determined to within three unknown constants. Using the approximation in eq. (8.20), solving the equilibrium equations (8.21), and satisfying the boundary conditions (8.23), one can find the stress functions Z Z 1 3 pb .0/ .0/ pl dx C c3 ; xy D  ps dx  C c2 ; x D  h 2h 4 “ Z 12l 6l 8lc2 x C c1 ; ps dxdx  2 pb dx C x.1/ D  3 h h h2 4l 2 .0/ 2l 2 pb lpl .2/ .1/ xy ; D  2 xy C ; xy D 2 h h h h dpl pt 4l 2 2l 2 y.0/ D C ; y.2/ D  2 y.0/ C 2 p t ; 2 8 dx h h 2 l 3lp 4l 4l 3 dp s b C ; y.3/ D  2 y.1/ C 3 ps : (8.39) y.1/ D 2h 4 dx h h

188

Chapter 8 An asymptotic approach

By utilizing the approximation in eq. (8.20) for the displacements, the functions x , y , and xy can be expressed as follows: ! .0/ .0/ .1/ .1/ du.1/ x  y du.0/ x  y  C  x D Y dx E dx E ! ! .2/ .3/ y y du.3/ du.2/ 2 C C Y C Y 3; C dx E dx E ! .0/ .0/ .1/ .1/ 2v .2/ y  x v .1/ y  x  C  Y y D l E l E ! ! .2/ .3/ 4v .4/ y 3v .3/ y 2   C Y C Y 3; l E l E ! .0/ .1/ u.2/ 1 dv .0/ xy 1 dv .1/ xy u.1/ C  C C  xy D Y 2l 2 dx 2G l 2 dx 2G ! .2/ 1 dv .2/ xy 3u.3/ 1 dv .3/ 3 1 dv .4/ 4 C  Y2 C Y C Y : (8.40) C 2l 2 dx 2G 2 dx 2 dx Having adopted the hypothesis of plane sections discussed above, one can write the beam equilibrium equation in the form x.0/ .x/  y.0/ .x/ du.0/ D ; dx E .1/ .1/ x .x/  y .x/ du.1/ D ; dx E

.0/

xy .x/ dv .0/ u.1/ C D : dx l G

(8.41)

It is apparent from (8.41) that the beam equations represent the relations of Hooke’s law for the respective components in terms of the displacement and stress approximations (8.20). Therefore, these equations can be readily generalized to the case of an anisotropic beam. For example, for an orthotropic material oriented along the x-axis, the system of differential equations (8.41) transforms as .0/

.0/ x .x/ 12 y .x/ du.0/ D  ; dx E1 E2 .1/ .1/ du.1/ x .x/ 12 y .x/ D  ; dx E1 E2 .0/

xy .x/ u.1/ dv .0/ C D ; dx l G12

E2 21 D E1 12 :

(8.42)

Here, E1 and E2 are Young’s moduli along the principal axes of elasticity (x and y), G12 is the shear modulus, and 12 and 21 are Poisson’s ratios.

189

Section 8.2 Integrodifferential approach

In all of the following examples in this section, apart from the last one, it is assumed that the beam has the length l D 10 , the height h D 1, and thickness equal to the height (the cross section is a square); Young’s modulus E D 106 . Example 8.4. Consider a cantilever beam (see Figure 8.1) that is clamped on one edge (u.0; y/ D v.0; y/ D 0) and is loaded by distributed shearing forces ps .x/ D q D const. The other functions and constants determining the boundary conditions are equal to zero, i.e., P D F D M D 0;

pl .x/ D pb .x/ D p t .x/  0 :

The equilibrium stress field is uniquely defined by the expressions in eq. (8.20) and (8.39). The vertical displacement of the beam v .0/ .x/ is determined by solving the system of equations (8.42). The solution for a homogeneous and isotropic body yields  qx 2  2 x  4lx C 6l 2 ; v .0/ D vc C v; vc D 24EJ   3qx x 2l  x v D C ; 4h E G where vc is the function obtained on the basis of the Bernoulli beam equation. If q > 0 then the additive term v is positive for all x > 0. For q D 1, the classical deflection of the beam end is equal to vc .l/ D 0:015 and the additional displacement is given by v.l/ D .1:5 C 2:25 /  104 . Note that the deflection of the beam end is a linear expression of Poisson’s ratio. The value of v.l/ normalized by the total displacement for D 0:3 is measured by ˇ v.l/  100% ˇˇ  1:43% : vc .l/ C v.l/ ˇD0:3 Example 8.5. Let the beam be clamped at both edges according to u.0; y/ D v.0; y/ D u.l; y/ D v.l; y/ D 0 and loaded by a distributed shearing force ps .x/ D q D const. The other functions occurring in the boundary conditions are zero, i.e., pl .x/ D pb .x/ D p t .x/  0: The lateral deflection of the beam v .0/ .x/ for these boundary conditions has the form v .0/ D vc C v;

vc D

qx 2 .l  x/2 ; 24EJ

v D

3qx.l  x/ : 4Gh

190

Chapter 8 An asymptotic approach

As in Example 8.4, for q > 0 the additive term v is positive for all x > 0. For q D 1, the displacement at the beam midpoint is vc .0:5l/ D 3:125  104 and the additional displacement is defined by v.0:5l/ D .1 C /  3:75  105 . The value of v .0/ .0:5l/ for D 0:3 is measured by ˇ v.0:5l/  100% ˇˇ  13:49% : vc .0:5l/ C v.0:5l/ ˇD0:3 Note that the type of the boundary conditions significantly influences the accuracy of the beam solutions. Example 8.6. Consider a cantilever orthotropic beam with the square cross section that is clamped along one edge, u.0; y/ D v.0; y/ D 0, and is loaded by distributed shearing forces with ps .x/ D q D const. The other functions and constants in the boundary conditions are equal to zero, as in Example 8.4. The parameters of an orthotropic material are chosen as follows: E1 D 106 ;

E2 D 4  105 ;

G12 D 2  105 ;

12 D 0:12 :

The orientation of principal Young’s modulus E1 coincides with the x-axis. The overall sizes of the beam coincide with those of Example 8.4. The lateral displacement v .0/ .x/ is determined by solving the system of equations (8.42), which yields  qx 2  2 x  4lx C 6l 2 ; 24E1 J 3q 12 2 3q.2l  x/ xC x ; v D 4G12 h 4E2 h

v .0/ D vc C v;

vc D

where vc is the displacement function obtained on the basis of the beam bending equation for the case of E D E1 . For q > 0, the additive term v is positive for all x > 0. For q D 1, the deflection of the free beam end is defined by vc .l/ D 0:015. The additional displacement is equal to v.l/ D 3:975  104 ; its relative value is v.l/  100%  2:58% : vc .l/ C v.l/

8.3 Elastic beam vibrations 8.3.1 Statement of an eigenvalue problem An elastic rectangular beam of the unit thickness with the sizes h  l is considered. The origin of the Cartesian system of coordinates Oxy is placed at the middle point of a beam side with height h, and the x-axis is parallel to the sides with lengths l (see

191

Section 8.3 Elastic beam vibrations

Figure 8.1). It is assumed that the stress-strain state of the isotropic plate is described by 2D equations of the linear theory of elasticity according to eqs. (7.4) – (7.6). Suppose that the beam boundary is free of loading. Then, the boundary conditions are given by xy D y D 0 for x D xy D 0 for

h yD˙ ; 2

(8.43)

x D 0; l :

(8.44)

It is considered that the beam can oscillate (free small elastic vibrations) with respect to its equilibrium state and the volume force vector f is defined by the inertia forces due to motions of internal points fx D 

@2 u ; @t 2

fy D 

@2 v ; @t 2

(8.45)

where is a volume density of the body. When the beam length l is much greater than its height h, the stress-strain state of the plate is described by the approximate equations of the classical beam theory. In agreement with eqs. (2.96) and (2.120), the equations for longitudinal and lateral beam vibrations have the form E

@2 u @2 u 

D 0; @x 2 @t 2

EJ

@4 v @2 v C

A D 0; @x 4 @t 2

(8.46)

respectively, where EJ is a bending stiffness and A is the area of the beam cross section. To find the plate eigenvibrations, the method of separation of variables is used and the unknown stresses and displacements are chosen as the infinite power series with respect to Y D l 1 y x .x; y/ D e i!t

1 X

x.n/ .x/Y n ;

nD0 1  X y.n/ .x/Y n ; y .x; y/ D e i!t 1  2 Y 2

  xy .x; y/ D e i!t 1  2 Y 2

nD0 1 X

.n/ xy .x/Y n ;

nD0

u.x; y/ D e i!t v.x; y/ D e i!t

1 X nD0 1 X nD0

u.n/ .x/Y n ; v .n/ .x/Y n ;

D

h ; 2l

(8.47)

192

Chapter 8 An asymptotic approach

where ! is the unknown eigenfrequency and  D h=.2l/ is the geometrical plate parameter. In accordance with the integrodifferential approach, the boundary value problem (7.4) – (7.6), (8.43), (8.44) can be reduced, for example, to the minimization of the functional (8.16) under the differential constraints (7.5), (7.6) and boundary conditions (8.43), (8.44). By taking into account the expansions of the stresses and displacements (8.47), the components of the strain error tensor  can be presented as follows: " # 1 X 1  2 Y 2 .n/ du.n/ x.n/ x D  C y Y n; dx E E nD0 1  X .n/ n C 1 .nC1/ 1  2 Y 2 .n/ v y C x Y n ;  y D l E E nD0  1 1 X n C 1 .nC1/ dvn 1  2 Y 2 .n/ u  xy Y n : xy D C (8.48) 2 l dx G nD0

The equality of the integral ˆ" in eq. (8.27) to zero denotes that the functions x , y , and xy are equal to zero in the domain  occupied by the beam everywhere excluding a set of measure zero [34]. So, it is assumed that all the coefficients at the powers of Y in relations (8.48) must be equal to zero. By substituting eq. (8.47) in the equilibrium equations, it is possible to resolve eq. (7.4) with respect to displacement functions u.k/ .x/, v .k/ .x/ for k D 0; 1; 2; : : : according to " .k/  # d k C 1 1 1 x .kC1/ .k1/ C xy  2 xy u.k/ D  2

! dx l  " .k/ #  .k2/ d  d  1 k C 1 1 1 xy xy  2 C : (8.49) v .k/ D  2 y.kC1/  2 y.k1/

! dx  dx l  Here, the void functions .1/ .2/ D xy D y.1/ D y.2/  0 xy

are introduced for uniformity of eq. (8.49). By taking into account relations (8.48) and (8.49), the components of the strain error tensor can be rewritten as x D

1 X kD0

x.k/ Y k ;

y D

1 X kD0

y.k/ Y k ;

xy D

1 X kD0

.k/ k xy Y ;

(8.50)

193

Section 8.3 Elastic beam vibrations

where x.k/

1 D 2

!

"

.k/

d 2 x dx 2

.kC1/ .k1/ d xy 1 d xy  2 dx  dx

kC1 C l

.k/

!#

.k2/

.k/ y 1 y x C  2 ; E E  E " .kC1/ !# .k1/ .k/ y 1 d xy kC2 k C 1 d xy .kC2/  2 C y D  2

! 2 l dx  dx l 



y.k/

y.k/  C E " .kC1/ k C 1 dx D

! 2 l dx " .k/ d 2 xy 1 C 

! 2 dx 2 .k/

C

.k/ 2xy

x E

.k/



.k2/

1 y 2 E

;

.k/ xy kC2 .kC2/ C  2 xy l  .k2/

1 d 2 xy 2 dx 2

kC1 C l

!#

.kC1/

dy dx

.k1/

1 dy  2  dx

!#

.k2/

xy 1 xy C 2 G  G

;

k D 0; 1; 2; : : : :

(8.51)

By analyzing the structure of eq. (8.51), it is possible to select two subsystems. It has been noted in Section 7.3, that this decomposition is typical for an arbitrary degree of the stress and displacement approximations and arises due to the beam symmetry .2j / .2j / .2j C1/ with respect to the x-axis. The functions x , y , xy , u.2j / , v .2j C1/ for .2j C1/ .2j C1/ .2j / , y , xy , j D 0; 1; : : : describe tension and compression. In contrast, x .2j / u.2j C1/ , v .2j / characterize bending of the beam. Similarly, the coefficients x , .2j / .2j C1/ .2j C1/ .2j C1/ .2j / y , xy correspond to longitudinal vibrations, whereas x , y , xy describe the lateral motions of the body. The finite dimensional representations of the stresses and displacements are considered to find the approximate solution of the problem (7.4) – (7.6), (8.43), (8.44). .j / It means that the functions x.j / , y.j / , xy are equal to zero at j > N , where N is maximal degree of expansion in eq. (8.47). Then, the number of nonzero coefficients in eq. (8.51) is also finite and equal to 3.N C 3/ (k D 0; 1; : : : ; N C 2). Since there are only 3.N C 1/ unknown stress functions, in general, it is not possible to resolve exactly the system of 3.N C 3/ ODEs x.k/ D 0;

y.k/ D 0;

.k/ xy D 0:

To solve approximately this overdetermined system of equations, various optimization methods can be applied, for example, by using the variational formulation given in Section 7.3. In this section, an approach is discussed in which the only equations corresponding to the lower degrees of the expansions for the strain error components

194

Chapter 8 An asymptotic approach

x , y , and xy with respect to the variable Y are satisfied. After that, the unsolved .j / for j D N C 1; N C 2 are used to qualify the nonzero coefficients x.j /, y.j / , and xy approximate solution of the integrodiffeerential problem (8.16), (7.5), (7.6), (8.43), (8.44).

8.3.2 Longitudinal vibrations The finite dimensional system of ODEs x.0/ D 0;

y.0/ D 0;

x.2j / D 0; y.2j / D 0;

.2j 1/ xy D 0;

j D 1; 2; : : : ; N1 C 1;

(8.52)

and corresponding boundary conditions x.0/ .0/ D x.0/ .l/ D 0; .2j 1/ .2j 1/ .0/ D yx .l/ D 0; x.2j / .0/ D x.2j / .l/ D yx

j D 1; 2; : : : ; N1 ; (8.53)

describing longitudinal vibrations of the beam are obtained from eqs. (8.44) and (8.51) .2j / by using the decomposition of the unknown stresses. Note that all functions y can .2j / .2j 1/ , and their derivatives. be eliminated from the system (8.52) through x , xy The second-order ODEs x.0/ D 0;

x.2k/ D 0;

.2k1/ xy D 0;

k D 1; 2; : : : ; N1

(8.54)

.2k1/ after taking into are resolved with respect to 2N1 C 1 variables x.0/ , x.2k/ , xy .2j / account the explicit expressions for y . The general solution of the system (8.54) under the boundary conditions (8.53) can be used to find the eigenvalues !. In the sequel, the system of equations (8.54) jointly with the boundary conditions (8.53) is named the N1 -th approximation of the origin problem.

Example 8.7. Consider the zeroth approximation (N1 D 0) for the problem of free longitudinal beam vibrations. In this case, it is necessary to take only two equations from the system (8.52) into account .0/

E d 2 x C x.0/  y.0/ D 0 ;

! 2 dx 2

.0/

8E y  y.0/ C x.0/ D 0 :

! 2 h2

(8.55)

The equations (8.55) are reduced to an ODE of the second order with respect to unknown function x.0/ .0/

d 2 x C 2 x.0/ D 0 ; dx 2

(8.56)

195

Section 8.3 Elastic beam vibrations

where !2  D 2 2 !0 h 2

! 2 2 1 2 !  8!02

! :

(8.57)

Here, !0 is the characteristic frequency defined by the relation !02 D .0/

The function y

E :

h2

(8.58)

is found from the second equation of eq. (8.55) y.0/ D x.0/

with  D

! 2 : ! 2  8!02

(8.59)

The boundary conditions are written in the form x.0/ .0/ D x.0/ .l/ D 0 :

(8.60)

In contrast to the Bernoulli model, the eigenvalue  in eq. (8.56) depends nonlinearly on the eigenfrequency ! as well as parameters , h, and !02 . Note that the wavenumbers 2 .!/ are positive at ! 2 .0; !1 / [ .!2 ; 1/ and negative at ! 2 .!1 ; !2 /, where 8!02 : (8.61) !12 D 8!02 ; !22 D 1  2 One is able to show that there is only trivial solution of the eigenvalue problem (8.56), (8.60) at 2 .!/ 0. When the inequality 2 .!/ > 0 is valid the solution has the form (8.62) x.0/ D c sin.x/ : To find the eigenvalues !, the characteristic equation can be presented as !p 1  .!/ D  m; !0

D

h ; l

m 2 Z:

(8.63)

The two positive roots !C and ! for eq. (8.63) are explicitly defined as the functions of m 0 v q u u 8 C  2 m2 ˙ 64 C  2 m2  2 m2  16.1  2 2 / t : (8.64) !˙ .m/ D !0 2.1  2 / The functions !˙ .m/ are monotonically increasing, ! .0/ D 0, !C .0/ D !2 , and the following asymptotic formulae are valid: ! jm!1 ! !1 ;

!0  m !C jm!1 ! p : 1  2

196

Chapter 8 An asymptotic approach

Figure 8.4. Longitudinal eigenfrequencies ! vs. the mode number m.

Figure 8.4 shows the eigenfrequencies !˙ .m/ (solid lines) for the following values of the parameters: !0 D 1,  D =10, D 0:3. The critical values ! D !1  2:828 and ! D !2  2:965 are denoted by the dashed lines. The inclined dash-dot line corresponds to the classical beam solution at the same parameters. The ratio of the vibration amplitudes  introduced in eq. (8.59) versus the number of eigenfrequency m is shown in Figure 8.5. The solid curve corresponds to the lower branch ! .m/ of the solution, and the dashed curve indicates the upper branch !C .m/. It is the important feature for the longitudinal motions that the following inequalities hold:  0 for the lower branch and  > 0 for the upper branch. Note that .0/ .0/ the maximal values of the functions x and y at m D 9 are approximately equal to each other for both roots of eq. (8.64). .0/ .0/ Forms of x , y for different values !˙ .m/ and the given beam parameters are shown in Figure 8.6 (lower branch ! ) and Figure 8.7 (upper solution branch !C ). .0/ The dashed curve in Figure 8.6 corresponds to x , defined with eq. (8.62) at c D 1, for the first vibration mode m D 1 (! .1/ D 0:3140). In this case, the stress ratio  of eq. (8.59) is a rather small value (jj 1). The eigenfunctions x.0/ and y.0/ at m D 9 .! .9/ D 2:4803/ are presented by solid and dash-dot curves, respectively. .0/ .0/ .0/ .0/ In Figure 8.7 the eigenforms x , y for !C .1/ D 2:9667 and x  y for !C .9/ D 3:3800 are shown respectively by dashed, dash-dot, and solid curves. The eigenshapes of the beam boundary for ! .1/ and !C .1/ are shown by solid lines in Figure 8.8 and Figure 8.9, respectively. The undeformed beam is presented in both figures by dashed lines.

197

Section 8.3 Elastic beam vibrations

Figure 8.5. Longitudinal eigenstress ratio  vs. the mode number m.

.0/

.0/

Figure 8.6. Longitudinal stresses x and y for the lower frequency branch ! .

Note that if the eigenfrequency belongs to the lower branch ! , then the displacement vector is approximately collinear to the x-axis (u  v for m D 1). In contrast, the vibrations corresponding to the upper branch !C (so-called breathing mode) are characterized by large lateral displacements (v  u for m D 1).

198

Chapter 8 An asymptotic approach

.0/

.0/

Figure 8.7. Longitudinal stresses x and y for the upper frequency branch !C .

Figure 8.8. Longitudinal eigenshape for ! .1/.

Figure 8.9. Longitudinal eigenshape for !C .1/.

199

Section 8.3 Elastic beam vibrations

8.3.3 Lateral vibrations Similarly to the case of the beam longitudinal motions, the finite dimensional system of ODEs x.2j C1/ D 0;

y.2j C1/ D 0;

.2j / xy D 0;

j D 0; 1; : : : ; N2 C 1;

(8.65)

with the boundary conditions .2k/ .2k/ .0/ D xy .l/ D 0; x.2kC1/ .0/ D x.2kC1/ .l/ D xy

k D 0; 1; : : : ; N2 ; (8.66) for lateral vibrations is obtained by decomposing the unknown stresses. The variables .2kC1/ .2kC1/ y are explicitly expressed from system (8.65) through the coefficients x , .2k/ xy and their derivatives. The first 2N2 C 2 ODEs of the second order x.2j C1/ D 0;

.2j / xy D 0;

j D 0; 1; : : : ; N2 ;

(8.67)

.2k/ are resolved with respect to 2N2 C 2 unknown functions x.2kC1/ and xy by tak.2kC1/ . The general solution of sysing into account the explicit expressions for y tem (8.67) jointly with the boundary conditions (8.66) is used to find the eigenvalues !.

Example 8.8. When N2 D 0 (the zeroth approximation), the system of equations in eq. (8.67) has the form

.0/ d 2 xy dx 2

where y.1/

.0/ .1/ 

! 2  .1/ 8l d xy d 2 x .1/ D0; C     x y dx 2 h2 dx ! E   2 .1/ dy.1/ 8 1 dx

! .0/ C  2 xy C D0; C l dx dx G h

" # .0/ d xy 1 2 2 .1/ D

! h x C 8El :

! 2 h2  24E dx

.0/ Resolve explicitly the function xy from system (8.68) as .1/

.0/ xy

!2 h2 B3 D 2.1 C /  24 2!32 4.1 C

.1/

d 3 x dx D C.!/ B3 .!/ C B1 .!/ 3 dx dx

with

C D

(8.68)

! ;

B1 D 1 C C

3h2 !34 ;  ! 2 /.6!32  ! 2 /

/l.!32

!32 D

! (8.69)

1  2 ! 2 1  ! 4  ; 6 !32 12 !34 4!02 : 1C

200

Chapter 8 An asymptotic approach

Then, one ODE of the fourth order is obtained A4 .!/ where

d 4 x.1/ d 2 x.1/ C A .!/ C A0 .!/x.1/ D 0; 2 dx 4 dx 2

(8.70)

! ! 4.1  /! 2 !2 12! 2 !2 A2 D 1 C 2 2 1 .1 C /h2 !32 4!32 h !3 24!02 ! ! 24!02 !2 !2 1 2 : 1  2 x.1/ D 0; !42 D 16!02 ; !52 D 1  2 !3 !5

!2 A4 D 1  2 ; !4 A0 D

! 2 EJ

Note that the denominator of the parameter C.!/ in eq. (8.69) is equal to zero at p ! D !3 and ! D 6!3 . These cases should be considered separately. The boundary conditions for eq. (8.70) have the form .0/ .0/ .0/ D xy .l/ D 0 x.1/ .0/ D x.1/ .l/ D xy

(8.71)

.0/

with xy given by eq. (8.69). The roots j .!/, j D 1; 2; 3; 4, of the biquadratic characteristic equation for ODE (8.70) are analytically found as   p ! D3 ! 3 C D1 ! ˙ D6 ! 6 C D4 ! 4 C D2 ! 2 C D0 : (8.72)

2 D  2.! 2  16!02 /!02 h2 The constants D3 D .1 C /.  3/;

D1 D 32.2 C /!02 ;

D6 D .1 C /4 ;

D4 D 32.1 C 2 /.1 C /!02 ;

D2 D 256.6 2 C 4  1/!04 ; D0 D 3  84 !06 are introduced in eq. (8.72). There always exist two imaginary complex conjugate roots for every !. Let us consider the following frequency zones: 1/ ! 2 .0; !3 / ;

2/ ! 2 .!3 ; !4 / ;

3/ ! 2 .!4 ; !5 / ;

4/ ! 2 .!5 ; C1/ :

In the first and third intervals, the two other roots have real values, but for second and fourth zones these roots are purely imaginary. Hence, for the frequencies which belong to the first and third intervals, the general solution of eq. (8.70) has the form x.1/ D c1 sin.j 1 jx/ C c2 cos.j 1 jx/ C c3 sh.j 2 jx/ C c4 ch.j 2 jx/;

(8.73)

Section 8.3 Elastic beam vibrations

201

Figure 8.10. Lateral eigenfrequencies ! vs. the wavenumber j j.

but for the other zones x.1/ D c1 sin.j 1 jx/ C c2 cos.j 1 jx/ C c3 sin.j 2 jx/ C c4 cos.j 2 jx/:

(8.74)

Here, j 1 j and j 2 j are referred to as the wavenumbers. The eigenvalues ! are found when nontrivial solutions of either eqs. (8.73) or eq. (8.74) under the homogeneous boundary conditions (8.71) exist. The correlation between the roots j of the characteristic equation and the eigenvalues ! is presented in Figure 8.10 at the dimensionless values of the parameters !0 D 1, h D 1, and D 0:3. The dependence of the frequency ! on the wavenumber of purely imaginary roots j is shown by the solid curves. The dash-dot lines correspond to the real roots. The critical values ! D !k (k D 3; 4; 5) are marked by the dotted lines. The dashed curve characterizes the solution of the classical eigenvalue problem (the second equation of (8.46)). The lower frequencies (m 12) correspond to the solution (8.73) (the first zone). For chosen values of the system parameters, the critical frequency is approximately equal to !3  1:75. It is worth noting that the number of the eigenvalues corresponding to solution (8.73) is always finite and is increasing when the value of the parameter h  D 2l is decreasing. The plot of the eigenvalues versus the mode number is presented in Figure 8.11 by the solid curve. The dashed curve shows the classic frequencies !c of the lateral vibrations obtained in accordance with the Euler–Bernoulli beam theory (the second equation in eq. (8.46)). Note that the classic eigenvalues as a function of the number m are quadratic, whereas this dependence obtained by the MIDR is more or less linear.

202

Chapter 8 An asymptotic approach

Figure 8.11. Lateral eigenfrequencies ! and !c vs. the mode number m.

The eigenforms u.x; h=2/ and v.x; 0/ given by relations (8.49) for different values m are shown in Figure 8.12 (! < !3 ). The displacements v.x; 0/ of the beam reference line are denoted by solid curves in this figure. These functions correspond .1/ to the eigenstresses x defined by solution (8.73) at c1 D 1 for m D 1 and m D 12. The curve with one maximum is plotted for m D 1 while the multiextremal curve corresponds to the case m D 12. The variables u.x; h=2/ for m D 1 and m D 12 are shown by the dashed curves. The monotonic curve corresponds to m D 1. It is worth noting that the eigenshapes at m D 1 are characterized by bending (the shear deformation is nearly absent), whereas the influence of the shear deformations on the eigenshapes is increasing when the number m is growing. At m D 12 the function u.x; h=2/ is positive everywhere. The shape of the beam changes dramatically when the value of eigenfrequency overpasses the bifurcational point !3 . In Figure 8.13 the eigenforms u.x; h=2/ (dash curve) and v.x; 0/ (solid curve) are plotted for the second frequency zone (! > !3 with m D 13). The eigenshapes of lateral beam vibrations for !.1/  0:061 and !.12/  1:741 are depicted in Figure 8.14 and 8.15, respectively. The beam vibration form is shown by solid curves; the undeformed beam shape is presented by dashed lines. It is an important feature of the lateral motions that the shear deformations significantly influence the eigenforms of beam vibrations for higher modes. It is seen in Figure 8.15 that all beam cross sections (thin solid lines) have slopes in one direction.

Section 8.3 Elastic beam vibrations

203

Figure 8.12. Lateral eigenforms u and v for ! < !3 .

Figure 8.13. Lateral eigenforms u and v for ! > !3 , n D 13.

Note that the difference between the eigenvalues obtained by the MIDR (!) and the classical approach (!c ) is diminishing if the parameter  is decreasing. The corresponding beam eigenvalues ! and !c for the natural lateral vibrations for the parameters !0 D 1, h D 1, l D 100, D 0:3 are shown in Table 8.3.

204

Chapter 8 An asymptotic approach

Figure 8.14. Lateral beam deformation for the first eigenfrequency !.1/.

Figure 8.15. Lateral beam deformation for the eigenfrequency !.12/.

Table 8.1. The eigenfrequencies ! and !c for lateral vibrations at  D 0:005.

8.4

m

1

2

3

4

5

6

! !c !; %

0.000645 0.000646 0.0598

0.00178 0.00178 0.192

0.00348 0.00349 0.390

0.00573 0.00577 0.658

0.00853 0.00862 0.992

0.0118 0.0120 1.390

3D static problem

Consider an elastic body (beam), depicted in Figure 8.16, which occupies the 3D domain  D ¹x; y; z W

0 < x < l;

jyj < h;

jzj < bº;

205

Section 8.4 3D static problem

Figure 8.16. Prismatic beam with the rectangular cross section.

where l is the beam length and 2h  2b is the sizes of its rectangular cross section. A stress-strain state of the body is described by the PDE system  D "0  C 1 W  D 0; r D r   D 0I   @v @u @v @w 1 @u "0x D ; "y0 D ; "0z D ; "0xy D C ; @x @y @z 2 @y @x     @w @w 1 @u 1 @v 0 0 C C "xz D ; "yz D ; 2 @z @x 2 @z @y

(8.75)

and the boundary conditions q D nD0

for

u D v D w D 0 for for q D ql

y D ˙h and z D ˙b ; x D 0; xDl:

(8.76)

Here, ql .y; z/ is a given vector function of external load; r is the equilibrium vector; u, v, and w are the displacements along the axes x, y, and z, respectively; "˛ and "˛ˇ , ˛; ˇ D x; y; z, are the components of the strain tensor ". The problem is to find the displacements u , v  , and w  and the stress tensor   which satisfy the equations (8.75) and the boundary conditions (8.76). The asymptotic approach is based on the MIDR and the expansion of the displacement and stress functions in power series with respect to coordinates y and z. After implementation of the boundary conditions at y D ˙h and z D ˙b, finite dimension approximations

206

Chapter 8 An asymptotic approach

u and  can be presented in the form uD

N X

u

.i;j / i j

vD

y z ;

iCj D0

wD

N X

N X

v .i;j / .x/y i z j ;

iCj D0

w .i;j / .x/y i z j I

iCj D0

x D

N X

x.i;j / y i z j ;

iCj D0

y D

N X

N X

yz D

.i;j / 2 yz .h  y 2 /.b 2  z 2 /y i z j ;

iCj D0

y.i;j / .h2  y 2 /y i z j ;

iCj D0

xy D

N X

N X

z D

z.i;j / .b 2  z 2 /y i z j ;

iCj D0 .i;j / 2 xy .h

2

i j

 y /y z ;

N X

xz D

iCj D0

.i;j / 2 xz .b  z 2 /y i z j :

(8.77)

iCj D0 .i;j /

.i;j /

Here, u.i;j / , v .i;j / , w .i;j / , ˛ , and ˛ˇ for ˛; ˇ D x; y; z are the unknown functions of the coordinate x. Then, it follows from eqs. (8.75) and (8.77) that the vector r and tensor  are described by the finite sums rD

N C3 X

r .k;m/ y k z m ;

D

kCmD0

N C4 X

 .k;m/ y k z m ;

(8.78)

kCmD0

where r .k;m/ and  .k;m/ are linear differential relations expressed through the unknown functions of eq. (8.77). It is also supposed that the vector ql is polynomial with respect to coordinates y and z. Note that the boundary conditions (8.76) must be consistent. .i;j / .i;j / The algorithm of finding u.i;j / , v .i;j / , w .i;j / , ˛ , and ˛ˇ consists of successive steps. In the first step, it is necessary to satisfy the boundary conditions q D 0 on the faces y D ˙h and z D ˙b. After that the system of ODEs is composed r .k;m/ D 0;

 .k;m/ D 0;

kCm N C4

(8.79)

with the boundary conditions u.k;m/ D v .k;m/ D w .k;m/ D 0 for q

.k;m/

D

.k;m/ ql

for

x D l;

ql D

x D 0; M X

.i;j / i j

ql

y z ;

iCj D0

which follow from eqs. (8.76) and (8.77). It is considered that N M .

(8.80)

207

Section 8.4 3D static problem

Note that the finite dimension system (8.79) is overdetermined, since the number of equations exceeds the number of variables. In order to compose the consistent boundary-value problem, let us equate to zero all the coefficients r .k;m/ and the required number of  .k;m/ at the lowest degrees of the polynomial expansions  in eq. (8.78). After solving this reduced system and taking into account the boundary conditions (8.80), the remaining undefined terms in eq. (8.78) can be used to estimate the quality of the obtained approximate displacement and stress fields. Analyzing the system of equations (8.79) and boundary conditions (8.80), one can show that there are four independent subsystems describing the compression-tension, bending with respect to the y-axis and the z-axis, as well as the torsion of the elastic beam. Such decomposition is typical for an arbitrary polynomial approximation of displacements and stresses and is due to the symmetry of the domain  with respect to the coordinate planes xy and xz. The symmetry properties (even or odd powers of the coordinates y and z) and the minimum required degree of approximations are presented in Table 8.2 for the different stress-strain states of the beam. Here, m is a positive integer indicating the order of the problem and the index ˛ takes the letters x; y; z. The corresponding functions are absent if N < 0. Example 8.9. As an example, consider a homogeneous and isotropic cantilever beam with a square cross section and size a D h. The left end of the beam is clamped according to eq. (8.76). All the lateral faces are free of load, volume forces are absent, and the tensile stress ql D ¹1; 0; 0º T is uniformly distributed on the end cross section at x D l. This means that the tensile force Px D 4h is applied to the free beam end (see Figure 8.16).

Table 8.2. Symmetry properties of the approximate functions (i D 2m; m D 1; 2; : : :).

Tension

Bending, y-axis

Bending z-axis

Torsion

y j ,N

z k ,N

y j ,N

z k ,N

y j ,N

z k ,N

y j ,N

z k ,N

u, rx , ˛ , ˛

even i

even i

even i

odd i C1

odd i C1

even i

odd i 1

odd i 1

v, ry , xy , xy

odd i 1

even i 2

odd i 1

odd i 1

even i

even i

even i 2

odd i 1

w, rz , xz , xz

even i 2

odd i 1

even i

even i

odd i 1

odd i 1

odd i 1

even i 2

yz , yz

odd i 3

odd i 3

odd i 1

even i 2

even i 2

odd i 1

even i 2

even i 2

208

Chapter 8 An asymptotic approach

To compose and solve a consistent system of ODEs, define the degree of approximation N D 2m D 2 for the function u in accordance with Table 8.2. Then, the finite dimensional displacements of eq. (8.77) can be written as u D u.0;0/ .x/ C u.0;2/ .x/z 2 C u.2;0/ .x/y 2 ; v D v .1;0/ .x/y;

w D w .0;1/ .x/z:

(8.81)

The components of the stress tensor  satisfying exactly the homogeneous boundary conditions on the lateral faces have the form x D x.0;0/ .x/ C x.2;0/ .x/y 2 C x.0;2/ .x/z 2 ;   y D y.0;0/ .x/ C y.2;0/ .x/y 2 C y.0;2/ .x/z 2 .y 2  h2 /;   z D z.0;0/ .x/ C z.2;0/ .x/y 2 C z.0;2/ .x/z 2 .z 2  h2 /; .1;0/ xy D xy .x/y.y 2  h2 /;

yz D 0;

.1;0/ xz D xy .x/z.z 2  h2 /:

(8.82)

The equilibrium equations of eq. (8.75) taking into account eq. (8.82) can be rewritten as dx.0;0/ .0;1/ 2 .1;0/ 2  xy h  xz h dx ! ! .2;0/ .0;2/ dx dx .1;0/ 2 .0;1/ C 3xy C 3xz C y C z 2 D 0; dx dx ! .1;0/ d xy 2 .0;0/ .2;0/ 2 h  2y C 2y h y ry D  dx ! .1;0/ d xy .2;0/ C 4y y 3 C 2y.0;2/ yz 2 D 0; C dx ! .0;1/ d xz h2  2z.0;0/ C 2z.0;2/ h2 z rz D  dx ! .0;1/ d  xz C 4z.0;2/ z 3 D 0 C 2z.2;0/ y 2 z C dx

rx D

(8.83)

The solution of eq. (8.83) is found from the condition that all the coefficients at the corresponding powers of the variables y and z are equal to zero. All the relations (8.83) except one are resolved algebraically with respect to the unknown stress functions. As a result, a first-order ODE remains, for instance,  1 d  .0;2/ dx.0;0/ C x C x.2;0/ h2 D 0 : dx 3 dx

(8.84)

209

Section 8.4 3D static problem

By taking into account eq. (8.83), the components of the stress tensor introduced in eq. (8.82) take the form x D x.0;0/ C x.2;0/ y 2 C x.0;2/ z 2 ; .2;0/

yz D 0 .0;2/

1 d 2 x 1 d 2 x 2 2 2 .y  h / ;  D .z 2  h2 /2 ; z 12 dx 2 12 dx 2 .2;0/ .0;2/ 1 dx 1 dx D y.y 2  h2 /; xz D  z.z 2  h2 / : 3 dx 3 dx

y D xy

(8.85)

In order to satisfy the constitutive relation of eq. (8.75), the six following equations are composed componentwise after taking into account eqs. (8.81) and (8.85): x D x.0;0/ C x.2;0/ y 2 C x.0;2/ z 2 C x.4;0/ y 4 C x.0;4/ z 4 D 0; y D y.0;0/ C y.2;0/ y 2 C y.0;2/ z 2 C y.4;0/ y 4 C y.0;4/ z 4 D 0; z D z.0;0/ C z.2;0/ y 2 C z.0;2/ z 2 C z.4;0/ y 4 C z.0;4/ z 4 D 0; .1;0/ .3;0/ 3 y C xy y D 0; xy D xy

.0;1/ .0;3/ 3 xz D xz z C xz z D 0;

yz D 0

(8.86)

with .2;0/

y.4;0/ x.0;0/ x.2;0/ x.0;2/ y.0;0/ y.2;0/ z.0;0/ z.2;0/ .1;0/ xy .0;1/ xz

.0;2/

1 d 2 x 1 d 2 x .0;4/ D ;  D ; z 12E dx 2 12E dx 2    .0;0/ du.0;0/ x  h4 y.4;0/ C z.0;4/ C ; D dx E .2;0/ du.2;0/ x C 2 h2 y.4;0/ C ; x.4;0/ D  y.4;0/; D dx E .0;2/ du.0;2/ x C 2 h2 z.0;4/ C ; x.0;4/ D  z.0;4/ I D dx E D h4 y.4;0/  h4 z.0;4/  x.0;0/  v .1;0/ ; y.0;4/ D  z.0;4/ ; E D 2h2 y.4;0/  x.2;0/ ; y.0;2/ D 2h2 z.0;4/  x.0;2/ ; E E D h4 z.0;4/  h4 y.4;0/  x.0;0/  w .0;1/ ; z.4;0/ D  y.4;0/; E .2;0/ 2 .4;0/ D 2h y  x ; z.0;2/ D 2h2 z.0;4/  x.0;2/ I E E .2;0/ .2;0/ .1;0/ 2 h dx 1 dv 1 dx .3;0/ C  u.2;0/ ; xy ; D D 2 dx 6G dx 6G dx .0;2/ .0;2/ h2 dx 1 dw .0;1/ 1 dx .0;2/ .0;3/ C  u1 ; xz : D D 2 dx 6G dx 6G dx

210

Chapter 8 An asymptotic approach

Note that the eight unknown functions ¹x.0;0/ ; x.2;0/ ; x.0;2/ ; u.0;0/ ; u.2;0/ ; u.0;2/ ; v .1;0/ ; w .0;1/ º

(8.87)

are only available to satisfy exactly the system of equations (8.86). At the same time, it is required to ensure the vanishing of 19 expressions, i.e., the coefficients at the corresponding powers of the coordinates in the equations (8.86). To compose the consistent boundary value problem, choose the following coefficients for the lower powers of y and z in expansions (8.86). From the expression for the component x take three coefficients (zeroth and quadratic degree)  du.0;0/  .0;0/ h4 d 2  .2;0/ x .0;2/  D 0;  C C  x 12E dx 2 x dx E .2;0/ .2;0/ h2 d 2 x x du.2;0/ C D 0;  6E dx 2 dx E x.0;2/ h2 d 2 x.0;2/ du.0;2/ C D 0:  6E dx 2 dx E

(8.88)

From the remaining components of the tensor , only the coefficients of the zeroth and first degrees are chosen  h4 d 2  .2;0/ .0;2/     x.0;0/  u.1;0/ x 12E dx 2 x E  h4 d 2  .0;2/   x.2;0/  x.0;0/  w .0;1/ 12E dx 2 x E 1 dv .1;0/ h2 dx.2;0/  u.2;0/  6G dx 2 dx .0;2/ 2 h dx 1 dw .0;1/  u.0;2/  6G dx 2 dx

D 0; D 0; D 0; D 0:

(8.89)

Thus, there are 8 relations (8.84), (8.88), (8.89) to find the unknown functions (8.87). The system of equations (8.84), (8.88), (8.89) can be reduced to five ODEs of the second order. Therefore, it is necessary to specify 10 boundary conditions at the ends of the beam. The first eight boundary conditions follow from the statement of the problem. Since the left end of the beam is clamped, then u.0;0/ D u.2;0/ D u.0;2/ D v .1;0/ D w .0;1/ D 0;

x D 0:

(8.90)

The conditions of uniform tension on the right end of the beam determine the functions of normal stresses at this point x.0;0/ D 1;

x.2;0/ D x.0;2/ D 0;

x D l:

(8.91)

211

Section 8.4 3D static problem

The two remaining conditions (zero shear stresses at the free beam end) follow from eq. (8.85) according to .0;2/

dx dx

.2;0/

D

dx dx

D 0 for

x D l:

(8.92)

For this particular case, the system of linear differential equations (8.84), (8.88), (8.89) can be reduced to one tenth order equation with respect to, for example, the .0;0/ function x (in view of the cumbersome form, this equation is not given). The 1 .0;0/ solution of this equation is sought in the form x D e xh . Then, the following characteristic equation:    2 .1 C /4 C 4.2  /2  24 .1 C /4  4.2 C /2 C 24 D 0 (8.93) is to determine the unknown eigenvalue . General solution of the ordinary differential equation for x.0;0/ can be written as x.0;0/ D C1 C C2 x 1 h  x  x  x  x X sin ˇj C C4j C4 sh ˛j cos ˇj C4j C3 sh ˛j C h h h h j D0  x  x i  x  x C C4j C5 ch ˛j sin ˇj C C4j C6 ch ˛j cos ˇj (8.94) h h h h Here, ˛j and ˇj are the real and imaginary parts of the roots of the characteristic equation (8.93) with s s p p 2 C C 6 C 6 2  C 6 C 6 ; ˇ0 D ; ˛0 D 1C 1C s s p p 2  C 6  6 2 C C 6  6 ˛1 D ; ˇ1 D : (8.95) 1 1 The unknown constants Cj , j D 1; : : : ; 10 are found from the boundary conditions (8.90) – (8.92). The following mechanical and geometrical dimensionless parameters have been chosen: Young’s modulus E D 1, Poison’s ratio D 0:3, beam length l D 10, size of the cross section h D b D 1. For arbitrary approximation order m introduced in Table 8.2, the system of ODEs with respect to the displacement and stress functions is composed of all the equilibrium relations rx.2i;2j / D ry.2iC1;2j / D rz.2i;2j C1/ D 0, i Cj m (3.mC1/.mC2/=2 equations) and the part of constitutive relations of eq. (8.79) corresponding to a lower degree of polynomial expansions for the tensor .

212

Chapter 8 An asymptotic approach

The following equations are taken: x.2i;2j / D 0; y.2i;2j / .2iC1;2j C1/ yz

D

i C j mI

z.2i;2j /

D 0;

.2iC1;2j / .2i;2j C1/ D xy D xz D 0;

i C j m  1I

i C j m  2:

This system of equations together with boundary conditions (8.80) given at the beam ends is a correctly formulated boundary value problem for an arbitrary integer m > 0. The approximate boundary value problems related to the number m are characterized by the following parameters:  total number of variables is .9m2 C 15m C 8/=2;  the number of boundary conditions on the beam ends is .m C 1/.3m C 2/;  the total number of the constitutive relations in eq. (8.79) is 3m2 C 12m C 8. Unsatisfied coefficients of eq. (8.79) (the total number 9m C 7) are used to analyze the quality of the numerical solutions based on Theorem 5.2 given in Section 5.2. The distribution of the stress function x along the beam axis x at y D z D 0 for N D 8 is shown in Figure 8.17 by solid curve. The elementary solution of a simple tension of prismatic rod (homogeneous state x D const, see [70]) is well known in the linear theory of elasticity. As it can be seen from Figure 8.17, the clamping of the beam end has a significant influence on the distribution of tensile stresses along the beam. This influence extends to a considerable distance (more than 20% of the beam length for given parameters). In this area, the distribution of displacements and stresses change noticeably in the cross section. A characteristic deplanation of a beam cross section at x D 1 is depicted in Figure 8.18. It may be noted that the Bernoulli hypothesis of plane sections [18] is significantly violated in the vicinity of the clamped cross section. An important characteristic that can serve as a local measure of the solution quality is the spatial distribution of the energy error 1 ' D  WC W: 2 The values of this function at the clamped cross section are shown in Figure 8.19. It is worth noting that ' is close to zero almost everywhere, except for a narrow layer along the cross-sectional boundary, and reaches its maximum at the corners. This nature of the local error distribution is not contrary to modern ideas about the type of singularity in three-dimensional problems of linear elasticity. As shown, for example, in [67], singularities in stress-strain state of a body may arise at some boundary lines and points.

Section 8.4 3D static problem

213

Figure 8.17. Distribution of normal stress for tension, x .x; 0; 0/, bending, x .x; 0; h/, and torsion x .x; 0; 0/.

Figure 8.18. Deformation of the beam cross section at x D 1.

The convergence of the asymptotic approach is illustrated in Figure 8.20, which shows the bilateral energy estimates ….u/ W .u ;   / …c . /;

214

Chapter 8 An asymptotic approach

Figure 8.19. Distribution of the local error ' in the clamped cross section at x D 0.

depending on the approximation degree N of the longitudinal displacements u. Here, the following notation is introduced: Z 1 … D W"  W; …c D W ; W D  W "0 d ; 2  Z Z 1 1 1 W D  W C W  d ; W" D "0 W C W "0 d  : 2  2  The following energy characteristics of the approximate solutions have been obtained for N D 8: the functional value ˆ D … C …c D 0:03 (integral quality), the D magnitude of the potential energy … D 19:86, the relative error  D ˆ…1 c 0:08%. Similarly, the consistent systems of ODEs and corresponding boundary conditions can be derived for the beam bending and torsion by using the symmetry properties presented in Table 8.2. Let us consider only the bending around the y-axis and give the following shear load on the free beam end at x D l: ² ³T 3 ql D 0; 0; .1  z 2 / : 8 The resulting solution characteristics have been obtained for the approximation parameter N D 9: ˆ D 0:0369;

… D 12:66;

 D 0:29% :

215

Section 8.4 3D static problem

Figure 8.20. Bilateral estimates of the elastic energy W by total potential, …, and complementary, …c , energies.

In the case of beam torsion, the stresses, which do not cause any shear forces, are applied at x D l as follows: ²

9y.1  z 2 / 9z.1  y 2 / ql D 0; ;  16 16

³T

to create the torque on this face. The calculations have been performed for different polynomial degrees N . The solution corresponding to N D 10 is characterized by the following integral parameters: ˆ D 0:0041;

… D 5:757;

 D 0:07% :

The normal components x .x; 0; h/ on the lateral face for the bending and x .x; 0; 0/ at the midpoint of the cross section for the torsion are shown in Figure 8.17 by dashed and dash-dot curves, respectively. As it can be seen in Figure 8.17, the stress fields change significantly near the clamped end for all considered cases. At that, these fields calculated far from the outer cross section correspond to those predicted by classical beam models. This behavior can be explained by Saint-Venant’s principle. Quantitative analysis of the Saint-Venant effect can be performed in the frame of the asymptotic model discussed. Figure 8.21 presents convergency rates with respect to the approximation order N for three different types of beam loading, namely, tension, bending, and torsion. For the given parameters, the functions .N / are strictly decreasing in these cases.

216

Chapter 8 An asymptotic approach

Figure 8.21. Relative energy error  for tension, bending, and torsion.

Figure 8.22. Deplanation u of the cross section at x D 5 for the beam torsion.

Section 8.4 3D static problem

217

The deplanaition u of the cross section at x D 5 is shown in Figure 8.22 in the case of torsion. This function is antisymmetric with respect to not only the axes y and z but also the diagonals of the section [44]. This means that the deplanation is absent along the lines y D 0, z D 0, y D ˙z. Due to the complex shape of displacement field, it is necessary to take a polynomial approximation at least of the fourth degree, 2  z 2 /, to reproduce only the characteristic features for example u D h4 u.x/yz.y Q of such distribution. Then, it is possible to attain acceptable accuracy (  1%) as seen in Figure 8.21.

Chapter 9

A projection approach

Various projection approaches that rely on the MIDR can also be developed and applied effectively to reliable numerical modeling of physical processes. In contrast to the variational technique, integral projections of constitutive relations on a functional space chosen in a special way are used to compose a consistent system of equations. A modification of the MIDR, which is based on such technique and an ansatz representation of unknown functions, was derived in [54] to define the temperature profile and heat flux density for one-dimensional heat transfer problems. In this chapter, a variant of the Petrov–Galerkin method is described in which integral projections of the strain error tensor and discretization of admissible stress and displacement fields are applied. Two numerical projection algorithms are considered to design either algebraic or ODE systems that guarantee given solution quality. The relations among the projection, asymptotic, and variational approaches are discussed.

9.1 Projection formulation of linear elasticity problems Instead of the variational formulation (5.1), (5.2), (5.7), and (5.14) given in Chapter 5, a projection approach can be applied to develop various numerical algorithms for solving static and dynamic problems. For definiteness, the plane stress state, which is described by eqs. (7.4) – (7.9) for a rectangular elastic body (see Figure 7.1), is considered. In this case, a formulation of the linear elasticity problem is as follows: find the displacement components u and v  as well as the stress tensor   satisfying the equilibrium equation (7.4) and boundary conditions (7.7) – (7.9) such that the following integral relation holds: BŒu ; v  ;   ;  D 0 Z BŒu; v; ;  D

for 8 2 L2 ./ ; h Z l  W  dxdy :

h

l

(9.1)

Here,  is the strain error tensor introduced in eq. (7.6),  is a tensor of the second rank the components of which are defined on the body domain  given in eq. (7.1). As it has been discussed in Section 3.4, the main ideas of the Petrov–Galerkin method and its modifications are that the unknown functions are sought in some finitedimensional trial space and that the resulting approximation is a solution of a finite system of equations. These equations in the current realization are composed as zero

219

Section 9.1 Projection formulation of linear elasticity problems

integral projections of the strain error tensor  on a subspace of the test tensor space .x; y/. First, consider an algorithm based on polynomial approximations of displacements and stresses. To be more specific, the static state of a homogeneous isotropic rectilinear plate is under study. The origin O of the coordinate system Oxy is placed at the center of the domain . The axes x and y is parallel to the body sides with length 2l and 2h, respectively, as in Figure 7.1. The following boundary conditions: xy D y D 0

for y D ˙h ;

(9.2)

u D v D 0 for x D 0 ; x D qx .y/ and xy D qy .y/

for

xDl

(9.3)

are only considered. Here, the components qx and qy are known polynomials of the degree Nq . The external load is supposed to be admissible, i.e., qy .h/ D qy .h/ D 0. The volume forces are absent. Algorithm 9.1. Let the displacement and stress fields be approximated by the polynomials obeying all the boundary conditions (9.2) and (9.3) according to NX  1

uQ D .1 C X/

u.ij / X i Y j ;

vQ D .1 C X/

iCj D0

NX  1

v .ij / X i Y j ;

iCj D0 N X

Q x D qx .Y / C .1  X/

x.ij / X i Y j ;

iCj D0



Qxy D qy .Y / C .1  X/ 1  Y

2

 2  NX

.ij / i j xy X Y ;

iCj D0

  Qy D 1  Y 2

NX  1

y.ij / X i Y j :

(9.4)

iCj D0

Here, the dimensionless parameters Y D

y h

and X D

x l

(9.5)

are introduced for a more compact representation, the number N Nq is the order of approximation (9.4).

220

Chapter 9 A projection approach

After implementation of the equilibrium equation (7.4), the following integral projections on the polynomial space are equated to zero: Z 1Z 1 Z 1Z 1 Qx X i Y j dXd Y D 0; Qy X i Y j dXd Y D 0; 1 1 1 1 Z 1Z 1 (9.6) Qxy X i Y j dXd Y D 0 1

1

for all the powers i and j such that 0 i C j N  2. In this case, the components of the strain error tensor Q are given by Q x Qy Qy Q x 1 @uQ 1 @vQ  C ; Qy D  C ; Qx D l @X E E h @Y E E  1 @uQ Qxy 1 1 @vQ C  : Qxy D 2 l @X h @Y 2G

(9.7)

The total number of unknown coefficients of displacements and equilibrium stresses in eq. (9.4) is greater than the number of the algebraic equations (9.6). But these coefficients are not enough to satisfy all the projections on the next polynomial level for i C j D N  1. Note that the algebraic equations with respect to the parameters remaining after resolving eq. (9.6) can be constituted in various ways. Let these unknown constants minimize the energy error functional Z 1Z 1  1  Q2 2 2 Q Q Q Q (9.8) ˆDG x C 2 x y C y C 2xy dXd Y D 0 : 1 1 1  Now, proceed to the next algorithm which relies on the semi-discretization discussed in Chapters 7 and 8. Algorithm 9.2. As it has been proposed in Chapter 7, the displacement and stress fields in  are approximated by polynomial functions with respect to the variable Y of eq. (9.5): uQ D

Nu X

u.k/ .x/Y k ;

kD0

Q x D

Nu X

vQ D

NX u 1

v .k/ .x/Y k I

kD0

x.k/ .x/Y k ;

u 1  NX  .k/ Qxy D 1  Y 2 xy .x/Y k ;

kD0

  Qy D 1  Y 2

kD0 Nu X

y.k/ .x/Y k :

(9.9)

kD0

The coefficients in these expansions are unknown functions of the spatial coordinate x. These semi-discrete approximations automatically obey the boundary conditions (9.2).

221

Section 9.1 Projection formulation of linear elasticity problems

In the projection approach, one does not need to satisfy the equilibrium equation before solving the constitutive relations as in the variational algorithms of Section 7.2. The consistent system of DAEs can be assembled simultaneously. More particularly, this system consists of 2Nu C 3 first-order ODEs   dx .j C1/ .j 1/ C .j C 1/ xy D 0 for  xy dx .k/ .k2/   d xy d xy  C .k C 1/ y.kC1/  y.k1/ D 0 for dx dx .j /

j D 0; : : : ; Nu ; k D 0; : : : ; Nu C 1; (9.10)

which follow the body equilibrium, and of 2Nu C 1 differentials as well as Nu  1 algebraic projections of the constitutive relations Z 1 Qx Y i dX D 0 for i D 0; : : : ; Nu ; 1 Z 1 (9.11) Qxy Y j dX D 0 for j D 0; : : : ; Nu  1 ; 1 Z 1 Qy Y k dX D 0 for k D 0; : : : ; Nu  2 : 1

Here, the components of the strain error tensor  are used after substituting the approximations (9.9) into eq. (9.7). In this case, the total number N t D 5Nu C 3 of the equations (9.10) and (9.11) coincides with the dimension of the design parameter vector a.x/ D ¹au .x/; a .x/ºT 2 RN t ; ± ° au D u.0/ ; : : : ; u.Nu / ; v .0/ ; : : : ; v .Nu 1/ ; ±T ° .0/ .Nu 1/ ; : : : ; xy ; x.0/ ; : : : ; x.Nu / : a D x.0/ ; : : : ; x.Nu / ; xy

(9.12)

The differential order of system (9.10) and (9.11) equals to Nd D 4Nu C 2. This order is in agreement with the total number of boundary conditions u.j / .0/ D 0 and x.j / .l/ D qx.j / v

.k/

.j /

.0/ D 0 and .k/

where qx and qy

qx D

.k/ xy .l/

D

qy.k/

for

j D 0; : : : ; Nu ;

for

k D 0; : : : ; Nu  1;

(9.13)

are the coefficients of external polynomial load Nu X iD0

qx.i/ .x/Y k ;

qy D .1  Y 2 /

NX u 1

qy.i/ .x/Y k :

(9.14)

iD0

The approximated displacement and stress fields are found as the solution of the boundary value problem (9.10), (9.11), (9.13).

222

Chapter 9 A projection approach

9.2 Projections vs. variations and asymptotics The modification of the Galerkin method discussed in the previous section has a close relation with methods which are based on the variational technique as in Chapter 4 or 7. Indeed, compare the minimization algorithm for the energy error functional ˆ described in Chapter 4 (Problem 3) and Algorithm 9.1. For the displacements and stresses (9.4), the first variation of the minimized functional ˆ in form (9.8) can be presented as follows: Z 1Z 1  ıˆ D Qx ı &Q x C Qy ı &Qy C 2Qxy ı &Q xy dXd Y D 0 (9.15) 1

1

with the components of the stress error tensor   @vQ E 1 @uQ &Q x D Q x  C ; 1  2 l @X h @Y   @uQ E 1 @vQ &Qy D Qy  C ; 1  2 h @Y l @X   1 @vQ 1 @uQ C : &Q xy D Qxy  G h @Y l @X

(9.16)

Let the vector of design parameters b D ¹b1 ; : : : ; bN ºT 2 RN composed of in.ij / dependent constants u.ij / , v .ij / , x.ij / , y.ij / , xy be incorporated in Algorithm 9.1. Then, the integral relation (9.15) is equivalent to the system of integral projections of the strain error tensor Q on a special tensor space of polynomials Z 1Z 1 (9.17) Q W #j .X; Y / dXd Y D 0 for j D 1; : : : ; N; 1

1

where #j D

@&Q .X; Y; b/ @bj

(9.18)

are the basis tensors of this finite dimensional space. The projections chosen in such a way are optimal. This means that, among all values of the vector b, the solution of system (9.17) minimizes the energy error functional ˆ. On the other hand, the projection approach under consideration has a direct intersection with the asymptotic algorithms described in Chapter 8. For example, the strain error tensor Q in the polynomial presentation (9.4) takes the form Q D

NX  C1

 .ij / .b/X i Y j ;

iCj D0

where  .ij / are linear algebraic expressions of the vector components bk .

(9.19)

223

Section 9.2 Projections vs. variations and asymptotics

If the projections (9.6) used in Algorithm 9.1 are replaced for the following equations (9.20)  .ij / D 0 for 0 i C j N  2 then a modification of the asymptotic approach outlined in Chapter 8 is obtained. Moreover, the expansion (9.19) is not unique and the tensor Q allows one to take the equivalent form NX  C1 Q D (9.21) Q .ij / .ij / .X; Y / : iCj D0

Here, .ij / are polynomial functions, which can be chosen in a rather arbitrary way, in particular, as the products of normalized Legendre polynomials .ij /

.X; Y / D Li .X/Lj .Y /;

(9.22)

p m 4m C 2 d m  2 z 1 for z 2 Œ1; 1 : (9.23) Lm .z/ D mC1 m 2 mŠ dz Due to orthogonal properties of these polynomials, the asymptotic system of equations is identical to the projection one

where

Z

1 1

Z

1 1

Q

.ij /

.X; Y /dXd Y D Q .ij / D 0 for

0 i C j N  2 :

(9.24)

In turn, all the functions .ij / for i C j N  2 compose a basis for the complete bivariate polynomials of degree N  2. This means that the system of algebraic equations (9.24) is equivalent to the system of integral projections (9.6). Example 9.3. To demonstrate the properties of the projection algorithms introduced in the previous section and compare them with variational and asymptotic ones, consider again the problem of beam bending (see Figure 7.1) investigated in Example 7.1. Remember the parameters that have been chosen in that example: beam sizes l D 5 and 2h D 1, Young’s modulus E D 104 , Poison’s ratio D 0:3, the boundary stresses qx D 0;

qy .y/ D

 6 1Y2 : 8

(9.25)

The same approximations (9.4) have been used in calculations for the projection, asymptotic, and variational approaches discussed above. The convergence rates for these methods are shown in Figure 9.1. Regardless of the approaches, the dimension of approximating systems is identical for the fixed degree N . As it has been mentioned in Chapter 1, only the Ritz method provides the symmetric matrix of such a system. Naturally, this approach also gives the minimum relative error  D ˆ…1 c , where …c is the value of the total complementary energy.

224

Chapter 9 A projection approach

Figure 9.1. The relative error  vs. the polynomial degree N of stresses in Example 9.3 for the projection, asymptotic, and variational approaches.

Similar arguments can be given to reveal the relationship between Algorithm 9.2 and variational or asymptotic techniques based on the semi-discrete approximations. In this case, the main distinction between the listed approaches is that the differential order of the approximating ODE system for both asymptotic and projection algorithms is two times less than for variational one. Thus, the algorithms based on variational principles and semi-discretization might be not so effective despite of their optimality. To adapt Algorithm 9.2 based on the Galerkin method to the asymptotic one, the integral equations (9.11) are replaced with the following relations: x.i/ D 0 for

i D 0; : : : ; Nu ;

.j / xy y.k/

D 0 for

j D 0; : : : ; Nu  1 ;

D 0 for

k D 0; : : : ; N  2 :

(9.26)

by using the expansions Q D

NX u C2

 .i/ #i .Y / :

(9.27)

iD0

x.i/ , y.i/ ,

.i/ and xy of the tensor  .i/ are differential or algebraic Here, the components expressions of the displacements and stresses given in eq. (9.9). Note that the DAE system (9.26) coincides with the projection system (9.11) if this basis consists of the Legendre polynomials #i D Li .Y / defined in eq. (9.23). In the following example, we constrain ourselves to the case when #i D .Y  Y0 /i , where Y0 is a parameter of expansion (9.27).

Section 9.2 Projections vs. variations and asymptotics

225

Figure 9.2. The potential energy … and complementary energy …c vs. the approximation order Nu for the projection (dash-dot lines), asymptotic (dashed lines, Y0 D 0), and variational approaches (solid lines) in Example 9.4.

Example 9.4. The same boundary value problem as in Example 9.3 is analyzed. The approximation orders Nu D 3; 5; 7; 9 in eq. (9.9) have been taken in numerical calculations in accordance with the corresponding variational, asymptotic, and projection algorithms. It is worth noting a high sensitivity of solution accuracy, which is observed in the asymptotic approach, with respect to the coordinate Y0 . Thus, for instance, the integral relative error .Y0 / for Nu D 5 changes dramatically jY D0 D 0:13%;

jY D0:1 D 0:08%;

jY D0:2 D 9:42%;

jY D0:3 > 632%:

In contrast to the asymptotic technique, the projection algorithm allows one to avoid this instability due to its unambiguity in assembling the approximating ODE system. As shown in Figure 9.2 for all the three methods, the bilateral energy estimates are provided by the potential and complementary energies in accordance with Theorem 5.4. For the approximation (9.9) at any order Nu , the variational approach (solid lines) definetely gives, while minimizing the energy error, the best interval estimating the elastic energy stored by the body. For the other approaches, the accuracy obtained for the given parameters is approximately the same, but the numerical estimates have different upper and lower bounds. The solution quality improves simultaneously with the system dimension. It can be seen in Figure 9.3, where the relative error  is presented. The differential order of ODE system obtained in the projection and asymptotic approaches is Nd D 4Nu C 2 in contrast to 2Nd in frame of the variational algorithm at the same degree Nu in

226

Chapter 9 A projection approach

Figure 9.3. Relative energy error  vs. the approximation order Nu for projection, asymptotic (Y0 D 0), and variational approaches in Example 9.4.

eq. (9.9). For example, the projection system at Nu D 7 and the variational one at Nu D 3 have roughly the same differential orders and relative errors. So, an additional comparative analysis is required to evaluate the effectiveness of both schemes.

Chapter 10

3D static beam modeling

10.1 Projection algorithms Consider a long rectilinear prismatic body (beam) with a triangular cross section as in Figure 10.1. The origin of the Cartesian coordinate system is placed at the barycenter of one prism base. The axis x is directed to the other base along the beam length, and, hence, the axes y and z are parallel to the body cross sections. It is assumed that the beam is made of homogeneous isotropic material with Young’s modulus E and Poisson’s ratio . The choice of the object under study is stipulated by the following circumstances. First of all, such an elastic prism contains specific features which are typical for linear elasticity problems associated with various types of boundary conditions (clamping, etc.) and with the presence of corner points. Secondly, it allows to use polynomial representations of stresses and displacements to approximate an elastic state of the prism. Another reason is that such beam elements can be used to compose structures with more complicated cross-sectional shapes (e.g., thin-walled beams) and applied to specific FEM algorithms. The stress-strain state of the body is described by the PDE system (8.75). The boundary constraints can be divided into two groups. The conditions on the lateral sides of the beam can be attributed to the first part. These relations must be satisfied before constituting an approximating ODE system in frame of the projection approach which will be discussed in this chapter. The second group consists of the conditions on the prism bases that are implemented together with the system of ODEs. Only three types of boundary conditions are considered, namely, it is assumed that displacements, stresses, or an elastic foundation can be defined on the lateral faces and bases of the prism. First, consider the boundary conditions which are referred to as the first group. It is supposed that the displacements may be given on beam lateral faces. Then, these conditions are written as u D uN .i/ .x; y; z/;

v D vN .i/ .x; y; z/;

w D wN .i/ .x; y; z/;

¹x; y; zº T 2 i ; i D 1; : : : ; M1 :

(10.1)

Here, uN .i/ , vN .i/ , wN .i/ are known functions, i are the prism faces on which these displacements are defined. If the stresses are specified on some beam faces i , M1 < i M2 3, then these conditions can be divided into two subgroups due to the fact that these sides are

228

Chapter 10 3D static beam modeling

y

O z

x Figure 10.1. A prismatic beam with the triangular cross section.

parallel to the x-axis. The equations which relate to shear stresses xy and xz have the form qx.i/ D xy ny.i/ C xz n.i/ N x.i/ : (10.2) z Dq The other relations combine components y , z , and yz qy.i/ D y ny.i/ C yz n.i/ Ny.i/ ; z Dq N z.i/ : qz.i/ D yz ny.i/ C z n.i/ z Dq .i/

.i/

.i/

.i/

(10.3) .i/

.i/

Here, qN x , qNy , qN z are the given external load; nx D 0, ny , nz are the components of the normal vectors n.i/ to the prism faces i . In terms of eqs. (10.2) and (10.3), the last type of boundary conditions has the form qx.i/ C x.i/ u D 0; .i/

.i/

qy.i/ C y.i/ v D 0;

qz.i/ C z.i/ w D 0:

(10.4)

.i/

Here, x , y , z are the stiffness coefficients of the elastic foundation. The following boundary conditions are given on the bases of the prism (the second group) similar to eqs. (10.1) – (10.4): u D uN .0;l/ .y; z/;

v D vN .0;l/ .y; z/;

w D wN .0;l/ .y; z/;

(10.5)

or x D qN x.0;l/ .y; z/;

xy D qNy.0;l/ .y; z/;

xz D qN z.0;l/ .y; z/;

(10.6)

x C x.0;l/ u D 0;

xy C y.0;l/ v D 0;

xz C z.0;l/ w D 0

(10.7)

or at x D 0 and x D l, respectively. Here, l is the length of the beam.

229

Section 10.1 Projection algorithms

Approximations of the displacement and stress fields are taken in the form uD

N1 X

u.ij / .x/y i z j ;

vD

iCj D0

wD

N3 X

N5 X

w .ij / .x/y i z j ;

x D

y D

.ij / xy .x/y i z j ;

yz D

x.ij / .x/y i z j ;

xz D

N6 X

.ij / xz .x/y i z j ;

(10.8)

iCj D0

y.ij / .x/y i z j ;

iCj D0 N9 X

N4 X iCj D0

iCj D0 N7 X

v .ij / .x/y i z j ;

iCj D0

iCj D0

xy D

N2 X

z D

N8 X

z.ij / .x/y i z j ;

iCj D0 .ij / yz .x/y i z j :

iCj D0

The basic idea for solving the problem (8.75), (10.1) – (10.7) is to use the approximations (10.8) and the projection approach discussed in the previous chapter. These approximations must exactly satisfy the boundary conditions (10.1) – (10.4) and the equilibrium equations @xy @xz @x C C D 0; @x @y @z @y @yz @xy ry D C C D 0; @x @y @z @yz @z @xz rz D C C D 0: @x @y @z

rx D

(10.9)

After that, an ODE system with respect to the unknown coefficients introduced in eq. (10.8) is composed through Hooke’s relations   @u x  C y C z ; @x E E   @v y @w z  C .x C z / ; z D  C x C y ; y D @y E E @z E E     @v xy @w xz 1 @u 1 @u xy D C  ; xz D C  ; 2 @y @x 2G 2 @z @x 2G   @w 1 @v yz C ; (10.10) yz D  2 @z @y 2G x D

Note that the choice of polynomial approximations (10.8) designates the way to determine a subspace of test functions for which the integral projections should be

230

Chapter 10 3D static beam modeling

calculated. The space of complete polynomials of degree K 0 8 9 K < X = PK D cij pij ; pij D y i z j ; cij 2 R : ;

(10.11)

iCj D0

is suitable for this purpose. Let us give the following two definitions which will be useful. By analogy with the notion of complete polynomials discussed in Section 4.1, the set of inner products Z r.y; z/pij .y; z/dydz; 0 i C j K; (10.12) S

is called complete projection of degree K on the space PK for a function r.y; z/. The set which includes the complete projection of degree K  1 as well as some but not all projections on the subspace ´K μ X ci pi;Ki ; pi;Ki D y i z Ki ; ci 2 R ; (10.13) PQK D iD0

are referred to as the incomplete projection of degree K. For our purposes, the polynomial approximations of degrees Ni , i D 1; : : : ; 9, in eq. (10.10) have to possess the following properties. Firstly, it is necessary that such an approximation is able to satisfy exactly the equilibrium equations (10.9) and the boundary conditions (10.1) – (10.4). Secondly, it is important to select correctly the corresponding polynomial spaces of test functions (10.11) of degrees Ki > 0 for i D 1; : : : ; 6. This choice must guarantee that the system of differential-algebraic equations (DAEs), ¯ or incomplete projections of the strain ® which results from complete error functions x ; y ; z ; xy ; xz ; yz on these spaces, is consistent. It is also desirable in a numerical algorithm that the structure of these DAE systems does not change with the approximation order. Note that the rather complicated form of the equilibrium (10.9) and constitutive equations (10.10) as well as the boundary conditions (10.1) – (10.4) implies that, in general case, the parameters Ni may be different from each other. Restrict ourselves to the case when only the boundary conditions (10.1) – (10.3), (10.5), and (10.6) are considered. After equa° their implementation, the equilibrium ± .kl/ .kl/ .x/; xz .x/ and their deritions contain NQ d independent stresses x.kl/ .x/; xy Q vatives ° with respect to ±the spatial coordinate x. Similarly, Nu independent derivatives of u.ij / ; v .ij / ; w .ij / are included in the constitutive relations (10.10). To design an effective algorithm, the differential order Nd of a desirable system should be chosen according to ¯ ® Nd D 2 min NQ d ; NQ u : (10.14)

231

Section 10.1 Projection algorithms

Condition (10.14) brings some complicities in the composing such a system. Nevertheless, as it is shown below, these difficulties can be eliminated by choosing appropriate displacements and stresses as well as corresponding projections of the constitutive relations. In the proposed approach, the degrees of approximations ® (10.8) are fixed first, and¯ then, generally incomplete projections of the functions x ; y ; z ; xy ; xz ; yz on the corresponding polynomial subspaces are constructed. Practical implementation depends on the type of boundary conditions. For greater clarity, consider the case when the prism is loaded only by forces on the lateral faces in accordance with conditions (10.2) and (10.3). Without loss of generality, it is assumed that the equations of the prism faces are explicitly resolved with respect to, e.g., the coordinate z ° ± .i/ .i/ i D y D y .i/ D k0 C k1 z : (10.15) This property can be always achieved with respective rotation of the coordinate system about the x-axis. Here, k0i and k1i are real coefficients. To satisfy exactly the boundary conditions (10.2) and (10.3) by using approximation (10.8), the external loads are assumed to be polynomial qN˛.i/ D

M0 X

q˛.ij k/ .x/y j z k ;

˛ D x; y; z :

(10.16)

kCl

Let N5 D N6 D N0 C 1 M0

and N7 D N8 D N9 D N0 C 2:

Then, the boundary conditions (10.2) and (10.3) have the form qx.i/

D

NX 0 C1

  j .j k/ .j k/ .i/ xy y .i/ .z/ z k .x/ny.i/ C xz nz

j CkD0

D

M0 X

 j qx.ij k/ y .i/ .z/ z k ;

j CkD0

qy.i/ D

NX 0 C2

  j .j k/ .i/ y.j k/ .x/ny.i/ C yz y .i/ .z/ z k nz

j CkD0

D

M0 X j CkD0

 j qy.ij k/ y .i/ .z/ z k ;

(10.17)

232

Chapter 10 3D static beam modeling

qz.i/

D

NX 0 C2

  j .j k/ .j k/ .i/ yz y .i/ .z/ z k .x/ny.i/ C xz nz

j CkD0

D

M0 X

 j qz.ij k/ y .i/ .z/ z k :

(10.18)

j CkD0

It is worth noting that the external loads qNy.i/ and qN z.i/ have to be consistent with each other, since only three of four required conditions can be met at the vertices of a cross section. Algorithm 10.1. In the first step, a positive integer N0 > 0 and the corresponding complete polynomial subspace PN0 are taken in accordance with eq. (10.11). Let this number be simultaneously the degree N4 D N0 of approximations of x in eq. (10.8). Note that the differentiation of a linear combination of functions (10.8) with respect to x does not change the degree of polynomial terms. In contrast, differentiation with respect to y and z reduces the degree of complete polynomials by unit. Thus, the order of approximations for other stresses of eq. (10.8) should be chosen according to eq. (10.17) to solve the equilibrium equations (10.9). The expressions (10.10) suggest that the degrees of displacement approximation order can be taken as N1 N0

and N2 D N3 N0 C 1:

(10.19)

In the second step, the boundary conditions (10.18) on the lateral faces of the beam are satisfied. Consider the first equations of (10.18). Since these boundary conditions are univariate polynomials with respect to spatial coordinate z, it is necessary to fulfill .j k/ .x/, N0 C2 conditions on each beam side, or 3.N0 C2/ as a whole, with respect to xy .j k/ xz .x/ (linear relations at each monomial). Consequently, the approximations of xy and xz , contain only NQ D .N0 C 2/ .N0 C 3/  3 .N0 C 2/ .j k/

(10.20)

.j k/

independent coefficients xy .x/, xz .x/ after implementation of these equations. By introducing a new notation Qi .x/, i D 1; : : : ; NQ , for the remaining coefficients .j k/ .j k/ xy .x/ and xz .x/, the approximation of the shear stresses satisfying the boundary conditions (10.18) can be presented as ´ xy μ ² ³ X NQ  #Q i .y; z/ xy Qi .x/ D : xz #Q xz .y; z/ iD0

(10.21)

i

Here, #Q i xy .y; z/ and #Q i xz .y; z/ are basis functions obtained in agreement with these boundary conditions.

233

Section 10.1 Projection algorithms

After resolving the second and third relations in eq. (10.18), the approximations of y , z , and yz contain 3 NQ  D .N0 C 3/ .N0 C 4/  3 .2N0 C 5/ 2 .j k/

.j k/

(10.22)

.j k/

independent coefficients y .x/, z .x/, yz .x/. Renumbering these coefficients can provide the relevant stress approximation in the form 8 y 9 8 9 ˆ #Q i .y; z/ > ˆ Q > N   < y= X < = z Q (10.23) z D Q i .x/ #i .y; z/ : : ; > ˆ > ˆ yz iD0 ; : Q yz #i .y; z/ 

Here, #Q i y .y; z/, #Q iz .y; z/, and #Q i yz .y; z/ are new basis functions consistent with the boundary conditions (10.18). Implementation of the equilibrium equations implies the vanishing of the following complete projections of the vector components rx , ry , rz in eq. (10.9): Z .ij / rx pij dydz D 0 for i C j N0 ; Qx D ZS Qy.ij / D ry pij dydz D 0 for i C j N0 C 1; S Z rz pij dydz D 0 for i C j N0 C 1: (10.24) Qz.ij / D S

By using identical transformation, the DAE system (10.24) can be reduced to a system of .N0 C 1/.N0 C 2/ NQ d D NQ C (10.25) 2 2 .kl/ ODEs with the first derivatives of Qi .x/, i D 1; : : : ; NQ , and x .x/, k C l N0 , as well as NQyz D 3 .N0 C 2/ (10.26) algebraic equations which are linear combinations of the coefficients Q i .x/, i D 1; : : : ; NQ  . Note that only the first derivative of u with respect to x is presented in the component x of eq. (10.10). Since the displacements u are not subject to any boundary conditions on the lateral beam sides, the approximation degree N1 in eq. (10.8) should be chosen as for x , that is, N1 D N0 . Thus, the subspace of the test functions is the space of complete polynomials PN0 . Then, the system of ODEs which follows from the corresponding projection has the form Z .ij / Qxx D x pij dydz D 0; i C j N0 : (10.27) S

234

Chapter 10 3D static beam modeling

The approximation degrees for v and w can be chosen ambiguously. So, two branches of the algorithm are selected here. This is due to the fact that it is not possible to extract the complete polynomial subspaces the dimensions of which are equal to the total number NQ of Qi . In the first approach, a complete projection of xy and xz with maximum admissible degrees is taken. At that, some functions Qi are excluded from consideration. In the other approach, all the variables i available are used to build the projections of the components xy and xz on the specifically assembling polynomial spaces. In this case, the projections are incomplete. Algorithm 10.2. Initially, it is necessary to define the degree N 0 of the complete polynomial to project the components xy and xz . This parameter is defined as the maximum positive integer for which the following inequality holds: NQ N ;

N D .N 0 C 1/.N 0 C 2/:

(10.28)

To reduce the number of variables Qi .x/, the approximation (10.21) is transformed in the following way. First, the complete projections of the functions xy and xz on the subspace PN 0 are calculated Z Z xy pj k dydz D 0; xz pj k dydz D 0; k C l N 0 : (10.29) S

S

After that, system (10.29) is resolved with respect to N coefficients Qi .x/ selected arbitrarily. In composing a consistent ODEs, it is necessary to solve underdetermined systems of algebraic equations with respect to Qi .x/. If the calculations are performed analytically, then the choice of variables for which the equations are resolved is not so essential. But in approximate computations, a special numerical approach should be applied, e.g., based on the Gauss elimination method, to diminish computational errors. At the beginning of this successive process, an equation is chosen which contains a coefficient of maximum absolute value. After that, the variable at this maximum coefficient is expressed from this equation. This procedure accompanied with an appropriate transformation is repeated N  1 times. By solving eq. (10.29) and substituting the result into eq. (10.21), the following expression is obtained: ´

O xy O xz

μ D

Q N X iDN C1

´ i .x/



#i xy .y; z/ #i xz .y; z/

μ :

(10.30)

Here, i .x/, i D N C 1; : : : ; NQ , are new coefficients, which are linear combina

tions of Qj .x/, j D 1; : : : ; NQ ; #i xy .y; z/ and #i xz .y; z/ are new basis functions orthogonal to the polynomial space PN 0 .

235

Section 10.1 Projection algorithms

Find a representation of xy and xz equivalent to eq. (10.21) through a new basis

including #i xy .y; z/ and #i xz .y; z/. For this purpose, let us compose the following system of equations: Z

xy #i xy .y; z/ C xz #i xz .y; z/dydz D 0; i D N C 1; : : : ; NQ

(10.31) S

and resolve it with respect to some coefficients Qi .x/ by the Gauss elimination method. After substituting the solution of eq. (10.31) into eq. (10.21) and collecting similar terms, the following expression is obtained: μ ´ μ ´ xy N P X xy #i .y; z/ : (10.32) i .x/ D

xz P # xz .y; z/ i iD1

Here, similarly to eq. (10.30), i .x/ for i D 1; : : : ; N are new coefficients; #i xy .y; z/ O O and xz . It is and #i xz .y; z/ are new basis components, which are orthogonal to xy possible to verify that the obtained approximations ´ μ ´ Pμ ´ Oμ xy xy xy D C (10.33) P O xz xz xz satisfy the boundary conditions (10.2). Fix N2 D N3 D N 0 in the approximations (10.8) and calculate the complete projection of the functions xy and xz in eq. (10.10) on the space PN 0 . Then, a system of N ODEs with the first derivatives of the coefficients v .kl/ .x/ and w .kl/ .x/ is obtained Z Z .j k/ .j k/ Qxy D xy pj k dydz D 0; Qxz D xz pj k dydz D 0 (10.34) S

S

for j C k

As a result, the coefficients i .x/, i D N C 1; : : : ; NQ , are not included in the relation (10.34). These variables as their first derivatives remain only in eq. (10.24). Since only one of the boundary conditions on the bases of the prism can be satisfied via each of these functions, they should be excluded from the approximation (10.33), i.e., (10.35) i .x/ D 0; i D N C 1; : : : ; NQ : N 0.

In the next steps of the algorithm, new approximations of y , z , yz are found; the corresponding polynomial subspace is determined; projections of the components y , z , yz of eq. (10.10) on this subspace are calculated. First, note that the components y , z , yz of the strain error tensor do not contain derivatives with respect to the coordinate x. The corresponding relations following from eq. (10.10) can be satisfied through the variables Q i .x/, which are not subjected to the boundary conditions at the beam ends.

236

Chapter 10 3D static beam modeling

It follows from eqs. (10.20), (10.22), (10.26), (10.28), and (10.35) that the number of these functions is equal to Nyz D

.N0 C 2/.N0  3/ C .N 0 C 1/.N 0 C 2/: 2

(10.36)

Then, the degree N 1 of the polynomial subspace PN 1 , on which y , z , yz are projected, is defined through the maximum integer N that obeys the inequality Nyz N ;

3 N D .N 1 C 1/.N 1 C 2/: 2

(10.37)

After that, the orthogonalization of the functions (10.23) is performed similarly to eqs. (10.29) – (10.33). First, equate to zero the complete projections of y , z and yz Z Z y pj k dydz D 0; z pj k dydz D 0; S ZS yz pj k dydz D 0; j C k N 1 : (10.38) S

Then, resolve this system with respect to some of the coefficients Q i .x/, iD1; : : : ; NQ  . After solving eq. (10.38) and transforming (10.23), the following expression is obtained: 9 8 9 8  ˆ ˆ O> #i y .y; z/ > ˆ Q ˆ > N < y > < = = X zO D i .x/ #iz .y; z/ : (10.39) ˆ > > ˆ > > ˆ ˆ

: # yz .y; z/ ; : O ; iDN C1 yz

i

Here, i .x/, i D N C 1; : : : ; NQ  , are new variables; #i y .y; z/, #iz .y; z/ and

#i yz .y; z/ are new basis functions orthogonal to the subspace PN 1 . To express y , z , and yz through a new basis, the following system of equations is built Z 

y #i y .y; z/ C z #iz .y; z/ C yz #i yz .y; z/dydz D 0; 

S

i D N C 1; : : : ; NQ  : (10.40)

As previously, system (10.40) is resolved with respect to some of the parameters Qj .x/, j D 1; : : : ; NQ  . By substituting the solution of eq. (10.40) into eq. (10.23), the stress field is obtained 9 8  8 9 ˆ ˆ yP > #i y .y; z/ > ˆ > > ˆ N  < = < = X z P z D i .x/ #i .y; z/ : (10.41) > ˆ > ˆ ˆ ˆ : # yz .y; z/ > ; iD1 : P > ; yz

i

237

Section 10.1 Projection algorithms 

Here, i .x/, i D 1; : : : ; N , are new stress coefficients; #i y .y; z/, #iz .y; z/,

#i yz .y; z/ are new basis functions which are orthogonal to the subspace (10.39). It is also important that the approximations 8 9 8 9 8 9 ˆ P > ˆ O > < y > < y > < y = ˆ = ˆ = P z D z C yO (10.42) > ˆ > : ; ˆ ˆ yz : O > ; ˆ ; : P > yz

yz

exactly satisfy the boundary conditions (10.3). .j k/ The first equations Qx D 0 in eq. (10.24) can be explicitly resolved with respect to the first derivatives of the stresses as follows: dx.j k/ D fx.j k/ .1 .x/; : : : ; N .x// C X .j k/ .x/; dx

j C k N0 :

(10.43)

.j k/

are linear combinations of m ; the functions X .j k/ .x/ follow from the Here, fx .i/ representation of external load qN x in eq. (10.16) for i D 1; 2; 3. The total number of relations (10.43), which are part of the desired system of ODEs, is equal to Nxe D

.N0 C 1/ .N0 C 2/ : 2

.j k/

(10.44)

.ij /

Similarly, some the relations Qy D 0 and Qz D 0 in eq. (10.24) are resolved with respect to the differential terms of shear stresses: d i .i/ D fyz .1 .x/; : : : ; NQ  .x// C Y .i/ .x/; dx

i D 1; : : : ; N :

(10.45)

.i/

Here, fyz are linear combinations of m ; the corresponding functions Y .i/ .x/ result .i/ from the expressions for external loads qN ˛ for ˛ D x; y; z and i D 1; 2; 3. With account of eq. (10.45), the remaining equations in eq. (10.24), the number of which is .N0 C 2/ .N0 C 3/  N , are linear relations with respect to the coefficients i .x/, i D 1; : : : ; NQ  . Another part of the algebraic equations is derived from the projection of y , z , and yz on the space PN 1 Z Z .j k/ .j k/ Qyy D y pj k dydz D 0; Qzz D z pj k dydz D 0; S ZS .j k/ Qyz D yz pj k dydz D 0; j C k N 1 : (10.46) S

Note that the number of algebraic equations obtained is less than the number of coefficients i .x/. To complete the system of algebraic equations and express all the

238

Chapter 10 3D static beam modeling

components i .x/ through the variables v, w, x , let all the obtained algebraic equations of eqs. (10.24) and (10.46) be resolved with respect to some of the coefficients i .x/. This solution can be presented as follows: 0

j0 .x/

D

N X

cj k k0 .x/ C fj C Zj .x/;

j D N0 C 1; : : : ; NQ  :

(10.47)

kD1

Here,

N0 D NQ   N  .N0 C 2/ .N0 C 3/ C N

(10.48)

is the number of superfluous coefficients i .x/ renumbered according to j0 .x/ D k.j / .x/

for

j; k D 1; : : : ; NQ  I

(10.49)

cj k are real coefficients; fj are linear combinations of the variables v .im/ , w .im/ , x.im/ introduced in eq. (10.8); Zj are functions of external load. Then, by taking into account eq. (10.47), relation (10.42) can be written as follows: 8 y 9 9 8 9 8 9 N .y; z/ > 8 ˆ # 0 ˆ > i N  < = ˆ yz fyz Yyz iD1 ; : N yz # .y; z/ i

By substituting eq. (10.50) into the expressions for y , z and yz , the missing algebraic equations for i .x/ are obtained Z  

 y #N i y .y; z/ C z #N iz C yz #N i yz dydz D 0; i D 1; : : : ; N0 : (10.51) S

The systems of differential equations (10.27) and (10.34) can be resolved with respect to the first derivatives of the corresponding variables u.j k/ for j C k N0 , v .j k/ and w .j k/ for j C k N 0 . It is necessary to do so by taking into account the solution of the algebraic system (10.24), (10.46), and (10.51) with respect to i .x/. All independent variables

i .x/;

i D 1; : : : ; N I

x.k;l/ .x/;

u.k;l/ .x/;

k C l N0 I

v .kl/ .x/;

w .kl/ .x/;

k C l N0

can be collected into a vector a.x/ 2 RNd of design parameters with the dimension Nd D .N0 C 1/.N0 C 2/ C 2.N 0 C 1/.N 0 C 2/ :

(10.52)

After assembling the differential equations, this system can be written in the vector form   d C K a C F .x/ D 0 : (10.53) RD I dx

239

Section 10.1 Projection algorithms

Here, K 2 RNd Nd is a quadratic matrix composed from the elements Kij D

@Ri ; @aj

(10.54)

I 2 RNd Nd is the unit matrix, F .x/ 2 RNd is a vector of external loads applied to the lateral sides of the beam. Algorithm 10.3. In contrast to Algorithm 10.2, the approximation of xy and xz does not change after satisfying the boundary conditions (10.2) on the lateral sides of the prism. Instead, the approximation degree for v and w is raised by unit: N2 D N3 D N 0 C 1. Therefore, the procedure assembling the ODE system is modified. Similarly to Algorithm 10.2, the orthogonalization of xy , xz is performed in the first step in accordance with eqs. (10.29) – (10.33). The system of polynomials of eq. (10.33) is chosen μ ´ xy Q N X #i .y; z/ : (10.55) ci P D #i xz .y; z/ iD1 as a basis on which xy and xz are projected. By using the basis of eq. (10.55), the orthogonalization of v and w is fulfilled. For this purpose, the complete projection of these functions on the space P is calculated as follows: Z Z

xy v#i .y; z/dydz C w#i xz .y; z/dydz D 0; i D 1; : : : ; NQ : (10.56) S

S

Then, this system is resolved with respect to NQ variables of v .j k/ .x/ and w .j k/ .x/ for k C l N 0 C 1. After solving system (10.56) and performing corresponding transformations of eq. (10.8), the following expression is obtained: μ ´ uy Qu ² O³ N X #i .y; z/ v .i/ uyz .x/ D : (10.57) uz wO # .y; z/ i Q iDN C1

.i/ Here, new coefficients uyz .x/, i D NQ C 1; : : : ; NQ u , are linear combinations of u v .j k/ .x/, w .j k/ .x/; #i y .y; z/ and #iuz .y; z/ are new basis functions orthogonal to the polynomial vector P ; NQ u D .N 0 C 2/.N 0 C 3/ is the number of coefficients in the approximations for v and w. To find a representation of v and w equivalent to that in eq. (10.8), the followu ing system of equations is composed by using the new basis functions #i y .y; z/, uz #i .y; z/: Z u u v#i y .y; z/ C w#i y .y; z/dydz D 0; i D NQ C 1; : : : ; NQ u : (10.58) S

240

Chapter 10 3D static beam modeling

As for eq. (10.56), resolve system (10.58) with respect to some coefficients v .k;l/ .x/, w .k;l/ .x/. After substituting the solution of system (10.58) into the approximations (10.8) and grouping there similar terms, the following relation is obtained: ²

vP wP

³ D

Q N X

´ .i/ uyz .x/

iD1

u

#i y .y; z/

μ

#iuz .y; z/

:

(10.59)

u .i/ Here, uyz .x/, i D 1; : : : ; NQ , are new variables; the basis functions #i y .y; z/ and u uz #i .y; z/ for i D 1; : : : ; NQ are orthogonal to the polynomials #j y .y; z/ and #juz .y; z/ in eq. (10.57) for j D NQ C 1; : : : ; NQ u . As a result, new approximations for v and w can be presented according to ² ³ ² P ³ ² O³ v v v D C : (10.60) P w w wO

Then, the system of NQ first-order ODEs with respect coefficients uyz .x/ are derived by calculating all the projections of xy and xz on the corresponding subspace as follows: Z   u xy #i y C xz #iuz dydz D 0; i D 1; : : : ; NQ : (10.61) .i/

S

The relations (10.61) do not contain the coefficients j D NQ C 1; : : : ; NQ u :

j .x/; uyz

So, these functions should be excluded from the consideration, i.e., i uyz .x/ D 0;

i D NQ C 1; : : : ; NQ u :

(10.62)

In assembling the algebraic projections for y , z and yz , Algorithm 10.3 coincides with Algorithm 10.2. In accordance with the current procedure, the dimension of the vector of design parameters a.x/ is Nd D .N0 C 1/.N0 C 2/ C 2NQ :

(10.63)

To derive a general solution of the approximating ODE system similar to eq. (10.53), it is necessary to solve the eigenvalue problem K C I D 0; which is obtained after substitution of the vectors a D a.i/ e i x ;

i D 1; : : : ; Nd ;

(10.64)

241

Section 10.2 Cantilever beam with the triangular cross section

into eq. (10.53). Here, i and a.i/ are eigenvalues and eigenvectors for which the system (10.53) has nontrivial solutions. The determinant of system (10.64) has the following structure: det .K C E/ D 12 PNd 12 ./ D 0 :

(10.65)

Here, PNd 12 is a polynomial of degree Nd 12 with respect to the parameter . The root  D 0 with the multiplicity equal to 12 reflects a polynomial part of the general solution, while the roots of the equation PNd 12 ./ D 0

(10.66)

form a solution consisting of exponential terms. Note that the solutions of eq. (10.66), in general case, are complex valued.

10.2 Cantilever beam with the triangular cross section Let us demonstrate the effectiveness of Algorithms 10.2 and 10.3 on an example of the following model problem. Consider a rectilinear beam (see Figure 10.1) with the length l D 10 and a cross section in the form of an isosceles triangle with the base b D 1 parallel to the y-axis and the height h D 1 oriented along the z-axis. The origin of the Cartesian coordinate system is placed in the barycenter of the prism cross section. It is considered that the lateral faces of the beam are free of loads, .i/ .i/ .i/ i.e., qN x D qNy D qN z D 0 for i D 1; 2; 3 in eq. (10.2). The beam is made of homogeneous isotropic material with E D 1 and D 0:3. Define, for example, the following degrees of approximations N4 D N0 D 2, N5 D N6 D 2 and N7 D N8 D N9 D 3 for the stresses in eq. (10.8). If the Algorithm 10.2 is used, then it follows from eqs. (10.20), (10.28), and (10.52) that NQ D 8;

N 0 D 1;

N D 6;

Nd D 24:

(10.67)

For Algorithm 10.3, the ODE system is characterized by the following parameters: N D 8;

Nd D 28:

(10.68)

The calculations have been performed at different orders of approximation N0 D 2; 3; 4. In this case, the maximum orders of ODEs systems are equal to Nd D 70 and Nd D 78 for the Algorithms 10.2 and 10.3, respectively. It is supposed that one of the beam ends is clamped u D v D w D 0 for

x D 0:

(10.69)

242

Chapter 10 3D static beam modeling

The other end (x D l) is subjected to action of various factors:  tensile force

Z Fx D

 shearing forces

S

x dS I

Z Fy D

(10.70)

Z xy dS; S

Fz D

S

xz dS I

(10.71)

 bending moments Z Mz D

Z x ydS; S

 torque

Z Mx D

S



My D

S

x zdS I

 xy z  xz y dS:

(10.72)

(10.73)

Numerical simulation has been carried out for four different problems: Problem 1. Tension of the cantilever beam is specified by the following external load: Fx D 1;

Fy D Fz D Mx D My D Mz D 0:

(10.74)

Problem 2. Bending of the beam by the shearing force with respect to axes y is provided by (10.75) Fy D 1; Fx D Fz D Mx D My D Mz D 0 Problem 3. Bending of the consol with respect to axes z is defined by Fz D 1;

Fx D Fy D Mx D My D Mz D 0:

(10.76)

Problem 4. Torsion of the beam with respect to the longitudinal axis x is given by Mx D 1;

Fx D Fy D Fz D My D Mz D 0:

(10.77)

Bilateral convergence of the algorithms discussed in this chapter is shown in Figure 10.2 for Problem 1. The energy estimates obtained by Algoritm 10.3 are shown by the solid lines in this figure. The dashed lines correspond to Algorithm 10.2. Similar dependencies have been also obtained for Problems 2–4. It is worth noting that both algorithms are in a good agreement with each other. At the same time, the relative error  D ˆ…1 c of less than one percent has been reached for Problems 1–4 at N0 D 4 (see Figure 10.3). The best solution has been obtained for Problem 1. At that, the behavior of the function .N0 / for problems 2–4 is approximately the same. Its characteristic feature is that the magnitude of the

Section 10.2 Cantilever beam with the triangular cross section

243

Figure 10.2. Bilateral estimates of the solution quality by the potential energy … (lower lines) and the complementary energy …c (upper lines) for Problem 1.

Figure 10.3. Relative energy error .N0 /.

relative errors at N0 D 2 exceeds 15% for these problems. This can be explained by rather complicated stressed-strain state of the beam caused by bending or torsion. To illustrate this fact, the deplanation of the cross section at x D 5 for Problem 4 is depicted in Figure 10.4.

244

Chapter 10 3D static beam modeling

Figure 10.4. The deplanation of the cross section at x D 5 for Problem 4.

The qualitative distribution of the local errors 1 'D  WC W 2

(10.78)

on the clamped beam face x D 0 is presented in Figure 10.5 for Problem 1. Note that the values inside the triangle are close to zero except for a narrow region near the edges and corners of the cross section. Influence of boundary conditions on the solution of elasticity problems is an important aspect in reliable beam modeling. Special features of the solution due to this influence can be explained qualitatively by Saint-Venant’s principle. But a quantitative estimation of boundary effects can be only performed based on a detailed analysis of 3D stress-strain state. For example, the minimum absolute value  D min jRe.m /j mNd

for

m ¤ 0

gives one a possibility to calculate approximately the characteristic length lS , in which the end effect is noticeable. Here, m are the roots of eq. (10.66). The parameter lS is determined from the condition that the amplitude of the exponential, which corresponds to the eigenvalue m , decreases, for instance, hundredfold e lS  0:01 D 0 :

(10.79)

Section 10.2 Cantilever beam with the triangular cross section

245

Figure 10.5. Distribution of the energy error '.0; y; z/ for Problem 1.

The quantity  is equal to  D 4:66 at given beam parameters. Then, the solution of eq. (10.79) is about (10.80) lS  1 : Figure 10.6 presents the distributions of x along the x-axis. The stresses have been calculated at the cross-sectional points which are defined by the coordinates y D 12 and z D  13 (dash-dot curve), y D  12 and z D  13 (dot curve), y D 0 and z D  13 (dashed curve), y D 12 and z D  13 (solid curve) for problems 1–4, respectively. To present all curves on one figure, the normal components x .x/ have been multiplied by the factor 100 for Problems 1 and 4. As it has been estimated in eq. (10.80), noticeable perturbations of stress fields occur near the end cross sections at a distance less than unit only. This length corresponds to the parameter lS given by formula (10.74). Note that the disturbances of the stress state can occur not only near the clamped part of the beam. Typical disturbances may be caused by a specific kind of applied load. As can be seen in Figure 10.6 for Problem 4, the normal component x .x/ calculated near the right edge of the beam has a considerable change at a distance which does not exceed lS .

246

Chapter 10 3D static beam modeling

Figure 10.6. Typical distributions of the normal stresses x along the beam length for Problems 1-4.

10.3 Projection beam model The case when N0 D 1 has not been considered yet. Note that no solution of Problems 1–4 can be obtained based on Algorithm 10.2 in this case. When applying the Algorithm 10.3, the determinant in eq. (10.65) can be presented as det .K C E/ D c12 ; where c is some real constant. In other words, the resulting displacements and stresses are described by polynomial functions only. Since the polynomials are always presented in the approximations for any degree N0 , it seems important to study this type of general solution separately. In what follows, the polynomial part of the approximations is referred to as projection beam solution. Note that such polynomials can be separated for any degree N0 and always depend on 12 undetermined constants. Therefore, it is natural to formulate a new boundary value problem for the projection beam solutions only. First, it is necessary to fix the position of the beam as a rigid body in the 3D space. To do so, the following integral displacements are defined: Z Z Z u.x0 ; y; z/dS D v.x0 ; y; z/dS D w.x0 ; y; z/dS D 0 : (10.81) S

S

S

247

Section 10.3 Projection beam model

The zero integral angles of rotation also have to be given    Z  Z  Z  @u @v @u @w @v @w    dS D dS D dS D 0 @x @z @x @y S @y S S @z

(10.82)

at x D x0 . The point x0 , at which conditions (10.81) and (10.82) are specified, can be chosen arbitrarily. For example, if this point belongs to the end cross section, then it can be assumed that this beam is consol. Similarly to the displacement conditions, six loading factors, namely, forces and moments in accordance with eqs. (10.70) – (10.73), can be integrally taken at another point x1 . If the lateral sides of the beam are free of loading, i.e., qN x.i/ D qNy.i/ D qN z.i/ D 0 for

i D 1; 2; 3

in eq. (10.2), then the solution in accordance with the approximations (10.8) has the following structure of stresses: .kl/ .kl/ x.kl/ D cxx;1 C cxx;2 x; .kl/

.kl/ .kl/ y.kl/ D cyy;1 C cyy;2 x;

.kl/

z.kl/ D czz;1 C czz;2 x; .kl/ .kl/ D cxy;1 ; xy

.kl/ .kl/ xz D cxz;1 ;

.kl/ .kl/ .kl/ yz D cyz;1 C cyz;2 x

(10.83)

and displacements .kl/

.kl/

.kl/

.kl/

.kl/

.kl/

u.kl/ D cu;1 C cu;2 x C cu;3 x 2 ; .kl/

v .kl/ D cv;1 C cv;2 x C cv;3 x 2 C cv;4 x 3 ; .kl/ .kl/ .kl/ 2 .kl/ 3 C cw;2 x C cw;3 x C cw;4 x : w .kl/ D cw;1

(10.84)

Here, the letter c with upper and lower indices denotes some constants depending on boundary conditions. To compare different models, the displacements v.x; 0; 0/ near the clamped beam end for Problem 2 (bending of the console under the action of the unit shearing force) are shown in Figure 10.7. In this plot, the dot curve corresponds to the projection beam model studied in this section with x0 D 0; the solid curve represents the solution obtained on the basis of Algorithm 10.3 at N0 D 3; the dashed curve is the Bernoulli beam deflections, which has the form vc .x/ D

x 2 .3l  x/ : 6EI

(10.85)

It should be noted that all three polynomial solutions are approximately the same at points which are located far from the clamped beam end (x > lS ). The difference between the solutions near the clamped part is quite small and can be interpreted as some disturbances in boundary displacements.

248

Chapter 10 3D static beam modeling

Figure 10.7. Displacements v.x; 0; 0/ for different models.

10.4 Characteristics of a beam with the triangular cross section It has been shown in the previous section that the beam equation based on the Bernoulli model describes accurately the bending deflections of the cantilever beam. It is possible to say the same about the beam torsion equation. This section is devoted to the question how to determine efficiently the flexural center and torsional stiffness of a rectilinear beam with a triangular cross section in accordance with the projection algorithms discussed in this chapter. Consider again the cantilever beam with the geometrical and mechanical parameters given in Section 10.2. Let this body be loaded on free end by shearing force Fy D 1 and unknown torque Mx . The other loading factors are zero: Fx D Fz D My D Mz D 0: Let us fix the approximation order N0 and solve the above formulated problem in the frame of the projection beam model discussed in Section 10.3. Then, the magnitude of the torque Mx can be found from the condition that the beam torsion does not occur  Z  1 @v @w  dS  0; (10.86) .x/ D 2S S @z @y where .x/ is the integral angle of cross-sectional rotation. It can be shown that eq. (10.86) is reduced to a linear algebraic relation with respect to unknown torque

Section 10.4 Characteristics of a beam with the triangular cross section

249

Mx . Indeed, if the origin of force is moved along the z axis at a distance zb D

Mx D Mx ; Fy

(10.87)

where the torque Z Mxb D

S



 xy .z  zb /  xz y dS

is equal to zero, then the torsion is absent from this beam model. By using similar calculation, it is possible to find the coordinate yb for which there is no integral rotation of the cross section if the shearing force applied is parallel to the z-axis. The resulting point with coordinates y D yb and z D zb is of great importance. If the force acting perpendicular to the x-axis is attached at this point then this force causes no rotation of the cross-sectional elements and the point is called the flexural center of the beam. If the beam cross section has two axes of symmetry, the flexural center coincides with the barycenter of this section. When there is only one axis of symmetry, this point must lie on this axis. The coordinates y D yb and z D zb have been found based on the projection beam model for the different orders N0 D 1; : : : ; 5. Figure 10.8 plots the coordinate zb as the function of the number N0 . It is worth noting that the value zb is quite accurately determined in frame of this model at N0 3. If the precise location of the flexural center is not so important for practical purposes, then its coordinates can be obtained from the simplest model (N0 D 1). In this case, the position of this point is found with relative error less than 3%. Another important characteristic, which is used in the simplest equation of torsion Mx d.x/ D ; dx

t

t D kGJ

(10.88)

is the torsional rigidity. Here, .x/ is the angle of beam torsion with respect to x-axis, J is the polar moment of inertia, G is the shear modulus, k is a coefficient which depends on the shape of the cross section. The quantity t is called torsional stiffness and its inverse value 1 ˇt D (10.89) kGJ is named torsional flexibility. It is known (see, for example, [70]) that the Poisson’s equation have to be solved to determine the beam torsional stiffness. For beams with a complex cross-sectional shape, this problem does not seem too simple. In the frame of the projection beam model, the procedure for determining the torsional rigidity is as follows. First, Problem 4 formulated in Section 10.2 is solved in

250

Chapter 10 3D static beam modeling

Figure 10.8. Flexural center coordinate zb vs. the approximation order N0 .

Figure 10.9. Torsional flexibility ˇ t for beam with a triangular cross section vs. the approximation order N0 .

Section 10.4 Characteristics of a beam with the triangular cross section

251

polynomials for a given degree N0 . After that, the torsional flexibility is found from the following equation: ˇt D

d.x/ dx

for

Mx D 1:

Calculations have been performed for different values of the parameter N0 D 1; : : : ; 5. The dependence of the torsional flexibility ˇ t for beam with a triangular cross section on N0 is shown in Figure 10.9. This beam model at N0 2 gives quite accurate values for torsional flexibility ˇ t . It is interesting to point out that the magnitude ˇ t corresponds to the value of the coefficient k D 1 at N0 D 1.

Chapter 11

3D beam vibrations

11.1 Integral projections in eigenvalue problems Consider a long rectilinear prism (beam) with a triangular cross section (see Figure 10.1). It is assumed, as in Chapter 10, that the beam is made of homogeneous isotropic material with the volume density , Young’s modulus E, and Poisson’s ratio . The elastic vibrations of the beam are described by system (8.75), except the vector equilibrium that must be written componentwise as follows: @xy @xz @x C C C ! 2 u D 0 ; @x @y @z @y @yz @xy C C C ! 2 v D 0 ; ry D @x @y @z @yz @z @xz C C C ! 2 w D 0: rz D @x @y @z

rx D

(11.1)

In particular, only a free beam is studied in this section. This means that no displacements are given on the prism faces. The boundary conditions in the stresses (10.2), (10.3), and (10.6), defined on the lateral sides and bases of the beam, can be rewritten as qx.i/ D xy ny.i/ C xz n.i/ z D 0; qy.i/ D y ny.i/ C yz n.i/ z D 0; qz.i/ D yz ny.i/ C xz n.i/ z D 0;

(11.2)

x .0; y; z/ D xy .0; y; z/ D xz .0; y; z/ D 0; x .l; y; z/ D xy .l; y; z/ D xz .l; y; z/ D 0;

(11.3)

for i , i D 1; 2; 3, and

respectively. It is also supposed that the approximation of displacements and stresses is given by eq. (10.8). It follows from the structure of eqs. (11.1) and (10.8) that the equilibrium conditions can be satisfied in accordance, for example, with Algorithm 10.1. Therefore, in composing the approximating ODEs on the basis of the projection approach, it is possible to use all the algorithms discussed in Chapter 10 with minor changes. However,

253

Section 11.1 Integral projections in eigenvalue problems

as v and w are not defined on the body boundary, it is no need to eliminate some of the variables v .ij / .x/, w .ij / .x/ in the approximations of the displacement components as proposed, e.g., in Algorithm 10.3. Hence, these functions can be used to make some additional integral zero projections of the constitutive relations. Algorithm 11.1. Let us describe schematically a modified algorithm for this dynamic problem based on Algorithms 10.1 – 10.3 to constitute the governing ODEs. Step 1. In this step, the sequence of operations is performed, which completely corresponds to the algorithm 10.1. Namely, the degree of approximations is taken in accordance with eqs. (10.14) and (10.17). After that, the homogeneous boundary conditions on the lateral sides of the prism are satisfied and reduced approximations of the stresses (10.21) and (10.23) are constructed. Step 2. First, an algebraic system of equations is separated from eq. (10.24) and resolved with respect to variables Q i , i D 1; : : : ; NQ  , introduced in eq. (10.23). The differential equations obtained from eq. (10.24) are added to subsystem (10.27). Then, the value of the parameter N 0 is determined from eq. (10.28) and an orthogonalization of the approximation for xy and xz is carried out in accordance with eqs. (10.29) – (10.33). Step 3. The complete projections of y , z , and yz defined in eq. (10.10) on the corresponding polynomial subspaces are calculated. These equations are resolved with respect to all remaining coefficients Q i .x/ and a part of v .ij / .x/, w .ij / .x/ of the approximations (10.8). Step 4. Finally, the zero projections of xy and xz on the subspace PN0 C1 are added to ODE system. Consider the last step in more detail. A part of the differential equations is obtained from the following conditions: Z Z .j k/ .j k/ Qxy D xy pj k dydz D 0; Qxz D xz pj k dydz D 0; j C k N 0 : S

S

O xy

O xz

(11.4) of eq. (10.30), since they

In this case, the relation (11.4) does not contain and are orthogonal to the subspace PN 0 . To take into consideration these variables, a partial orthogonalization of the following monomials: pi D y i z N

0 C1i

for i D 0; : : : N 0 C 1:

(11.5)

is performed. Introduce the polynomials by D

0 C1 NX

iD0

cy.i/ pi .y; z/;

bz D

0 C1 NX

iD0

cz.i/ pi .y; z/;

(11.6)

254

Chapter 11 3D beam vibrations

where cy.i/ and cz.i/ are undefined constants. To determine which linear combinaO O and xz of tions of monomials of eq. (11.5) are orthogonal to the components xy eq. (10.30), the following system of equations is composed: Z Z

xy by #i dydz D 0; bz #i xz dydz D 0; i D N C 1; : : : ; NQ : (11.7) S

S

By substituting the solution of system (11.7) into eq. (11.6) and performing appropriate transformations, the following expressions are obtained: 0

byO D

Np X

0

cy.0i/ py.0i/ .y; z/;

iD0 .0i/

bzO D

Np X

cz.0i/ pz.0i/ .y; z/;

(11.8)

iD0

.0i/

O and  O . The dimenwhere py and pz are new basis functions orthogonal to xy xz 0 sion Np of these polynomials is introduced in eq. (11.8) according to

Np0 D N C N 0 C 2  NQ : .1i/ .1i/ The basis functions py .y; z/ and pz .y; z/ for i D 1; : : : ; NQ  N , which give O O , are similarly found from the linear nontrivial projections on stresses xy and xz system of equations Z Z by py.0i/ dydz D 0; bz pz.0i/ dydz D 0; i D 1; : : : ; Np0 : (11.9) S

S

Then, the additional ODEs are obtained as follows: Z   xy py.1i/ C xz pz.1i/ dydz D 0; i D 1; : : : ; NQ  N :

(11.10)

S

In the same way as it has been done in Chapter 10, the general solution is constructed for the resulting system of equations I

da C K.!/a D 0 ; dx

(11.11)

where a.x/ 2 RNd is the vector of design parameters. In this case, the characteristic equation takes the following form: det .K.!/ C I / D PNd ./ D 0 ;

(11.12)

where Nd is the differential order of the system. In contrast to the static case, eq. (11.12) does not contain the zero root .!/ D 0 at ! ¤ 0. In other words, the general solution of the eigenvalue problem is a linear combination of exponentials only. The natural frequencies !i for i D 1; : : : ; Nd and corresponding eigenforms of stresses and displacements are found from the condition that the determinant D of the algebraic system, which is obtained after substituting the general solution of system (11.11) into the boundary conditions (11.3), is equal to zero.

255

Section 11.2 Natural vibrations of a beam with the triangular cross section

11.2 Natural vibrations of a beam with the triangular cross section Consider the rectilinear beam shown in Figure 10.1 with the the parameters given in Section 10.2. Due to the symmetry of the cross section with respect to the z-axis, the governing ODE system can be decomposed into two independent subsystems in the way discussed in Chapter 8. At that, one of the subsystems describes the bendingtorsional (bt ) and torsional-bending (t b) motions of the beam. This system includes only the even functions of the components x with respect to the variable z. The other subsystem describes the bending-longitudinal (bl) and longitudinal-bending (lb) beam vibrations. It includes only the odd functions x of eq. (10.8). The coupling of bending with either tension or torsion is caused by an asymmetry of the beam cross section with respect to the y-axis. In this case, natural vibrations cannot be separated into four independent types of longitudinal, bending, and torsional motions as it has been supposed in Table 8.2. Nevertheless, only one type of displacement and stress fields described in this table makes the largest contribution in the corresponding amplitudes of vibrations. This is a reason to introduce the classification of eigenfrequencies and attendant eigenforms with two letters abbreviating corresponding fields. The first letter denotes the dominant type of motions.

.i /

Figure 11.1. System determinant D vs. frequency ! for the bending-torsion (!bt ) and .i / torsion-bending (! t b ) motions.

256

Chapter 11 3D beam vibrations

.i / Figure 11.2. System determinant D vs. frequency ! for the bending-longitudinal (!bl ) and .i / longitudinal-bending (!lb ) beam motions.

Let us analyze the determinant D of the boundary system following from eq. (11.3). Its behavior as a function of ! for these two subsystems is represented in Figures 11.1 .i/ and 11.2 in a logarithmic scale. The eigenfrequencies are marked by the symbols !bt , .i/

.i/

.i/

! tb , !bl , !lb , which relate to the appropriate types of beam motions. The numerical values of the natural frequencies for corresponding eigenmodes, obtained in accordance with Algorithm 11.1 at N0 D 3, are given in Table 11.1. The convergence rate of approximate eigenfrequencies to their exact values is an important characteristic of algorithm efficiency. Figure 11.3 shows the change in .1/ magnitude of the first bending-torsional natural frequency !bt versus the degree of

.1/ approximation N0 . It should be noted that the value !bt is decreasing with increasing

Table 11.1. Eigenfrequencies for the beam with triangular cross section.

i

1

2

3

4

.i/ !bt .i/ ! tb .i/ !bl .i/ !lb

0:0448 0:1498 0:0515 0:3140

0:1194 0:2995 0:1359 –

0:2250 – 0:2521 –

0:3535 – 0:3908 –

Section 11.2 Natural vibrations of a beam with the triangular cross section

257

.1/

Figure 11.3. The eigenfrequency !bt as a function of the approximation degree N0 .

N0 and the frequencies calculated at N0 D 3 and N0 D 4 differ from each other only in the sixth decimal place. Another characteristic of the numerical algorithm efficiency is the relative error  which can be defined as follows: ˆ  100%: (11.13) D W Here, ˆ is the energy error functional in eq. (8.6), W is the stress energy expressed through the tensor  according to Z 1  W C 1 W  d  0 : (11.14) W D 2  As both quadratic functionals ˆ and W are homogeneously dependent only on one undefined constant in the eigenforms, the ratio of these values (11.13) does not depend on this constant. Consequently,  is a parameter which unambiguously reflects the quality of approximate solutions. Dependence of the error  on the approximation parameter N0 is presented in .1/ Figure 11.4 in a logarithmic scale for !bt . This chart shows that the quantity  decreases rapidly with increasing the degree of approximation and attains the value approximately equal to  D 104 % when N0 D 4. The distribution of the energy error Z 'x .x/ D '.x; y; z/dS (11.15) S

258

Chapter 11 3D beam vibrations

.1/

Figure 11.4. Relative error log  vs. the approximation degree N0 for !bt .

along the beam length can serve as local characteristics of the approximate solution. Here, the energy error density '.x; y; z/ is given by eq. (10.78) and the integration performs over the beam cross-sectional area S . The function 'x .x/ is shown in Fig.1/ ure 11.5 for the first bending-torsional frequency !bt at N0 D 3. It is seen that trigonometric functions, which are present in the general solution, significantly influence the form of this error. The local error '.x0 ; y; z/ calculated at x0 D l=2 is shown in Figure 11.6. Its characteristic feature for the dynamic as well as static case (see Figure 10.5) is that the values of ' are close to zero almost everywhere, except for a narrow area near the edges and corners of the cross section. The eigenfrequencies presented in Table 11.1 can be compared with the natural frequencies that are derived from the classical beam equations (Euler–Bernoulli model) d 4 vc  S !z2 vc D 0 dx 4 d 4 wc EJy  S !y2 wc D 0: dx 4 EJz

(11.16)

Here, the lateral displacements of the beam midline vc and wc are directed along coordinate axes y and z, respectively; !y and !z are unknown frequencies; EJy and EJz are bending stiffnesses with respect to the corresponding axes, which are equal to EJy D for given parameters.

1 36

and

EJz D

1 48

Section 11.2 Natural vibrations of a beam with the triangular cross section

259

.1/ Figure 11.5. Distribution of the linear error density 'x .x/ along the beam length for ! D !bt at N0 D 3.

Figure 11.6. Energy error density '.x0 ; y; z/ in the beam cross section at x0 D l=2.

260

Chapter 11 3D beam vibrations

Figure 11.7. Classical eigenfrequencies !y.m/ , !z.m/ (dashed lines) as well as bending.m/ .m/ torsional, !bt , and bending-longitudinal, !bl , eigenfrequencies (solid lines) vs. their serial number m.

The following frequency mismatches: .1/ !y.1/  !bl .1/

!y

D 2:2%

and

.1/ !z.1/  !bt .1/

!z

.1/

D 1:9%

(11.17)

.1/

are obtained for the first classical bending frequencies !y and !z with respect to .1/ .1/ and !bt . It is possible to conclude, taking into account eq. (11.17), that the pro!bl jection and classical beam models are in good agreement with each other. However, this difference is increasing rapidly when the serial number of natural frequency m is growing so that even when m D 3, the contrast is about 20%. This fact is reflected in Figure 11.7. A characteristic feature of these dependences is that the following inequalities: !y.m/ > !z.m/

.m/

.m/

and !bl > !bt

are always valid for given beam parameters. However, even if m > 3, the following .m/ .m/ inequality is correct: !y > !bt : This property can be explained by the fact that the Euler–Bernoulli model overestimates the natural frequencies of the 3D elastic body. For example, this model does not take into account the rotational inertia of the beam as well as the motions associated with the deformation of the cross sections.

Section 11.2 Natural vibrations of a beam with the triangular cross section

261

Figure 11.8. Bending-longitudinal eigenform: lateral (solid curve) and longitudinal (dashed .1/ . curve) displacements for the frequency !lb

.1/

The first longitudinal-bending mode of natural vibrations with the frequency !lb is shown in Figure 11.8. In contrast to the form of longitudinal vibrations corresponding the Euler–Bernoulli model, the longitudinal-bending motions are characterized not only by the longitudinal displacements u0 .x/ but also by the transverse displacement w0 .x/. Here, u0 and w0 are the following integral characteristics: Z Z 1 1 u.x; y; z/dS; w0 .x/ D w.x; y; z/dS; (11.18) u0 .x/ D S S S S The amplitudes u0 .x/ and w0 .x/ calculated at N0 D 3 can be compared by the following ratio: max ju0 j D 48674: (11.19) ˇ1 D max jw0 j This value shows that the relationship between longitudinal and lateral vibrations is quite weak and can be neglected under certain assumptions. For better visualization, both forms are placed on the same figure and scaled appropriately by the factor ˇ1 . The first mode of bending-longitudinal vibrations corresponding to the frequency .1/ !bl is shown in Figure 11.9. This form includes not only the component of bending w0 , as follows from the classical concept, but also the longitudinal displacements u0 . At that, the deflections w0 are dominant over u0 . The inverse amplitude ratio ˇ2 D

max jw0 j D 68900: max ju0 j

(11.20)

262

Chapter 11 3D beam vibrations

Figure 11.9. Bending-longitudinal eigenform: lateral (solid curve) and longitudinal (dashed .1/ curve) displacements corresponding to the frequency !bl .

.1/ .1/ at !bl is quite large in contrast to the longitudinal-bending vibrations at !lb . By taking into account eq. (11.20), the influence of the longitudinal displacements can be neglected in most cases. Amplitude relations change appreciably for torsional-bending and bending-torsional vibrations. The first torsional-bending eigenform is shown in Figure 11.10. To compare the bending and torsional motions, the functions  Z  1 @v.x; y; z/ @w.x; y; z/  dS (11.21) # D h.x/; .x/ D 2S S @z @y

and

1 v0 .x/ D S

Z v.x; y; z/dS

(11.22)

S

(dashed and solid curves in Figure 11.10, respectively) are introduced. Here, v0 .x/ is the integral lateral displacements; .x/ is the average angle of cross-sectional rotation with respect to the x-axis. The function #.x/ is dominant for this type of natural motions. However, the amplitude ratio ˇ3 D

max j#j  214 max jv0 j

(11.23)

is not so large as in the previous two examples. Therefore, it may not be so reasonable to neglect the transverse displacements in some cases.

Section 11.2 Natural vibrations of a beam with the triangular cross section

263

Figure 11.10. Torsional-bending eigenform for the eigenfrequency ! t.1/ . b

The first bending-torsional form of natural beam vibrations is shown in Figure 11.11. The integral transverse displacements v0 .x/ (solid curve) can characterize this type of motions. However, the amplitude ratio ˇ4 D

max jv0 j  288 max j#j

(11.24)

is not large enough to neglect shear deformations #.x/ (dashed curve in Figure 11.11). Note that if the eigenvalue number m is increasing, then the values of amplitude ratios are decreasing. So, this ratio is max jv0 j  11 max j#j

(11.25)

.6/

for the sixth bending-torsional frequency (!bt D 0:6455). The deformed shape of .6/

the beam corresponding to the natural frequency !bt is shown schematically in Figure 11.12. This figure shows five characteristic cross sections with the coordinates x D 0; 2:7; 5; 7:3; 10 as well as the twisted curves of prism edges. It is seen in the picture that the cross sections have noticeable deformations in their planes and, moreover, are quite significantly rotated around z-axis (this is particularly considerable for the end sections). In-plane deformation of the cross section is shown in Figure 11.13 at x D 7:3. It may be observed that the Bernoulli hypothesis about beam straight lines is considerably disturbed. This is especially significant for the triangle height shown in the figure by dashed curve.

264

Chapter 11 3D beam vibrations

.1/

Figure 11.11. Bending-torsion eigenform corresponding to the eigenfrequency !bt .

Figure 11.12. Deformed beam shape corresponding to the bending-torsion eigenfrequen.6/ . cy !bt

It is worth noting that the Bernoulli hypothesis about plane beam cross sections is also violated. The distribution of u.x; y; z/ at x D 7:3, cross-sectional deplanation, is shown in Figure 11.14. Thus, significant deformation of the beam cross section in the plane and its deplanation lead to the fact that the natural frequencies obtained from the Euler–Bernoulli model are quite different from the frequencies found by the projection approach in the framework of linear elasticity. Moreover, this difference increases while the number N of natural frequencies grows. It may be noted that for a sufficiently long beam (l= h > 100) the classical theory allows us to obtain several natural frequencies with a high degree of reliability. However, to determine a priori the number of such frequencies is rather difficult.

Section 11.2 Natural vibrations of a beam with the triangular cross section

265

Figure 11.13. Deformed shape of the beam cross section at x D 7:3 for the bending-torsion .6/ . eigenfrequency !bt

Figure 11.14. Deplanation of the beam cross section at x D 7:3 for the bending-torsional .6/ eigenfrquency !bt .

266

Chapter 11 3D beam vibrations

11.3 Forced vibrations of a beam with the triangular cross section The influence of cross-sectional asymmetry on the natural frequencies and forms of beam vibrations has been discussed in the previous section. It has also been mentioned that the eigenmotions of these beams cannot be divided into purely longitudinal, bending, or torsional. Due to this asymmetry, such motions are coupled to each other. In applications, it is also rather important to analyze the vibrations occurring when beams are loaded by periodic forces with frequencies close to beam eigenfrequencies. Consider again a rectilinear beam with a triangular cross section. The mechanical and geometrical parameters are taken as in Section 10.2. It is supposed that the beam faces are free of loading and two types of loads on the beam bases are given. Forced beam motions are excited by the bending moment MQ z0 .t / D ˙Mz .0/ cos !f t;

MQ zl .t / D Mz .l/ cos !f t:

(11.26)

Other loading factors are absent Fx .0/ D Fx .l/ D Fy .0/ D Fy .l/ D Fz .0/ D Fz .l/ D 0 ; Mx .0/ D Mx .l/ D My .0/ D My .l/ D 0:

(11.27)

Another type of beam loading is characterized only by the torque MQ x0 .t / D ˙Mx .0/ cos !f t;

MQ xl .t / D Mx .l/ cos !f t

(11.28)

under conditions Fx .0/ D Fx .l/ D Fy .0/ D Fy .l/ D Fz .0/ D Fz .l/ D 0 ; My .0/ D My .l/ D Mz .0/ D Mz .l/ D 0:

(11.29)

Here, the integral forces and moments have the following form: Z Z Z x dS ; Fy .x/ D xy dS ; Fz .x/ D xz dS; Fx .x/ D S S ZS Z   zxy  yxz dS ; My .x/ D Mx .x/ D zx dS ; S S Z yx dS I (11.30) Mz .x/ D S

!f is frequency of external excitation. Two values .1/ !f0 D 0:0448  !bt ;

.1/ !f1 D 0:1498  ! tb

close to the resonance frequencies obtained in Section 11.2 are taken.

(11.31)

Section 11.3 Forced vibrations of a beam with the triangular cross section

267

Figure 11.15. Steady-state form of beam vibrations at !f0 .

First, consider the case when the beam is loaded only by torque according to condition (11.28) with amplitudes Mx .0/ D Mx .l/ D 1 and frequency !f0 . This value is approximately equal to the first bending-torsional frequency of beam natural vibrations. In agreement with classical beam theory, such loading must excite only small rotations of the beam cross sections around the x-axis and does not cause a resonance twisting of the beams. The form of the steady-state vibrations for given loading is shown in Figure 11.15. The dashed curve corresponds to #.x/ (beam torsion). The solid curve represents the beam bending (integral lateral displacements v0 .x/). In this case, the amplitude ratio corresponds to eq. (11.24). Note that quite limited periodic rotations of the beam excite large flexural vibrations. In this case, the beam with a triangular cross section can be considered as a mechanical amplifier that transfers a significant part of torsional energy into the bending one. Of course, this conclusion corresponds to an ideal case. In reality, such external and internal factors as friction, viscosity, etc. can significantly influence this process. However, this phenomenon can be observed experimentally for elastic structures with a high quality factor. While prism vibrations are excited by the asymmetric torques Mx .0/ D Mx .l/ D 1

268

Chapter 11 3D beam vibrations

Figure 11.16. Steady-state form of vibrations at the external frequency !f1 .

no resonance phenomena are observed. This can be explained by the fact that the first mode of beam natural bending-torsional vibrations is symmetric with respect to the middle point of the beam (x D 5). The asymmetric external torque can cause, for .2/ example, large amplitude of the second mode corresponding to !bt . For vibrations excited by the bending moment of eq. (11.28), the effect is similar to the previous case. Due to the antisymmetry of the first torsional-bending mode of the beam vibrations with respect to its middle point at x D 5, steady-state vibrations corresponding to this form are realized under the bending moments Mz .0/ D Mz .l/ D 1 with the frequency !f1 . Figure 11.16 shows the shape of steady-state vibrations under given external loading. At the same time, the amplitude ratio corresponds to the formula (11.23). In contrast to the previous case, the beam transfers a significant portion of the bending energy in the torsional motion. Under symmetric loading Mz .0/ D Mz .l/ D 1, the resonance effect is absent. Resonance phenomena of the beam under this type of loading may only occur for a torsion-bending eigenform with an even number.

Appendix A

Vectors and tensors

Some specific characteristics of tensors and vectors are brought below. The thorough description of these objects can be found, e.g., in [68]. Let a and b be spatial Euclidean vectors which are defined by their components as column arrays of real numbers a D ¹ai ºT D ¹a1 ; a2 ; a3 ºT 2 R3

and b D ¹bi ºT D ¹b1 ; b2 ; b3 ºT 2 R3

in a Cartesian coordinate system x D ¹x1 ; x2 ; x3 ºT with the orts e .1/ , e .2/ , and e .3/ . This means that a D a1 e

.1/

C a2 e

.2/

C a3 e

.3/

D ai e

.i/

WD

b D b1 e .1/ C b2 e .2/ C b3 e .3/ D bi e .i/ WD

3 X iD1 3 X

ai e .i/ ; bi e .i/ :

iD1

Here, the summation is assumed over repeated indexes. The dot product or scalar product is an algebraic operation that takes two vectors and returns a scalar number obtained by multiplying corresponding components and then summing the products as a  b D ai bi WD a1 b1 C a2 b2 C a3 b3 :   The basis vectors are orthonormal according to e .i/  e .j / D ıij with the Kronecker delta ıij . The cross product of two vectors is expressed as a  b D .a2 b3  a3 b2 / e .1/ C .a3 b1  a1 b3 / e .2/ C .a1 b2  a2 b1 / e .3/ : If tensors of second rank  and  are given with their components defined for the same coordinate system in the matrix form as 9 9 8 8 H k ./ D

k Z X jijD0

D i uD i v d  

for

u; v 2 H k ./ :

The spaces introduced here are embedded as it follows from Table B.1.

273

Appendix B Sobolev spaces

Table B.1. Inclusions of the Banach spaces W k;s .

1 0 1 2 :: : k :: :



W k;1 [ :: :



 

L [ H1 [ H2 [ :: :













Hk [ :: :



 

2

1

s

2

L [ W 1;1 [ W 2;1 [ :: : 1



L [ W 1;s [ W 2;s [ :: :













W k;s [ :: :



:: : 

s

:: :

Ls [ W 1;1 [ W 2;1 [ :: : W k;1 [ :: :

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Index

action functional 51 aeolotropy 24 Argyris triangle 127 asymptotic approximations 177, 223 barycentric coordinates 68 beam bending 168, 177, 207 beam tension 207 beam torsion 207 Beltrami–Michell equations 27 bending moment 36, 242, 266 Bernoulli beam 35, 179, 247, 258 bilateral energy estimates 108, 118, 136, 166 Biot strain tensor 16 Boltzmann’s theorem 13 boundary conditions 11, 26, 32, 36 boundary element method (BEM) 62 boundary value problem 27 Cauchy strain tensor 17 Cauchy stress tensor 9 Clapeyron’s formula 23 Clapeyron’s theorem 29 complete polynomials 75, 230 complete projection 230, 253 constitutive relations 20 differential equilibrium equations 26 Dirichlet problem 27 displacement vector 14, 26 dynamic equilibrium 30, 31 eigenfrequency 31, 90, 170, 192, 255 eigenvalue problem 31, 40 elastic (Winkler) foundation 26, 119, 142 elastic compliance tensor 23 elastic energy 87 elastic modulus tensor 22 elastic potential 21

energy 21, 22, 43 energy error density 82, 142, 166 energy error functional 82, 106, 114, 141, 166, 222 engineering shear strains 18 Euler coordinates 13 Euler–Almansi strain tensor 15 finite difference method 62 finite element method (FEM) 62, 124 flexibility matrix 138 flexural center 248 force strain tensor 79 forced vibrations 31, 266 g-functions 68, 125 Galerkin method 64 Gauss theorem 11 geometric stress tensor 79 Green–Lagrange strain tensor 15 Hamilton principle 51 Hellinger–Reissner principle 58 Hooke’s law 22 Hu–Washizu principle 56 incomplete projection 230 initial-boundary value problem 33 integral equilibrium equations 11 integral projections 218, 229, 253 internal elastic forces 7 isotropy 24 kinematic relations 17, 26, 32 kinetic energy 50 kinetic energy density 50 Lagrange coordinates 13 Lagrange–Euler equations 115, 161, 165 Lame moduli 24 Legendre polynomials 223 linear element 14

280 material gradient tensor 15 Maxwell’s stress functions 115 membrane 38, 90 membrane tension 38 mesh adaptation 145, 152 mesh nodes 69 mesh refinement 145 micromorphic theory of elasticity 7 minimizing sequence 63 natural vibrations 31, 90, 171, 191, 255 Navier’s equations 27 Neumann problem 27 normal stresses 8 outward normal 7 Petrov–Galerkin method 2, 218 piecewise polynomial spline 124, 127 plane strain state 41 plane stress state 40, 218 Poisson’s ratio 25 polar decomposition theorem 15 polynomial approximations 76, 108, 219 potential energy 20, 29, 38 prestressing 29 principle of complementary work 48 principle of minimum complementary energy 48, 121, 162 principle of minimum energy error 114 principle of minimum potential energy 46, 66, 121, 160, 177 principle of virtual work 45, 55 projection beam solution 246 regular triangulation 67 relative integral error 83 resonance 266

Index right stretch tensor 16 Ritz method 63 Ritz parameters 63 rotational matrix 9 Saint–Venant’s equations 18 semi-discrete approximations 158, 177, 220 shear modulus 25 shearing force 36, 178, 242 Sobolev spaces 78 spatial gradient tensor 15 stiffness matrix 71, 132, 137 strain compatibility 18 strain energy density 21 strain error functional 81 strain error tensor 81, 167, 218 stress and strain principal axes 24 stress energy density 21 stress error functional 81 stress error tensor 81, 152, 172, 222 stress-strain state 7 surface loads 6 tangential (shear) stresses 8 test functions 65 torque 35, 215, 242, 266 torsional stiffness 35, 248 total potential energy 45 trial functions 65 variational derivatives 61 vector of external forces 71 volume forces 6 work 20, 29, 43 Young’s modulus 25