Fundamentals of Structural Optimization: Stability and Contact Mechanics (Mathematical Engineering) 3031346319, 9783031346316

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Mathematical Engineering

Vladimir Kobelev

Fundamentals of Structural Optimization Stability and Contact Mechanics

Mathematical Engineering Series Editors Bernhard Weigand, Institute of Aerospace Thermodynamics, University of Stuttgart, Stuttgart, Germany Jan-Philip Schmidt, Universität of Heidelberg, Heidelberg, Germany Advisory Editors Günter Brenn, Institut für Strömungslehre und Wärmeübertragung, TU Graz, Graz, Austria David Katoshevski, Ben-Gurion University of the Negev, Beer-Sheva, Israel Jean Levine, CAS- Mathematiques et Systemes, MINES-ParsTech, Fontainebleau, France Jörg Schröder, Institute of Mechanics, University of Duisburg-Essen, Essen, Germany Gabriel Wittum, Goethe-University Frankfurt am Main, Frankfurt am Main, Germany Bassam Younis, Civil and Environmental Engineering, University of California, Davis, Davis, CA, USA

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Vladimir Kobelev

Fundamentals of Structural Optimization Stability and Contact Mechanics

Vladimir Kobelev Department of Natural Sciences University of Siegen Siegen, Germany

ISSN 2192-4732 ISSN 2192-4740 (electronic) Mathematical Engineering ISBN 978-3-031-34631-6 ISBN 978-3-031-34632-3 (eBook) https://doi.org/10.1007/978-3-031-34632-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Foreword

This book began as the notes for the courses at the University of Siegen. The aim of the course was to give an introduction to structural optimization theory to senior-level undergraduates and graduate students from engineering, automotive, mathematics, and business schools. The only prerequisites for the course were the master-level courses in ordinary differential equations and mechanical engineering. Accordingly, the backgrounds of the students were widely dissimilar, and the common denominator was their interest in the applications of structural optimization. Now search for optimal solutions had started from times immemorial. Indeed, as Garrett N. Vanderplaats—one of the active developers of structural optimization, notes, “The concept of optimization is intrinsically tied to natural phenomena as well as to the human desire to excel…Oliver Wendall Holmes (1809–1894), in his classic verse, recorded the human desire to produce a uniformly strong, durable product” (Structural Optimization—Past, present, and Future”, AIAA Journal, Vol. 2-(7), 992–1000, 1982). This beautiful text differs from the standard ones in that the author—a recognized authority in optimization’s various aspects and especially known internationally for his search and success in obtaining closed-form solutions–has not attempted to provide complete proofs the analytical derivations, since this was beyond the background and interest of most of the students in the course. Instead, he has tried to show optimization theory’s strengths and limitations through practical examples. Since the optimality conditions arising in the structural design can habitually be solved only numerically, only relatively simple contact and stability problems are displayed. The manuscript could be roughly divided into two parts. The first part encompasses Chaps. 1 to 4 and studies various contact problems in an optimization context. In Chap. 1 author introduces the optimization of the rigid punches on the elastic foundations. The author studies the optimization of the axisymmetric rigid punches on the elastic foundations in Chap. 2. The problem of optimization of a stringer, or a stiffener, attached to an elastic infinite plate is investigated in Chap. 3. In Chap. 4, the problem of stiffening of the infinite plate with the periodically arranged inclusions (stiffeners) is examined.

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The second part contains the Chaps. 5 to 12. In these Chapters, the exact solvable optimization problems of structural stability are deliberated. The optimization problem for a column, loaded by compression forces for Sturm-Liouville boundary conditions is studied in Chap. 5. The bimodal optimization problem for a symmetrically clamped column, loaded by compression forces is studied in Chap. 6. The applications of the method for stability problems of torsion members are illustrated in Chap. 7. The periodic Greenhill problem, which describes the forming of a loop in an elastic bar under torsion, is solved in Chap. 8. Chapter 9 considers the stability problems in the context of simultaneously twisted and compressed rods. Chapter 10 deals with another problem of the optimal tapering of a conservative system of the second kind, namely the generalized Pflueger column. Chapter 11 displays the closed-form solution for optimization and isoperimetric inequalities for the stability of the circular ring. In Chap. 12, the optimization problem for an axially compressed strut on an elastic foundation is studied. We are all awaiting for next volume “Fundamentals of Structural Optimization. Topology and Anisotropy”. Both volumes could serve as an advanced introduction to research in this area. Isaac Elishakoff Distinguished Research Professor

Preface

Chapter 1 illustrates the optimization of the rigid punches on the elastic foundations. The indentation of a non-deformable punch into a half plane is considered for plane stress and plane strain problems of isotropic elasticity. The rigid indenters penetrate without friction into the surface of an isotropic elastic half plane. The elastic energy of the elastic body and the maximal pressure on the contact surface are considered as the principal functionals. The bottom shape of the rigid punch is primarily unknown and represents the examined function of the optimization problems. Optimal solutions are achieved for a single indenter and for periodically spaced equal indenters. The closed-form solutions of optimization problems are completed in terms of high transcendental functions. The method allows the finding of optimal shape for the non-deformable indenters in plane strain and plain stress elasticity problems The authors study the optimization of the axisymmetric rigid punches on the elastic foundations in Chap. 2. Considered is the contact problem of the theory of elasticity on indentation of an indenter with an unknown base into an elastic isotropic half plane. The authors consider the rigid axisymmetric punches of arbitrary profiles indenting without friction into the surface of an isotropic elastic half plane. The bottom represents the searchable function in the optimization problems. In optimal formulations, the elastic energy of the elastic body and the maximal pressure on the contact surface represent the principal functionals. Singular stress analysis in the punch problem of half plane is carried out. The distribution for the stress components is determined in an explicit form. Solutions in a closed form are obtained for the problems of a singular and periodically spaced concentric punches. Followed a similar study in the punch analysis, the punch singular stress factor is defined from the singular stress distribution. For the problems of a singular and periodically spaced concentric punches, solutions in a closed form are obtained. The problem of optimization of a stringer, or a stiffener, attached to an elastic infinite plate is investigated in Chap. 3. The plate is exposed to tensioning and to contact stresses with the reinforcing stiffener. The stringer is of an elongated, needle-shaped form of the variable cross-section along the axis of the stiffener. The cross-section, which is primarily unknown, represents the searchable function in the optimization problems. Two optimization problems: minimization of the ultimate vii

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stress along the stringer and minimization of the stringer mass under the constraints on the integral compliance of the reinforced body are solved. A flattened, plateshaped stiffener that supports a semi-infinite prismatic body is also briefly pursued. Optimization problems of the flattened stiffener are proved to be quite similar to those of the elongated one. The governing equations of both studied cases transform into each other by means of alternation of the elastic constants. Consequently, the optimal cross-sections of both problems turn out to be identical after the appropriate choice of material parameters. Notably, that contact problems in elasticity theory are completely mathematically equivalent to the tasks of electrostatics and magnetostatics of disks. Chapter 3 studies two optimization problems: minimization of the ultimate stress along the stringer and minimization of the stringer mass under the constraints on the integral compliance of the reinforced body. The shape optimization is studied in the case of an isolated stiffener. The core of the proposed method is based on the representation of the stiffener as an infinitesimally thin rod with a variable crosssection. The stringer tightened on the edge of the semi-infinite thin plate (case A). The semi-infinite plate is in the plane stress state. Thus, in case A the needle-form inclusions that support the thin plate are considered. The analogous formulation concerns the stiffening of the elastic bodies in plane strain state. The flattened stiffener is tightened to the surface of the semi-infinite, elongated in z direction prismatic body (case B). Both cases transform immediately to each other by means of recalculation of elastic constants and lead to the identical principal equations. The method allows the finding of optimal shape for thin stiffeners subjected to the variable loads along the stiffener axis. The governing equations in the considered examples are singular integral equations. The idealized problem in the optimal case leads to singular integral equations for the desired optimal cross-section of the needleform inclusion. The solutions of the optimization equations are achieved in analytical form. In Chap. 4, the problem of stiffening of the infinite plate with the periodically arranged inclusions (stiffeners) is examined. The plate is exposed to tensioning by uniformly distributed forces and also to contact stresses due to forces as a result of enforced continuity with the reinforcing stiffener. The optimization problem is emphasized in an elongated, needle-shaped form of the inclusion. The cross-section together with its axial stiffness is variable along the axis of the stiffener. An infinite chessboard array of inclusions, or stiffeners, attached to an elastic infinite plate is discussed as well. The optimization is applied to the infinite arrays of inclusions, arranged in rectangular and square lattices. In Chaps. 5–8, the isoperimetric inequalities arising in exactly solvable structural optimization problems of stability are discussed. It is not necessary to consider complex mechanical systems in order to identify most of the characteristic features of elastic stability theory problems. This can be done by limiting the study to the simplest mechanical systems that allow for an elementary analytical description. The most important notions of the theory of elastic stability—the bifurcation point, the

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critical load, the linearized equation, the stability boundary, and the energy criterion—are introduced and illustrated by examples. The applications of this method for stability problems are illustrated in this manuscript. Approved are the inequalities for Euler’s column with boundary conditions of mixed type, for a twisted rod with periodic simple supports and for a ring acted upon by a uniformly distributed, compressive hydrostatic load. The optimal design in stability problems of elastic structural elements has the peculiarity that the “control” function describing the thickness distribution is included in the coefficients of the equation. This is related to the approximate nature of the constitutive equations used, in which averaging over one of the spatial variables (thicknesses) is performed. The rigorous solutions of such problems are demonstrated in this manuscript. Rigorously verified are the inequalities for Euler’s column with boundary conditions of mixed type, for a twisted rod with periodic simple supports and for a ring acted upon by a uniformly distributed, compressive hydrostatic load. The optimization problem for a column, loaded by compression forces for Sturm– Liouville boundary conditions is studied in Chap. 5. The direction of the applied forces coincides until buckling moment with the axis of the column. The critical value of buckling is equal among all competitive designs of the columns. The dimensional analysis eases the mathematical technique for the optimization problem. The dimensional analysis introduces two dimensionless factors, one for the total material volume and one for the total stiffness of the columns. The principal results are the closed-form solution in terms of the hypergeometric and elliptic functions, the analysis of single regime, and the exact bounds for the masses of the optimal columns. The isoperimetric inequality was formulated as the strict inequality sign, because the optimal solution could not be attained for any finite setting of the design parameter. The additional restriction on the minimal area of the cross-section regularizes the optimization problem and leads to the definite attainable shape of the optimal column. The optimization of a column, compressed by axial forces was solved in closed form. The alternative designs were characterized by positive cross-sectional area functions. The critical values of buckling were equal for all alternative designs of the columns. The solution of the alternative, regularized Nikolai problem with the additional restrictions to the cross-sections is appropriate for the engineering applications, because the stress in the narrowest cross-section remains limited. The solution of optimization problem in Nikolai sense leads to the sharp isoperimetric inequality. The optimization problem for a column, loaded by compression forces is studied in Chap. 6. The closed-form solutions for mixed boundary conditions are derived. The solutions are expressed in terms of the higher transcendental functions. The principal results are the closed-form solution in terms of the hypergeometric and elliptic functions, the analysis of single and bimodal regimes, and the exact bounds for the masses of the optimal columns. The isoperimetric inequality was formulated as the strict inequality sign, because the optimal solution could not be attained for any finite setting of the design parameter. The additional restriction on the minimal area of the cross-section regularizes the optimization problem and leads to the definite attainable shape of the optimal column. The alternative designs were characterized by

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positive cross-sectional area functions. The critical values of buckling were equal for all alternative designs of the columns. The optimization problem in Lagrange sense searches the minimal mass of the column. The method of dimensionless factors was used for the optimization analysis. The closed-form solution contains one auxiliary parameter µ. For each positive value of the parameter µ, the necessary optimality conditions are resolved in closed form. Using the method of Hölder inequalities, for each given positive value of parameter µ the isoperimetric inequality for the buckling eigenvalue is uniquely specified. The statement of the isoperimetric inequality is based on the customary optimality conditions for multiple eigenvalues. Accordingly, the existence of the optimal solution is guaranteed for each positive value of the auxiliary parameter. The mass of the established optimal design is the monotonically decreasing function of the auxiliary parameter µ. Because there is no smallest positive real number, there is also no attainable solution of the optimization problem in Lagrange sense. In this problem, the authors search the solution among the positive functions. With the disappearing parameter µ, the optimal cross-sections asymptotically adjust to its limit shape. The solution of the alternative, regularized Nikolai problem with the additional restrictions to the cross-sections is appropriate for the engineering applications, because the stress in the narrowest cross-section remains limited. The applications of the method for stability problems of torsion members are illustrated in Chap. 7. In the context of twisted rods, the counterpart for Euler’s buckling problem is Greenhill’s problem, which studies the forming of a loop in an elastic bar under pure torsion. The general non-conservative systems are described by non-self-adjoint equations of motion and lose stability by bifurcation (divergence) and/or flutter. The stability of such systems is studied by the dynamic method. It is well known that certain non-self-adjoint systems have only a bifurcation type instability, despite the presence of non-conservative forces. This system is referred to as a conservative system of the second kind. The conservative system of the second kind is self-adjoint in a generalized sense with respect to an assigned operator. For this system, both Lyapunov functional and generalized Rayleigh quotient could be defined. The optimal shape of the rod along its axis is searched. The crosssections are similar geometric figures related by a homothetic transformation with respect to a homothetic center on the axis of the rod and vary along its axis. The distribution of material along the length of a twisted rod is optimized so that the rod is of the certain constant volume and will support the maximal moment without spatial buckling. The cross-section that delivers the maximum or the minimum for the critical eigenvalue must be determined among all convex, simply connected domains. The applied method for integration of the optimization criteria delivers different length and volumes of the optimal twisted rods. The solution of optimization problem for twisted rod is stated in closed form in terms of the higher transcendental functions. In Chap. 8, the authors demonstrate the isoperimetric inequality arising in exactly solvable structural optimization problem of stability under torque load. The periodic Greenhill problem describes the forming of a loop in an elastic bar under torsion. The inequality for infinite rod with periodical cross-section with two types of supports is

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rigorously verified. The optimal shape of the twisted rod is constant along its length, and the optimal shape of cross-section is the equilateral triangle. The technique to demonstrate of isoperimetric inequalities exploits the variational method and the Hölder inequality Chapter 9 considers the stability problems in the context of twisted and compressed rods are demonstrated. The complement for Euler’s compressional buckling and Greenhill torsional buckling problems is the generalized Greenhill’s problem. The generalized Greenhill’s problem studies the forming of a loop in an elastic bar under simultaneous torsion and compression. The authors search the optimal distribution of bending flexure along the axis of the rod. The distribution of material along the length of a twisted and compressed rod is optimized so that the rod must support the maximal moment without spatial buckling, presuming its volume remains constant among all admissible rods. The static Euler’s approach is applicable for simply supported rod (hinged), twisted by the conservative moment and axial compressing force. For determining the optimal solution, the authors directly compare the twisted rods with the different lengths and cross-sections using the invariant factors. The solution of optimization problem for simultaneously twisted and compressed rod is stated in closed form in terms of the higher transcendental functions. The final formulas involve the length of the rod, its volume, and critical torque and axial compression force. Remarkable that in the torsion stability problem, the optimal shape of the rod is roughly parabolic along its length and the optimal shape of cross-section is the equilateral triangle. Chapter 10 deals with another problem of the optimal tapering of a conservative system of the second kind, namely the generalized Pfluger column. The conservative systems of the second kind are truly non-self-adjoint and non-conservative systems in a classical sense, but self-adjoint in a generalized sense. These systems buckle by divergence and possess a generalized conservation theorem and a generalized Rayleigh quotient. Optimization problems for the generalized Pfluger column with pinned-pinned or sliding ends subjected to distributed compressive follower forces are considered. It is shown that by means of a special transformation of independent variables, the problem is reduced to a classical conservative bifurcation problem for the column loaded by the axially distributed load. The optimal solutions for some load distributions are found in closed form. The closed-form solution for optimization and isoperimetric inequalities for stability of the circular ring are illustrated in Chap. 11. The problem of maximizing the critical load causing a loss of stability in an elastic inextensible circular ring under hydrostatic pressure is studied. The undeformed ring has the form of a circle of unit radius, and its thickness, and hence the flexural rigidity, varies along the arc. The thickness distribution must be determined from the condition of the maximum critical load causing the loss of stability, under the condition that the mass of the ring remains constant. It is rigorously shown that of all circular rings of the same mass, the ring of constant thickness can bear the greatest load before losing stability. The optimization problem for an axially compressed strut on an elastic foundation is studied in Chap. 12. The axis of the rod coincides with the boundary of the elastic

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half-space. The direction of the applied forces coincides with the axis of the undeformed rod. The critical values of buckling are equal among all competitive designs of the rods. The closed-form solutions for the simply supported conditions on the ends of the rod are derived. Several models of elastic foundations are implemented, namely Winkler, Pasternak, Filonenko-Borodich, and Reissner models. The critical values of buckling remain equal among all competitive designs of the rods. The solutions are expressed in terms of the higher transcendental functions. The principal results are the closed form and the exact bounds for the masses of the optimal struts. Siegen, Germany

Vladimir Kobelev

About This Book

Concept and Main Ideas of the Manuscript Optimization problems are of importance in many scientific disciplines, for instance in engineering, physics, chemistry and economics. Among them, the problems of structural optimization have recently attracted a lot of attention. A significant number of papers have been published, mainly in the last decades [1–6]. The interest in research in the arena of optimal design has increased significantly due to the rapid development of automotive engineering, aviation and space technology [7, 8], ship building, and manufacturing machinery. Significant reductions in the weight of vehicles and improvements in the mechanical characteristics of structures are achieved on the basis of optimal design [9]. Optimization problems also arise in the design of industrial equipment [10]. Thus, research in this area is of definite applied significance. In structural optimization, the majority of problems could be solved only numerically. A numerical solution means making conjectures of the solution and checking whether the problem is solved well enough to stop. Almost all practically important solutions are obtained with the aid of numerical methods. At present, the majority of studies on structural optimization are carried out using powerful computers. Thus, numeric methods of solution are principally necessary for practical purposes. In this context, a number of works develop computational algorithms designed to solve certain classes of optimal design problems. There exist several acknowledged, commercial codes of structural analysis and optimization. The majority of tasks on engineering optimization cover the numerical methods for optimization problems [11]. Classical books on this subject are [12, 13]. Several established algorithms can be used to compute optimal solutions. The foundations for creating computational algorithms are provided by the theory of optimal control, nonlinear programming, calculus of variations, and chiefly by the numerical optimization methods. In a number of cases, optimal design comes down to solving variational problems with unknown bounds and game problems of optimization, for which no regular research methods are available. Most of the theoretical research in the field of optimization of elastic structures has been carried out using classical

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methods of calculus of variations. In addition to the widespread application of these methods, there are also works based on the use of control theory methods. Methods of optimum control theory, in particular the Pontryagin maximum principle [14], were also used for structure optimization [15]. Methods of control theory for systems with distributed parameters were systematically applied [16, 17]. The nonlinearity of the optimization problems is determined by the nonlinearity of the optimality conditions. The resolution of optimal shapes of elastic bodies comes down to solving nonlinear boundary value problems for systems of differential equations. The nonlinearity of the problems strongly confines the applicability of known analytical methods. Thus, the optimization problems belong to nonlinear problems of mechanics. Finding optimal shapes and structures of elastic bodies using traditional analytical methods faces serious mathematical difficulties. Though, for the modeling of technical systems, analytical solutions repeatedly offer noteworthy advantages. First, we point to the evidence of analytical solutions. Because the closed-form solutions are exhibited as mathematical expressions, they provide an understandable sight of how variables and relations between variables affect the result. Second, their ability: the results expressed with analytical solutions are often more effective than the corresponding numerical models. For example, to compute the solution of an ordinary differential equation for different values of its parametric inputs, it is often faster, more accurate, and more suitable to assess an analytical solution than to integrate it numerically. Third, numerical solutions are sometimes extremely time and resource consuming. The main reason is that sometimes we either don’t have an analytical approach, or that the analytical solution is too slow and instead of computing for hours and getting an exact solution, we rather compute for seconds and get a good approximation. In conclusion, numerical solutions can rarely contribute to the proofs of new ideas. For these reasons, the treatment of material in this book resolves the studied tasks to the solutions in the form of mathematical expressions. The problems of optimal design are also of the theoretical implication. It is academically valuable that the study of new classes of mathematical problems account of various physical factors in optimal design and the development of effective analytical and numerical optimization methods. . Only certain selected problems allow the closed-form analytical solutions. An analytical solution involves bordering the problem in an accurate-understanding form and demonstrating the exact solution. We prefer the analytical method in general because the solution is exact and it allows an instant and exact sensitivity analysis with respect to all involved parameters. In this volume, all solutions are obtained in closed form and the exact inequalities are proved. A remark about the solution methods in this manuscript. This monograph studies optimization for contact and stability problems. The book summarizes the analytical methods for solution of various optimization problems. The principal task is to review the mathematical methods for the structural optimization problems with the control function of one variable. The authors present only exactly solvable structural optimization problems, which require no numerical procedures. Optimal design, as mentioned above, leads to complex nonlinear problems. The notable feature is the following. Generally saying, each closed-form solution

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of the nonlinear problem appears unexpectedly. Several problems in mathematical physics and the processes in applied sciences are descripted with the ordinary differential equations. The optimization methods for such processes require some grades of variability. The variability assumes that the process is controllable and there are control functions. The control functions make the coefficients of the differential equations’ variable. If the coefficients are constant, the solution of the differential equation is occasionally reduced to integrals. If the coefficients are arbitrary, there is commonly no closed-form solutions. Remarkably, in the optimal situations the closed-form solutions appear once again. It is hard to explain, why the boundary value problem or eigenvalue problem for the one or two simultaneous differential equations of the second order with variable coefficients could be solvable. The central effort in this manuscript is to explain the sources of the solvability of the equations of structural optimization. The explanation is the following. The necessary optimally conditions could be considered as the first integral of the static equations of the second order [18]. This first integral is analogous to the first integral of energy, which is familiar in the theory of dynamic systems. The first integral preserves symmetry of the solution and allows to reduce the dimension of the system. The generality destroys the symmetry and solvability, and the optimality restores the lost symmetry. The system of the second order turns into the system of the first order, which could be solvable to integrals. The integrals in their terms reduce occasionally to the special functions. The representation in terms of special functions finalizes the closed-form solutions of the optimization problems. The analogous situation appears in the contact problems. The common contact problems use the apparatus of integral or integral-differential equations with the variable coefficients. These equations with arbitrary variable coefficients possess only minor symmetries. With the minor grad of symmetry, no conservation laws exist. The solution of these equations is possible only with the application of the numerical methods. Fortunately, the optimization conditions cardinally improve the symmetry of the governing equations. The type of the boundary value problem for the integraldifferential equations changes. The integral-differential equation degenerates to the ordinary mixed problem for the integral equation. This closed-form solution for the mixed problem for the integral equation was established in the classic works [19–21]. Consequently, the optimization problem solves analytically in exceptional cases. The correct formulation of design optimization problems represents a considerable effort. Therefore, it is of great importance to identify the “simplest” problems that allow for analytical investigation and the search for exact solutions. In a number of cases, it is possible to carry out these investigations efficiently for onedimensional structural elements (beams, columns, curvilinear rods, and rod systems) and to distinguish essential limitations, to study qualitative features of optimal forms and internal structure of structures, to compare effectiveness of various methods of optimization, and, what is important, to obtain tests necessary for approbation of computational algorithms and approximate techniques designed for two-dimensional problems. Further purpose of the following chapters is also to establish the special types of inequalities that are regarded as “isoperimetric”. This type of inequalities is long known in geometry and physics [22–25]. The variational method is a powerful

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way to prove isoperimetric inequalities for systems, which is described by ordinary differential equations. The theoretical importance of the isoperimetric inequalities rested on the fact that the approved isoperimetric inequality finalizes the optimality of design, leaving no questions about possible further improvements and on the uniqueness of the found solution.

Structure of the Book Part I: Optimization in Contact Problems Part I of the book analyzes the optimal solutions for contact problems and encloses Chaps. 1–4. It is well known that the transfer of forces and pressures from some parts and assemblies of mechanisms and machines to others is due to contact interactions. In recent years, this problem has become particularly important in mechanical engineering and construction. It is contact interaction that determines wear processes in modern mechanical engineering, and, due to the increase in relative velocities, the magnitude of contact pressures is becoming a determining factor influencing the durability of structures. The material failure factor in the contact zone, which is also determined by the contact conditions, is becoming increasingly important. Finally, contact phenomena become particularly important in calculating the strength and fracture resistance of foundation structures. All these circumstances make the development of methods for evaluation and optimization contact interactions and the investigation of specific contact problems particularly important. In mathematical terms, contact problems belong to the class of continuum mechanics problems with mixed boundary conditions and are generally reduced to the necessity of solving integral equations [26–29]. It is often possible to eventually obtain closed-form solutions for significant contact problems and determine the optimal shapes of the elements in contact.

Part II: Optimization in Stability Problems Part II of the book deals with the closed-form solutions of optimization problems for buckling members and encompasses the Chaps. 5–12. Improvements in the strength properties of traditional construction materials and the use of new high-strength composite materials have resulted in the widespread use of lightweight, sleek, and economical thin-walled structures in modern mechanical engineering. In the general cycle of strength calculations, the role of calculations for general stability in thinwalled structures has increased dramatically, because failure of any thin-walled structure is in most cases related to the loss of its overall stability or the stability of some of its individual elements. The book does not include a number of practically important

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problems of optimization of thin-walled elements of structures, for example stability of flat bending form of beams, stability of twisted springs and naturally twisted rods, thin-walled shells, thin-walled rods, etc. Most of such problems could be optimized only with numerical methods. The classical book [30] studies the eigenvalues of inhomogeneous structures analytically. It investigates the unusual closed-form solutions, offering simple solutions for vibrating bars, beams, and plates. The closed solutions in this book not only have applications that allow for the design of tailored structures, but also transcend mechanical engineering to generalize into other fields of engineering. Some of the cited studied stability problems will be investigated below in their optimal cases.

Target Audience of the Book This book was written as a complementary script for the courses “Advanced structural optimization” and “Structural optimization in automotive engineering”, delivered by the author in the University of Siegen, North-Rhine-Westphalia, Germany, since 2001. Each chapter starts with the brief account of the classical results and derivation of the governing equations. The solutions of optimization problems are rigorously derived in the closed form. The readers of the book will be guided over several analytical methods for solution of stability and contact tasks. The author has tried to make the conclusion of each relation clear even to the unexperienced in numerical computations reader. Several basic problems on the optimization have been selected from a large number of the already found results. The specific features of optimization problems have been emphasized. The author hopes that the reader, having become acquainted with the analytical solutions set out in the book, will find it easier and more profound to understand other known optimization problems and, most importantly, will rather learn to set and solve new problems independently. This book is recommended for qualified engineers dealing with development of optimization methods and their application in the industry, graduated from mechanical engineering or engineering computation courses in technical high school, or in other higher engineering schools. The developers of optimization codes will also find a general review for analytical investigation and modeling. The second volume of the book “Fundamentals of Structural Optimization. Shape, Anisotropy, Topology” will cover the following subjects: variation methods for shape optimization, local variations and topological derivatives, methods of tensor transformations for anisotropic medium. The discussion will follow the same principles, namely the application of the analytical methods, the search for the closed-form solutions, and validation of the exact bounds for the principal design objectives. The content of the present book is logically related to the works “Durability of Springs” and “Design and Analysis of Composite Structures for Automotive Applications”. The former book [31] represents the actual tendencies and the state of the art of the spring’ modeling, highlights the mechanics of the elastic elements

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About This Book

made of steel alloys with focus on the metal springs for the automotive industry. The latter book [32] extends the actual book and covers the subject of composite materials. The manuscript investigates the distinctive features of composites, such as their anisotropy, inhomogeneity, and stress-coupling and stacking capabilities.

References 1. Rozvany, G. I. N. (1989) Structural design via optimality criteria: The Prager approach to structural optimization, 101 Philip Drive, Norwell, MA 02061, USA: Kluwer Academic Publishers. 2. Vasiliev, V. V. & Gürdal, Z. (1999) Optimal design: Theory and applications to materials and structures. Technomic Publishing Company, Inc. 3. Banichuk, N. V. (1990) Introduction to optimization of structures, New York Berlin Heidelberg: Springer-Verlag. 4. Arora, J. S. (2017) Introduction to optimum design (4th ed.), Academic Press, ISBN 9780128008065, https://doi.org/10.1016/B978-0-12-800806-5.00025-1. 5. Optimization of large structural systems, Proceedings of the NATO/DFG Advanced Study Institute. Berchtesgaden. Germany. 23 September–4 October 1991, ed. G. I. N. Rozvany. (NATO ASI series. Series E. vol. 231). 6. Papalambros, P. Y. & Wilde, D. J. (2000) Principles of optimal design: Modeling and computation. Cambridge University Press. 7. Grihon, S., Krog, L., & Bassir, D. (2009) Numerical optimization applied to structure sizing at AIRBUS: A multi-step process, International journal for simulation and multidisciplinary design optimization 3, 432–442, https://doi.org/10.1051/ijsmdo/2009020. 8. Zhu, J. -H., Zhang, W.-H., & Xia, L. (2016) Topology optimization in aircraft and aerospace structures design, Archives of Computational Methods in Engineering, 23:595–622, https:// doi.org/10.1007/s11831-015-9151-2. 9. Rao, S. S. (2019) Engineering optimization theory and practice, 5th ed, Print ISBN:9781119454717 | Online ISBN: 978-1-119-45481-6 |https://doi.org/10.1002/978111 9454816. John Wiley & Sons, Inc. 10. Arora, J. S., Ed. (1997) Guide to structural optimization, ASCE Manuals and Reports on Engineering Practice No. 90, Reston, VA 20191: American Society of Civil Engineers. 11. Encyclopedia of Optimization (2009) 2nd ed., Floudas, C. A., Pardalos, P. M. (Eds.), Springer, https://doi.org/10.1007/978-0-387-74759-0, ISBN: 978-0-387-74758-3. 12. Haug, E. J. & Arora, J. A. (1979) Applied optimal design. Chichester: John Wiley & Sons Ltd. 13. Fletcher, R. (2000) Practical methods of optimization, 2nd edn, Chichester: John Wiley & Sons Ltd. 14. Bolibruch, A. A. et al. eds. (2006). Mathematical events of the twentieth century (pp. 85–99) ISBN 3-540-23235-4. Berlin: Springer. 15. Armand, J. -L. (1975) Applications of optimal control theory to structural optimization: Analytical and numerical approach, Optimization in Structural Design, ISBN: 978-3-64280897-5. 16. Lions, J. L. (1968) Contrôle optimal do systèmes gouvernes par des équations aux dérivées partielles. Dunod Gauthier-Villars, Paris. 17. Rodrigues, L. (2018) A unified optimal control approach for maximum endurance and maximum range, IEEE Transactions on Aerospace and Electronic Systems, 54(1), February. https://doi.org/10.1109/TAES.2017.2760538 18. Olver, P. J. (2012) Applications of lie groups to differential equations, Band 107 von Graduate Texts in Mathematics, Springer Science & Business Media, 497 p. ISBN 1468402749, 9781468402742 19. Muskhelishvili, N. I. (2013) Singular integral equations: Boundary problems in the theory of functions and some of their applications to mathematical physics, Springer, ISBN: 9400999968

About This Book

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20. Sneddon, I. N. (2003) Fourier transforms, Dover Books on Mathematics, Dover Publications Inc. 21. Green, A. E. & Zerna, W. (1954). Theoretical elasticity. London: Oxford University Press. 22. Pólya, G. & Szeg˝o, G. (1951). Isoperimetric Inequalities in Mathematical Physics. Annals of Mathematics Studies. Princeton, NJ.: Princeton University Press. ISBN 9780691079882. ISSN 0066-2313. 23. Burago Yu. D. (2001), “Isoperimetric inequality”, Encyclopedia of Mathematics, EMS Press, https://encyclopediaofmath.org/index.php?title=Isoperimetric_inequality 24. Chavel, I. (2001) Isoperimetric inequalities: Differential geometric and analytic perspectives (Cambridge Tracts in Mathematics, Series Number 145) Cambridge. 25. Bandle, C. (1980) Isoperimetric inequalities, Pitman Publishing, ISBN-13: 978-0273084235 26. Muskhelishvili N. I. (1975) Some basic problems of the mathematical theory elasticity. Leyden: Noordhoff. 27. Barber, J. R. (2018) Contact mechanics, Cham, Switzerland: Springer. 28. Popov, V. L. & Heß, M., Willert, E. (2019) Handbook of contact mechanics, Exact solutions of axisymmetric contact problems. Springer-Verlag GmbH Deutschland 29. Alexandrov, V. M. & Chebakov, M. I. (2007) Introduction to contact mechanics, 2nd edn, Publisher “ZVVR”, Moscow, Rostov on Don 30. Elishakoff, I. (2005) Eigenvalues of inhomogeneous structures: Unusual closed-form solutions of semi-inverse problems; ISBN 0-8493-2892-6. Boca Raton: CRC Press 31. Kobelev, V. (2021) Durability of springs. Springer Cham, https://doi.org/10.1007/978-3-03059253-0 32. Kobelev, V. (2019) Design and analysis of composite structures for automotive applications: Chassis and drivetrain, Print ISBN: 9781119513858 | Online ISBN: 9781119513889 | https:// doi.org/10.1002/9781119513889, John Wiley & Sons Lt.

Contents

Part I 1

Optimization in Contact Problems

Optimization and Inverse Solutions for Plane Contacts . . . . . . . . . . . 1.1 Plane Elasticity Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Plane Stress Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Plane Strain Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Airy Stress Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Equations in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . 1.1.5 Boundary Forces on Half-Space . . . . . . . . . . . . . . . . . . . . 1.2 Direct and Inverse Plane Contact Problems . . . . . . . . . . . . . . . . . . . 1.2.1 Equilibrium Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Solutions with Chebyshev Polynomials, or Base Functions of First Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Solutions with Base Functions of Second Type . . . . . . . . 1.2.4 Solutions with Base Functions of Third Type . . . . . . . . . 1.2.5 Integral Formulations of Inverse Problem with Given Normal Stress and Friction Free Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Optimal Shape of Rigid Punch Penetrating into Elastic Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Optimal Shapes of Periodically Arranged Indenters . . . . . . . . . . . 1.3.1 Normal Displacement Under the Action of Periodical Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Optimal Shapes of Periodically Spaced Penetrators . . . . 1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Summary of Principal Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 3 6 7 9 10 12 12 13 16 21

25 28 29 29 34 38 38 38 39

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Optimization for Axisymmetric Contacts, Charged and Conducting Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Axisymmetric Elastostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Boussinesq-Papkovich Solution . . . . . . . . . . . . . . . . . . . . . 2.1.2 Integral Equation of Axisymmetric Contact Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Green and Collins Solution . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Series Solutions of Contact Equation . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Form Factor and Shape Function . . . . . . . . . . . . . . . . . . . . 2.2.2 Direct Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Reaction Force as Function of Form Factor . . . . . . . . . . . 2.2.4 Elastic Energy as Function of Form Factor . . . . . . . . . . . 2.2.5 Dependence of Reaction Force and Elastic Energy Upon the Indenter Radius . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Optimization of Maximal Stress for Fixed Contact Force in Circular Contact Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Solutions with Form Factor p = 0 . . . . . . . . . . . . . . . . . . . 2.3.2 Solutions with Form Factor p = 1/2 . . . . . . . . . . . . . . . . 2.3.3 Solutions with Form Factor p = 1 . . . . . . . . . . . . . . . . . . . 2.3.4 Optimization of Total Force and Contact Stress . . . . . . . 2.3.5 Optimization of Stiffness of Contact Region . . . . . . . . . . 2.4 Optimization of Ring-Shape Indenters . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Stored Elastic Energy, Spring Rate and Contact Force of Concentric Ring-Shaped Indenters . . . . . . . . . . . 2.4.2 Multiple Concentric Indenters . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Optimization of Ring-Shaped Indenters . . . . . . . . . . . . . . 2.5 Electromagnetic Potentials of Disk with Radially-Variable Charge or Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Summary of Principal Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimization of Needle-Shaped Stiffeners . . . . . . . . . . . . . . . . . . . . . . . 3.1 Load Diffusion and Load Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Stringer with Variable Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Compliance and Deformation Energy of Stiffened Elastic Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Minimization of the Maximum Stress . . . . . . . . . . . . . . . . . . . . . . . 3.5 Mass Optimization of a Stiffener . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Method of Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . 3.5.2 Alternative Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 41 47 50 53 53 55 56 56 58 58 58 60 65 67 71 76 76 82 83 89 93 93 95 95 96 99 99 100 104 105 108 108 111

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3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Summary of Principal Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 113 114 115

Optimization for Periodic Arrays of Needle-Shaped Stiffeners . . . . . 4.1 Optimal Load-Transfer for Periodically Arranged of Stiffeners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Equilibrium Equations for Periodic Array of Inclusions or Stiffeners . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Lagrange Multipliers Method for Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Optimal Forms of Periodic Rows of Coaxial Stiffeners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Optimal Forms of Periodically Located, Parallel Stiffeners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Character of Boundary Value Problems for Periodically Located Optimal Stiffeners . . . . . . . . . . . 4.2 Optimization of Double-Periodic Array of Inclusions or Stiffeners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Necessary Optimality Conditions for Chess-Board Lattices of Elastic Stiffeners . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Rectangular and Upright Square Lattice . . . . . . . . . . . . . . 4.2.3 Optimization Problem for Double Periodic Arrays of Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Shapes of Double-Periodic Arrays of Inclusions . . . . . . . 4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Summary of Principal Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part II 5

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117 117 126 128 135 140 141 141 146 149 150 158 159 160 161

Optimization in Stability Problems

Optimization of Compressed Rods with Sturm Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Stability of Axially Compressed Rod . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Efficiency Approach for Optimization in Stability Problems . . . . 5.4 Optimization Problem of Sturm Type . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Auxiliary Solution of Generalized Emden–Fowler Equation . . . . 5.6 Closed-Form Solution of Optimization Problem . . . . . . . . . . . . . . 5.7 Isoperimetric Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Summary of Principal Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165 165 169 174 178 180 182 186 190 190 191 192

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Optimization of Axially Compressed Rods with Mixed Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Optimization of Compressed Rods with Mixed Type Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Optimality Conditions for Mixed Type Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Isoperimetric Inequality for Mixed Type Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Equations of Optimization Problem with Mixed Type Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Shape of Optimal Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Length, Volume and Total Stiffness of Optimal Column . . . . . . . . 6.7 Fundamental Functions for Buckling Moments . . . . . . . . . . . . . . . 6.8 Fundamental Functions for Buckling Displacements . . . . . . . . . . . 6.9 Asymptotic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Isoperimetric Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 Summary of Principal Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.13 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195 195 199 200 203 207 212 214 215 217 221 224 225 226 228

Stability Optimization of Twisted Rods . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Isoperimetric Inequality for Twisted Rod with Arbitrary Convex, Simply-Connected Cross-Section . . . . . . . . . . . . . . . . . . . 7.2 Optimization Problem and Isoperimetric Inequality for Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Closed-Form Solution of Optimization Problem . . . . . . . . . . . . . . 7.4 Effectiveness of Optimal Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Summary of Principal Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Periodic Greenhill’s Problem for Twisted Elastic Rod . . . . . . . . . . . . . 8.1 Periodic Greenhill’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Periodic Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Stability of Twisted, Periodically Supported Rod with Varying Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Optimization Problem for Periodically Supported Twisted Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Isoperimetric Inequality for Periodically Supported Twisted Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Summary of Principal Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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232 236 238 242 247 248 250 250

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261

Optimization of Concurrently Compressed and Torqued Rod . . . . . 9.1 Twisted and Axially Compressed Shafts with Convex Simply-Connected Cross-Sections . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Optimization Problem and Isoperimetric Inequality for Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Closed-Form Solution of Optimization Task . . . . . . . . . . . . . . . . . . 9.4 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Optimal Rod for Greenhill Torsion . . . . . . . . . . . . . . . . . . 9.4.2 Optimal Strut for Euler Compression . . . . . . . . . . . . . . . . 9.5 Arbitrary Relation Compression to Torque . . . . . . . . . . . . . . . . . . . 9.5.1 Optimal Rod for Shape Exponent α = 1 . . . . . . . . . . . . . 9.5.2 Optimal Rod for Shape Exponent α = 2 . . . . . . . . . . . . . 9.5.3 Optimal Rod for Shape Exponent α = 3 . . . . . . . . . . . . . 9.6 Mass Comparisons of Optimal Shafts to Constant-Cross-Section Shafts . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Summary of Principal Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Optimization for Buckling of Conservative Systems of Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Pfluger Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Stability Optimization for “Pfluger Column” . . . . . . . . . . . . . . . . . 10.3 Auxiliary Conservative System of First Kind: “Generalized Euler Column” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Isoperimetric Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Optimal Shapes of “Generalized Pfluger Columns” . . . . . . . . . . . . 10.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Summary of Principal Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Structural Optimization for Stability of Circular Rings . . . . . . . . . . . 11.1 Stability of Circular Rings and Arches . . . . . . . . . . . . . . . . . . . . . . . 11.2 Optimization for Stability of Circular Rings and Arches . . . . . . . . 11.3 Basic Equations and Formulation of Optimization . . . . . . . . . . . . . 11.4 Transforming of Variational Formulation and General Properties of Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . 11.5 The Proof of Isoperimetric Inequality . . . . . . . . . . . . . . . . . . . . . . . 11.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Summary of Principal Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

262 263 266 269 269 270 270 270 272 275 278 281 282 282 283 285 285 289 290 291 293 298 300 300 301 303 303 307 308 309 311 313 314 314 315

xxvi

Contents

12 Stability Optimization of Axially Compressed Rods on Elastic Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Stability for Axially Compressed Rods on Elastic Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Deformation of an Infinite Elastic Layer . . . . . . . . . . . . . . . . . . . . . 12.2.1 Elastic Layer Under the Surface Load . . . . . . . . . . . . . . . . 12.2.2 Elastic Layer of Intermediate Thickness . . . . . . . . . . . . . . 12.2.3 Limit Case of Half-Infinite Elastic Medium . . . . . . . . . . . 12.2.4 Limit Case of Thin Elastic Layer . . . . . . . . . . . . . . . . . . . . 12.3 Stability of Infinitely Long, Homogeneous Struts . . . . . . . . . . . . . 12.3.1 Stability of Homogeneous Infinite Strut on Winkler Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 Stability of Homogeneous Strut on Semi-infinite Elastic Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.3 Stability of Homogeneous Strut on Elastic Layer . . . . . . 12.4 Optimal Strut on Elastic Foundations . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Formulation of Optimization Problem . . . . . . . . . . . . . . . 12.4.2 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.3 Optimal Strut on Winkler Foundation . . . . . . . . . . . . . . . . 12.4.4 Optimal Strut on Reissner Foundation . . . . . . . . . . . . . . . 12.4.5 Optimal Strut on Half-Infinite Elastic Space . . . . . . . . . . 12.5 Optimal Strut on an Elastic Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.1 Optimization of Compressed Strut on Elastic Layer . . . . 12.6 Appendix. Direct Calculation of Hilbert Integrals . . . . . . . . . . . . . 12.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8 Summary of Principal Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

317 317 320 320 322 326 327 328 328 332 333 335 335 336 337 338 338 340 340 343 344 345 346 346

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

Part I

Optimization in Contact Problems

Chapter 1

Optimization and Inverse Solutions for Plane Contacts

Abstract In this chapter the authors deliberate the optimization of the rigid punches on the elastic foundations. The contact problem for an indentation of a nondeformable punch with an unknown base surface into a half-plane is considered. The plane stress and plane strain problems of isotropic elasticity are briefly reviewed. No friction occurs for the rigid punches of arbitrary profiles indenting into the surface of an isotropic elastic half-plane. The bottom shape of the rigid punch is primarily unknown and represents the examined function of the optimization problems. The elastic energy of the elastic body and the maximal pressure on the contact surface are considered as the principal functionals of the optimization problem. Optimal solutions are achieved for a single indenter and for periodically spaced equal indenters. The closed form solutions of optimization problems are completed in terms of high transcendental functions. Keywords Shape optimization · Contact problem · Plane elasticity

1.1 Plane Elasticity Problems 1.1.1 Plane Stress Problem The theory of contact tasks is widely used in mechanical engineering. It is well known that the transmission of forces in machines involves the contact of parts. In most cases, the latter can be considered as elastic bodies. The methods developed in the theory of contact problems allow us to determine the distribution of pressures at contact points. This allows us to address the important question of the locations of the concentrations. More recently, contact stiffness issues have been developed where the deformation of irregularities on the surface of an elastic body must be considered.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Kobelev, Fundamentals of Structural Optimization, Mathematical Engineering, https://doi.org/10.1007/978-3-031-34632-3_1

3

4

1 Optimization and Inverse Solutions for Plane Contacts

With the advent of structural materials containing polymers, contact problems have become very relevant for viscoelastic bodies. It also allows to obtain results for the important problem that important problems in engineering, such as rolling friction. The determination of the stresses that occur under foundations and footings, even when consolidation of the soil occurs, also leads to contact problems. The mechanics of contact interactions of solid deformable bodies is currently a broad and actively progressing field of continuum mechanics. It remains at the focus of attention of researchers. This is attributed to the fact that majority of mechanisms and structures are made up of interacting parts and the distribution of contact forces between these parts is not known in advance and can only be found as a result of solving specific problems, which are called contact problems. The derivation of the law of the variations of the contact pressure on the contact area then allows us to formulate the boundary conditions of the stresses on the surfaces of the bodies and to deal with simpler problems of determining the stress–strain state within of the stress–strain state inside the interacting bodies. The characteristic feature of the contact problem is that mathematically they are mostly problems of mixed boundary conditions, which, as a rule, reduce to integral equations of the first kind and require special methods solution methods. As we can see further, the optimization tasks in the contact mechanics differ the type of the mixed formulations. Remarkably, that the optimization problems could be solved in closed form. The equations of the plane elasticity for a plate of an isotropic material will be briefly reported [1, 2]. The thin plate lies in the plane (x, y) and is subjected to loads in its plane. The top and the bottom surfaces of the plate is traction-free. The edge of the plate may have two kinds of the boundary conditions, namely displacements prescribed. Alternatively, traction is prescribed: σx x n x + τx y n y = tx , τx y n x + σ yy n y = t y .

(1.1)

In Eq. (1.1) tx and t y are components of the traction vector prescribed on the edge of the plate, and n x and n y are the components of the unit vector normal to the edge of the plate. The above two equations provide two conditions for the components of the stress tensor along the edge. If on the whole boundary of the elastic body the tractions are prescribed, the traction problem [3], or the first boundary value problem (Muskhelishvili [4]). Otherwise, if on the whole boundary of the elastic body the displacements are given, the displacement problem, or the second boundary value problem). Occasionally the tractions are given on one part of the boundary, and the displacements on the complementary part. The mixed problem of elasticity consists in finding the corresponding elastic state. Semi-inverse method. In the plane elasticity problems is assumed that the stress field in the plate only has nonzero components in its plane: σx x , σ yy , τx y , and the components out of the plane vanish: σzz = τx z = τ yz = 0. It is conjectured, that

1.1 Plane Elasticity Problems

5

in-plane stress components vary with x and y, but are independent of z. That is, the stress field in the plate is described by three functions: σx x (x, y), σ yy (x, y), τx y (x, y). Equilibrium equations. The three equilibrium equations to two equations in plane elasticity are: ∂τx y ∂σx x + = 0, ∂x ∂y ∂τx y ∂σ yy + = 0. ∂x ∂y

(1.2)

Stress–strain relations. Given the guessed stress field, the 6 components of the strain field express: ) σ yy σ yy σx x σx x ν( −ν , ε yy = −ν , εzz = − σx x + σ yy , E E E E E 2(1 + ν) = τx y , γx z = γ yz = 0. E

εx x = γx y

(1.3)

For the isotropic elastic material, the elasticity constants are: E

Young modulus

[Pa]

Plane strain modulus

[Pa]

E E = 1−ν 2 E −1 ν = 2μ ν ν = 1−ν 2μν λ = 1−2ν

First Lamé parameter

[Pa]

μ=G=

Shear modulus, or second Lamé parameter

[Pa]

Poisson’s ratio Plane strain Poisson’s ratio

E 2(1+ν)

Strain–displacement relations. Further we recall the 6 strain–displacement relations: εx x = γx z =

∂u ∂v ∂w , ε yy = , εzz = , ∂x ∂y ∂z

(1.4)

∂w ∂w ∂v ∂v ∂u ∂u + , γ yz = + , γx y = + . ∂z ∂y ∂z ∂y ∂y ∂x

The independence of stresses upon the normal to the plane was assumed above. Accordingly, the in-plane displacements u and v vary only with x and y, but not with z. Consequently, together with the conditions γx z = γ yz = 0, we find that

6

1 Optimization and Inverse Solutions for Plane Contacts

∂w ∂w = = 0. ∂x ∂y Consequently, w is independent of x and y, and can only be a function of z. Set of equations for plane elasticity problems. The following set of equations are self-consistent for the plane stress problem: ∂σx x ∂x

εx x ε yy γx y

∂τ

∂τ

∂σ

+ ∂ yx y = 0, ∂ xx y + ∂ yyy = 0, σ = σEx x − ν Eyy , εx x = ∂∂ux , σ yy σx x = E − ν E , ε yy = ∂v , ∂y ∂u 1 γx y = ∂ y + ∂∂vx . = G τx y ,

(1.5)

Equation (1.5) display 8 equations for 8 functions.

1.1.2 Plane Strain Problem Alternative plane problem deliberates an infinitely long cylinder with the axis in the z-direction, and a cross section in the (x, y) plane. The loading of the cylinder is invariant along the z-direction. Consequently, the displacement field takes the form: u = u(x, y), v = v(x, y), w = 0. From the strain displacement relations, we find that only the three in-plane strains are nonzero: εx x (x, y), ε yy (x, y), γx y (x, y). The three out-of-plane strains components vanish: εzz = γx z = γ yz = 0. Since γx z = γ yz( = 0, the stress–strain ) relations imply that τx z = τ yz = 0. From εzz = 0 and εzz = σzz − νσx x − νσ yy /E, we obtain for the stress component ) ( σzz = ν σx x + σ yy . Consequently, the remained three stress–strain relations are:

1.1 Plane Elasticity Problems

7

) ( ) 1 − ν2 ν 1( σ yy , σx x − νσ yy − νσzz = σx x − E E 1−ν ) 2( ( ) 1 ν 1−ν σ yy − σx x , σ yy − νσx x − νσzz = = E E 1−ν 1 = τx y . G

εx x = ε yy γx y

(1.6)

The three stress–strain relations (1.6) are equivalent to those under the plane stress conditions, provided the following substitutions: E → E, ν → ν.

1.1.3 Airy Stress Function If two functions f (x, y) and g(x, y) satisfy ∂g ∂f = , ∂x ∂y there exists a function α(x, y), such that f =

∂α ∂α ,g = . ∂y ∂x

We now apply this preposition to the equilibrium equations. From the first equilibrium equation ∂τx y ∂σx x + = 0, ∂x ∂y one can appreciate that there exists a function α(x, y), such that σx x =

∂α ∂α , τx y = − . ∂y ∂x

Correspondingly, from the second equilibrium equation ∂σ yy ∂τx y + = 0, ∂x ∂y there exists a function β(x, y), such that

8

1 Optimization and Inverse Solutions for Plane Contacts

σ yy =

∂β ∂β , τx y = − . ∂x ∂y

Finally, it follows from ∂β ∂α = , ∂x ∂y that there exists a function ϕ(x, y), such that α=

∂ϕ ∂ϕ , β= . ∂y ∂x

The function ϕ(x, y) is known as the Airy stress function. The three components of the stress field can now be represented by the stress function: σx x =

∂ 2ϕ ∂ 2ϕ ∂ 2ϕ , σ = , τ = − . yy x y ∂ y2 ∂ y2 ∂ y∂ x

(1.7)

Using the stress–strain relations (1.6), we can also express the three components of strain field in terms of the Airy stress function: εx x =

1 E

(

) ) ( ∂ 2ϕ ∂ 2ϕ 1 ∂ 2ϕ ∂ 2ϕ 2(1 + ν) ∂ 2 ϕ , ε , γx y = − . − ν = − ν yy 2 2 2 2 ∂y ∂x E ∂x ∂y E ∂ x∂ y

Compatibility equation. Recall the strain–displacement relations: εzz =

∂w ∂w ∂w ∂u ∂v , γx z = + , γ yz = + . ∂z ∂z ∂x ∂z ∂y

Elimination of the two displacements in the three strain–displacement (1.6) relations leads to the compatibility equation: ∂ 2 ε yy ∂ 2 γx y ∂ 2 εx x . + = ∂ y2 ∂x2 ∂ x∂ y

(1.8)

Biharmonic equation. Inserting the expressions of the strains in terms of Airy stress function ϕ(x, y) into the compatibility equation, and we obtain that ϕ obeys the bi-harmonic equation in Cartesian coordinates: ΔΔϕ = 0, Δ =

∂2 ∂2 + 2. 2 ∂x ∂y

(1.9)

For a plane problem with traction-prescribe boundary conditions, both the governing equation and the boundary conditions can be expressed in terms of ϕ. All these equations are independent of elastic constants. Consequently, the stress field in such

1.1 Plane Elasticity Problems

9

a boundary value problem is independent of the elastic constants. This statement is correct for boundary value problems in simply connected regions.

1.1.4 Equations in Polar Coordinates Transformation of stress components due to change of coordinates. A material particle is in a state of plane stress. If we represent the material particle by a square in the (x, y) coordinates, the components of the stress state are σx x , σ yy , τx y . If we represent the same material particle under the same state of stress by a square in the (r, θ ) coordinates, the components of the stress state are: σx x − σ yy σx x + σ yy + cos 2θ + τx y sin 2θ, 2 2 σx x − σ yy σx x + σ yy − cos 2θ − τx y sin 2θ, = 2 2 σx x − σ yy sin 2θ + τx y cos 2θ. =− 2

σrr = σθθ τr θ

(1.10)

The Airy stress function ϕ(r, θ ) is a function of the polar coordinates. The stresses are expressed in terms of the Airy stress function: σrr =

( ) 1 ∂ϕ ∂ 2ϕ ∂ ∂ϕ ∂ 2ϕ , σ . + = , τ = − θθ rθ r 2 ∂θ 2 r ∂r ∂r 2 ∂r r ∂θ

(1.11)

The bi-harmonic Eq. (1.9) in polar coordinates reads: ΔΔϕ = 0, Δ =

∂2 ∂ ∂2 + + . ∂r 2 r ∂r r 2 ∂θ 2

(1.12)

The stress–strain relations in polar coordinates are similar to those in the rectangular coordinates: εrr =

σθθ σrr σθθ 2(1 + ν) σrr −ν , εθθ = −ν , γr θ = τr θ . E E E E E

(1.13)

The strain–displacement relations are: εrr =

∂u θ ∂u θ uθ ∂u r ur ∂u r , εθθ = + , γr θ = + − . ∂r r r ∂θ r ∂θ ∂r r

(1.14)

10

1 Optimization and Inverse Solutions for Plane Contacts

1.1.5 Boundary Forces on Half-Space A half space of an elastic material is subject to a force on its surface. The half space lies in z > 0. Let S be the local force per unit length, which points in the direction of z normal to the straight boundary. The local force T points parallel to the boundary in the direction of x. The stress function takes the form: rθ (S sin θ + T cos θ ), π √ x z sin θ = , cos θ = . r = x 2 + z 2 . r r

ϕ(r, θ ) = −

(1.15)

The displacement field in polar coordinates reads: S [(K − 1)θ sin θ − cos θ + (1 + K) cos θ ln r ] 2π T − [(K − 1)θ cos θ + sin θ − (1 + K) sin θ ln r ] + C1 , 2π

2μu r = −

S [(K − 1)θ cos θ − sin θ − (1 + K) sin θ ln r ] 2π T + [−(K − 1)θ sin θ − cos θ − (1 + K) cos θ ln r ] + C2 . 2π

(1.16)

2μu θ =

(1.17)

The dimensionless parameter in Eqs. (1.16) and (1.17) is1 : ( K=

3 − 4ν, for plane strain, 3−ν , for plane stress. 1−ν

The stress components in the Cartesian coordinates are: ) 2 ( S sin2 θ cos2 θ − T sin3 θ cos θ , πz ) 2 ( S cos4 θ − T sin θ cos3 θ , =− πz ) 2 ( S sin θ cos3 θ − T sin2 θ cos2 θ . =− πz

σx x = − σx x τx z

(1.18)

In Cartesian coordinates, the displacements on the surface θ = ±π/2 result from (1.17):

1

Another parameter is also frequently used in textbooks [1]: E G 4G = 1−ν = 1+κ . o = 2 1−ν ( 2)

1.1 Plane Elasticity Problems

11

1+K Cu K−1 sgn x + T ln r + , 8μ 4π μ 8μ K−1 1+K Cv w = −T sgn x − S ln r + , 8μ 4π μ 8μ ( 1, x > 0 sgn x = . −1, x < 0

u = −S

(1.19)

It is suitable to handle the displacement derivatives ∂u/∂ x and ∂w/∂ x. In the expressions for displacement derivatives, the arbitrary constants Cu and Cv vanish. Consider the element dt of the line on the surface of the half-space. We calculate the actions of the pressure and shear load on the displacement derivatives ∂u/∂ x and ∂w/∂ x. The partial influences of the pressure and shear tractions, which act on the line element dt, are S(t)dt and T (t)dt correspondingly. The loads are applied over the certain segment Gl = [−l, l] and the rest of the surface is load-free. The total effects of distributed pressure and shear tractions, are the sums of partial actions over the segment Gl . For each point of the segment x ⊂ Gl , the total displacement derivatives are: K−1 1+K ∂u = S(x) + ∂x 4μ 4π μ 1+K ∂w = ∂x 4π μ

∫ Gl

∫ Gl

T (t) dt, x −t

K−1 S(t) dt − T (x). x −t 4μ

(1.20)

The multiplier in Eq. (1.20) will be symbolized as:

k = K4π+μ1 l. The singular integrals are interpreted hereafter as the Cauchy principal value [5, 6]. The integrals (1.20) with Cauchy kernel provide the basis for computing the displacements from the given surface tractions. In reverse, these equations must be resolved for S(x), T (x) for x ∈ Gl . calculating of surface tractions from the given the displacements. The corresponding integral equations of the contact problem follow from (1.20) along with both the boundary and loading conditions [7]. Various loading conditions were considered, such as frictionless contact, sliding contact, complete stick, and partial slip. Solutions for both elastically similar and dissimilar materials of the coupling surfaces were evaluated assuming Coulomb friction. Although a theoretical resolution is difficult

12

1 Optimization and Inverse Solutions for Plane Contacts

due to the general conditions, essential practical understanding can be gained from certain comprehensible borderline cases.

1.2 Direct and Inverse Plane Contact Problems 1.2.1 Equilibrium Equations The indentation of a punch with an a-priory unknown base into an elastic isotropic half-plane is considered in this chapter. As deliberated in the previous section, the rigid body is pressed against the elastic half-space [8]. The indentation of the contact body is normal to boundary of the half-space with no friction occurring. The slump and the rigid rotation of one body relative to another are defined. The length of the contact is fixed; thus, the contact region extends along the known interval Gl = [−l, l]. We set for briefness of the formulas l = 1. The specified local displacement is −W(x). The shear stress vanishes for the friction free contact, T (t) = 0. The normal reaction stress on the contact region S(x) results from the summation of partial loads using Eq. (1.20). The external actions on an unrestrained body reduce to the principal vector Ftotal and moment Mtotal pro unit of body width: ∫1 Ftotal =

∫1 S(y)dy, Mtotal =

−1

S(y)y dy. −1

The summation leads to the equation with the logarithmic kernel reads:

k

∫1

−1

ln

1 S(y)dy = W(x) for |y − x|

x ∈ G1 = [−1, 1].

(1.21)

The eventual rotation of the contact body occurs about the axis, which locates in the half-plane boundary and normal to X axis. The normal displacement W(x), x ∈ G1 is the sum of local elastic indentation w(x) the rigid body translation C0 and inclination C1 x:

w

W(x) = (x) + C1 x + C0 . The unknown constants C0 , C1 could be determined from the symmetry considerations or from the static conditions. To avoid the rigid-body movement, the derivative of the elastic indentation (x) is usually taken as an unknown variable. Differentiation of the Eq. (1.21) reduces it to the singular integral equation with Cauchy kernel [9] :

w

1.2 Direct and Inverse Plane Contact Problems

k

∫1

−1

w

S(y) d (x) dy = + C1 , y−x dx

13

w

d (x) = Θ(x) for dx

x ∈ G1 .

(1.22)

The function Θ(x) is the local inclination angle of the distorted upper surface of the half-space. In the direct contact problem, the shape of the punch and consequently indentation into the elastic half-space is given. The stress in the contact region S(x) have to be resolved from the solution of Eq. (1.22). If the rigid body translation and inclination vanish, both functions coincide: W(x) = (x). This case in relevant for the optimization and will be considered hereafter. Contrarily, if the normal stresses S in the contact region are given, the normal displacements follow from the direct integration in Eq. (1.22). This problem is the inversion of the common contact problem. The normal stress S in one certain state is assumed as the design load, which is known a priori. The penetration, which corresponds to the unknown shape of the punch W, follows from the solution of the contact problem. This formulation is referred to as the “inverse contact problem”. There are some other different formulations of the inverse problems, which require specific notations for the design loads. We adopt the next notations for the succeeding conversions. The whole numbers are the numbers without fractions and it is a collection of positive integers and zero [10]. It is signified by the symbol W . The even whole numbers are frequently used below. We symbolizes the even whole numbers, which include even positive integers along with 0, namely m = 0, 2, 4, . . .:

w

def

We ={m|m = 2k, k is a whole number, k ∈ W }.

1.2.2 Solutions with Chebyshev Polynomials, or Base Functions of First Type Chebyshev polynomials For the beginning, we briefly summarize the standard method of solution of the contact problems, which is based on the first type of base functions. As the first type of base functions serve Chebyshev polynomials. The standard functions Tn (x), Pn (x) are defined as the real and imaginary parts of the complex function O(x):

14

1 Optimization and Inverse Solutions for Plane Contacts

)n ( √ def Tn (x) + i Pn (x) = O(x), O(x) ≡ x + i 1 − x 2 , n ∈ W, T0 (x) = 1, T1 (x) = x, T2 (x) = 2x 2 − 1. x = cos θ, Tn (cos θ ) = cos(nθ ),

−1 ≤ x ≤ 1. P0 (x) = √ 0, P1 (x) = 1√− x 2 , P2 (x) = 2x 1 − x 2 , 0 ≤ θ ≤ π. Pn (cos θ ) = sin(nθ ),

(1.23)

The functions Tn are Chebyshev polynomials of the first kind [11]. The Chebyshev polynomials of the second kind Un (x) are commonly used instead of the functions Pn (x): def Pn+1 (x) √ , 1−x 2

Un (x) =

Un−1 (x) = U0 (x) = 1,

1 dTn (x) , n dx

Un (cos θ ) ≡ Tn (x) = n ·

sin((n+1)θ ) , sin(θ ) ∫x

−1

Un−1 (y)dy,

(1.24)

U1 (x) = 2x, U2 (x) = 4x 2 − 1.

The orthogonality conditions for Chebyshev polynomials read: ∫1

∫1 Tn (x)Tm (x) Pn (x)Pm (x) dx = dx = An,m , √ √ 1 − x2 1 − x2 −1 −1 ⎧ ⎨ π, m = n = 0, def An,m = π/2, m = n /= 0, ⎩ 0 n /= m.

(1.25)

The following singular integrals of the Chebyshev polynomials express in closed form for n ≥ 0: 1 π

∫π 0

sin(nθ ) cos(nϕ)dϕ = , cos(ϕ) − cos(θ ) sin(θ )

1 π

∫π 0

sin(nϕ)dϕ = − cos(nθ ). (1.26) cos(ϕ) − cos(θ )

With Eq. (1.24), the singular integrals (1.26) redraft to: 1 π

∫1 −1

Tn (y)dy √ = Un−1 (x), (y − x) 1 − y 2

1 π

∫1 √ −1

1 − y 2 Un−1 (y)dy = −Tn (x). y−x (1.27)

Additionally, the following formula is valid for n = 0:

1.2 Direct and Inverse Plane Contact Problems

1 π

∫1 −1

15

T0 (y)dy √ = 0. (y − x) 1 − y 2

(1.28)

The logarithm function could be represented as the infinite sum: ln

∞ ∑ Tn (x) · Tn (ξ ) 1 = ln(2T0 (x)T0 (ξ )) + 2 . |x − ξ | n n=1

(1.29)

The equality (1.29) paws the way for the solution of the integral Eq. (1.23). Multiplication of (1.29) by Tn (y) √ 1 − y2 and integration over the interval G, leads to the expression: ∫1 −1

(

Tn (y) 1 √ ln dy = 2 |x − y| 1−y

T0 (x)π ln 2 ≡ π ln 2, for n = 0, π T for n /= 0. n n (x),

(1.30)

As follows from (1.30), the Chebyshev polynomials of the first kind are fundamental functions of an integral equation with a logarithmic kernel. The inverse formula is also valid: ∫1 −1

Pn (y) 1 √ ln dy ≡ 2 |x − y| 1−y =

∫1 Un (y) ln −1

1 dy |x − y|

( π ln 2 − 1 T (x), for n = 1, · Tn−1 (x) 2 2Tn+1 (x) − n+1 , for n > 1. 2 n−1

(1.31)

Contact problem with Chebyshev polynomials The common method for solving singular integral equations of the first kind is based on the application of Eqs. (1.27) and (1.28). With the methods of Fourier analysis, continuous functions transform into convergent series of Chebyshev polynomials. S(y) =

∞ ∑ n=0

∞

Tn (y) pn sn , sn (y) = √ , 1 − y2

∑ dW vn u n , u n (x) = = dx n=0

(

0, for n = 0, Un−1 (x), for n ≥ 1;

(1.32)

(1.33)

16

1 Optimization and Inverse Solutions for Plane Contacts

f

(

∫1 =−

n

sn (y)dy =

−1

m

(

∫1 =−

n

−π, for n = 0, 0, for n > 0;

ysn (y)dy =

−1

−π/2, for n = 1, 0 for n /= 1.

(1.34)

For the solution, the infinite sums (1.32) and (1.33) are substituted into Eq. (1.23):

k

∫1

−1

1 π π

dW S(y) dy = + C1 ⇒ y−x dx

∫1 ∑ ∞ −1 n=0 ∞ ∑

k

k

∫1 ∑ ∞

−1 n=0

∞

pn

∑ sn dy = vn u n + C1 , y−x n=0

∞

∑ pn Tn (y) dy √ = u n pn , 1 − y2 y − x n=0

u n pn =

∞ ∑

n=0

vn u n + C1 ≡ (v1 + C1 )u 1 +

n=0

∞ ∑

vn u n .

(1.35)

n=1

Note, that u n = U0 (x) = 1. The comparison of the coefficients delivers the solution of the contact problem:

k

k

π p1 = (v1 + C1 ), π pn = vn for n > 1.

(1.36)

Inappropriately, all base functions sn are singular on both ends of the interval G1 . Thus, for an arbitrary sequence of the coefficients pn , the function S also contain singularity on the ends. With this disadvantage, the common method, which uses Chebyshev polynomials as base functions, is not ideally suited for the solutions of optimization problems.

1.2.3 Solutions with Base Functions of Second Type Second type of base functions To avoid the singularities on the ends of the interval G1 , we introduce other types of base functions. The base functions of the second type could be defined analogously to Eq. (1.21):

1.2 Direct and Inverse Plane Contact Problems

17

def √ ~ ~ Tn (x) + i√ Pn (x) = 1 − x 2 O(x) for n ∈ W, √ −1 ≤ x ≤ 1, ( ) ~ ~ Tn (x) = 1 − x 2 Tn (x), Pn (x) = 1 − x 2 Pn (x) ≡ 1 − x 2 Un−1 (x).

(1.37) Tn are not the polynomials. Note, that the functions Oppositely to Tn , the functions ~ ~ Pn are polynomials. The functions ~ Tn and ~ Pn vanish on both ends of the interval Un is defined thru the integration of the function ~ T n: G1 = [−1, 1]. The function ~ def 1 ~ Un−1 (x) = π

∫1 −1

∫1 ~ Tn (y)dy Tn (y)dy 1 √ . ≡ 2 π y−x (y − x) 1 − y

(1.38)

−1

Tn and ~ Pn is The major reason of the introduction of the second type functions ~ the following. The integrand (1.38) in the case n = 0 is the product of the constant function T0 (y) = 1 and the singular weight functions (y − x)−1 . All other integrands are products of Chebyshev polynomials Tn (y) with the weight functions. The only singularity of the integrand comes also from the weight function. This feature allows to find the optimization solutions for the contact problems. Un−1 (x) and ~ Pn (x), which looks However, there is no trivial relation between ~ Un (x) contain the similar to the shown in Eq. (1.22). In its place, the functions ~ products of polynomials and logarithms: ( ln

) 1−x . 1+x

Tn and ~ Un−1 , it is possible to find the Based on the modified Chebyshev functions ~ closed form solutions for the inverse and optimization problems. The integrals for an arbitrary integer n ≥ 2 the function Un (x) read: ) ( 1−x ~ , π Un−1 (x) = H(n) + G(n, x) + Tn (x) · ln 1+x H(n) =

G(n, x) =

2 B(x) · ~ Tn (x), π2

⎧ [n/2] ∑ T2l−1 (x) ⎪ ⎪ , ⎨4 n−2l+1

for even n,

[n/2] ∑ ⎪ ⎪ ⎩ n2 + 4

for odd n.

l=1

l=1

(1.39) (1.40)

(1.41) T2l (x) , n−2l

In Eq. (1.41), the integer part of a real number corresponds to rounding towards zero. Thus, the integer part of a positive number n/2 is its floor: [n/2]. The functions B(x) in (1.40) are the real parts of the sum of complex dilogarithms Q(x) [12]:

18

1 Optimization and Inverse Solutions for Plane Contacts

B(x) = Re[Q(x)], Q(x) = −Li 2 (α1 ) + Li 2 (α2 ) − Li 2 (α3 ) + Li 2 (α4 ), ∫0

∞

∑ zn ln(1 − t) , dt ≡ t n2 n=1 z √ √ √ √ 1−x 1−x 1+x 1−x , α2 = −i , α3 = −i , α4 = i . (1.42) α1 = i 1+x 1+x 1−x 1+x def

Li 2 (z) =

The dilogarithms Li 2 (x) are displayed in NIST Handbook of Mathematical Functions, 2010, 25.12(i) [13]. Notably, that the function Q(x) is pure imaginary on the interval G1 . To show this, one differentiates the function Q(x). Its derivative is an imarinary function: πi dQ(x) = . dx 1 − x2

(1.43)

The integration of the differential Eq. (1.43) with the initial condition Q(0) = 0 reads: Q(x) = π i arctanh(x). The function Q is a pure imaginary. Its real part ReQ(x) disappears. Consequently, the functions B(x) and H(n) in Eq. (1.42) vanish: Re Q(x) ≡ B(x) = 0, H(n) = 0.

(1.44)

Finally, tor an arbitrary integer n ≥ 2 the function Un (x) reads: ) ( 1−x ~ . π Un−1 (x) = G(n, x) + Tn (x) · ln 1+x

(1.45)

The above expression could be proved by the method of induction. Some functions ~ Un are exposed below: ) 1−x , 1+x ) ( 1−x , π~ U0 (x) = 2 + x ln 1+x ) ( ) ( 1−x , π~ U1 (x) = 4x + 2x 2 − 1 ln 1+x ) ( ( ) 1−x 10 + x 4x 2 − 3 ln . π~ U2 (x) = 8x 2 − 3 1+x

π~ U−1 (x) = ln

(

(1.46)

1.2 Direct and Inverse Plane Contact Problems

19

R n (x) is defined analogously to the differential relation between the The function ~ Chebyshev polynomials of the first and second kinds in Eq. (1.22): x

def ~ Rn (x) = ∫ ~ Un−1 (y)dy, −1

(1 − x)(x−1) + 2 ln 2, (1 + x)(x+1) 1 − x2 x2 − 1 ln(1 − x) + ln(1 + x) + 1 + x, π~ R1 (x) = 2 2 −2x 3 + 3x + 1 2x 3 − 3x + 1 ln(1 − x) + ln(1 + x) π~ R2 (x) = 3 ( 23 ) 4 x − 1 − 2 ln 2 + , 3 −2x 4 + 3x 2 + 1 2x 4 − 3x 2 + 1 ln(1 − x) + ln(1 + x) π~ R3 (x) = 2 2 6x 3 − 7x − 1 + , (1.47) 3

π~ R0 (x) = ln

The orthogonality conditions for the base functions of the second type read: ∫1 ~ ∫1 ~ Pn (x)~ Tn (x)~ Tm (x) Pm (x) ( )3/2 dx = ( )3/2 dx = An,m , 1 − x2 1 − x2 −1 −1 ⎧ for m = n = 0, ⎨ π, An,m ≡ π/2, for m = n /= 0 ⎩ 0 for n /= m.

(1.48)

The inversion of the polynomials could be stated as: 1 π

∫1 −1

√ 1 − y 2 dy ~ = −~ Tn (x) − dn , Un−1 (y) y−x ( dn =

0, 2 , π (n−1)(n+1)

(1.49)

for odd n, for even n.

For the following, we need the solution of the simultaneous equation for an arbitrary N > 1, x ∈ G1 : N ∑ n=1

T n (x) = f (x). cn~

(1.50)

20

1 Optimization and Inverse Solutions for Plane Contacts

For solution, the Eq. (1.50) are multiplied by: )−3/2 ( ~ T m (x). 1 − x2 The succeeding integrations of the right side of Eq. (1.50) over the interval G result the coefficients: f˜m =

∫1 −1

f (x)~ Tm (x) ( )3/2 dx. 1 − x2

Assumed, that for m > N all above coefficients vanish f˜m = 0. In this case, the Eq. (1.50) reduces to simultaneous linear algebraic equations with unknown coefficients cn : N ∑

An,m cn = ~ fm .

n=1

According to Eq. (1.48), the matrix An,m is diagonal. Consequently, the solution of (1.50) reads as: ( 1 ~ f n , for n = 0, . cn = f n , for n = 1, 2, . . . π 2~

(1.51)

A polynomial is identically equal to zero if and only if all of its coefficients are equal to zero. Thus, the sole solution of the homogeneous equation: N ∑

Tn (x) = 0 cn~

n=1

is cn = 0. Contact problem with second type of base functions S(y) =

∞ ∑ n=0

~ Tn (y) ≡ Tn (y), pn~ sn , ~ sn = √ 1 − y2

(1.52)

∞

dW(x) ∑ = vn u˜ n , u˜ n = ~ Un−1 (x), dx n=0

(1.53)

1.2 Direct and Inverse Plane Contact Problems

f

~

n

21

(

∫1 =−

0,

sn (y)dy = π dn ≡

2 , (n−1)(n+1)

−1

m~

(

∫1 n

=−

y~ s n (y)dy =

0, 2 , (n−2)(n+2)

−1

for odd n . for even n

(1.54)

for even n . for odd n

With the above definitions (1.52) and (1.54), we perform the solution of the principal integral equation:

k

∫1

k

−1

dw(x) S(y) dy = + C1 ⇒ y−x dx

1 π

∫1 ∑ ∞ −1 n=0

1 dy = pn~ sn y−x π

∫1 ∑ ∞ −1 n=0

∫1 ∑ ∞ −1 n=0 n pn

kπ p

1

= (v1 + C1 ),

(1.55)

∞

=

n=0

∑ dy = vn~ u n + C1 , y−x n=0

∑ Tn (x) dy pn ~ = ~ un pn , √ 1 − x2 y − x n=0

∞

kπ ∑ ~u

∞

pn~ sn

∞ ∑

vn~ u n + C1 ,

(1.56)

(1.57)

n=0

kπ p

n

= vn for n /= 1.

(1.58)

Equations (1.57) and (1.58) could be satisfied for C1 = 0. In this case the inclination of the contact body vanishes.

1.2.4 Solutions with Base Functions of Third Type Base functions of third type The third type of the base functions on the interval could be defined analogously to the definitions (1.21): def ~ Pn (x) = eiπ nx Tn (x) + i ~

for n ∈ W, −1 ≤ x ≤ 1.

(1.59)

Tn and ~ Pn are not the polynomials. The functions ~ Tn vanish on The functions ~ ~ both ends of the interval G1 . The function Un is defined thru the integration of the Tn with cosine and sine integrals [14]: function ~

22

1 Optimization and Inverse Solutions for Plane Contacts def 1 ~ Un−1 (x) = π

[ ] ( π Re ~ Un−1 (x) = [ ] ( ~ π Im Un−1 (x) =

∫1 −1

ei πny

dy , y−x

(1.60)

, for n = 0, ln 1−x 1+x cos(π nx)ϒ(n, x) − sin(π nx)Ξ(n, x), for n = 1, 2, . . . (1.61)

0, n = 0, cos(π nx)Ξ(n, x) + sin(π nx)ϒ(n, x), n = 1, 2, . . .

(1.62)

The functions ϒ(n, x) and Ξ(n, x) in Eqs. (1.62) and (1.63) are the combinations of integral sine and integral cosine: ϒ(n, x) = Ci(π n(1 − x)) − Ci(π n(1 + x)), Ci(x) = − Ξ(n, x) = Si(π n(1 − x)) + Si(π n(1 + x)), Si(x) =

∫x 0

∫∞ cos y x

y

dy, (1.63)

sin y dy. y

Rn (x) is defined as an integral of the function ~ Un−1 . For the definition The function ~ ~ of the function Rn (x) we use the formula, which is similar to the relation between the Chebyshev polynomials of the first and second kinds in Eq. (1.22): def ~ Rn =

∫x

~ Un−1 (y)dy,

(1.64)

−1

] [ (1 − x)x−1 ~ . π R0 (x) = 2 ln 2 + ln (1 + x)x+1

(1.65)

Tn and ~ Pn is comparable to those The reason of the introduction of the functions ~ Tn and of the functions of the second type. Obviously, based on the base functions ~ ~ Un−1 , it is possible to find the closed form solutions for the inverse and optimization problems. Notable from the viewpoint of handling, that the base functions of the third type allow the direct summation of the series. This makes the solution of the optimization problems simpler. The orthogonality conditions for modified Chebyshev functions read: ∫1

~ Tm (x)dx = Tn (x)~

−1

∫1 −1

2 An,m π

2 ~ Pm (x)dx = An,m , Pn (x)~ π

⎧ ⎨ 2, for m = n = 0, ≡ 1, for m = n /= 0 ⎩ 0 for n /= m.

(1.66)

1.2 Direct and Inverse Plane Contact Problems

23

For the following, we need the solution of the simultaneous equation for an arbitrary N > 1, x ∈ G1 : N ∑

Tn (x) = f (x). cn~

(1.67)

n=1

Tm (x). The succeeding integrations of For solution, the Eq. (1.67) are multiplied by ~ the right side of Eq. (1.67) over the interval G1 result the coefficients: ∫1

~ ~ fm =

Tn (x)dx. f (x)~

−1

f m . = 0. In this case, Assumed, that for m > N all above coefficients vanish ~ the Eq. (1.67) reduces to simultaneous linear algebraic equations with unknown coefficients cn : N 2∑ An,m cn = ~ fm . π n=1

(1.68)

According to Eq. (1.66), the matrix An,m is diagonal. Consequently, the solution of (1.53) reads as: ⎧ 1~ ⎪ ⎨ 2 f 0 , for n = 0, . (1.69) cn = ⎪ ⎩~ f n , for n = 1, 2, . . . The sole solution of the homogeneous equation

∑N

~

n=1 cn Tn (x)

= 0 is cn = 0.

Series solution of an inverse problem for given normal stress and friction free contact The series solution could be developed with the base functions of the third type. A guess for the contact stress and the derivative of the normal displacement in the contact region is: ∞ ∑

sn , ~ pn~ sn = cos(π ny),

(1.70)

dw(x) ∑ Un−1 (x), = un , ~ vn~ un = ~ dx n=0

(1.71)

S(y) =

n=0 ∞

f

~ n

∫1 =− −1

~ sn (y)dy =

(

2, for n = 0, 0, for n > 0;

m~

n

∫1 =− −1

y~ sn (y)dy = 0.

(1.72)

24

1 Optimization and Inverse Solutions for Plane Contacts

Plugging this series into the singular integral Eq. (1.23), we obtain:

k

∫1

−1

S(y) dW(x) dy = + C1 ⇒ y−x dx 1 π

∫1 ∑ ∞

sn pn~

−1 n=0

k

∫1 ∑ ∞

sn pn~

−1 n=0

∞

∑ dy un + C1 , (1.73) vn~ = y−x n=0

∞

∞

∑ ∑ dy ~ ~ Un−1 (x) pn ≡ = un pn , y−x n=0 n=0 ∞

kπ ∑ ~u p

n n

=

n=0

∞ ∑

un + C1 , vn~

(1.74)

(1.75)

n=0

kπ p

n

= vn for n ≥ 0.

(1.76)

The solution (1.76) is valid for an additional condition C1 = 0. This means, that the inclination of the rigid contact body must disappear. With the Eqs. (1.72) and (1.76) the following optimization problem receives the solution. The authors examine the shape of the rigid contact body, which leads to the minimum normal stress everywhere in contact region. The rigid contact body is forced without inclination into the deformable half-space. The friction in the contact region lacks. The moment is absent and total force on the contact body is fixed: Ftotal =

∞ ∑ ~

f

n.

(1.77)

n=0

Because all higher members do not contribute to the total force due to Eq. (1.72), the only distribution of normal stress withstand the vertical force: Ftotal =

∞ ∑ ~

f

n=0

n

=−

∞ ∑

∫1 pn

n=0

~ sn (y)dy = 2 p0 .

(1.78)

−1

The optimization problem examines the shape of the rigid indenter, which minimizes the contact stress under the requirements of the given total normal force F0 and the absence of the total moment, which act on the indenter: Wopt = optimum W

[ ] max S Wopt ≤ max S [W ], −1≤x≤1 [ ] Mtotal Wopt = Mtotal [W ] = 0, [ ] Ftotal Wopt = Ftotal [W ] = F0 . −1≤x≤1

Accordingly, the sum of the members with the numbers from 1 to infinity induces the zero force:

1.2 Direct and Inverse Plane Contact Problems

f (x) ≡

∞ ∑ n=1

25

∫1 pn

~ sn (y)dy = 0.

(1.79)

−1

There are two possibilities. One possibility is covered by Eqs. (1.67) and (1.69). In this case all coefficients disappear (cn = 0); consequently, the shape of the rigid contact body causes only the constant stress: S(y) = p0 =

1 F0 . 2

(1.80)

in the entire contact region. Otherwise, the function f (x) changes the sign in the interval G1 = [−1, 1] and the total force in the positive regions compensate the total force in the negative regions of pressure distribution: f+ =

∫

f (y)dy, f (y) > 0 on G+ ,

G+

f− =

∫

G−

f (y)dy, f (y) ≤ 0 on G− ,

f − + f + = 0,

(1.81)

G+ ∪ G− = G1 .

If G+ does not disappear, it means that the contact pressure exceeds its average value: f >

p0 1 on G+ /= ∅, S(y) ≥ p0 = F0 . 2 2

(1.82)

The minimal achievable contact pressure is equal to its mean value (1.80). Finally, R0 u0 or ~ the only shape that delivers the minimum of the normal stress corresponds ~ from Eq. (1.65).

1.2.5 Integral Formulations of Inverse Problem with Given Normal Stress and Friction Free Contact The above presented result reduces the inverse contact problem to the exhaustively studied Flamant problem. On the author knowledge, such inverse contact problem was studied in [15]. In this article the problem earth engineering mechanics was solved. The question of how the load of a structure is transferred to the soil by its foundation was analyzed. If the soil is plastically very compliant with respect to the foundation, the load will be distributed approximately uniformly over the base of the foundation. As a rule, however, large structures are built on firmer ground, which probably undergoes a certain permanent change in shape when subjected

26

1 Optimization and Inverse Solutions for Plane Contacts

Fig. 1.1 Two different indenters with the constant pressure in the contact regions. The solid lines mark the shape of indenter. The dashed lines show the deformed free boundary due to the pressure on indenters

to strong surface pressure, but is for the most part elastically deformed. In these cases, the distribution of the pressure over the foundation base and thus the stress on the foundation and soil is by no means clear. The constant normal stress is also advantageous from the tribological considerations (Fig. 1.1). The shear stress vanishes everywhere on the boundary of the half-space due to the lack of friction on the contact boundary. The normal penetration W in the design state is the unknown function. This function describes the shape of the punch, which must guarantee the constant normal stress. The even distribution of contact reactions R0 R0 (1.47) and ~ The solution for the constant normal stress gives both formulas ~ (1.65). Another closed form solution S(x, n) = x 2n

for n = 0, 2, 4, . . .

(1.83)

The total force for the above base functions reads: ∫1 Ftotal (n) =

S(x, n)dx = −1

2 . 2n + 1

With the Eq. (1.83) the total moment on the indenter disappears:

(1.84)

1.2 Direct and Inverse Plane Contact Problems

27

∫1 Mtotal (n) =

x S(x, n)dx = 0.

(1.85)

−1

The shape of the indenter follows from Eq. (1.22): W(x, n) =

k

∫1 S(t, n) ln(|t − x|)dt

−1

=

k

⎡

2n + 1

⎣ln(1 − x) + ln(1 + x) −

∫1 −1

⎤ t 2n+1 ⎦ dt . t−x

(1.86)

The integrals in Eq. (1.86) could be evaluated in closed form with the higher transcendent functions: [ ( ) ( x 1 ln(1 + x) + ln(1 − x) + Φ , 1, 2n W(x, n) = · 2n + 1 2n + 1 x ) )] ( 2 1 (1.87) −Φ − , 1, 2n − x (2n + 1)2

k

In Eq. (1.87) the Lerch function [16] is: Φ(z, a, ν) =

∞ ∑ n=0

zn (n + ν)a

For n = 0 results the above solution with the constant contact pressure. The odd distribution of contact reactions At first, consider the odd stress distribution S(x) = −S(−x) in contact region. The base functions are in this case: ( H (x) − H (−x) ≡ sgn(x), for n = 0 . (1.88) S(x, n) = for n > 0 x 2n+1 , The base function for n = 0 is the sign function: ( S(x, 0) =

1, for x ≥ 0, −1, for 0 > x.

With the Eq. (1.88) the total force disappears:

28

1 Optimization and Inverse Solutions for Plane Contacts

∫1 Ftotal (n) =

S(x, n)dx = 0.

(1.89)

−1

The total moment reads: ∫1 Mtotal (n) =

x S(x, n)dx = −1

2 . 2n + 3

(1.90)

The shape of the indenter follows from Eq. (1.22): W(x, n) =

k

∫1

−1

k

⎡

(

⎣ln 1 − x S(t, n) ln(|t − x|)dt = 2n + 2 1+x

)

∫1 − −1

⎤ t 2n+2 ⎦ dt t−x (1.91)

The integrals in Eq. (1.91) reduce to Lerch functions, Eqs. (1.87) (Figs. 1.2, 1.3 and 1.4): ( ) [ ( ) ( )] 1−x x2 1 1 1 ln + Φ , 1, 2n − Φ − , 1, 2n W(x, n) = · 2n + 2 1+x 2n + 2 x x ) x . (1.92) − 2 2n + 3n + 1

k

(

1.2.6 Optimal Shape of Rigid Punch Penetrating into Elastic Layer The shape of the rigid punch, which guarantee the constant normal pressure on an elastic strip could be determined with the method of integral transforms [17]. This elastic layer rests in turn on a rigid frictionless foundation. The dimensionless length of the contact region is w . This parameter is the ratio of the half-length of the contact region l to the depth of the elastic layer h: def l w = . h

The contact stress in the moment of penetration into the elastic layer must be constant. The optimal shape of the rigid punch, which causes the constant contact stress, is:

1.3 Optimal Shapes of Periodically Arranged Indenters

29

Fig. 1.2 Optimal shapes of the pressure punch (blue line) and of the moment punch (red line) W (x, 0)

W(ξ ) =

k

∫∞ Θ(ξ, ζ )dζ, Θ(ξ, ζ ) 0

=F

) sin(w ζ ) cos(ξ ζ ) ( 2ζ ν−1 · e + e−2ζ − 2 . · 2 ζ π ζ + sinh(ζ ) cosh(ζ )

1.3 Optimal Shapes of Periodically Arranged Indenters 1.3.1 Normal Displacement Under the Action of Periodical Forces In this section the authors analyze the optimal shape of the periodically arranged punches on the boundary of the elastic half-space. For the analysis of periodic contact problems in plane elasticity, the method is based upon the summation of evenly spaced Flamant solutions. The solutions are studied in a direct routine without necessity of the advanced mathematical theory. More general theory was developed in [18]. Various methods for solving the partial contact of surfaces with regularly periodic profiles. Analogous to the contact of a single indenter, the preparation produces coupled Cauchy singular integral equations of the second kind upon transforming variables.

30

1 Optimization and Inverse Solutions for Plane Contacts

Fig. 1.3 Shapes of the pressure punch (picture above) and contact stress (picture below). The shapes of the indenters are symmetric to y-axis. Optimal indenter W (x, 0) causes the constant stress in the contact region S (x, 0) = 1

For the beginning, the normal displacement of the boundary must be determined. The Flamant solution [19] provides expressions for the stresses and displacements in a linear elastic half-space, loaded by an infinite system of periodically applied point forces. The Flamant solution follows from the general solution to the elasticity equations in polar coordinates of Michell [20]. For the solution consider 2N + 1 local equally spaced forces with the identical intensities F (Fig. 1.5). In coordinates X , the distance between the local forces is

1.3 Optimal Shapes of Periodically Arranged Indenters

31

Fig. 1.4 Shapes of the pressure punch (picture above) and contact stress (picture below). The shapes of the indenters are mirror-symmetric to y-axis

lΔ. It is assumed, that Δ > 2. The dimensionless coordinates x = X/l will be used for the formal solution. In dimensionless coordinates the period of local forces is Δ. The total displacement function due to 2N + 1 local forces is denoted as 2N +1 (x). The derivative of the displacement function 2N +1 is the inclination of the boundary:

w

w

32

1 Optimization and Inverse Solutions for Plane Contacts

Fig. 1.5 The optimal punches for an elastic layer of the unit depth and different dimensionless lengths of the contact region 2w = 1, 2, 4

w

d

2N +1 (x) def

dx

= Θ2N +1 .

(1.93)

The derivative Θ2N +1 in Eq. (1.93) must be assessed in closed form. For this purpose, all 2N + 1 forces are divided in three groups. In the first group is only one central force with the number “0”. This central force is applied exactly at the point x = 0. Another group with the index “N−” comprises N forces to the left of the central force with the numbers from −N to −1. The third group with the index “N+” includes N forces to the right of the central force with the numbers from 1 to N . The normal displacements of the boundary due to the action of each group are 0 , N − , N + correspondingly. Respectively, the inclinations groups of forces are signified as Θ0 , Θ N − , Θ N + :

w w

w

d

w (x) = Θ (x), dw 0

dx

0

N − (x)

dx

= Θ N − (x).

w

d

N + (x)

dx

= Θ N + (x).

(1.94)

The inclinations (1.94) have to be determined for each group at the point with the coordinate x. The corresponding integrals evaluate with the higher transcendent functions. For each of three groups the inclinations are equal correspondingly to: Θ0 =

F , x

−1 ( ∑

) 1 1 − x +Δn Δn n=−N ( ] x) F[ (x ) Ψ + Ψ(N + 1) − Ψ −N + +γ , = Δ Δ Δ

ΘN − ≡ F

(1.95)

(1.96)

1.3 Optimal Shapes of Periodically Arranged Indenters

ΘN +

) 1 1 − ≡F x +Δn Δn n=1 [ ( ) ( ] x x) F Ψ N +1+ − Ψ(N + 1) − Ψ 1 + −γ . = Δ Δ Δ

33

N ( ∑

(1.97)

The functions in Eq. (1.92) are [21]: Ψ(x) = γ

d(ln [(x)) dx

digamma function, Euler' sconstant.

In Eqs. (1.95) and (1.97) the sums are corrected to avoid the total displacement of the body. For this purpose, the sum of displacements is subtracted. The remained sums converge for an infinitely increasing number of forces. The total displacement is the addition of the partial displacements of all three groups (1.95) and (1.97): Θ2N +1 = Θ0 + Θ N + + Θ N − . Using the expressions (1.95) and (1.97), the total displacement simplifies to: Θ2N +1 =

] [ ( (x )) ( ( x) x) 2xΔ F . π cot π − N +Ψ N + −Ψ N − − 2 2 Δ Δ Δ Δ Δ N − x2 (1.98)

At fourth, we express the limit of the series Eq. (1.98) at infinity N → ∞. With the infinitely increasing number of forces in each group, the asymptotic formulas could be derived in closed form. The inclination due to an infinite half-row of forces to the left of the central force is: ] [ 1 1 (x) γ ˜ Θ N − = lim Θ N − = −F · Ψ + + . (1.99) N →∞ Δ Δ Δ x The inclination due to an infinite half-row of forces to the right of the central force is: ˜ N + = lim Θ N + = F · Θ N →∞

[

] 1 1 ( x) γ Ψ − + − . Δ Δ Δ x

(1.100)

˜ is the sum of formulas for the left Θ ˜ N −, The formula for the total inclination Θ ˜ central Θ0 and right Θ N + contributions: ˜ = Θ0 + Θ ˜ N− + Θ ˜ N +. Θ The total inclination, which is caused by the infinite row of equally spaced in both direction forces simplifies to the trigonometric function:

34

1 Optimization and Inverse Solutions for Plane Contacts

˜ = Θ0 + Θ ˜ N− + Θ ˜ N+ Θ

[ ( ) ] ( ) (πx ) Ψ − Δx Ψ Δx 1 Fπ − − cot . =F· = Δ Δ x Δ Δ (1.101)

˜ is determined, the displacement results by the inteOnce the total inclination Θ gration of Eq. (1.101). The displacement (x), which is caused by an infinite number of equally spaced forces, is equal to:

w

w(x) =

∫x 0

˜ =− Θdt

[ ( π x )] ( π x )] F [ ln 1 + cot 2 ≡ F ln sin . 2 Δ Δ

(1.102)

With the periodic Flamant solution Eq. (1.102), one can determine the displacement of the half-space due to the action of the periodical normal force S(t) with the period Δ: S(t) = S(t + Δ).

1.3.2 Optimal Shapes of Periodically Spaced Penetrators The shape of the penetrator and normal stress in the contact region are the symmetric function with respect to the center of the period. Thus, the normal force is an even function in every period: S(t) = S(−t) for

0 ≤ t ≤ Δ/2.

The total displacement is the sum of the displacements, which affected by the local normal stress in each point of the interval. Consequently, the normal displacement is the integral:

W(x) =

k

∫Δ/2

w

S(t) (x − t)dt.

(1.103)

−Δ/2

The total rigid-body displacement was eliminated with the subtraction of the solid displacements in Eqs. (1.96) and (1.97). Substitution of Eq. (1.102) into (1.103) leads to the desired expression of the displacement of the half-space boundary under the action of the periodic force:

1.3 Optimal Shapes of Periodically Arranged Indenters

W(x) =

∫Δ/2

k

−Δ/2

[ (π )] S(t) ln sin (x − t) dt. Δ

35

(1.104)

For example, if the periodic stress functions vanish in every period outside the interval x ∈ G1 , then the normal force the reads: ( S=

1, for x ∈ G1 , 0, for x ∈ / G1 and x ∈ GΔ/2 .

The displacement due to the periodic functions S(t), which periodically vanish outside the interval reads: W(x) =

k

∫1

−1

[ (π )] S(t) ln sin (x − t) dt. Δ

(1.105)

This equation is the periodic equivalent to Eq. (1.22). For the infinitely increasing period Δ the Eq. (1.105) reduces to Eq. (1.22). Either the normal displacement W (x) is given along the contact interval G in the period and the contact stress S(t) will be determined, or the contact stress σ (t) is given and one look for the unknown normal displacement W(x). The first formulation is the direct contact problem and the second is the inverse contact problem. In current context, the design stress is prescribed and the shape of the indenter, which leads to this contact stress, will be determined in closed form. Particularly, the stress function in form of an ideal square wave function will be presumed. Formally, the square wave is a non-sinusoidal periodic waveform in which the amplitude alternates with the fixed period between fixed minimum and maximum values. In an ideal square wave, the transitions between minimum and maximum are instantaneous. This means for our applications, that the deformation of the half-plane caused be the periodical series of the identical rigid indenters. Each indenter has the width 2. The period is equal to Δ > 2, such that the distance between indenters is positive and is equal to Δ − 2. The normal displacement under the action of the above normal stress (1.105) reduces for x ∈ G1 to (Fig. 1.6): W(x) =

k

∫1

−1

[ (π )] ln sin (x − t) dt. Δ

The integral (1.106) evaluates in the closed-form:

k

−1

W(x) = −2 ln(2) +

Δ Im[Li 2 (α1 ) − Li 2 (α2 )]. 2π

(1.106)

36

1 Optimization and Inverse Solutions for Plane Contacts

Fig. 1.6 Periodically spaced forces under the punch and caused vertical displacements. The distance between each neighboring indenter is Δ

( α1 = exp

) ) ( 2πi(x − 1) 2πi (x + 1) , α2 = exp . Δ Δ

(1.107)

The functions in Eq. (1.107) are the dilogarithms Li 2 (x), see Eq. (1.42) [22, 23]. Equation (1.107) provides the shape of the periodically arranged indenters. Each indenter has the length 2. The central indenter acts in the contact region x ∈ G1 . The distance between each two neighboring indenters is Δ > 2. For plotting of the results, the authors use the normalized function W(x), vanishing in zero point: def

W(x) = W(x) − W(0).

(1.108)

The function W(x) reads with Eq. (1.107) as: W(x) =

kΔ Im[Li (α ) − Li (α ) − Li (α ) + Li (α )], 2

1

2

2

2

3

2

) ) ( (2π 2πi (x + 1) 2πi(x − 1) , α2 = exp , α1 = exp Δ Δ ) ) ( ( 2πi 2πi , α4 = exp . α3 = exp − Δ Δ

4

(1.109)

We can answer the question, what happens to the indenters, if the distance Δ increases infinitely. The asymptotic expansion of the function W(x) for an infinitely growing period Δ → ∞ reads: ~ W(x) → W(x). Δ→∞

(1.110)

This formula gives once again the solution for the single constant normal stress R 0 (1.65). R0 (1.47) and ~ indenter, which was given by each of formulas ~

1.3 Optimal Shapes of Periodically Arranged Indenters

37

~ The normalized expression for the optimal shape of an indenter W(x), which fades on the point x = 0, is: def ~ ~ ~ = W(x) − W(0), W(x) ~ W(x) → W(x).

(1.111)

Δ→∞

For an infinitely growing period Δ → ∞, the normalized function for the optimal ~ appears as: shape of an indenter W(x) ~ W(x) =

k·

[

( x · ln

1+x 1−x

)

] ( ) + ln 1 − x 2 .

(1.112)

~ The normalized functions W(x), W(x) make apparent the comparison of the optimal punches. The shapes of the optimal forms for the indenters of different length are shown on Fig. 1.7.

Fig. 1.7 The normalized functions W(x) which describe the optimal shapes W(x) of the periodi22π 23π cally located systems of punches for different periods = 21π 10 , Δ = 10 , Δ = 10 . The black line ~ shows the normalized optimal shape of the single indenter W(x) for an infinitely increasing period Δ→∞

38

1 Optimization and Inverse Solutions for Plane Contacts

1.4 Conclusions This chapter studies the optimal design for contact problems of maximal stiffness and minimal normal stress. The method allows the finding of optimal shape for the non-deformable indenters in plane strain and plain stress elasticity problems. The optimal problems for the sole indenter and multiple optimal indenters are solved in analytical form. The technique developed in this chapter is appropriate for a board class of optimal problems in contact mechanics.

1.5 Summary of Principal Results • The solution of contact problem uses the series expansions; • The series expansions with standard Chebyshev polynomials leads to stress singularities on the corners of the indenter; • The series expansions with the base polynomials of the second type are regular on the corners; • The series expansions with the base polynomials of the third type are also regular on the corners; • The base functions of the third type allow the direct summation of the series and simplify the solution of the optimization problems; • Optimal shapes of the sole indenters are presented in closed form; • Exact solutions for the periodically spaced indenters are displayed.

1.6 List of Symbols (

3 − 4ν,

f or plane strain

K

Dimensionless parameter K =

k

Dimensionless parameter

W(x)

Displacement, normalized W(x) = W (x) − W (0),

S (t)dt

The pressure traction, which act on the line element dt

T (t)dt

The shear traction, which act on the line element dt

G1 = [−1, 1]

The interval of the contact region in dimensionless variable, x ∈ G1

Gl = [−l, l]

The interval of the contact region of length 2l, x ∈ Gl

Tn , Pn

Chebyshev polynomials, Tn (x) + i Pn (x) = O(x) def √ Second type base functions, ~ Tn (x) + i ~ Pn (x) = 1 − x 2 O(x)

~ Tn , ~ Pn

k=

3−ν 1−ν ,

f or plane str ess

K+1

4π μ l

def

(continued)

References

39

(continued) ~ Tn , ~ Pn

Third type base functions, ~ Tn (x) + i ~ Pn (x) = eiπ nx

W

Normal displacement, x ∈ G1

S

Normal stress, x ∈ G1

Ftotal

Total force pro unit of body width

Mtotal

Total moment pro unit of body width

def

References 1. Barber, J. R. (2022). Elasticity. Springer Nature Switzerland AG 2. Sadd, M. (2014). Elasticity (3 ed., p. 600). Academic Press. ISBN-13: 978-0124081369. 3. Gurtin, M. E. (1972). The linear theory of elasticity. In: S. Flugge (Ed.) Encyclopedia of physics. Mechanics of solids II (Vol. VIa/2). Springer. 4. Muskhelishvili, N. I. (1975). Some basic problems of the mathematical theory elasticity. Noordhoff. 5. Muskhelishvili, N. I. (2013). Singular integral equations. Boundary problems in the theory of functions and some of their applications to mathematical physics. Springer. ISBN 9400999968. 6. Gakhov, F. D. (1990). Boundary value problems (p. 561). Dover. ISBN 9780486662756. 7. Barber, J. R. (2018). Contact mechanics. Springer. 8. Galin, L. A. (2008). Contact problems. The legacy of L.A. Galin. Solid mechanics and its applications (Vol. 155). Springer Science+Business Media, B.V. https://doi.org/10.1007/9781-4020-9043-1 9. Mandal, B. N., & Chakrabarti, A. (2011). Applied singular integral equations (p. 270). CRC Press. ISBN 9781578087105. 10. Weisstein, E. W. (2022). Whole number. From MathWorld—A Wolfram Web Resource. https:// mathworld.wolfram.com/WholeNumber.html 11. Mason, J. C., & Handscomb, D. C. (2002). Chebyshev polynomials (p. 360). CRC Press. 12. Zagier, D. (2006). The dilogarithm function. In P. Cartier (Ed.) Frontiers in number theory, physics and geometry II. Springer. 13. Olver, F., Lozier, D., Boisvert, R., & Clark, C. (2010). The NIST Handbook of Mathematical Functions, 25.12 Polylogarithms. Cambridge University Press, New York, NY, p. 610 14. Olver, F. W. J., et al. (2010). NIST handbook of mathematical functions (p. 967). Cambridge University Press. 15. Föppl, L. (1941). Elastische Beanspruchung des Erdbodens unter Fundamenten. Forschung, 12, Jan./Febr. 1941, S.31 16. Bateman, H., & Erdélyi, A. (1953). Higher transcendental functions (Vol. I). McGraw-Hill. 17. Sneddon, I. N. (2003).Fourier transforms. Dover books on mathematics. Dover Publications Inc. 18. Block, J. M., & Keer, L. M. (2008). Periodic contact problems in plane elasticity. Journal of Mechanics of Materials and Structures, 3(7), 1207–1237. https://doi.org/10.2140/jomms.2008. 3.1207 19. Flamant, A. (1892). Sur la répartition des pressions dans un solide rectangulaire chargé transversalement. Compte. Rendu. Acad. Sci. Paris, 114, 1465. 20. Michell, J. H. (1899). On the direct determination of stress in an elastic solid, with application to the theory of plates. Proceedings of the London Mathematical Society, 31(1), 100–124. https:// doi.org/10.1112/plms/s1-31.1.100 21. Abramowitz, M., & Stegun, I. A. (Eds.) (1972). Handbook of mathematical functions with formulas, graphs, and mathematical tables (10th ed.). 6.3 psi (Digamma) Function. Dover.

40

1 Optimization and Inverse Solutions for Plane Contacts

22. Temme, N. M. (2010). Exponential, logarithmic, sine, and cosine integrals. In F. W. J. Olver, D. M. Lozier, R. F. Boisvert, & C. W. Clark (Eds.), NIST handbook of mathematical functions. Cambridge University Press. ISBN 978-0-521-19225-5, MR 2723248. 23. Lewin, L. (1958). Dilogarithms and associated functions. Foreword by J. C. P. Miller. Macdonald. MR 0105524.

Chapter 2

Optimization for Axisymmetric Contacts, Charged and Conducting Disks

Abstract In this Chapter the authors deliberate the axisymmetric problem of optimization of the rigid indenters. Considered is the contact problem of the theory of elasticity on indentation of an indenter with an unknown base into an elastic isotropic half-plane. The authors consider the rigid axisymmetric punches of arbitrary profiles indenting without friction into the surface of an isotropic elastic half-plane. The bottom of the punch is the surface of revolution, which is created by rotating a curve (the generatrix) around an axis of rotation. The generatrix is not a priori given. Namely, the generatrix represents the searchable function in the optimization problems. In optimal formulations, the elastic energy of the elastic body and the maximal pressure on the contact surface represent as the principal functionals. Singular stress analysis in the punch problem of half-plane is carried out. The distribution for the stress components is determined in an explicit form. Solutions in a closed form are obtained for the problems of a singular and periodically spaced concentric punches. Keywords Shape optimization · Contact problem · Axisymmetric elasticity problem

2.1 Axisymmetric Elastostatics 2.1.1 Boussinesq-Papkovich Solution 1. As shown in Chap. 1, the planar problems were reduced to the search for a biharmonic function or a function of various complex arguments. Investigations of axisymmetric and, in the more general case, spatial problems are relatively complicated. In this case it is not possible to use the apparatus of functions of a complex variable. For the beginning, we summarize the axisymmetric elastostatics for an isotropic body. The components of the displacement vector u→ in a rectangular coordinate system x, y, z are denoted by u, v, w. In the absence of the volume forces, the equilibrium equation of elasticity theory expresses as [1]:

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Kobelev, Fundamentals of Structural Optimization, Mathematical Engineering, https://doi.org/10.1007/978-3-031-34632-3_2

41

42

2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks

(1 − 2ν)Δu→ + ∇di v u→ = 0, ∇ 2 ≡ Δ, ν =

λ . 2(λ + μ)

(2.1)

In the absence of volume forces, a solution of the vector equilibrium Eq. (2.1) expresses as: ( ) u→ = ∇ ϕ + r→ · ψ→ − 4(1 − ν)ψ→ = 0.

(2.2)

) ( The function ϕ and each component of the function ψ→ = ψx , ψ y , ψz are the harmonic functions: Δϕ = 0, Δψ→ = 0.

(2.3)

From the Boussinesq-Papkovich solution Equations (2.1)–(2.3) it is easy to derive a simpler solution including only one harmonic function [2]. This harmonic function can be chosen so that the tangential stress is zero on the plane z = 0. The solution of (2.3) could be expressed with a harmonic function Θ(x, y, z): ) ( ∂Θ , ΔΘ = 0. ϕ = (1 − 2ν)Θ, ψ→ = 0, 0, ∂z

(2.4)

The components of the displacement vector (2.2) could be expressed in terms of the harmonic function Θ(x, y, z): ) ∂ ∂2 + (1 − 2ν) Θ, u= z ∂ x∂z ∂x ) ( ∂ ∂2 + (1 − 2ν) Θ, v= z ∂ y∂z ∂y ( 2 ) ∂ ∂ w = z 2 − 2(1 − ν) Θ. ∂z ∂z (

(2.5)

In their turn, the relations (2.5) lead to the components of stress: ) ( ∂2 ∂2 ∂3 σx x = 2μ z 2 + (1 − 2ν) 2 − 2ν 2 Θ, ∂ x ∂z ∂x ∂z ) ( ∂2 ∂2 ∂3 σ yy = 2μ z 2 + (1 − 2ν) 2 − 2ν 2 Θ, ∂ y ∂z ∂y ∂z ) ( 3 ∂2 ∂ σzz = 2μ z 3 − 2 Θ, ∂z ∂z ) ( ∂3 ∂2 Θ, τx y = 2μ z + (1 − 2ν) ∂ x∂ y∂ z ∂ x∂ y

(2.6)

(2.7)

(2.8)

(2.9)

2.1 Axisymmetric Elastostatics

43

τ yz = 2μz

∂ 3Θ , ∂z 2 ∂ y

(2.10)

τx z = 2μz

∂ 3Θ . ∂ z2∂ x

(2.11)

From formulas (2.10) and (2.11) follows that if the derivatives. ∂ 3Θ ∂ 3Θ , . ∂z 2 ∂ y ∂z 2 ∂ x both remain finite at z → 0, then: τ yz = τx z = 0

at z = 0.

Similarly, if the derivatives ∂ 3Θ , ∂x3

∂ 2Θ ∂z 2

do not tend to infinity at z → 0, then the normal stress and displacement on the boundary z = 0 are defined by the following equations: | ∂ 2 Θ || σzz = −2μ 2 | , ∂z z=0 | ∂Θ || w = −2(1 − ν) . ∂z |z=0

(2.12) (2.13)

Consequently, the normal stress σzz and normal displacement w relate to each other by means of second and first derivatives of a harmonic function Θ(x, y, z). This condition plays the central role for the establishing of the governing equations on the boundary z = 0. 2. In the cylindrical coordinate system, the equation for harmonic function Θ(r, φ, z) reads: ΔΘ = 0, Δ ≡

1 ∂2 ∂2 1 ∂ ∂2 + + + . ∂r 2 r ∂r r 2 ∂φ 2 ∂z 2

(2.14)

With this function, the solution will have the form: ) ( ∂2 ∂2 ∂3 σrr = 2μ z 2 + (1 − 2ν) 2 − 2ν 2 Θ, ∂r ∂ z ∂r ∂z

(2.15)

44

2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks

) ( ∂3 ∂3 ∂2 ∂2 σφφ = −2μ z 2 + z 3 + 2 + (1 − 2ν) 2 Θ, ∂r ∂z ∂z ∂z ∂r ) ( 3 ∂ ∂2 σzz = 2μ z 3 − 2 Θ, ∂z ∂z ) ( 2z ∂ 3 z ∂2 1 − 2ν ∂ Θ, τr φ = μ − 2 + r ∂r ∂φ∂z r ∂r ∂φ r 2 ∂φ ∂ 3Θ , ∂z 2 ∂r 2μz ∂ 3 Θ , τφz = r ∂z 2 ∂φ ∂Θ ∂ 2Θ + (1 − 2ν) , ur = z ∂r ∂ z ∂r ) ( 1 − 2ν ∂ z ∂2 + Θ, uφ = r ∂φ∂ z r ∂φ ) ( 2 ∂ ∂ u z = z 2 − 2(1 − ν) Θ. ∂z ∂z

(2.16)

(2.17)

τr z = 2μz

(2.18)

In problems with axial symmetry the angular derivatives vanish. The explored solution Θ is the function of radius. 3. According to [3], we examine the expression for the harmonic function as the product of Bessel function J0 (r ξ ) and the exponent with an arbitrary function ζ(z): Θ(r, z) = ζ(z)J0 (r ξ ) exp(−ξ z).

(2.19)

The value ξ is a positive parameter. The function Θ(r, z) in (2.19) will be harmonic, if ζ(z) = C1 + C2 exp(2ξ z). The function Θ(r, z) fades at infinity → ∞, if C2 = 0. We put C1 = 1 and get with ζ(z) = 1 the desirable harmonic function: Θ(r, z) = J0 (r ξ ) exp(−ξ z).

(2.20)

2.1 Axisymmetric Elastostatics

45

To evaluate the stress and displacement components, the expression (2.20) is substituted in Eqs. (2.17) and (2.18). The necessary components of stress and displacement follow as: σzz = −2μξ 2 · (1 + ξ z) J0 (r ξ ) exp(−ξ z), τr z = −2μzξ 3 J1 (r ξ ) exp(−ξ z), u z = ξ · (2 − 2ν + ξ z) J0 (r ξ ) exp(−ξ z).

(2.21)

From Eq. (2.21) follows, that at z = 0 the shear stress fades, and normal stress and displacement shrink to: σzz = −2μξ 2 J0 (r ξ ),

(2.22)

τr z = 0,

(2.23)

u z = 2ξ · (1 − ν) J0 (r ξ ).

(2.24)

4. Now we multiply (2.20) by

P(ξ ) 2μξ

1 Θ(r, z) = 2μ

and integrate over ξ from 0 to infinity: ∫∞ 0

p(ξ ) J0 (r ξ ) exp(−ξ z)dξ . ξ

(2.25)

The function (2.25) is the harmonic function for an arbitrary auxiliary function p(ξ ) and give rise with Eq. (2.21) the integrals for the displacements and stresses: uz =

1 2μ

∫∞ (2 − 2ν + ξ z)p(ξ )ex p(−ξ z)J0 (r ξ )dξ,

(2.26)

0

∫∞ σzz = −

ξ (1 + ξ z)p(ξ )ex p(−ξ z)J0 (r ξ )dξ, 0

The formulas for displacement and for stress (2.26) on the surface z = 0 reduce to: def

W(r ) = u z (z = 0) =

1−ν μ

∫∞ p(ξ )J0 (r ξ )dξ. 0

(2.27)

46

2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks

def

S(r ) = σzz (z = 0) = −

∫∞ ξ p(ξ )J0 (r ξ )dξ,

(2.28)

0

Notable, that the functions p(ξ ) and S(r ) in (2.28) are the pair of two Hankel inverses [4]: ∫∞ p(ξ ) =

ρS(ρ)J0 (ρξ )dρ.

(2.29)

0

For the upcoming computations, the elastic constant E ∗ will be used: E∗ = def

5. Now we multiply (2.25) by

1−ν 2 = ∗. μ E

2μ , 1−ν

q(ξ ) 2(1−ν)

and integrate over ξ from 0 to infinity:

1 Θ(r, z) = 2(1 − ν)

∫∞ q(ξ )J0 (r ξ )ex p(−ξ z)dξ.

(2.30)

0

The function (2.30) is the harmonic function for an arbitrary q(ξ ) . Together with Eq. (2.21) it results the integrals for the displacements and stresses: 1 uz = 2(1 − ν)

∫∞ (2 − 2ν + ξ z)ξ q(ξ )ex p(−ξ z)J0 (r ξ )dξ,

(2.31)

0

μ σzz = − 1−ν

∫∞ (1 + ξ z)ξ 2 q(ξ )ex p(−ξ z)J0 (r ξ )dξ. 0

The formulas for displacement and stress (2.31) on the surface z = 0 reduce to: def

∫∞

W(r ) = u z (z = 0) =

ξ q(ξ )J0 (r ξ )dξ.

(2.32)

0

μ S(r ) = σzz (z = 0) = − 1−ν def

∫∞ ξ 2 q(ξ )J0 (r ξ )dξ. 0

(2.33)

2.1 Axisymmetric Elastostatics

47

Table 2.1 Elastostatics of a disk: normal displacement W , normal stress S and the auxiliary functions P and Q Input quantity 0 0.

(2.36) With Eq. (2.36), the integral Eq. (2.35) if the traction problem condenses to the final form [9]: 1−ν μ

∫∞ ρS(ρ)KW (r, ρ)dρ = W(r ).

(2.37)

0

The integral Eq. (2.37) has the kernel with the complete elliptic integral. In the problems with the given normal stress and with outer radius of the indenter of one, the function S(ρ) vanishes for ρ > 1. In this case, the upper limit in Eq. (2.35) will be 1. With the Eq. (2.37), for the given stress S(ρ) follows the unknown normal displacement W(r ) in the region ρ < 1 (inverse punch problem, or optimization task). The Eq. (2.37) will be applied below to the solution of the contact direct and inverse optimization problems, leading to the closed form solutions for the principal cases. 3. The formal inversion of the integral Eq. (2.37) accomplishes similarly. In this case, the displacements W(ρ) are given. For computation of the normal stress S(r ) we substitute Q(ξ ) from Eq. (2.34) into Eq. (2.33) and change the integration order: ⎡ ⎤ ∫∞ ∫∞ μ ⎣ ρW(ρ)J0 (ρξ )dρ ⎦ξ 2 J0 (r ξ )dξ, S(r ) = 1−ν 0 0 ⎡∞ ⎤ ∫∞ ∫ μ ρW(ρ)⎣ ξ 2 J0 (ρξ )J0 (r ξ )dξ ⎦dρ (2.38) S(r ) = 1−ν 0

0

The integral over ξ in the Eq. (2.38) reduces formally with Sonine and Schafheitlin formula to the hypergeometric function. For the formal inversion the equation from the Appendix leads to [10], Eq. 2.12.31:

50

2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks

KS (r, ρ)

⎧ ⎨−

1 ρ3 = ⎩ − 13 r

def

([ ) ] 2 · 2 F2 23 , 23 , [1], ρr 2 , for r > ρ > 0, ([ ) ] 2 · 2 F2 23 , 23 , [1], ρr 2 , for ρ > r > 0.

(2.39)

The hypergeometric function in Eq. (2.39) expresses with the complete elliptic integrals: KS (r, ρ) =

⎧ 2⎨ π⎩

( )

( )

1 K ρr − 2 2ρ 2 2 E ρr ρ (ρ 2 −r 2 )π (ρ −r ) ( ) (ρ ) 1 2r − K E ρ , 2 2 r 2 r (r −ρ ) (r −ρ 2 )2 r

, for r > ρ > 0, for ρ > r > 0.

(2.40)

With Eq. (2.40), the integral Eq. (2.38) for the displacement problem reduces to the concluding form: μ 1−ν

∫∞ ρW(ρ)KS (r, ρ)dρ = S(r ).

(2.41)

0

Unfortunately, the integral diverges for all functions W(ρ) and the Eq. (2.41) could not be applied for any practical calculation. 4. For the circular contact region Ω1 with the radius of one length unit, the total contact force is: ∫1 Ftotal = 2π

r S(r )dr.

(2.42)

0

2.1.3 Green and Collins Solution 1. In the technical problems, the shape of the indenter W(r ) is usually given a priori. Thus, the normal displacement in the contact region W(r ) is given by the shape of the indenter. On the complementary part of the boundary, both components of tractions (shear and normal stress) are specified. For the given normal displacement in the contact region W(r ), the unknown stress S(ρ) will be determined (mixed problem of elastostatics, or direct punch problem). The common way to study the mixed problem was deliberated in [11–13]. The task is to determine the stress S(ρ) in the contact region. The standard approach uses the Hankel transform P(ζ ) of the function: ∫∞ p(ζ ) =

ρS(ρ)J0 (ζρ)dρ, S(ρ) = 0 for ρ > 1 : 0

(2.43)

2.1 Axisymmetric Elastostatics

51

Because the function vanishes outside the domain Ω1 , the upper limit of integration will be 1 instead of ∞: ∫1 p(ζ ) =

ρS(ρ)J0 (ζρ)dρ

for 0 ≤ ρ ≤ 1.

(2.44)

0

The inversion of the Hankel transforms (2.44) expresses as: ∫∞ S(r ) =

ζ p(ζ )J0 (r ζ )dζ

0 ≤ r ≤ 1.

(2.45)

p(ξ ), for 0 ≤ ξ ≤ 1, 0, for 1 < ξ < ∞,

(2.46)

for

0

The auxiliary function is defined as: ∫∞

def

p(ζ ) cos(ξ ζ )dζ =

0

(

∫1 p(ζ ) =

P(ξ ) cos(ξ ζ )dξ.

(2.47)

0

The substitution of (2.44) in (2.46) delivers: ⎡∞ ⎡ ⎤ ⎤ ∫∞ ∫1 ∫1 ∫ ⎣ ρS(ρ)J0 (ζρ)dρ ⎦ cos(ξ ζ )dζ = ρS(ρ)⎣ J0 (ζρ) cos(ξ ζ )dζ ⎦dρ, 0

∫1 0

0

0

0

⎡∞ ⎤ ( ∫ def P(ξ ), for 0 ≤ ξ ≤ 1, ⎣ ⎦ ρS(ρ) J0 (ζρ) cos(ξ ζ )dζ dρ = 0, for 1 < ξ < ∞.

(2.48)

0

The evaluation of the inner integral in ](2.48) ] [ ) leads to expression in termsG of 1,0 ([ Meijer G-function G0,1 [a1 ] [] , [] [b2 ] |z [14, 15]. ∫∞ 0

| ) ( ] [ ]| ρ 2 π 1,0 [ G cos(ξ ζ )J0 (ρζ )dζ = [1/2] [] , [] [0] || 2 . ξ 0,1 ξ √

(2.49)

For the positive ρ, ζ the Meijer G-function in Eq. (2.49) reduces to the expression with the Heaviside step function (x): ∫∞ 0

H (ρ − ξ ) . cos(ξ ζ )J0 (ρζ )dζ = √ ρ2 − ξ 2

(2.50)

52

2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks

Using Eq. (2.50) in (2.48) provides the formulas for H(ξ ) in terms of the given S(ρ): ∫1 0

H (ρ − ξ ) dρ ≡ ρS(ρ) √ ρ2 − ξ 2 ∫1 0

∫1 ξ

ρS(ρ) √ dρ = P(ξ ) ρ2 − ξ 2

H (ρ − ξ ) dρ ≡ 0 ρS(ρ) √ ρ2 − ξ 2

for 0 ≤ ξ ≤ 1.

for 1 < ξ < ∞.

(2.51)

(2.52)

If the auxiliary function P(ξ ) is given, its inversion of (2.51) provides the normal stress on the contact surface: P(1) S(r ) = √ − 1 − r2

∫1 r

dξ dP(ξ ) √ 2 dξ ξ − r2

for 0 ≤ r ≤ 1.

(2.53)

2. The total force for the circular contact region Ω1 with the radius of one length unit was given in Eq. (2.42). The expression of stress in terms of the auxiliary function H(ξ ) and integration by parts leads to an alternative formula for the total contact force: ∫1 Ftotal = 2π

P(r )dr.

(2.54)

0

3. The expression for the function W(r ) could be arrived as well in terms of the auxiliary function P(ξ ). With Eq. (2.44), we combine the terms ρS(ρ)J0 (ζρ). The replacement of the resulting integral with p(ζ ) from Eq. (2.47) gives: 2 W(r ) = ∗ E

∫1 S(ρ)ρdρ 0

W(r ) =

2 E∗

∫∞

∫∞

2 J0 (ρζ )J0 (rζ )dζ → ∗ E (2.44)

0

p(ζ )J0 (r ζ )dζ , 0

(2.47)

p(ζ )J0 (r ζ )dζ → 0

∫∞

2 E∗

∫∞ ∫1 P(ξ ) cos(ξ ζ )dξ J0 (r ζ )dζ . 0

0

(2.55) The substitution of Eq. (2.50) into Eq. (2.55) expresses the normal displacement in the contact region over the auxiliary function (Barber, 28 [16]):

2.2 Series Solutions of Contact Equation

W(r ) =

2 E∗

∫1 0

53

⎧ ∫r P(ζ )dζ ⎪ ⎪ √ , for 0 ≤ r ≤ 1, H (r − ζ )P(ζ )dζ 2 ⎨ 0 r 2 −ζ 2 √ ≡ ∗ ∫1 E ⎪ P(ζ )dζ r2 − ζ2 ⎪ ⎩ √ , for r > 1. 2 2 0

(2.56)

r −ζ

The Eq. (2.56) is the integral equation of the Carleman type and allows the closed form solution for P(ζ ), if W(r ) is given by the indenter shape. This closed form solution finalizes the study of the direct contact problem. One remark about the solution of the inverse contact problem. Equation (2.56) provides the identical solution, if P(ζ ) is given by an already known S(ρ). The substitution (2.44) in (2.52) gives: 2 W(r ) = ∗ E

⎡ ⎤ ∫∞ ∫1 ⎣ ρS(ρ)J0 (ζρ)dρ ⎦ J0 (r ζ )dζ. 0

(2.57)

0

In the contact problems with the given outer radius of the indenter of one, the function S(ρ) vanishes for ρ > 1. In this case, the upper limit in Eq. (2.37) will be 1. The interchange of the order of integration and application of Sonine and Schafheitlin formula (2.40) gives: 2 W(r ) = ∗ E

∫1 0

4 ρS(ρ)KW (r, ρ)dρ ≡ π E∗

∫1 0

( ) r dρ. S(ρ)K ρ

(2.58)

Accordingly, the formula (2.58) will match the formula (2.37). Consequently, the inverse contact problem reduces to the traction problem.

2.2 Series Solutions of Contact Equation 2.2.1 Form Factor and Shape Function For briefness, the outer radius of the indenter will be temporarily put to one. The expansion for the normal stress is assumed of the form: S=

∞ ∑

Cm Sm( p) (r )

m=0

(

Sm( p) =

2σc [ (m+1+ 2p ) √ π [ (m+ 21 + 2p )

0,

for m ∈ W, 0 ≤ p; √( )2m−1+ p 1 − r2 , for 0 ≤ r ≤ 1, for r > 1.

(2.59)

54

2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks

The parameter p is referred to as the form factor. The value σc has the dimension of stress. The substitution of Eq. (2.59) into (2.44) and (2.46) leads to the expressions for the auxiliary shape function: ( ) 2m+2p+1 ( σc 2 p) J 2m+ p+1 (ζ ). p(ζ ) = √ [ m+1+ 2 2 π ζ √ ( ) 2σc 2m+ p P(ζ ) = 1 − ζ2 . π

(2.60)

The function Jv (ζ ) is the Bessel function. With the (2.58) expression and normal displacement on the contact surface. The normal displacement on the contact surface is for 0 ≤ r ≤ 1: W(r ) =

∞ ∑

Cm , Wm( p) (r ), Wm( p) (r )

m=0

~ = W

∞ ∑

~m( p) , W ~m( p) Cm W

∫1 =2

m=0

2σc = ∗ 2 F1 E

([

r Wm( p) (r )dr

0

) ] 1 p 2 , −m − , [1], r , 2 2

) . ( p 2σc 2[ m + 2 + 23 ( ) = ∗ √ E π [ m + 2p + 2 (2.61)

~ W ~m symbolize the total and the partial average displaceThe values W, ment correspondingly. For future integrations, we use the representation of the hypergeometric function as an infinite series: ( p)

2 F 1 ([a, b], [c], x) =

∞ ∑ (a)k (b)k x k , (c)k k! k=0

where (a)k is the Pochhammer symbol, or falling factorial [17]: def

(a)m =

[(a + m) . [(a)

( p)

Alternatively, the expression for Wm (r ) could be expressed in (

Wm( p) (r )

(α,β)

where Jacobi polynomials Pn def

Pn(α,β) (z) =

)

p ( ) 2σc 0,− 2n+1+ 2 1 − 2r 2 , = ∗ P− 1 2 E

are defined as [18]:

) ( 1−z (α + 1)n . F n + 1 + α + β], + 1], [−n, [α 1 n! 2 2

2.2 Series Solutions of Contact Equation

55 ( p)

For positive m and even p, the formula (2.61) for Wm (r ) could be expressed equivalently: Wm( p) (r )

2σc = ∗ E

( ) √ ( )m+ 2p 2 − r2 2 for 0 ≤ r ≤ 1, 1−r · Pm+ p √ 2 2 1 − r2

where the function Pm (z) is the Legendre polynomial [19]: def

Pn (z) =

( ) )n 1 dn ( 2 1−z z − 1 ≡ 2 F1 [−n, n + 1], [1], . 2m n! dz n 2

The above expressions will be used for the calculations of elastic energy in the contact region.

2.2.2 Direct Integration The normal displacement in the contact region could be calculated after the direct application of (2.58). The integration over ρ of (2.37) with the expression for the stress ( p) ( p) ( p) Sm from Eq. (2.58) leads to two dimensionless functions ψm , Φm for 0 ≤ r ≤ 1: ( ) ( ) ∫1 ∫1 ) 2m+ p−1 [ m + 2p + 1 ( 2σc 4 4 r ( p) ( ) 1 − ρ 2 2 dρ, dρ = ∗ Sm (ρ)K √ p 1 ∗ πE ρ E π π [ m+ 2 +2 0 0 ( ) ] r 2σc [ K dρ = ∗ ψm( p) (r ) + iΦ(mp) (r ) , (2.62) ρ E ( ([ ) ) √ ] ( )m+ 2p 2 − r2 1 p , , −m − , [1], r 2 ≡ 1 − r2 · Pm+ p √ ψm( p) (r ) = 2 F2 2 2 2 2 1 − r2 (2.63) ( ) ([ ) ] [ ] 4[ m + p + 1 r 1 p 3 3 , , , r 2 . (2.64) Φ(mp) (r ) = − 3/2 ( 2 p 1 ) · 2 F2 1, 1, − m − 2 2 2 2 π [ m+ 2 +2 The function Pm (z) is the Legendre polynomial. The real part of the expression (2.62) in identically equal to the normal displacement from Eq. (2.61): 2σc ( p) ψ (r ) ≡ Wm( p) (r ). E∗ m The imaginary part plays no role for the solutions of mechanical problems and for the optimization purposes.

56

2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks

2.2.3 Reaction Force as Function of Form Factor The partial contact force over the circular contact region Ω1 with the radius of one ( p)

length unit will be determined for each term Sm . The partial contact force the term with index m evaluates after the substitution (2.59) into (2.42) as:

f

( p) def

m

= 2π

∫1

r Sm( p) (r )dr

0

Ftotal =

( √ [ m+ = 2σc π ( [ m+ ∞ ∑

Cm

m=0

f

( p) m

p 2 p 2

) +1 ), + 23

f

( p)

m

for

(2.65)

.

Magnification factor is defined as the ratio of contact forces for two succeeding terms:

f f

( p)

m ( p)

2m + p ≡ , F(m, p) = 2m + p + 1

m−1

f f

( p) 1 ( p)

( √ p) 3 π [ 1+m+ 2 ). ( ≡ 4 [ m + 2p + 23

(2.66)

m

The magnification factor allows to evaluate contact forces as the function of m. As Eq. (2.66) shows, the contact forces reduce with the increasing index m. This remark is essential for the optimization procedure in the following paragraphs.

2.2.4 Elastic Energy as Function of Form Factor In the absence of any energy dissipation, this work is stored in the elastic half-space in the form of strain energy. Consider the contact region Ω1 for 0 ≤ r ≤ 1. For any fixed form factor p, the elastic energy reads: Etotal

8π σc2 = E∗

∫1 e(r )r dr, 0

e=

(2.67)

∗

E W(r )S(r ) . 4σc2 2

The total strain energy Etotal for the series (2.59) and (2.61) is the sum of partial ( p) strain energy Em for each term of the sum. The total strain energy and the strain energy for each term with the number m correspondingly are:

2.2 Series Solutions of Contact Equation

Etotal =

∞ ∑

57

Cm Em( p) ,

m=0

Em( p)

8π σc2 = E∗

∫1 0

( p)

( p)

E ∗ Wm Sm . r em( p) dr = 4σc2 2

(2.68)

For the calculation of the energy for each term m the series integration method is applied. where (a)k is the Pochhammer symbol, or falling factorial [17]. With (2.13), the normal displacement for each term m is: Wm( p) =

4σc2 F2 E∗ 2

( p)

wm,k =

([

) ] ∞ 1 4σ 2 ∑ ( p) , −M , [1], r 2 = ∗c w 2 E k=0 m,k

with M = m +

(2k)! def [(k − M) . (−M)k r 2k , (−M)k = 3 [(−M) · (k!)

4k

p ; 2 (2.69)

( p)

The term wm,k in Eq. (2.69) can be calculated using Eqs. (2.11) and (2.31). With gamma function, the terms of the infinite series in Eq. (2.70) read: ( p)

wm,k =

[(2k + 1) · [(k − M) 2k [(2k + 1) · (−M)k 2k r , r = k 4k · ([(k + 1))3 4 · ([(k + 1))3 · [(−M) √ ( p) ( )2M−1 Sm [(M + 1) ) =√ ( . 1 − r2 1 σ0 π[ M + 2 ( p)

(2.70)

( p)

Integration of the product wm,k and Sm delivers: ( p) em,k

E∗ = 2π 4σc2

∫1

( p)

r

( p)

Sm wm,k 2

dr

0

) ( sin(π (M + 1)) [ 2 (M + 1) · [(k − M)[ k + 21 ) ( = · . 2π [(k + 1) · [ M + k + 23

(2.71)

( p)

Now the infinite series of the expression Em can be calculated with Eq. (2.71): Em( p) =

√ ∞ 4σc2 ∑ ( p) 4σc2 π [(2m + p + 1) ). · ( e = E ∗ k=0 m,k E ∗ 2 [ 2m + p + 23

(2.72)

For the contact region Ω1 with the radius of one length unit. For a fixed form factor p, we get the expressions:

58

2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks

Etotal =

∞ ∑ m=0

Cm Em( p)

√ ∞ [(2m + p + 1) 2 π σc2 ∑ ). Cm ( = ∗ E [ 2m + p + 23 m=0

(2.73)

The stiffness, or spring rate of the contact region could be calculated with the above formulas: −1 ctotal =

2 2 Ftotal Ctotal Ftotal d 2 Etotal −1 , Ctotal = ctotal , E = = . total 2 2ctotal 2 dFtotal

(2.74)

The energy capacity of the contact Ctotal is the reciprocal spring rate. This value is essential for the technical applications. The seal or gaskets with the lower stiffness and higher energy capacity are customarily advantageous because of their insensitivity to wear.

2.2.5 Dependence of Reaction Force and Elastic Energy Upon the Indenter Radius The above formulas were derived for the compactness for the radius one of contact region. The shape of the indenter could be proportionally radially scaled, such that for the same axial coordinate the corresponding radii will be increased proportionally to its radius R. In other words, the indenter will be homothetically radially scaled (Table 2.2). For example, for the contact areas of different radii R but with the same normal penetration, the force is proportional to the area π R 2 of the contact region. The stored elastic energy is proportional to the volume R 3 of the contact region. Thus, these values will get the scaling factor R 2 and R 3 correspondingly. The coefficients in the Eq. (2.65) are referred as magnification factors. The magnification factors are shown at (Fig. 2.2).

2.3 Optimization of Maximal Stress for Fixed Contact Force in Circular Contact Region 2.3.1 Solutions with Form Factor p = 0 According to the common method, we the power series solution Eq. (2.59). The behavior of the stress on the outer radius of the indenter decisively depends upon the form factor p. The outer radius of the circular contact region is R = 1. The first important case for the optimization is p = 0. With this value for form factor p follows that M = m. Accordingly, Eqs. (2.59) and (2.61) lead to the expressions for the normal stress and normal displacement on the contact surface:

2.3 Optimization of Maximal Stress for Fixed Contact Force in Circular …

59

Table 2.2 Reaction force and elastic energy as functions of indenter radius R Equations

(2.59)

( p) Sm (r )

(2.61)

Wm (r )

(2.42)

Ftotal = m=0

(2.65)

f

Cm

√ ) (

([ 1 2σc E ∗ 2 F1 2 , −m

f

2π

( p)

∫1

1 − r2

)2m−1+ p

) p] 2 2 , [1], r

−

r S (r )dr

2σc [ (m+1+ 2p ) ( ) √ π [ m+ 21 + 2p 2σc R E ∗ 2 F1

2π

0

∫R

([

√(

1 2 , −m

1−

−

r2 R2

)2m−1+ p

p] r2 2 , [1], R 2

)

,

r S (r )dr

0

m

√ [ (m+ p +1) 2 π ( p2 3 ) σc

( p)

√ [ (m+ p +1) 2 π ( p2 3 ) σc R 2

[ m+ 2 + 2

Etotal = ∞ ∑

0≤r ≤ R

(

m

(2.67)

Contact region Ω R

0≤r ≤1

2σc [ m+1+ 2p ) √ ( π[ m+ 21 + 2p

( p)

∞ ∑

Contact region Ω1 , R = 1

8π σc2 E∗

( p) C m Em

∫1

[ m+ 2 + 2

8π σc2 E∗

e(r )r dr

0

∫R

e(r )r dr

0

m=0

(2.72)

√ 2 π

( p)

Em

[(2m+ p+1)) ( [ 2m+ p+ 23

·

√ 2 π

σc2 E∗

[(2m+ p+1)) ( [ 2m+ p+ 23

·

σc2 E∗

R3

Fig. 2.2 Magnification factors √ p/2) ) ( F(m, p) = 3 4 π [(1+m+ 3+ p [ m+

2

( Sm(0) =

Wm(0)

2σc √π[(m+1) [ (m+ 21 ) 0

√(

1 − r2

)2m−1

, for 0 ≤ r ≤ 1, for r > 1; ( ) 2

⎧ ([ 1 ) ( )m ] 2σc ⎨ 2 F2 2 , −m , [1], r 2 ≡ 1 − r 2 2 · Pm ([ ] [ ] ) = ∗ E ⎩ √1πr [[(1+m) · F 1 , 1 , 23 + m , r12 , ( 23 +m ) 2 2 2 2

2−r √ 2 1−r 2

(2.75)

, for 0 ≤ r ≤ 1, for r > 1. (2.76)

60

2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks

The function Pm (z) is the Legendre polynomial. Equation (2.76) represents the normal displacement Wm for each term of (2.16). Therefore, the total normal displacement is the sum of partial normal displacements for each term. The substitution of Eq. (2.76) provides the total normal displacement as the linear combination: ([ ) ] ∞ 1 2σc ∑ 2 , −m , [1], r . W= ∗ Cm · 2 F2 E m=0 2

(2.77)

The contact force for each term Eq. (2.65) evaluates for s = 0 as:

f

(0) m

f f

√ [(m + 1) ), = 2σc π ( [ m + 23 (0)

=

m (0)

2m + 2 < 1. 2m + 3

(2.78)

(2.79)

m+1

The next step is the calculation of the total normal force Ftotal , which causes the contact. Evidently, the total normal force Ftotal for the series (2.59) is the sum of partial normal forces (2.78):

f

(0) m

for each term of the sum. The total normal force reads with

Ftotal =

∞ ∑ m=0

Cm

f

(0) m

= 2σc

∞ ∑

√ [(m + 1) ). Cn π ( [ m + 23 m=0

(2.80)

The expressions (2.77) and (2.80) will be used later for the solution of the optimization problem. Some first base functions are polynomials. The base functions for the form factor p = 0 are displayed in Table 2.3 (Figs. 2.3, 2.4, 2.5 and 2.6).

2.3.2 Solutions with Form Factor p = 1/2 The second important case for the optimization is the form factor p = 1/2. The expressions for the normal stress (2.59) and normal displacement (2.61) with this value of the form factor are: ( [ m+ 5 ( ( 4 ) 1 − r 2 )m−1/4 , for 0 ≤ r ≤ 1; √ (1/2) π[ (m+ 34 ) (2.81) Sm = 2σc 0, for r > 1,

2.3 Optimization of Maximal Stress for Fixed Contact Force in Circular … (0)

61

(0)

Table 2.3 Base stress functions Sm , partial normal displacements Wm and partial normal forces

f

(0) m

m

0

1

2

(0)

(0) E∗ 2σc Wm

(

(

(

Sm

2σc

f

(0)

E ∗ (0) E 4σc2 m

m

2σc

1, 2 π

arcsin

(1) r

√1 π 1−r 2

0 < r < 1;

2

1

4 3

8 15

16π 15

128 315

, r > 1.

r2 2 , √ 2 2 r −1 − r π−2 π

1−

3r 4 ( 2 )√ 8 2 3 r −2 r −1 4π

arcsin

(1) r

2 π

0 < r < 1; , r > 1.

1 − r2 +

0 < r < 1;

−

−3r 4 +8r 2 −8 4π

( )1/2 1 − r2

( ) arcsin r1 , r > 1.

8 3π

( )3/2 1 − r2

Form factor p = 0 Fig. 2.3 Dimensionless stress Sm(0) /2σc in the contact region (p = 0)

Wm(1/2)

2σc = ∗ E

(

([

]

)

1 1 2 2 F2 2 , −m − 4 , [1], r , 5 ([ ] +m [ (4 ) · F 1, 1 , [7 √1 4 πr [ ( 74 +m ) 2 2 2 2

for 0 ≤ r ≤ 1; ] 1) + m , r 2 , for r > 1.

(2.82)

The stress intensity exponent is equal for nonnegative m to m − 1/4. That is, for ( )−1/4 m = 0 the stresses on the outer radius r = 1 are infinite 1 − r 2 . All other stress terms vanish for the positive m on the outer radius. The contact force for each term (2.81) evaluates as:

62

2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks

Fig. 2.4 Dimensionless stress Sm(0) /2σc in the contact region (p = 0), for a unit force Ftotal = 1

n=1

n=0

Fig. 2.5 Indenters shapes (0) E∗ 2σc Wm in the contact region (p = 0) for E ∗ = 1

n=0 n=1

f

(1/2) m

) ( √ [ m + 45 ). ( = 2σc π [ m + 47

(2.83)

2.3 Optimization of Maximal Stress for Fixed Contact Force in Circular …

63

Fig. 2.6 Indenters shapes (0) E∗ 2σc Wm in the contact region (p = 0), for a unit force for E ∗ = 1

n=0 n=1

f f

(1/2) m (1/2)

=

8m + 5 < 1. 8m + 7

(2.84)

m+1

According to the common method, we seek the solution that is represented as a power series: S=

∞ ∑

Cm Sm(1/2)

m=0

( ) )m−1/4 [ m + 45 ( ) 1 − r2 ≡ 2σc Cm √ ( . 3 π[ m + 4 m=0 ∞ ∑

(2.85)

Supposed, that the radius of convergence of the power series (2.59) is at least (1/2) for each term of 1. Equation (2.61) represents the normal displacement Wm (2.59). Accordingly, the total normal displacement for the series is the sum of partial normal displacements for each term. The total normal displacement as the linear combination: W=

] ([ ) ∞ 1 1 4σc ∑ 2 C · F , −m − , r . [1], m 2 2 E ∗ m=0 2 4

(2.86)

The next step is the calculation of the total normal force Ftotal , which causes the contact. Evidently, the total normal force Ftotal is the sum of partial normal forces fM :

64

2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks

Ftotal =

∞ ∑

Cm

m=0

f

(1/2)

m

( ) √ [ m + 45 ). = 2σc Cn π ( [ m + 47 m=0

The base functions for the form factor p = 2.8, 2.9 and 2.10). (1/2)

Table 2.4 Base stress functions Sm forces m

0

f

m

1 2

(2.87)

are displayed in Table 2.4 (Figs. 2.7,

(1/2)

, partial normal displacements Wm

and partial normal

,M = m + 1/2 (1/2)

f

(1/2) E∗ 2σc Wm

Sm

⎧ ([ ] ) ⎨ 2 F2 21 , − 41 , [1], r 2 , 0 < r < 1; ([ √ ] [7] 1 ) 2π 1 ) 1 1 ( ⎩ 3r 2 3 · 2 F2 2 , 2 , 4 , r 2 r > 1.

√ ( ) 2π 1/4 3 2 4[ 4 (1−r 2 )

2σc √ 3/2 2π( ) 3[ 2 34

π 4

√ ( )3/4 5 2π 1−r 2 ( ) 12[ 2 43

√ 5 2π(3/2) 21[ 2 43

5π 32

[

1

∞ ∑

([

2σc

4

]

)

5 1 r2 , 2 F2 2 , − 4 , [1], ([ √ ] [ 11 ] 1 ) 5 2π 1 1 1 ( ) 21r [ 2 3 · 2 F2 2 , 2 , 4 , r2 , 4

0 < r < 1; r > 1.

(1/2)

E ∗ (1/2) E 4σc2 m

m

Form factor p = 1/2

(1/2)

Fig. 2.7 Dimensionless stress Sm compression)

/2σc in the contact region (p = 1/2) (Negative stress means

2.3 Optimization of Maximal Stress for Fixed Contact Force in Circular …

65

Fig. 2.8 Dimensionless (1/2) stress Sm /2σc in the contact region (p = 1/2), for a unit force

Fig. 2.9 Indenters shapes (1/2) E∗ in the contact 2σc Wm region (p = 1/2) for E ∗ = 1

2.3.3 Solutions with Form Factor p = 1 The second important case for the optimization is the form factor p = 1. In this case, the expressions for the normal stress and normal displacement on the contact surface are:

66

2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks

Fig. 2.10 Indenters shapes

(1/2) E∗ 2σc Wm

(

Wm(1)

2σc = ∗ E

Sm(1)

= 2σc

(

([

in the contact region (p = 1/2), for a unit force for E ∗ = 1

[ (m+ 23 ) ( √ 1 π [(m+2)

− r2

0, ]

)

1 1 2 2 F2 2 , −m − 2 , [1], r , 3 ([ ] +m [ ( ) 2 √1 · F 1 , 1 , [2 πr [(2+m) 2 2 2 2

)m

, for 0 ≤ r ≤ 1; for r > 1,

+ m],

1 r2

The contact force for each term evaluates as: ( ) (1) √ [ m + 23 , = 2σc π m [(m + 2)

f

f f

(1)

m (1)

=

2m + 3 < 1. 2m + 4

for 0 ≤ r ≤ 1; ) . , for r > 1.

(2.88)

(2.89)

(2.90)

(2.91)

m+1

According to the common method, we seek the solution that represents a power series: ) ( ∞ ∞ ∑ ∑ )m [ m + 23 ( (1) 1 − r2 . S= Cm Sm ≡ 2σc Cm √ (2.92) π [(m + 2) m=0 m=0 Supposed, that the radius of convergence of the power series is at least 1. The total normal displacement is the sum of partial normal displacements:

2.3 Optimization of Maximal Stress for Fixed Contact Force in Circular …

67 (1)

Table 2.5 Dimensionless stress functions Sm , partial normal displacements Wm and partial normal forces m

f

(1) m

M =m+1

f

(1)

(1) E∗ 2σc Wm

Sm

2σc

(1)

E ∗ (1) E 4σc2 m

m

0

1

(

(

2σc 2 π(E(r ), ) ( 1 ) 2r ( 1 ) 2 1−r 2 πr K r + π E r ,

1 2

for 0 < r < 1;

for r > 1. ) ( ) ( 2 4 2 2 1 K(r ), for 0 < r < 1; 3π 2 − r E(r ) + 3π ( r − ) 2 ( ( ) ) 4 2 4r 2−r 4r −10r +6 K r1 + 3π E r1 , for r > 1. 3πr

3 4

(

1 − r2

)

π 2

2 3

3π 8

16 35

Form factor p = 1

W=

([ ) ] ∞ 1 1 2σc ∑ 2 . , −m − , r C · F [1], m 2 2 E ∗ m=0 2 2

(2.93)

The next step is the calculation of the total normal force Ftotal , which causes the contact. Evidently, the total normal force Ftotal for the series (2.16) is the sum of partial normal forces form factor p = 1 is:

f

(1)

m

Ftotal =

for each term of the sum. The total normal force for the ∞ ∑ m=0

Cm

f

(1) m

) ( ∞ √ ∑ [ m + 23 . = 2σc π Cm [(m + 2) m=0

(2.94)

The base functions for the form factor p = 1 are displayed in Table 2.5. The base functions express over complete elliptic integral (Figs. 2.11, 2.12, 2.13 and 2.14).

2.3.4 Optimization of Total Force and Contact Stress 1. The first optimization problem is the maximization of total force Ftotal for the restricted maximal stress over the contact region: Ftotal → max, Cn

max S (r ) ≤ S0 .

0≤r ≤1

The ultimate admissible stress in the contact region is S0 > 0. The dual optimization [20] aim is the minimization of the maximal stress over the contact region for the given fixed total force F0 :

68

2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks

n=1

n=0

Fig. 2.11 Stresses Sm(1) /2σc in the contact region ( p = 1)

Fig. 2.12 Stresses Sm(1) /2σc in the contact region (p = 1), for a unit force

2.3 Optimization of Maximal Stress for Fixed Contact Force in Circular …

Fig. 2.13 Indenters shapes

(1) E∗ 2σc Wm

in the contact region ( p = 1, E ∗ = 1)

Fig. 2.14 Indenters shapes

(1) E∗ 2σc Wm

in the contact region (p = 1), for a unit force,E ∗ = 1

69

70

2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks

max S (r ) → min,

0≤r ≤1

Ftotal ≥ F0 .

Cn

The expressions (2.75), (2.81) and (2.88) allow to establish the inequalities for the stiffness of the contact, contact force and maximal stress in the circular contact region. For the optimization purposes, the expressions for the ratios of elastic energy to maximal stress and to contact force are essential. For each given p, we assume the total force (2.65), total energy of elastic deformation (2.73) and the stress in the contact region are the linear functions of the coefficients Cm : Ftotal = Etotal =

∞ ∑ m=0 ∞ ∑

Cm

f

( p) m

,

f

( p) m

Cm Em( p) , Em( p)

m=0

( ) √ [ m + 2p + 1 ), ≡ 2σc π ( [ m + 2p + 23 √ π 4σc2 Γ (2m + p + 1) ), ( = 2 E ∗ Γ 2m + p + 23

( p) p ) 2m−1+ 2σc [ m + 1 + 2 ( 2 ( p) ( p) 2 ) ( S(r ) = , Cm Sm (r ), Sm (r ) = √ 1 − r π [ m + 21 + 2p m=0 ( ∞ p) ∑ 2σc [ m + 1 + 2 ( p) ( p) ( ). S(0) = Cm Sm (0). Sm (0) = √ π [ m + 21 + 2p m=0

(2.95)

∞ ∑

As follows from (2.96),

f

( p)

m

( p)

, Em are the reducing functions of the indexm for ( p)

each value of parameter p, but Sm (0) is the increasing function:

f f

( p)

m ( p)

≡

2m + p < 1, 2m + p + 1

≡

2m + 2 p < 1, 2m + 2 p + 1

≡

2m + p > 1. 2m + p − 1

m−1 ( p)

Em

( p) Em−1 ( p) Sm (0) ( p) Sm−1 (0)

(2.96)

The first optimization problem is the linear programming problems with an infinite number of the design variables Cn , n = 0, 1... For briefness, we set Cm = 1 and treat

f

( p)

the , n = 0, 1 as the design variables. The equations of the linear programming n problem simplify to:

2.3 Optimization of Maximal Stress for Fixed Contact Force in Circular … ∞ ∑

Ftotal =

m=0

S(r ) =

f

( p)

→ max,

f

m

( p)

71

max S(r ) ≤ S0 ,

0≤r ≤1

(2.97)

m

∞ ∑

) 2m + 1 + p ( 1 − r2 2π m=0

2m−1+ p 2

f m( p) .

The solution of the forest optimization problem is trivial, because for any constrained function its integral is not greater than the product of the integration area and the constraint value. The lowest possible form factor for the optimal setting of the contact stresses is p = 1. The solution of linear programming problem leads to the trivial result:

f f

( p)

0 ( p) n

= F0 , = 0,

for n = 1, 2, ..

(2.98)

This means, that the stress distribution must be as smooth as possible. For the given total force Ftotal = for the given force:

f

(1) 0

, the choice (2.98) leads to the minimal contact stress

max S(r ) ≥

0≤r ≤1

F 1+ p F0 = 0 = σc . 2π π

As evident form Table 2.1, the optimal shape of the indenter must be (Fig. 2.15). Wopt = W0(1) =

2F0 E(r ). π2 E∗

2.3.5 Optimization of Stiffness of Contact Region 1. One remark about the relation of the stored deformation energy to the total force of the elastic contact. As follows from the Table 2.2, the ratio of the elastic energy to the total force is the function with the dimension of radius R. This function reduces with the increasing parameters p, m. ( p)

Em

f

( p)

( ) p Er e f 22m+1+ p [ 2 m + 2 + 23 σc E0(1) 8 σc ) ( = R ≤ ≡ = R, (1) 2m + p + 1 [ 2m + p + 23 E ∗ 3π E ∗

m

Fr e f ≡

f

f

(1) 0

= π σc R 2 ,

Er e f ≡ E0(1) =

0 8R 3 σc2 . 3 E∗

f

ref

(2.99)

72

2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks

Fig. 2.15 Solutions of the non-negative und constrained optimization problems (left, red— constrained optimization (2.98), green non-negative optimization). Left are the results of quadratic approximations with different stress functions. On all figures the abscise symbolizes the radius and the ordinate is the contact stress

2.3 Optimization of Maximal Stress for Fixed Contact Force in Circular …

73

Notably, that the maximal value for the ratio of the elastic energy to the total force is equal to: ( p)

Em

f

( p) m

≤

E0(1)

f

(1)

=

8 σc R. 3π E ∗

0

This result is important for the optimization. Namely, the zeroth term in the expansion of stress (2.95) provides the most effective solution. 2. To prevent the singularities on the edges of the indenter, we put the shape factor in all optimization tasks to p = 1. The design parameters are the coefficients Cn in the expansions of stresses (2.59). With the formulas from Table 2.2, one can state, that all optimization tasks are the linear mathematical programming problems. Keeping into account these relations, we can formulate several optimization problems. One task is the maximization of total force Ftotal for the restricted maximal stress Smax and minimal value energy of the elastic deformation Emin : Ftotal → max, Cn

Emin ≤ Etotal ,

max S (r ) ≤ Smax

0≤r ≤1

The solution of the second optimization problem gives once again Eq. (2.98). For the given radius, the optimal distribution of stress is constant. The value of the constant contact stress depends on the relation between the constraints Emin and Smax in the optimization problem: If Emin
0, Sult.l > 0 for pressure and tension: max Etotal → Emax Cn

Ftotal ≤ F0 ,

max S (r ) ≤ Sult. p ,

0≤r ≤1

min S (r ) ≥ −Sult.t .

0≤r ≤1

The inverse optimization task is the minimization energy of the elastic deformation Emin for given total force F0 , the ultimate stress Sult : min Etotal → Emin Cn

Ftotal ≥ F0 ,

max S (r ) ≤ Sult. p ,

0≤r ≤1

min S (r ) ≥ −Sult.t .

0≤r ≤1

The constraint Ftotal = F0 must be consistent with the restrictions for minimal and maximal stresses. Namely, if the radius of the contact region is R, the force constant must satisfy. −π R 2 Sult.t ≤ F0 ≤ π R 2 Sult. p . The simple numerical solution of the linear programming task was performed. The constants for the solution were: R = 1, E ∗ = 1, Sult. p = Sult.t = 1. The absolute values of tension and compression stresses match. As displayed on Fig. 2.16, to maximize the elastic energy for the given force, the optimal indenter must occupy the region around its axis. If for the given force resulting contact stress will be too high, the radius must be increased. This design guarantees the lowest stiffness of the contact pair. Alternatively, if the stiffness of contact must be as high as possible, the optimal indenter must contact in the outer region. If the contact stress appears too high, the indenter should be extended to its axis. If the full available space is filled, both designs match and we come to the edge points of the diagram, as shown on Fig. 2.17. On Fig. 2.18 are exhibited the results of the optimization problems for the constants R = 1, E ∗ = 1, Sult. p = 1, S ult.t = 0. No tension is allowed in the contact region.

2.3 Optimization of Maximal Stress for Fixed Contact Force in Circular …

75

Fig. 2.16 Stress patterns for energy optimization. For the different contact forces F0 , the figures present the radial distributions of contact stresses for minimal (above) and maximal (below) stored energies. The above figures show the stresses that correspond to the maximal stiffness of contact region. The stresses for the minimal stiffness are on figures below. On all figures the abscise symbolizes the radius and the ordinate is the contact stress

Figure 2.19 shows the relation between the compression force and stored energy of elastic deformation. The numerical results pave the way for an analytical treatment of the optimization problem. It allows to recognize the character features of the optimization process.

76

2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks

Fig. 2.17 Minimal and maximal elastic energies for different given contact forces −π ≤ F0 ≤ π . The red curve shows maximal possible energies as the function of the normal force. The blue curve demonstrates the dependency of minimal stored energy for different force. The abscise symbolizes the total force and the ordinate is the minimal and maximal possible elastic energies. Sult. p = Sult.t = 1

We demonstrate below in this Chapter the closed-form solution of the corresponding optimization task for the ring-shape indenters. The solution for the optimal circular indenter results as the special case and for briefness is omitted.

2.4 Optimization of Ring-Shape Indenters 2.4.1 Stored Elastic Energy, Spring Rate and Contact Force of Concentric Ring-Shaped Indenters For the optimization we study the circular ring region. The ring-shaped region is load by the rigid indenter with circular inner and outer boundaries of the contact region [21]. The form factor is p = 1. The radius of the inner circular edge of the contact region is R1 and outer radius is R2 . For the constant stress in contact region Ω R2 : ( S0 (r ) =

( ) σc , R1 ≤ r ≤ R2 , Ftotal = πσc R22 − R12 . 0, r > R1 > 0

(2.100)

To determine the shape of the single indenter, which causes the constant contact stress, we use the Eq. (2.58) with complete elliptic integral

2.4 Optimization of Ring-Shape Indenters

77

Fig. 2.18 Stress patterns for energy optimization. For the different contact forces F0 , the figures present the radial distributions of contact stresses for minimal (above) and maximal (below) stored energies. The above figures show the stresses that correspond to the maximal stiffness of contact region. The stresses for the minimal stiffness are on figures below. On all figures the abscise symbolizes the radius and the ordinate is the contact stress

78

2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks

Fig. 2.19 Minimal and maximal elastic energies for different given contact forces 0 ≤ F 0 ≤ π . The red curve shows maximal possible energies as the function of the normal force. The blue curve demonstrates the dependency of minimal stored energy for different force. The abscise symbolizes the total force and the ordinate is the minimal and maximal possible elastic energies. Sult. p = 1, Sult.t = 0

[ ( ) ( )] ( ) 4σc r r r R − R , dρ = E E 2 1 ρ π E∗ R2 R1 R1 ( ) ( ) 1 1 2σc σc r2 + o r2 . (2.101) − = ∗ (R2 − R1 ) + ∗ E 2E R1 R2

4 W(r ) = π E∗ W|r =0

∫R2

S0 (r )K

The total stored elastic energy Etotal is the product of the reference energy Er e f and the the dimensionless function Ξ(∈ ) . The parameter ∈ is the ratio of the outer radius R2 to the innere radius R1 of the ring-shaped region. The stored energy of elastic deformation due to the force (2.100) is equal to: Etotal = Er∈ e f Ξ(∈ ),

lim E ∈ →∞ total

= Er e f ,

) ( ( ) ( ) 1 1 1 + ck21 K(∈ ) + ce12 E + ce21 E(∈ ) + c0 , Ξ() = Re ck12 K 2 ∈ ∈

2.4 Optimization of Ring-Shape Indenters

79

R2 8σc2 R23 = > 1, 3E ∗ R1 ∈ 2 − 1 ck12 , ck12 = , ck21 = − 2 ∈ ∈ 2 + 1 ce12 ce12 = − 2 , ce21 = , ∈ ) ( c0 = 2∈ −3 (∈ + 1) ∈ 2 − ∈ + 1 . Er∈ e f =

(2.102)

If the inner radius vanishes and the outer radius remains constant R2 = R, we come to the single indenter. The parameter increases infinitely ∈→ ∞. For the elastic energy, the expression (2.102) coincides with formula (2.99) in the limit case: lim Etotal = Er e f =

R1 = 0 R2 = R

8R 3 σc2 . 3 E∗

Analogously, for the disappearing inner radius and given outer radius R2 = R, the force ) ( Ftotal = 1 − ∈ −2 Fr e f ,

lim Ftotal = Fr e f = πσc R22 .

R1 = 0 R2 = R

(2.103)

For the optimization purposes, we need the ratio of the following dimensionless functions: Etotal = Ξ(∈ ), Er e f E(∈ ) =

Ftotal = 1 − ∈ −2 , Fr e f

Fr e f Etotal Ftotal Er e f

=

(2.104)

Ξ(∈ ) . 1 − ∈ −2

The function varies from 0 to 1 for the values 1 < ∈ < ∞ : lim E(∈ ) = 0,

∈→1

lim E(∈ ) = 1, 0 < E(∈ ) < 1.

∈→∞

(2.105)

The case ∈ → 1 leads to the degeneration of the ring to an infinitely thin circle with approximately equal inner and outer radii. If ∈ → ∞ , the inner radius disappears and FFtotal are and the indenter turns into a single circular punch. The functions EEtotal ref ref shown on Fig. 2.20. The function E(∈ ) is displayed on Fig. 2.21. The inner and outer radii for the equidistantly spaced concentric rings are correspondingly: R1 = R N ,

R2 = 1 + R N ,

(2.106)

80

2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks

Fig. 2.20 Dimensionless elastic energy

E Er e f

of the ratio outer radius to inner radius ∈ =

, and dimensionless contact force

R2 R1

as functions

R2 R1

Fig. 2.21 Ratio of elastic energy to contact force to inner radius ∈ =

Ftotal Fr e f

Fr e f Etotal Ftotal Er e f

as the function of the ratio outer radius

2.4 Optimization of Ring-Shape Indenters

81

for R > 1, N = 1, 2, 3, . . .. The value R is the radial step, and the width of each indenter is one: The stress the ring-shaped region with the inner radius R N is given with the formula (2.100): ( S N (ρ) =

for N R < ρ ≤ 1 + N R; σc , 0, for 1 + N R < ρ < R + N R.

(2.107)

To determine the shape of the ring-shaped indenter with the number N , we use the Eqs. (2.101) and (2.103). [ ) ( ( r )] 4σc r . − NR ·E wN = (N R + 1) · E π E∗ NR +1 NR

(2.108)

The shapes and the stresses under the optimal indenters are shown on Fig. 2.22. Figure 2.23 demonstrates the shape of the optimal indenter and the deformation of the free surface under the action of the indenter. The maximal displacement of the indenter in the middle of the ring follows from (2.101): W|r =0 = w N (r = 0) =

) ( ( ) r2 2σc + o r2 1 + ∗ E 4N R(N R + 1)

for N ≥ 1. (2.109)

The total force is equal to:

Fig. 2.22 R = 2: concentric indenters σc = 1, E ∗ = 1

82

2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks

Fig. 2.23 Concentric indenters 1+N ∫ R

Ftotal = 2π

r S N (r )dr = π σc (1 + 2N )R 2 .

(2.110)

NR

The elastic energy follows from (2.102) after the substitution (2.103): EN =

8σc2 R23 1 + RN . Ξ(∈ ), ∈ = 3E ∗ RN

(2.111)

2.4.2 Multiple Concentric Indenters The multiple-rings indenter consists of several concentric rings, which simultaneously push towards the elastic foundation. All rings are rigidly connected to each other, such that the axial shift of all rigs is equal. The reaction force distributes between the rings proportionally to their contact areas, because the exerted stress under the rings is constant. For the contact stress, the expression for the stress function with Eq. (2.104) will be: S ~N (ρ) =

N ∑

Sk (ρ).

(2.112)

k=0

According to (2.105), the optimal shape of the multiple-rings indenter, which causes the constant pressure under all concentric rings is:

2.4 Optimization of Ring-Shape Indenters

83

Fig. 2.24 Multiple concentric rigidly connected indenters, σc = 1, E ∗ = 1

) ( N [ ( ρ )] 4σc ∑ ρ . − kR · E w˜ N (ρ) = wk (ρ) ≡ (k R + 1) · E π E ∗ k=0 kR + 1 kR k=0 N ∑

(2.113) Figure 2.24 presents the deformed free surfaces of the half-space under the action of a single indenter (red curve) and the multiple rigid-connected indenters. The black curve shows the deformed surface by the double-indenter. This indenter has the central punch with the outer radius one, and the ring punch with the inner radius two and the outer radius of three. The blue curve describes the triple punch with the inner radius of the second ring with the inner radius four and the outer radius of five and so on.

2.4.3 Optimization of Ring-Shaped Indenters 1. For the given outer radius R2 , the deformation energy the increasing function of the ratio ∈ . This means, that for the given force F0 in the ring-shaped region, the averaged normal displacement reduces with the increasing outer radius of the −1 = d 2 E/dF02 indenter. In other word, the normal stiffness, or spring rate ctotal increases with the outer radius of the indenter.

84

2 Optimization for Axisymmetric Contacts, Charged and Conducting Disks

With this observation, the solutions of two optimization problems become evident. The first optimization problem searches the shape of the indenter Wopt for the prescribed total force F0 and the limited attainable normal stress Smax , which delivers the maximal stiffness ctotal of the contact pair. In this case, the elastic energy E of the contact pair must be minimal. This kind of contact is advantageous form the viewpoint of its precision. The outer radius is restricted by its maximal margin. As the normal stiffness increases with the outer radius, the solution of the optimization problem is the constant-stress indenter. The outer radius is maximal admissible radius R2 . The inner radius R1 follows from the restriction of the given acceptable maximal stress. Evidently, that for the given parameters Smax , R2∗ , F0 , the Eq. (2.114) must lead to an admissible solution R1 > 0. The design parameters are R1 , R2 . For some technical applications, the formulation optimization problem the shape of the indenter Wopt for the prescribed total force F0 and the given maximal normal stress Sult , which delivers the maximal stiffness of the contact pair.. In this case, the elastic energy Etotal of the contact pair must be minimal. The contact pair accumulates the minimal possible elastic energy, because for the given force the normal deflection in minimal. The contact pair is advantageous from the viewpoint of its higher stiffness. For this problem, the outer radius is restricted by its maximal border R2∗ . As the normal stiffness increases with the outer radius, the solution of the optimization problem is once again the constant-stress indenter. The inner radius of this indenter R2∗ ≥ R1 ≥ 0 follows from the restriction of the given acceptable maximal stress, and the outer radius R2 = R2∗ . The first direct optimization task is the minimization energy of the elastic deformation Etotal for given total force F0 , the ultimate compression stress Sult : min

0≤R1 0, which delivers the minimal stiffness of the contact pair.. In this case, the elastic energy Etotal of the contact pair must be maximal. The contact pair accumulates the maximal possible elastic energy, if the stiffness is minimal and the normal travel is maximal. The contact pair is advantageous from the viewpoint of its resilience. The outer radius is restricted by its maximal border R2∗ . As the normal stiffness increases with the outer radius, the solution of the optimization problem is once again the constant-stress indenter. The outer radius of this indenter R2∗ ≥ R2 ≥ 0 comes from the restriction of the given ultimate stress, and the inner radius vanishes R1 = 0. The design parameters are R1 , R2 . The second direct optimization task is the maximization energy of the elastic deformation Emin for given total force F0 , the ultimate compression stress Sult : max

0≤R1 0 Consequently, the maximum of stress in the stiffener attains its minimum value if and only if for all n = 3, 5, ... the coefficients Cn vanish: Cn = 0.

(3.24)

108

3 Optimization of Needle-Shaped Stiffeners

We show finally, that the optimal stiffener A must be equally stressed. The shear stress on the stiffener surface T , the axial load p(x) in the stiffener cross-section and its optimal shape results from the Eqs. (3.18) and (3.19) after the substitution (3.22) with the constants (3.21) and (3.23): T (x) = C1 ∫x p(x) = −

cos θ x = −C1 √ > 0, sin θ l2 − x2 √ T (y)dy = −C1 l 2 − x 2 > 0,

(3.25)

−l

A(x) ≡

√ p(x) 4χ W0 = l 2 − x 2 > 0. 2 σ0 Eε0 l 2 − 4χ W0

Consequently, the optimal single stiffener A is possesses the elliptic shape.

3.5 Mass Optimization of a Stiffener 3.5.1 Method of Lagrange Multipliers In this section the optimization problem of determination of the stiffener cross-section distribution over the length (3.8) is considered. The volume of the stiffener

a

∫l

V[ ] =

a(x)d x → min S(x)

(3.26)

−l

is to be minimized while the elastic energy is equal to a specified value (3.15). For derivation of the optimality conditions the Lagrange multipliers method is employed. For this purpose, the constraint (3.15) and equilibrium Eq. (3.24) are added to the objective function (3.25): ∫l

a

J [ , σ, ψ] = −l

a

⎡ l ∫ ⎣ (x)d x + λ −l

⎤ 2W0 ⎦ (x)σ (x)d x − ε0

a

⎡ ⎤ ∫l ∫l d + ⎣ [ (y)σ (y)]KC (x − y)dy + σ (x) − Eε0 ⎦ψ(x)d x. dy

a

−l

Here

−l

(3.27)

3.5 Mass Optimization of a Stiffener

109

ψ(x) is an adjoint function, λ is a Lagrange multiplier. The boundary conditions for the adjoint function are: ψ(−l) = 0, ψ(l) = 0

(3.28)

The full variation of augmented functional δ J [a, σ, ψ] results from the partial variations of the cross-section area δa(x), stresses δσ (x) and adjoint function δψ(x):

a

a

a

a

δ J [ , σ, ψ] = δ S J [ , σ, ψ] + δσ J [ , σ, ψ] + δψ J [ , σ, ψ]

(3.29)

The terms on the right side of Eq. (3.29) are: ∫l

a

δ S J [ , σ, ψ] =

∫l

a

δ (x)d x + λ −l

a

δ (x)σ (x)d x −l

⎡ ⎤ ∫l ∫l d = ⎣ [δ (y)σ (y)]KC (x − y)dy ⎦ψ(x)d x, dy

a

−l

∫l

a

δσ J [ , σ, ψ] = λ

(3.30)

−l

a(x)δσ (x)d x

−l

⎡ ⎤ ∫l ∫l d = ⎣ [ (y)δσ (y)]KC (x − y)dy + δσ (x)⎦ψ(x)d x, (3.31) dy

a

−l

−l

⎡ ⎤ ∫l ∫l d δψ J [ , σ, ψ] = ⎣ [ (y)σ (y)]KC (x − y)dy + σ (x) − Eε0 ⎦ dy

a

a

−l

−l

δψ(x)d x. The variation δψ J [a, σ, ψ] vanishes due to equilibrium Eq. (3.24). The nullification of the first partial variations:

a

a

δ S J [ , σ, ψ] = 0, δσ J [ , σ, ψ] = 0 leads to the following necessary optimality conditions and adjoint equations: ⎡ 1 + σ (x)⎣λ +

∫l −l

⎤ dψ(y) ⎦ KC (x − y) dy = 0, dy

(3.32)

110

3 Optimization of Needle-Shaped Stiffeners

⎡

a

ψ(x) + (x)⎣λ +

⎤

∫l KC (x − y) −l

dψ(y) ⎦ dy = 0. dy

(3.33)

Equations (3.24), (3.31) and (3.32) together with the boundary conditions (3.27) and (3.5) form a closed nonlinear boundary value problem for the functions σ (x), ψ(x) and (x). The particular solution can be found by the substitution:

a

a(x) = σ ψ(x).

σ (x) = σ0 ,

0

(3.34)

The substitution (3.33) into Eqs. (3.31) and (3.32) reduces these equations into one singular integral equation: 1 κ

∫l KC (x − y) −l

da(y) dy = −Q dy

(3.35)

with the constant on the right side: Q=

Eε0 − σ0 . χ σ0

(3.36)

a

Equation (3.34) contains only one unknown function (y). The conditions (3.5) for the absence of load transfer through the end of faces of the inclusion read:

a(−l)σ

0

= 0,

a(l)σ

0

= 0.

(3.37)

For the solution of the Eq. (3.34) the Riemann-Hilbert method is employed [13, 14]. For the determination of the optimal inclusion shape, the equation must be solved: 1 πχ

∫l F(y)KC (x − y)dy = Q(x). −l

The solution of the singular integral equation 1 π is given by the formula:

∫l F(y) −l

dy = f(x) x−y

(3.38)

3.5 Mass Optimization of a Stiffener

F(x) =

111

∫l

1

√ π l2 − x2

−l

√ c1 f (y) l 2 − y 2 dy +√ . 2 y−x l − x2

(3.39)

For the solution, we replace in Eq. (3.38) the function f(y) = −Q = const. The Cauchy principal value of singular integral in Eq. (3.38) provides of the formula [15, p. 687]: ∫l √ 2 l − y 2 dy = −π x. y−x

−l

With the substitution F = da/d x, the ordinary differential equation for the crosssection of the stringer reads:

a

qx d (x) c1 =√ +√ . 2 2 2 dx l −x l − x2

(3.40)

The integration of (3.39) leads to: √ x l 2 − x 2 + c2 + c1 ar csin , l

a(x) = Q

(3.41)

where c1 , c2 are unknown constants. The boundary conditions (3.40) demand c1 = 0, c2 = 0. Accordingly, the optimal cross-section area of the stiffener is: √ A(x) = Q l 2 − x 2 .

(3.42)

Therefore, the highly elongated ellipse (the elliptical needle) represents the optimal shape of the stiffener in this optimization problem as well.

3.5.2 Alternative Solution The straightforward technique for the evaluation of the equally stressed stringer is based on direct integration of the equilibrium equations. If the axial deformation ε(y) is given, from (3.6) follows that the shear stress along the contact line reads: ∫l √ 2 l − y2 ε(y)dy. T (x) = √ x−y π l2 − x2 χ

−l

For the constant axial stress ε(y) = σ0 /E, from (3.43) follows:

(3.43)

112

3 Optimization of Needle-Shaped Stiffeners

T (x) =

χσ0 x . √ E l2 − x2

(3.44)

With Eqs. (3.25) and (3.44), the axial load along the stringer will be: ∫x p=−

T (y)dy = −l

χ σ0 √ 2 l − x 2. E

(3.45)

From Eqs. (3.4) and (3.45) follow the optimal cross-sectional area and the volume of the stringer:

A(x) =

χ√2 p = l − x 2 , V[A] = σ0 E

∫l A(x)d x = −l

π χl 2 . 2E

(3.46)

Consequently, the boundary value problem for the optimal stringer is a mixed problem of elasticity: ∂u = ε0 , S = 0 for x ∈ Gl . ∂x T = 0, S = 0 for x ∈ / Gl .

(3.47)

The mixed boundary value problems in elasticity allow different formulations. The previously mentioned mixed problem (3.5) for the stringer with an arbitrary cross-section leads to the integral-differential equation [16]. Notably, that the mixed boundary value problems (3.47) and (3.5) are different. The mixed boundary value problem (3.47) is mathematically equivalent to the mixed boundary value problem (1.23) for the rigid punch on the surface of the elastic half-plane [17]. ∂w = Θ(x), T = 0 for x ∈ Gl . ∂x S = 0, T = 0 for x ∈ / Gl .

(3.48)

The only difference is the following. The problem (3.47) the displacements parallel to the boundary and corresponding shear stresses are fixed. In the mixed boundary value problem for the rigid punch the displacements normal to the boundary and corresponding normal stresses are prescribed. Both mixed problems (3.47) and (3.48) lead to the integral equation [18].

3.7 Summary of Principal Results

113

3.6 Conclusions This chapter studied the optimal design problems for contact and load transfer in elastic bodies. The core of the proposed method was based on the representation of the stiffener as an infinitesimally thin rod with a variable cross-section. The stringer tightened on the edge of the semi-infinite thin plate (case A). The semi-infinite plate was in the plane stress state. Thus, in case A the needle-form inclusions that support the thin plate are considered. The analogous formulation concerns the stiffening of the elastic bodies in plane strain state. The flattened, thin-plate stiffener was tightened to the surface of the semi-infinite, elongated in z direction prismatic body (case B). Both cases transform immediately to each other by means of recalculation of elastic constants and lead to the identical principal equations. The method allows the finding of optimal shape for thin stiffeners subjected to the variable loads along the stiffener axis. The governing equations in the considered examples are singular integral equations. The idealized problem in the optimal case leads to singular integral equations for the desired optimal cross-section of the needle-form inclusion. The solutions of the optimization equations were achieved in analytical form. Firstly, the problem involving minimization of the maximum stress in a single stiffener is examined. Secondly, the solution for the stiffener mass minimization under the constraints on the magnitude of elastic energy was delivered. The cases of an isolated stiffener and the periodic array of inclusions were examined rigorously and the optimal shapes were presented. The method is directly applicable for the optimization of elastic bodies in plane strain state as well (case B). In the case B the semi-infinite, prismatic elastic body with a flattened, wave-shaped reinforcement on the surface could be accounted.

3.7 Summary of Principal Results • Fundamental equations load diffusion and load transfer for stringer of finite length are summarized • Expressions for compliance and deformation energy of stiffened elastic body are derived • Minimization of the maximum stress with restricted deformation energy performed • Mass optimal stiffener has an elongated elliptic form • The mathematical type of integral equation for an arbitrary stringer differs from the type of integral equation for the optimal stiffener, constantly stressed along its length • The equations for an arbitrary stringer are integral–differential of Melan’s type • The equations for the optimal stringer are integral of mixed type.

114

3 Optimization of Needle-Shaped Stiffeners

3.8 List of Symbols Em νm hm εx x (x, 0) u0 u ε0 ε 2l E ν

Elastic modulus of the semi-infinite elastic body Poisson’s ratio of the semi-infinite elastic body Thickness of the semi-infinite elastic body Strain x-component of contact points of the semi-infinite elastic body Displacements along x axis in the body without a stiffener Displacements along x axis in the body with the stiffener The constant tensile strain in the infinite plate without a stiffener The tensile strain in the body with the stiffener Length of the stiffener (stiffener, reinforcement, inhomogeneity) Elastic modulus of the stiffener Poisson’s ratio of the stiffener Variable area of an arbitrary stiffener cross-section A Area of the optimal stiffener cross-section p Force in the stiffener σ Axial stress in the stiffener T Contact shear force between the stiffener and the body KC Kernel of integral equation for a single stiffener (Cauchy kernel) χ = χ E Coefficient of the kernels of integral equation χ Coefficient of the Green function for plane Cerruti and Kelvin problems Wm0 Stored elastic energy in the plate without a stiffener Wm Stored elastic energy in the plate with the stiffener Wf Elastic energy stored in the stiffener W Total stored elastic energy in the inhomogeneous, reinforced body ΔWm Work of the shear force on the contact boundary between both elements ΔW An increment of stored energy of the body due to the stiffener Cn , C˜ Coefficients of Chebyshev polynomials Tn , Un Chebyshev polynomials of the first and second kinds σ0 Constant component of stress in the stiffener Δσ Oscillating component of stress in the stiffener I Volume of the stiffener (optimization functional) J Augmented functional ψ(x) Adjoint function λ Lagrange multiplier K2 Kernel of integral equation for the periodic array (Hilbert kernel) Q Constant on the right side of integral equation ω Period in the plane (distance between centers of the stiffeners)

a

References

115

References 1. EN 1993-1-8. (2005). Eurocode 3: Design of steel structures—Part 1–8: Design of joints [Authority: The European Union Per Regulation 305/2011, Directive 98/34/EC, Directive 2004/ 18/EC]. 2. Melan, E. (1932). Ein Beitrag zur Theorie geschweißter Verbindungen. Ingenieur Archiv, 3(2), 123–129. 3. Muki, R., & Sternberg, E. (1968). On the diffusion of load from a transverse tension bar to a semi-infinite elastic sheet. ASME Journal of Applied Mechanics, 35, 737–746. 4. Sternberg, E. (1970). Load-transfer and load-diffusion in elastostatics. In Proceedings of the sixth U.S. National congress of applied mechanics. Published by The American Society of Mechanical Engineers, United Engineering Center, New York, N.Y. 10017. 5. Wheeler, L. (2004). Inhomogeneities of minimum stress concentration. Mathematics and Mechanics of Solids, 9, 229–242. https://doi.org/10.1177/1081286504038468,p.229-242 6. Liu, L. (2014). Geometries of inhomogeneities with minimum field concentration. Mechanics of Materials, 75, 95–102. https://doi.org/10.1016/j.mechmat.2014.04.004 7. Muki, R., & Sternberg, E. (1969). Elastostatic load-transfer to a half-space from a partially embedded axially loaded rod. Office of Naval Research, Contract Nonr-220(58), NR-064-431, Technical Report No. 18. 8. Podio-Guidugli, P., & Favata, A. (2014).Elasticity for geotechnicians, a modern exposition of Kelvin, Boussinesq, Flamant, Cerruti, Melan, and Mindlin problems. Springer. 9. Sadd, M. (2014). Elasticity (3 ed.). Academic Press (p. 600). ISBN-13: 978-0124081369. 10. Barber, J. R. (2004). Elasticity. Kluwer Academic Publishers. 11. Mkhitaryan, S. M., Mkrtchyan, M. S., & Kanetsyan, E. G. (2017). On a method for solving Prandtl’s integro-differential equation applied to problems of continuum mechanics using polynomial approximations. ZAMM Journal of Applied Mathematics and Mechanics, 97(6). https:// doi.org/10.1002/zamm.201600025 12. Mason, J. C., & Handscomb, D. C. (2002). Chebyshev polynomials (p. 360). CRC Press. 13. Gakhov, F. D., & Chibrikova L. I. (1954). On some types of singular integral equations solvable in closed form. Mat. Sb. (N.S.), 35(77, 3), 395–436. 14. Estrada, R., & Kanwal, R. P. (2012). Singular integral equations. Springer. ISBN13:9781461271239. 15. Gradshteyn, I. S., & Ryzhik, I. M. (2014). Table of integrals, series, and products (8th ed.). Academic Press. https://doi.org/10.1016/C2010-0-64839-5, ISBN 978-0-12-384933-5. 16. Podio-Guidugli, P., & Favata, A. (2014). The Melan and Mindlin problems. In Elasticity for geotechnicians. Solid mechanics and its applications (Vol. 204). Springer. https://doi.org/10. 1007/978-3-319-01258-2_7 17. Muskhelishvili, N. I. (1975). Some basic problems of the mathematical theory elasticity. Noordhoff. 18. Sneddon, I. N. (1965). The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. International Journal of Engineering Science, 3, 47–57.

Chapter 4

Optimization for Periodic Arrays of Needle-Shaped Stiffeners

Abstract In this chapter, the problems of stiffening of the infinite plates with the periodically and double-periodically arranged inclusions (stiffeners) are explored. The periodically spaced, infinite array of the identical thin elastic inclusions will be imbedded in an infinite elastic plane. The cross-section of an inclusion and its axial stiffness are variable along the axis of the stiffener. The cross-section, which is primarily unknown, represents the searchable function in the optimization problems. The closed-form optimal solutions are found. The authors discuss the problem of optimization of an infinite chess-board array of inclusions, or stiffeners, attached to an elastic infinite plate. The chapter studies the optimization problems for the infinite arrays of inclusions, arranged in rectangular and square lattices. The plate is exposed to tensioning by uniformly distributed forces, and also to contact stresses due to forces as a result of enforced continuity with the reinforcing stiffener. The optimization problem is emphasized in an elongated, needle-shaped form of the inclusion. Keywords Stiffener optimization · Periodic contact problem · Load transfer

4.1 Optimal Load-Transfer for Periodically Arranged of Stiffeners 4.1.1 Equilibrium Equations for Periodic Array of Inclusions or Stiffeners 1°. A thin, infinite plate is exposed to tensioning by uniformly distributed forces, and also to contact stresses due to forces as a result of enforced continuity with the reinforcing stiffener (Fig. 4.1a). The corresponding task is known as the first Melan problem. The cross-section together with its axial stiffness is variable along the axis of the stiffener. The cross-section, which is primarily unknown, represents the searchable function in the optimization problems.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Kobelev, Fundamentals of Structural Optimization, Mathematical Engineering, https://doi.org/10.1007/978-3-031-34632-3_4

117

118

4 Optimization for Periodic Arrays of Needle-Shaped Stiffeners

Fig. 4.1 Infinite plate of the periodic first Melan problem. The array of elongated stringers of finite length (a) and with an array of plate-like stiffeners (b)

A flattened, plate-shaped stiffener that supports an infinite prismatic body is studied similarly (Fig. 4.1b). Optimization problems of the flattened stiffener are proved to be quite similar to those of the elongated one. The governing equations of both studied cases transform into each other by means of alternation of the elastic constants. Consequently, the optimal cross-sections of both problems turn out to be identical after the appropriate choice of material parameters. In this section the authors derive the governing equations for a periodically spaced, infinite array of the identical thin elastic inclusions, imbedded in an infinite elastic plane. The stiffeners are arranged periodically with the period δ. The length of the axially aligned stiffeners with a variable cross-sectional area must be shorter than the distance between the stiffeners: 2l < δ. Let an elastic semi-infinite elastic body related to the right rectangular coordinate O x y with the thickness h m . The Young modulus of the inclusion or stringer is E. The elastic constants of the plate are G m , Poisson’s ratio νm and E m = 2(1 + νm )G m .

4.1 Optimal Load-Transfer for Periodically Arranged of Stiffeners

119

The axes of the inclusions are parallel to the axis x. The centers of the inclusions located along one line. The inclination angle of the centers’ line to the x-axis is ϑ. The origin of the coordinate system is in the center of the inclusion with the number n = 0. The numbers of the inclusions are −∞ < n < ∞. The coordinates of the centers of the inclusions are xc = n · δ · cos ϑ,

z c = n · δ · sin ϑ

(4.1)

For the optimization the authors need the expression for the Green’s function (Lurie, 2005). If the vector of force [ F = hm Em

Fx Fz

]

acts on the point with the coordinates (x, z), the displacement in the observation point (0, 0) reads (Figs. 4.2 and 4.3): [

u U= w −

]

[

] ReU = , ImU

U=

) 3 − 4νm Fx + i Fz ( 2 ln x + z 2 1 − νm 8π

Fx − i Fz 1 (x + i z)2 . 8π 1 − νm x 2 + z 2

(4.2)

For the derivation, the authors apply the periodic system of equal unit forces. On each inclusion with the number n acts the local force in x-direction. The stresses and forces in the direction z fade. For any given inclusion, the distance between the application point of the local force and the center of the nth inclusion is t. Thus, the vectors of local force F n and coordinates of the application point X n are: [ ] 1 F n = hm Em , 0

[ X n = hm Em

] nδ cos ϑ − t . nδ sin ϑ

(4.3)

Consider the observation point t. With Eq. (4.3), the displacement in the direction x due to the local force F n with the number n is: ] [ ) ( ( 2 ) 2 cos ϑ nδt − n 2 δ 2 + n 2 δ 2 − t 2 1 u n (t) = − , (3 − 4νm ) ln r + 8π (1 − νm ) r2 r 2 = n 2 δ 2 − 2nδt cos ϑ + t 2 . (4.4) In the observation point t, the strain in the axial direction due to action of the local force F n is equal with ρ = t/δ to:

120

4 Optimization for Periodic Arrays of Needle-Shaped Stiffeners

Fig. 4.2 An infinite array of the identical thin elastic inclusions, periodically imbedded in an infinite elastic plate. The inclination angle of the center line to the x axis is ϑ, distance between the center is δ

t z

t

t t

t

Fig. 4.3 Periodical system of equal forces loading the identical thin elastic inclusions

x

4.1 Optimal Load-Transfer for Periodically Arranged of Stiffeners

121

( ) ( ) ( ) − 21 n 2 cos2 ϑ − 2 νm − 34 ρn cos ϑ + νm − 14 n 2 + ρ 2 νm − 34 ( )2 n 2 − 2nρ cos ϑ + ρ 2 n cos ϑ − ρ , (4.5) · 8π (1 − νm )

εx x (ρ, n) = −

The strain due to the action of all local forces is the sum of strains increments εx x (ρ, n) over −∞ < n < ∞ (Fig. 4.4). εx x (ρ, ϑ) =

∞ ∑

εx x (ρ, n) =

n=−∞ 3

N(ρ, ϑ) , with: D(ρ, ϑ)

N(ρ, ϑ) = 16 sin ϑ(c3 − c1 )(c3 + c1 )N0 (ϑ) − 32π sin4 ϑρN1 (ϑ), ( ( ) c1 c2 ) c3 c4 cos(ϑ) 2 cos3 ϑ − 4νm + 1 − sin(ϑ) cos3 ϑ − 2νm + 1 , N0 (ϑ) = − 4 2 ) ) ) ( ) 3 1( 2 2 2 2 2 2 , N1 (ϑ) = 2c1 c3 − c1 − c3 sin ϑ 4 cos ϑ − 1 + c1 c2 c3 c4 cos ϑ cos ϑ − 8 4 D(ρ, ϑ) = 16δ(1 − νm )(c1 + c3 )2 (c1 − c3 )2 sin3 ϑ, c2 = sinh(πρ sin(ϑ)), c1 = cosh(πρ sin(ϑ)), c3 = cos(πρ cos(ϑ)), c4 = sin(πρ cos(ϑ)), ) ) c3 c4 ( c1 c2 ( 2 2 cos2 ϑ − 4νm + 1 cos(ϑ) + cos ϑ − 2νm + 1 sin(ϑ). (4.6) c5 = 4 2

Fig. 4.4 Axial strain due to the action of the array of local forces for different distances to the center ρ = δt and an inclination angle ϑ

122

4 Optimization for Periodic Arrays of Needle-Shaped Stiffeners

For the small values of ρ the total deformation expresses as: 3 − 4νm 2π 2 ρ − ρ→0 4δπ (1 − νm )ρ 3δπ (1 − νm ) ) ] [ ) 1 1 νm 4 2 cos ϑ + . − · cos ϑ − νm + 2 2 8

εx x (ρ, ϑ) =

(4.7)

The dimensionless values are used for the following derivations: ϕ=

πl , δ

β=

πx , δ

γ =

πy . δ

The value ϕ proportional to the ratio of the length of the inclusion l to the distance between their centers δ. The values β and γ signify the dimensionless axial coordinate along the stiffener span, −ϕ ≤ β ≤ ϕ, −ϕ ≤ γ ≤ ϕ. 2°. The equilibrium equations of the periodic first Melan problem are again Eqs. (4.1)–(4.4). For the given area of cross-section > 0, the total force in the cross-section p and normal stress in the stiffener σ read:

a

∫x

a

p=σ ,

p=

T (t)dt.

(4.8)

−l

The infinite isotropic elastic plate is subjected to equal contact forces, which act periodically on the plate. The horizontal strain component εx x (x, 0) of points on the axis x of the elastic infinite plate is: )

∫l εx x (x, 0) = ε0 − κ˜

T (t)εx −l

) x −t , ϑ dt, δ

(4.9)

The coefficient in Eq. (4.9) for both plane strain and plane stress states reads: 1 κ˜ = 4π h m E m

(

(1+νm )(3−4νm ) (1−νm )

for I. Melan’s plane strain problem (case A), (3 − νm )(1 + νm ) for I. Melan’s plane stress problem (case B).

In the case of the coaxially oriented periodic array of stiffeners, the second, periodic Melan’s problem could be studied. The stiffeners will be periodically attached on the free boundary of the half space (Chap. 3, Fig. 3.1). For both plane strain and plane stress states, the coefficient for the second Melan’s problem correspondingly reads: ( 2 1 − νm2 for II. Melans plane strain problem (case A), κ˜ = for II. Melans plane stress problem (case B). π hm Em 1

4.1 Optimal Load-Transfer for Periodically Arranged of Stiffeners

123

We use later the related coefficient: def

κ = E κ. ˜ Eliminating the contact forces T (y) in Eq. (4.9) on the basis of (4.8), the singular integral equation in terms of σ (x) reduces to: ∫l σ (x) = Eε0 +

Kϑ (x − t) −l

(x ) d def ˜ xx , ϑ . [ (t)σ (t)]dt, Kϑ (x) = κε dt δ

a

(4.10)

The kernel Kϑ (x) will be referred to as a generalized Hilbert kernel. 3°. Two important cases arise, if the centers of the inclusions with the coincident axes locate on the axes x and z correspondingly. The inclination angle ϑ = 0 brings the inclusions into line, which coincides with the axis x. The contact region Pl is the union of all single contact intervals [nδ − l . . . nδ + l] for each stringer with the number n (Fig. 4.5): Pl =

∞ )

[nδ − l . . . nδ + l].

n=−∞

As the length of the axially aligned stiffeners with a variable cross-sectional area must be shorter than the distance between the stiffeners: 2l < δ, in this case: ϕ=

π πl < . δ 2

The strain in the axial direction in the observation point due to action of the family of horizontal local forces (4.3) is equal to

Fig. 4.5 An infinite row of inclusions with the coincident axes, oriented along the axis x

124

4 Optimization for Periodic Arrays of Needle-Shaped Stiffeners

Fig. 4.6 Axial strain due to the action of the array of horizontal local forces for different distances to the center ρ = δt . The inclination angle ϑ = 0

3 − 4νm . ϑ=0 4π δ(1 − νm )(n − ρ)

εn (ρ) =

(4.11)

The total strain is equal: εx x (ρ, ϑ = 0) =

∞ ∑

εn (ρ) =

n=−∞

3 − 4νm cot(πρ). 4δ(1 − νm )

(4.12)

The generalized Hilbert kernel Kϑ (x) reduces with the vanishing inclination angle ϑ to a common Hilbert kernel with the cotangent function (Fig. 4.6): Kϑ=0 (x) = κ˜ cot

(πx ) . δ

4°. Another case shows the inclination angle ϑ = π/2. In this case, the center points are placed on the axis z (Fig. 4.7). The strain in the axial direction in the observation point due to action of the family of the vertical local forces is equal to: εn (ρ) = ρ · ϑ=π/2

(4νm − 1)n 2 + ρ 2 (4νm − 3) )2 . ( 2π (νm − 1) n 2 + ρ 2

For infinite row of inclusions with the parallel axes, the total strain will be:

(4.13)

4.1 Optimal Load-Transfer for Periodically Arranged of Stiffeners

125

Fig. 4.7 An infinite row of inclusions with the coincident axes, oriented along the axis x

εx x (ρ, ϑ = π/2) =

∞ ∑

εx x (ρ)

n=−∞

=

[ ] πρ coth2 (πρ) − 1 + 2(1 − 2νm ) coth(πρ) . 4δ(1 − νm )

(4.14)

The generalized Hilbert kernel Kϑ (x) reduces with inclination angle ϑ = π/2 to the kernel with the cotangent function: Kϑ=π/2 (x) =

] (πx ) { [ ( π x )} κ˜ − 1 + 2(1 − 2νm ) coth . π x coth2 4(1 − νm ) δ δ

For νm → 0 and for νm → 1/2, the kernel reduces to the kernel with the product of linear and squared hyperbolic cotangent functions: ] ( π x )} (πx ) κ˜ { [ π x coth2 − 1 + 2 coth , 4 δ δ [ ) ] ( πx π νm →1/2 −1 . Kϑ=π/2 (x) −−−−→ κ˜ x coth2 2 δ νm →0

Kϑ=π/2 (x) −−−→

For νm → ∞, the kernel transforms to the Hilbert kernel with hyperbolic cotangent (Figs. 4.8, 4.9 and 4.10): νm →∞

Kϑ=π/2 (x) −−−−→ κ˜ coth

(πx ) δ

( πx ) ≡ i κ˜ cot i . δ

126

4 Optimization for Periodic Arrays of Needle-Shaped Stiffeners

Fig. 4.8 Axial strain due to the action of the array of vertical local forces for different distances to the center ρ = δt . The inclination angle ϑ = π/2

Fig. 4.9 Axial strain due to the action of the array of local forces for different distances to the center ρ = δt . The inclination angles ϑ are 0, π2 , π4 , π6

4.1.2 Lagrange Multipliers Method for Optimality Conditions The kernel of the integral equation (4.10) is of Hilbert type. This equation contains two unknown functions (t) and σ (t). The optimization problem for the single stiffener with the variable cross-section over the length was examined in Chap. 3. The analogous method is applied for the periodic array of inclusions or stiffeners. The inclusion volume is to be minimized (, keeping the elastic energy being equal to a specified value (3.15):

a

4.1 Optimal Load-Transfer for Periodically Arranged of Stiffeners

127

Fig. 4.10 Polar diagrams εx x (ρ, ϑ) as the function of the inclination angle ϑ for different values of ρ

∫l

a

I[ ] =

a

−l

ε0 (x)d x → min, ΔW = (x) 2

a

∫l

a(x)σ (x)d x = W

0

> 0.

(4.15)

−l

For the derivation of the necessary optimality conditions the Lagrange multipliers method is employed. For this purpose, the constraint (3.2) and equilibrium equation (5.3) are added to the objective function (5.4): ∫l

a

J [ , σ, ψ] = ∫l + −l

⎡ ⎣−

a

⎡ l ∫ ⎣ (x)d x + Λ

−l

∫l −l

−l

a

⎤ 2W0 ⎦ (x)σ (x)d x − ε0 ⎤

a

d K ϑ (x − t) [ (t)σ (t)]dt ⎦ψ(x)d x − dy

∫l Eε0 ψ(x)d x.

(4.16)

−l

The boundary conditions for the adjoint function are once again (4.3). The full variation of augmented functional δ J [ , σ, ψ] results from the variations of the cross-sectional area δ (x), stresses δσ (x) and adjoint function δψ(x). Eqs. (3.29)– (3.31) deliver the similar formulas for the partial variations. The generalized equation follows after the replacement of Cauchy kernel KC in Eqs. (3.32) and (3.33) by the generalized Hilbert kernel Kϑ . The first partial variations of the augmented functional must be zero. This requirement leads to the necessary optimality condition and adjoint equations:

a

a

128

4 Optimization for Periodic Arrays of Needle-Shaped Stiffeners

⎡ 1 + σ (x)⎣Λ +

−l

⎡

a

∫l

⎤ dψ(t) ⎦ dt = 0, Kϑ (x − t) dt ∫l

ψ(x) + (x)⎣Λ +

−l

(4.17)

⎤ dψ(t) ⎦ dt = 0. Kϑ (x − t) dt

(4.18)

The above equations together with the boundary conditions form the closed nonlinear boundary value problem for the functions ψ(x), σ (x) and (x). The solution (3.34) gives:

a

a(x) = σ ψ(x). This particular solution results to one equation for a(y): σ (x) = σ0 ,

1 κ

∫l Kϑ (x − t) −l def

Q=

0

a

d (t) dt = −Q, dt

Eε0 − σ0 . κσ0

(4.19)

The boundary conditions for the absence of load transfer through the end of faces of the inclusion are:

a(−l)σ

0

= 0,

a(l)σ

0

= 0.

The plane stress problem was transformed into the corresponding plane strain model [1] over the common transformation of the elastic modules: νm →

νm , 1 − νm

Em →

Em . 1 − νm2

4.1.3 Optimal Forms of Periodic Rows of Coaxial Stiffeners In this section the closed form solution of Eq. (4.19) is derived for the inclination angle ϑ = 0. An infinite row of inclusions or stiffeners with the coincident axes are aligned along the axis x. The minimization of the maximum stress in the periodic array of the elastic stiffeners is considered. The authors perform the reduction of the periodic problem to the previously studied, single inclusion problem. For this purpose, the transformation of variables [2] is applied:

4.1 Optimal Load-Transfer for Periodically Arranged of Stiffeners

x˜ =

tan β , tan ϕ

y˜ =

129

tan γ πl π , ϕ= < . tan ϕ δ 2

(4.20)

The Chebyshev polynomials in the coordinates β, γ read: )) ) ) tan β def ~ = = Tn (x), ˜ tn (β) cos n arccos tan ϕ )) ( ( β sin (n + 1) arccos tan tan ϕ def )) ( ( u~n (β) = ˜ = Un (x). β sin arccos tan tan ϕ

(4.21)

In terms of the transformed variables (4.20), the shear force in the periodic problem is represented in terms of an infinite series of Chebyshev polynomials [3]: τ˜ (β) =

∞ ∑ n=0

≡

∞ ∑

Cn

~ tn (β) )) ( ( β sin arccos tan tan ϕ

Cn √

n=0

~ tn (β) 1−

tan2 β tan2 ϕ

.

(4.22)

tn (β) and u~n (β), −ϕ ≤ The graphs of the transformed Chebyshev polynomials ~ β ≤ ϕ, are shown on Figs. 4.11 and 4.12 with the dashed lines. For comparison, the graphs of the ordinary Chebyshev polynomials are plotted with the solid lines: Tn (β) = cos(nβ),

Un (β) =

sin((n + 1)β) , sin(β)

−1 ≤ β ≤ 1

The red lines mark the polynomials with index 1, blue with index 2, green with index 3 and brown with index 4. The results on Fig. 4.11 correspond to the value of parameter ϕ = π/8. Figure 4.12 correspond to the value of parameter ϕ = π/4. The strain is evaluated in terms of the transformed variables by means of the identities: 1 1 1 = − tan β, tan γ − tan β cos2 β tan(γ − β)

d x˜ =

1 dβ. cos2 β tan ϕ

With these identities the integrand of Cauchy integral equation transforms to: 1 dβ dβ d x˜ = = − tan β dβ. 2 y˜ − x˜ tan γ − tan β cos β tan(γ − β) In the new variables ϕ, γ , β, the Hilbert integral (4.20) turns into Cauchy integral:

130

4 Optimization for Periodic Arrays of Needle-Shaped Stiffeners

Fig. 4.11 Transformed Chebyshev polynomials t˜n and u˜ n for ϕ = π/8

4.1 Optimal Load-Transfer for Periodically Arranged of Stiffeners

Fig. 4.12 Transformed Chebyshev polynomials t˜n and u˜ n for ϕ = π/4

131

132

4 Optimization for Periodic Arrays of Needle-Shaped Stiffeners

∫ϕ

∫1 cot(γ − β)τ˜ (β)dβ =

−ϕ

−1

τ ( y˜ ) d y˜ + ∈ , y˜ − x˜

∫ϕ ∈ =

tan(β)τ˜ (β)dβ.

(4.23)

−ϕ

Thus, the singular integral equation with the Hilbert kernel transforms into the singular integral equation with the Cauchy kernel [4]. In terms of transformed variables, the singular integral equation with the Cauchy kernel reads: ∫1 −1

τ ( y˜ ) d y˜ = y˜ − x˜

∫1 −1

∞ ∞ ∑ 1 ∑ Tn (x) ˜ Cn √ d y˜ = π Cn Un−1 (x). ˜ y˜ − x˜ n=0 1 − y˜ 2 n=1

(4.24)

The only difference to Eq. (3.5) is that the variable x˜ is used instead of x. In the new coordinates, the oscillating integral in (4.32) reduces to: ∫ϕ

∫ϕ cot(γ − β)τ˜ (β)dβ =

−ϕ

−ϕ def

σ˜ (β) = π

∞ ∑

∞ ∑ tn (β) Cn ~ √ dβ = σ˜ (β). cot(γ − β) 2β 1 − tan n=0 tan2 ϕ

Cn u~n (β) + ∈ ,

(4.25)

n=1

The stress in the stiffener (4.25) is represented as the sum of a constant σ0 and an ˜ oscillating function Δσ (x): σ˜ (β) = σ0 + Δσ˜ (β), σ0 = Eε0 + κπ (C1 + ∈ ), Δσ˜ (β) = π

∞ ∑

Cn u~n (β).

n=2

(4.26) Because the shear force τ˜ (β) is an odd function of coordinate β, the coefficients with even indices are all zero Cn = 0

for n = 2, 4, 6 . . .

The function Δσ˜ (β) in (4.26) is an oscillating function with the zero-mean value. As a final result, the maximum of stress in the stiffener in the periodic array attains its minimum value if and only if the condition fulfils: Cn = 0

for n = 3, 5, 7.

4.1 Optimal Load-Transfer for Periodically Arranged of Stiffeners

133

The above method enables the solution of the singular integral equation with the Hilbert kernel: π δ

∫l −l

a

) ) π (x − t) d (t) cot dt = −Q. dt δ

(4.27)

The solution of the equation 1 π

∫ϕ F(γ ) cot(β − γ )dγ = f(β)

for 0 < ϕ ≤ π/2.

(4.28)

−ϕ

was discovered by [5]: ⎤ ∫ϕ √ 2 2 sin ϕ − sin γ ⎣− f (γ )dγ + c1 cos β ⎦. (4.29) F(β) = √ sin(β − γ ) π sin2 ϕ − sin2 β ⎡

1

−ϕ

The methods of solution for Eq. (4.28) were referred also in [6, 7]. For the solution we use the formula: 1 π

∫ϕ √ 2 sin ϕ − sin2 γ dγ = sin β. sin(β − γ )

−ϕ

For the constant, pure imaginary function (γ ) = iQ, the solution (4.29) reduces to: F(β) =

√

Q

π sin ϕ − sin2 β 2

{−i sin β + c1 cos β}.

(4.30)

Application of Eq. (4.30) for Eq. (4.27) leads to an ordinary differential equation for :

a

a

sin β d = −Qδ √ with 2 dβ sin ϕ − sin2 β

a(β = ϕ) = 0,

δ=

πl . ϕ

(4.31)

The right side of the differential equation (4.31) proportional to the shear stress T in Eq. (4.8). The function vanishes on both endpoints of the interval −ϕ ≤ β ≤ ϕ. Thus, the function (β) is even due to symmetry of the boundary conditions. The corresponding solution of the ordinary differential equation Eq. (4.31) delivers the cross-section area of the optimal stiffener (4.27):

a

a

134

4 Optimization for Periodic Arrays of Needle-Shaped Stiffeners

( A = Qδ ln

cos(β) +

) √ ) ) cos2 (β) − cos2 (ϕ) cos(β) . (4.32) ≡ −i Qδ arccos cos(ϕ) cos(ϕ)

The same optimal shape follows from Eq. (4.8) from the condition of the constant normal stress σ along the stiffener. For comparison the dimensionless cross-section function is introduced: def A(β) ˆ . A(β) = A(0)

(4.33)

The plots of function (4.26) for different values of parameter 0 < ϕ < π/2 are shown at Fig. 4.13. The middle cross-section of all stiffeners is: ˆ A(0) = 1. As the parameter ϕ increases, the ends of the adjacent stiffeners nearly touch each other. The end sections of the optimal stiffeners gradually sharpen with the increasing values of parameter α. In the limit case ϕ → 2π the optimal shape of stiffener is of the constant cross-section. The other limit case δ → ∞ gives rise to the distant, weakly interacted inclusions. The optimal stiffener retains again the elliptical shape. The asymptotic expansion of Eq. (4.25) reduces in this limit case to the already known solution for an isolated stiffener:

Fig. 4.13 Optimal shapes of inclusions A(β), coaxially oriented along one line for different values of parameter 0 < ϕ < π/2. Inclination angle ϑ = 0 Δ

4.1 Optimal Load-Transfer for Periodically Arranged of Stiffeners

) ) πl cos(β) arccos ϕ cos(ϕ) ( ( )) √ cos πδx ( πl ) −−−→ Qπ l 2 − x 2 , = −i Q δ arccos δ→∞ cos δ ( ) √ arccos cos(β) cos(ϕ) A(β) x2 ˆ ) −−−→ 1 − 2 . ( = A(β) = δ→∞ A(0) l arccos 1

135

A = −i Q

(4.34)

cos(ϕ)

4.1.4 Optimal Forms of Periodically Located, Parallel Stiffeners In this section the approximate form solution of Eq. (4.19) is derived for the inclination angle ϑ = π/2. For the large values of νm , Eq. (4.19) turn asymptotically into the singular integral equation with the Hilbert kernel: π δ

∫l −l

a

) ) d (t) π (x − t) dt = −Q. coth dt δ

(4.35)

Because −1 < νm ≤ 1/2, the solution has no direct physical sense, but illustrates the optimal shape of the inclusions, stacked over each other. Equation (4.35) reduces after the substitution δ → i δ to the previously studied Eq. (4.20). For this purpose, the transformation of variables is applied: )

x =

tanh β , tanh ϕ

)

y=

tanh γ , tanh ϕ

ϕ=

πl < ∞. δ

(4.36)

The transformed Chebyshev polynomials in the coordinates β, γ read: )) ()) tanh β = Tn x , cos n arccos tanh ϕ )) ) ) ( )) ) sinh ϕ cosh β tanh β de f = Un x . (4.37) sin (n + 1) arccos t n (β) = √ tanh ϕ cosh2 ϕ − cosh2 β ) de f t n (β) =

)

)

)

)

The graphs of the functions t n (β) and u n (β), −ϕ ≤ β ≤ ϕ, are shown on Figs. 4.13, 4.14 and 4.15 with the dash-dotted lines. For comparison, the graphs of the ordinary Chebyshev polynomials Tn (β), Un (β), −1 ≤ β ≤ 1 are plotted with the solid lines. The red lines mark the polynomials with index 1, blue with index 2,

136

4 Optimization for Periodic Arrays of Needle-Shaped Stiffeners

green with index 3 and brown with index 4. The results on Fig. 4.14 correspond to the value of parameter ϕ = π/8. Figure 4.15 correspond to the value of parameter ϕ = π/4. Figure 4.16 are shown to the value of parameter ϕ = 2π . In terms of the transformed variables (4.28), the shear force in the periodic problem is represented in terms of an infinite series of Chebyshev polynomials: )

τ(β) =

∞ ∑

)

t n (β) Cn √ . 2 β 1 − tanh n=0 tanh2 ϕ

(4.38)

The stress in the inclusion is represented as the sum of a constant σ0 and an ˜ oscillating function Δσ (x): )

)

σ (β) = σ0 + Δσ (β),

σ0 = Eε0 + κπ (C1 + ∈ ),

)

Δσ (β) = π

∞ ∑

)

Cn u n (β).

n=2

(4.39) Because the shear force τ˜ (β) is an odd function of coordinate β, the coefficients with even indices are all zero Cn = 0

for n = 2, 4, 6 . . .

)

Important, that the function Δσ (β) in (4.39) is an oscillating function with the zero-mean value. As a final result, the maximum of stress in the stiffener in the periodic array attains its minimum value if and only if the condition fulfils: Cn = 0

for n = 3, 5, 7.

For the constant stress in the inclusion, the solution of (4.35) leads to an ordinary differential equation for the area of the optimal inclusion:

a

sinh β d = −Q δ √ 2 dβ sinh ϕ − sinh2 β

a

with

a

| d || = 0, dβ |β=0

a(β = ϕ) = 0.

(4.40)

The d /dβ must be an odd function due to symmetry of the boundary conditions. The even solution of Eq. (4.40) delivers the cross-section area of the optimal stiffener:

4.1 Optimal Load-Transfer for Periodically Arranged of Stiffeners

)

)

Fig. 4.14 Transformed Chebyshev polynomials t n and u for ϕ = π/8 n

137

138

4 Optimization for Periodic Arrays of Needle-Shaped Stiffeners

)

)

Fig. 4.15 Transformed Chebyshev polynomials t n and u for ϕ = π/4 n

4.1 Optimal Load-Transfer for Periodically Arranged of Stiffeners

)

139

)

Fig. 4.16 Transformed Chebyshev polynomials t n and u for ϕ = 2π n

[ ] cosh(β) πlQ arccos ϕ cosh(ϕ) [ ( ]) cosh(β) π − arctan √ ≡ Qδ . 2 cosh2 (ϕ) − cosh2 (β)

A(β) =

(4.41)

The limit case δ → ∞ gives rise to the distant, weakly interacted parallel inclusions. The stiffener retains again the optimal elliptical shape: ( A = Q δ arccos

cosh

( πx ) )

cosh

( πlδ ) δ

√ −−−→ Qπ l 2 − x 2 . δ→∞

(4.42)

140

4 Optimization for Periodic Arrays of Needle-Shaped Stiffeners

Fig. 4.17 Optimal shapes a of the inclusions, stacked along the vertical axis for different values of parameter 3 < ϕ < 10π, −10π < ϕ < 10π . The inclination angle ϑ = π/2

The function A(β) depends upon ϕ as a parameter. Its amplitude proportional to Q, δ. To avoid the dependence on Q, δ, the dimensionless cross-section function is used: [ ] √ cosh(β) arccos cosh(ϕ) A(β) x2 ˆ ] −−−→ 1 − 2 . [ = A(β) = δ→∞ 1 A(0) l arccos cosh(ϕ)

Δ

ˆ = 1. The plots of function A(β) The middle cross-section of all stiffeners is A(0) for different values of parameter ϕ are shown at Fig. 4.17.

4.1.5 Character of Boundary Value Problems for Periodically Located Optimal Stiffeners As shown above, the boundary value problem for the periodically located, optimal stringers is a periodic mixed problem of elasticity: ∂u = ε0 . ∂x T = 0,

S=0 S=0

for x ∈ Pl . for x ∈ / Pl .

(4.43) (4.44)

The mixed boundary value problem (4.43) is mathematically equivalent to another mixed boundary value problem. This mixed problem describes the periodically

4.2 Optimization of Double-Periodic Array of Inclusions or Stiffeners

141

arranged rigid punches on the surface of the elastic half-plane: ∂w = Θ(x), ∂x S = 0,

T =0 T =0

for x ∈ Pl . for x ∈ / Pl .

(4.45) (4.46)

The periodic mixed problem for the axially located inclusions was investigated in [8].

4.2 Optimization of Double-Periodic Array of Inclusions or Stiffeners 4.2.1 Necessary Optimality Conditions for Chess-Board Lattices of Elastic Stiffeners 1°. The double-periodic array of two-dimensional inclusions in an elastic plate was considered by Vigdergaus [9]. The new criterion was introduced for the shape optimization. The criterion consists in the minimizing of the variations of the contact stresses along the material interface. The numerical solver is based on genetic algorithms. The numerical treatment of system of linear algebraic equations in terms of Laurent coefficients for the first Kolosov–Muskhelishvili potential was implemented in the proposed solver. The system entries involve boundary integrals of a regular type. The shape parameterization scheme was performed with the aid of the genetic algorithm. The closed-form solution of the plane elasticity of a multi-inclusion regular structure was demonstrated using the method of Kolosov–Muskhelishvili potentials [10]. The constant stress shapes of two-dimensional inclusion were presented in closed form. 2°. An infinite elastic medium containing a doubly periodic array of elongated thin elastic inclusions (or stringers) under the far-field uniform load in the plane of the inclusions is considered. Employing oblique coordinates, the plane problem is formulated in the form of integral–differential equations with variable coefficients. Using a technique that is based on Weierstrass functions, the adjoint equations are derived. It is shown that the closed-form solution can be efficiently found. Simple formulas for the stress fields at the inclusions and cross-section-variation along the axis of the inclusion are derived. Exact solutions are established in a closed form for an array with a rectangular fundamental cell. Solutions for periodic arrays of collinear or slant parallel inclusions and for a single inclusion are found in the limiting cases when the distance between stacks or rows of the array is large.

142

4 Optimization for Periodic Arrays of Needle-Shaped Stiffeners

The primitive translation vectors are known also as the lattice generators Ω1 and Ω3 [11]. Consider an elastic isotropic plate, reinforced with the stiffeners arranged double periodically with two lattice generators. The unit cell is referred to as the fundamental period parallelogram (Fig. 4.18). The points of the lattice are given by the expressions: Pm,n = mΩ1 + nΩ3 ,

Pm,n = mΩ1 + nΩ3 .

(4.47)

The lattice generators Ω1 , Ω3 of the higher functions are Ω1 = 2ω1 ,

Ω3 = 2ω3 ,

Ω1 + Ω2 + Ω3 = 0,

ω1 + ω2 + ω3 = 0.

The infinite isotropic elastic plate is subjected to equal contact forces, which act double-periodically on the plate. We consider the rectangular lattices, such that Ω1 is a real, Ω3 is an imaginary value. The area of the unit cell is: D = Ω1 |Ω3 | = 4ω1 |ω3 |.

a

The length of the stiffeners is 2l < Ω1 . The cross-sectional area (x), −l < x < l of stiffeners is variable and is a priori unknown. The contact region Dl is the union of all single contact intervals [Ω1 n + Ω3 m − l . . . Ω1 n + Ω3 m + l] for each stringer

Fig. 4.18 Double-periodic lattice and fundamental period parallelogram

4.2 Optimization of Double-Periodic Array of Inclusions or Stiffeners

with the number n and m: ) Dl =

143

[Ω1 n + Ω3 m − l . . . Ω1 n + Ω3 m + l].

−∞ 0, μ˜ 1/3 18μ2 9μ2 9μ2 μ˜ 1/3 √ μ˜ = 972μ16/3 + 108( w + 1)μ8/3 + 8,

E 2 = q2/3 =

w = 81μ16/3 + 14μ8/3 + 1. Finally, two other roots of are the complex conjugated values: ˜ E 3 = a˜ + i · b,

˜ E 4 = a˜ − i · b,

−36μ8/3 − μ˜ 2/3 + 4μ˜ 1/3 − 4 , 36μ2 μ˜ 1/3 √ −36μ8/3 + μ˜ 2/3 − 4 . b˜ = 3 36μ2 μ˜ 1/3

a˜ =

6.5 Shape of Optimal Column

209

The value μ is the maximal bending moment in the column, which attains in the both end points of the rod. The value q is the minimal bending moment in the column, which reaches its value in the middle section of the single-length rod. The 3/2 3/2 values E 1 and E 2 express the cross-section areas at the end cross-sections of the half-length column. The authors approved, that two roots of the polynomial F(s) in the integrand (6.34) are real, positive and ordered: 0 < q < μ. Two other roots of the polynomial F(s) are the complex conjugated values for all positive values of μ. This observation leads to the appearance of the optimal solution as an hourglass figure, as exposed below. Hence, the improper integral in Eq. (6.34) converges and reduces to: √ ∫ 2/3 s 2 ds 3i 2 M x(M, q) = − √ √ (s − E 1 )(s − E 2 )(s − E 3 )(s − E 4 ) 4 Δ q2/3 for 0 < q ≤ M ≤ μ, E 1 = μ2/3 , E 2 = q2/3 .

(6.36)

According to our previous setting, the expression (6.36) calculates the axial coordinate from an intermediate cross-section, where the area is exactly equal to 1. Now the authors can correct the formula (6.36). The authors calculate the axial coordinate, fixing afterwards the new coordinate origin in the middle of the single-length column x = 0. For briefness of formulas, the authors use the same symbols for the coordinates, keeping the new origin of the coordinate system in mind. The coordinate system for the half-length column is shown in Chap. 5, Fig. 5.2, IV. The axial coordinate starts in the middle of the rod and runs to the end section l: M = M 2/3 , P = q2/3 . The second important step is the representation of the integral Eq. (6.36) in terms of the higher transcendental functions. The integral Eq. (6.36) reads for the right side of the column as: √ 3 2 ∑3 x(M, P) = f (i) (M, P) for 0 < P ≡ E 2 ≤ M ≤ E 1 . (6.37) i=0 4 where the auxiliary functions are [47] are given in the Tables 6.1, 6.2, 6.3 and 6.4. The auxiliary functions f (i) (M, P), Φi (M) in Eq. (6.37) express in terms of incomplete elliptic integrals [48, 49]. Table 6.1 Parameters P , M, q, μ, ˜ w of the solutions Eq. (6.37)

2

2

q 3 = P, 2μ2/3 μ˜ 1/3

M 3 = M, +

μ˜ 1/3 18μ2

+ 9μ1 2 + 9μ21μ˜ 1/3 √ μ˜ = 972μ16/3 + 108( w + 1)μ8/3 + 8

q2/3 =

w = 81μ16/3 + 14μ8/3 + 1

210

6 Optimization of Axially Compressed Rods with Mixed Boundary …

Table 6.2 Auxiliary functions Φi (M), i = 0..4, for 0 < P ≡ E 2 ≤ M ≤ E1

Φ0 (M) =

√

√ √ E 1 −M√ E 3 −M E 4 −M M−E 2

Φ1 (M) = E(ω1 , ω2 ) Φ2 (M) = F(ω1 , ω2 ) Φ3 (M) = ∏(ω1 , ω3 , ω2 ) Φ4 (M) = ∏(ω1 , ω4 , ω2 ) √

√

2 √ E 1 −M ω1 = i√ EE4 −E −E M−E 4

√

2

1

√

2 √ E 4 −E 1 ω2 = √ EE3 −E E −E −E 3

ω3 = ω4 =

Table 6.3 Auxiliary functions f (i) (M, P ), i = 0.3, for 0 < P ≡ E 2 ≤ M ≤ E1

E 4 −E 1 E 4 −E 2 E 4 −E 1 E 4 −E 2

4

1

·

2

E2 E1

f (0) (M, P ) = f(0) · [Φ0 (M) − Φ0 (P )] f (1) (M, P ) = f(1) · [Φ1 (M) − Φ1 (P )] f (2) (M, P ) = f(2) · [Φ2 (M) − Φ2 (P )] f (3) (M, P ) = f(3) · [Φ3 (M) − Φ3 (P )]

Table 6.4 Auxiliary constants f(0) , f(1) , f(2) , f(3)

f(0) = 1 √ √ f(1) = i E 3 − E 1 E 4 − E 2 f(2) = i

E 22 +(E 1 +E 3 )E 4 −E 1 E 4 √ √ E 3 −E 1 E 4 −E 3

√ 4 )(E 2 −E 1 ) √ 2 +E 3 +E f(3) = i (E 1 +E E −E E −E 3

1

4

3

The plots of the moments over the axial coordinate are shown on the Fig. 6.1. The closed-form analytical solution (6.37) is shown with the bold lines for the values of μ = 2, 3, 4. As already stated, the optimal cross-sectional area follows from the amplitude of the moment M with the necessary optimality condition Eq. (6.5). The area of the cross-section runs in the interval: E2 ≤ A ≤ E1. The cross-section reaches the upper and lower values in the points x(μ) = ±l and x(q) = 0 correspondingly. The second solution of Eq. (6.31) possesses the negative sign. Accordingly, the multiplication of Eq. (6.37) by − 1 provides the axial coordinate for the left part of the column −l ≤ x(M) ≤ 0. The plots of the areas

6.5 Shape of Optimal Column

211

Fig. 6.1 Optimal moments over the span for different values μ. The exact and asymptotical solutions, Mixed eigenvalue problem (M), bimodal case, α = 2. Optimization problem in Lagrange sense

of the cross-sections over the axis of the rod are shown on the Fig. 6.2. The results from closed-form analytical solution A = M P with c L = 1. are shown with the solid lines for the values of μ = 2, 3, 4. The second moments of inertia of the cross-section is shown on the Fig. 6.3. Once again, the analytical solution for the moment of inertia J = M 2P with c L = 1. is displayed with the solid lines.

212

6 Optimization of Axially Compressed Rods with Mixed Boundary …

Fig. 6.2 Optimal cross-section areas over the span for different values μ. The exact and asymptotical solutions. Mixed eigenvalue problem (M), bimodal case, α = 2. Optimization problem in Lagrange sense

6.6 Length, Volume and Total Stiffness of Optimal Column For the optimal column with boundary conditions of mixed type (M), the half-length l, half -volume = V/2 and the half-total stiffness ε = E/2 allow the representations in terms of integrals:

v

√ ∫E2 s 2 ds 3i 2 √ю , l=− √ 4 4 Δ − E (s ) i i=1 E1 s 2 ds dl = √ √ю , 4 Δ i=1 (s − E i )

(6.38)

√ ∫E2 s 3 ds 3i 2 √ю , v =− √ 4 4 Δ − E (s ) i i=1 E1

(6.39)

6.6 Length, Volume and Total Stiffness of Optimal Column

213

Fig. 6.3 Optimal geometric moment of the second order over the span for different values mu. The exact and asymptotical solutions. Mixed eigenvalue problem (M), bimodal case, α = 2. Optimization problem in Lagrange sense

√ ∫ s 2+α ds 3i 2 E2 ε=− √ . √ 4 4 Δ E1 ю (s − E i )

(6.40)

i=1

The formulas (6.38)–(6.40) are based on the implicit parametrization of the axial coordinate, volume and stiffness. The parametrization implies the area of the crosssection as an independent variable. To prove the above formulas, the authors can spot, that the dummy variable s indicates the area of the cross-section under the integral sign. Thus, the integrand in (6.38) represents the element of axial length dl. Therefore, if the authors multiply the element of length dl by area s and integrate it in the interval [E 2 , E 1 ], the authors get in Eq. (6.39) the single-volume of the rod: V = 2v . Integrals (6.38)–(6.40) express with the elliptic functions. The derivation of the algebraic formulas for these integrals is similar to the derivation Eq. (6.37). The evaluation of integral v (M, P) for the right side of the column reads [47]:

214

6 Optimization of Axially Compressed Rods with Mixed Boundary …

Table 6.5 Auxiliary functions g(0) (M, P ), g(1) (M, P ), g(2) (M, P ), g(3) (M, P ) for 0 < P ≡ E2 ≤ M ≤ E1

g(0) (M, P ) = g(0) (M)Φ0 (M) − g(0) (P )Φ0 (P ) g(1) (M, P ) = g(1) · [Φ1 (M) − Φ1 (P )] g(2) (M, P ) = g(2) · [Φ2 (M) − Φ2 (P )] g(3) (M, P ) = g(3) · [Φ3 (M) − Φ3 (P )]

Table 6.6 Auxiliary linear function g(1) (M) and constants g(2) , g(3) , g(4) g(0) (M) = E 1 + 13 E 2 + E 3 + E 4 + 23 M √ √ g(1) = i (E 1 + E 2 + E 3 + E 4 ) E 3 − E 1 E 4 − E 3 g(2) = i g(3) = i

3(E 4 −E 2 )(E 1 −E 2 )2 +(E 1 −E 2 )(E 4 −E 2 )(8E 2 +E 3 +3E 4 )−8E 23 √ √ E 3 −E 1 E 4 −E 3 ) ] [ ( 2 (E 1 −E 2 ) 3 E 1 +E 22 +E 32 +E 42 +2E 1 (E 2 +E 3 +E 4 )+2E 2 (E 3 +E 4 )+2E 3 E 4 √ √ 3 E 3 −E 1 E 4 −E 3

√ ∫M s 3 ds 9i 2 √ю , v (M, P) = − √ 4 16 Δ − E (s ) i i=1 P ) 9 √ ( v (M, P) ≡ √ 2 · g(0) + g(1) + g(2) + g(3) . 16 Δ

(6.41)

The auxiliary functions Eq. (6.41) are presented in Tables 6.5 and 6.6. The constants E 1 , E 2 , E 3 , E 4 are roots of the quadric equation F(s) = 0 (6.35). The authors study below the most interesting case for α0 ≡ 2. The local bending stiffness is the squared area s 2 . The integral of the squared area s 2 over dl delivers the total stiffness εm = 2εm in Eq. (6.40). The similar expressions result for the half-total stiffness as well: √ ∫M s 4 ds 3i 2 √ю ε(M, P) = − √ . 4 4 Δ − E (s ) i i=1 P The formulas for auxiliary functions of ε(M, P) are analogous to the recipes, given in Eqs. (6.37) and (6.41). The terms rather bulky and are not displayed.

6.7 Fundamental Functions for Buckling Moments Once the function x(M) is determined, the phase ϑ(M) in Eq. (6.27) follows from (6.28) after the swapping of independent variables. The improper integral converges and expresses in terms of elliptic functions [47]. Finally, the phase reads for q ≤ def def M ≤ μ, q = P 3/2 , μ = M3/2 , as follows:

6.8 Fundamental Functions for Buckling Displacements

9 ϑ(M) = 8

∫

M 2/3

q2/3

√ s

ds 4 ю

(s − E i )

215

√ 27i 2 ψ(M, P). = √ √ 16E 1 E 2 E 3 − E 1 E 4 − E 2

i=1

(6.42) In Eq. (6.42) the following notations are used for the auxiliary function ψ(M, P): ψ(M, P) = [Φ2 (P) − Φ2 (M)] · E 1 + [Φ4 (M) − Φ4 (P)] · (E 1 − E 2 ). The fundamental functions for moments are: ) ) ( ( ϑ ϑ , m 2 = M · sin π . m 1 = M · cos π ϑ˜ ϑ˜

(6.43)

The constant ϑ˜ is used for the normalization of the moments: 9 def ϑ˜ = lim ϑ(M) = M→μ 8

∫μ

2/3

q2/3

s

√ю

ds

4 i=1 (s

− Ei )

.

(6.44)

Finally, the formulas (6.42), (6.43) deliver the explicit fundamental functions of the moment M. The axial coordinate x depends implicitly on m, Eq. (6.37). With these expressions the authors arrived the parametric form for all necessary functions m 1 , m 2 , M, ϑ. The plot of eigenvalue solutions for moments m 1 , m 2 and for their complex amplitude M are shown on the Fig. 6.4 for μ = 2. The function m 2 is the even function of the axial coordinate x. On Figs. 6.5 and 6.6 the similar plots of eigenvalue solutions for moments and for their complex amplitude for μ = 3 and μ = 4. All plots are shown for comparison again on the Fig. 6.7.

6.8 Fundamental Functions for Buckling Displacements The next task is to determine the fundamental functions for the lateral displacements of the rod in the moment of the buckling. As an exception, the authors show the buckling functions of the double-length rod. The buckling displacements y1 (x), y2 (x) are the even and odd functions of the axial coordinate −L < x < L. These functions are the unknowns in the ordinary differential equations: (

)'' J y1'' = Δm ''1 with y1 (L) = 0, y1' (L) = 0, y1' (0) = 0, y1''' (0) = 0,

216

6 Optimization of Axially Compressed Rods with Mixed Boundary …

Fig. 6.4 Optimal moments over the quarter-length span for value = 2, Mixed eigenvalue problem (M), bimodal case, α = 2. Optimization problem in Lagrange sense

(

)'' J y2'' = Δm ''2 with y2 (L) = 0, y2' (L) = 0, y2 (0) = 0, y2'' (0) = 0.

(6.45)

Beside the unknown functions, Eq. (6.45) contain several functions of x: J = Aα with c L = 1, m 1 , m 2 .These functions were already determined in Sect. 5.6. It is necessary to build the rod of the double-length from the symmetry elements of the half-length, stacking them along the axis of the rod. The fundamental functions for the displacements z 1 , z 2 result from the solution of the eigenvalue problem (6.45). The functions for second moment of inertia and for the moments seem to be too bulky for the closed form solution. The solution of the boundary value problem was performed with the numerical algorithms MAPLE 2020 [50]. The results are displayed on Fig. 6.8. The dashed lines demonstrate the even fundamental functions z 2 for definite values of parameter μ = 2, 3 and 4. The dotted curves show the odd fundamental functions z 1 for the same values of the parameter μ.

6.9 Asymptotic Solutions

217

Fig. 6.5 Optimal moments over the quarter-length span for value μ = 3, Mixed eigenvalue problem (M), bimodal case, α = 2. Optimization problem in Lagrange sense

6.9 Asymptotic Solutions For the estimation of the optimization effects, the authors determine for α = 2 the limit form of the column with mixed boundary conditions. The authors consider one asymptotic case, for which the parameter μ indefinitely increases (A). The shape of the optimal column appears as an hourglass figure with the infinitely slender waist q → 0. For the formal description, the authors have to determine at first the asymptotic limits of the roots: ) ( E˜ 1 = O μ2/3 ; μ→∞

E˜ i = o(μ) for i = 2, 3, 4. μ→∞

For the vanishing roots E˜ i = 0 for i = 2, 3, 4, the integral from Eq. (6.37) as μ → ∞ reduces to:

218

6 Optimization of Axially Compressed Rods with Mixed Boundary …

Fig. 6.6 Optimal moments over the quarter-length span for value μ = 4 Mixed eigenvalue problem (M), bimodal case, α = 2. Optimization problem in Lagrange sense

√ 3 2 2 x(M) ˜ = √ μ3 4 Δ

√ sds for 0 ≤ M < μ, 0 ≤ x˜ < l. √ 1−s

2/3 (M/μ) ∫

0

(6.46)

The proper integral (6.46) expresses with the elementary functions: [ ( 0 ] √ 2 3 2 2 2M 3 ∼ − √ μ 3 2 arcsin 1 − x(M) ˜ −π 2 μ→∞ 16 Δ μ3 √ 1 √ 3 2M 3 2 2 − √ μ3 − M 3 4 Δ for 0 ≤ M ≤ μ, 0 ≤ x˜ ≤ l.

(6.47)

From Eq. (6.47) follows the length l of the optimal half-length column. The similar considerations lead to the asymptotical expansions for volume and total stiffness (6.40) and (6.41) of the half-length column. The asymptotic solutions for the moment M, area of cross-section A and second moment J are displayed on Figs. 6.1, 6.2 and 6.3. The functions M = x˜ are drawn for the same parameters, as the exact analytical

6.9 Asymptotic Solutions

219

Fig. 6.7 Optimal moments over the quarter-length span for value m = 2,3,4 Mixed eigenvalue problem (M), bimodal case, α = 2. Optimization problem in Lagrange sense

solutions. The colors correspond the colors of the lines for the exact solutions, but the approximate limit solutions are drawn with the dash style. The single-length column is an hourglass figure symmetric to its middle section. With the vanishing narrow cross-section, the effectiveness of the optimal solution continually increases and asymptotically approaches its upper limit. Finally, the authors get the estimations of the invariants in the limit case of hourglass figure with the infinitely narrow tackle for the mixed boundary conditions (A): √ 3π 2 2 √ μ 3 , L a = 2la , μ→∞ 8 Δ √ 9π 2 4 va ∼ √ μ 3 , Va = 2va , μ→∞ 32 Δ √ 15π 2 2 εa ∼ √ μ , Ea = 2εa . μ→∞ 64 Δ

la ∼

(6.48)

The values la , va , εa stay for the asymptotic expressions of length, volume and total stiffness of the half-length column, which was displayed on the Chap. 5, Fig. 5.2,

220

6 Optimization of Axially Compressed Rods with Mixed Boundary …

for

for

for

for

L -L

L for

for

Fig. 6.8 Optimal buckling displacements over the half-length span for value μ = 2,3,4, Mixed eigenvalue problem (M), bimodal case, α = 2. Optimization problem in Lagrange sense

case VI. The values L a , Va, Ea are the corresponding asymptotic formulas for length, volume and total stiffness of the single-length column, as shown on the Chap. 5, Fig. 5.2, case IV. With these values, the invariants FA and FB for the single-length column (case IV) are calculated to: FA ∼ FA.a = μ→∞

ΔL a2+α = 2π 2 , Vaα α=2

FB ∼ FB.a = μ→∞

9π 2 ΔL a3 = . Ea α=2 5

(6.49)

The suitable setting for the eigenvalue is once again Δ = 1. The length, volume and total stiffness of the double-length column (case V) are twice the length, volume and total stiffness of the single-length column Eq. (6.48). Accordingly, the invariants FA , FB of the double-length column are four times greater than the invariants of the single-length column, Eq. (6.49).

6.10 Isoperimetric Inequalities

221

6.10 Isoperimetric Inequalities The authors evaluate the invariant factors FAc , FB.C for the columns with the constant cross-section using the standard formulas of technical mechanics. Hereafter the authors consider the single-length columns. With expressions (6.39), (6.41) and (6.49) the authors can determine the invariant factors for the optimal columns with the mixed boundary conditions: FA.I V.a = 2π 2 ,

FB.I V .a =

9π 2 . 5

(6.50)

The comparison of the factors (6.50) leads to the estimation of the masses for the column with the same lengths and the same critical buckling load: V I V.a = VI V

(

F˜ A.I V a FA.I V

0−1/2

1 =√ , 2

E I V .a = E˜I V

(

F˜ B.I V.a FB.I V

0−1 =

9 . 20

(6.51)

The mass estimation (6.50) is in good accordance with the numerically evaluated value. The mass of the optimal column V I V in relation to the mass of the reference column V˜ I V from the exact solution is shown on Fig. 6.9 with the red color. This value reduces continuously with the increasing parameter μ. The asymptotical limit VI V for the infinitely high values of μ is: of the ratio V ˜ IV

( 0−1/2 VI V 1 F˜ A.I V lim = lim =√ , μ→∞ V μ→∞ ˜IV FI V 2

(6.52)

and is equal to the corresponding value from the approximate solution (6.50). The analogous tendency has the ratio of total stiffness ε of the optimal rod to the total stiffness of the constant-section rod: ( 0−1/2 EI V 9 F˜ B.I V . lim = lim = μ→∞ μ→∞ E˜ F 20 IV IV

(6.53)

Notable, that the mass of the optimal rod reduces continuously with the increasing value of μ. The optimum could not be attained with an any finite value of the parameter μ. The waist of the hourglass q decreases with the growing values of μ, but does not vanish for any finite value of the parameter μ. Another argument for this observation follows from the comparison of the invariant factors. Figure 6.10 displays the invariant mass parameter FA.I V for the optimal column as the function of the parameter μ. This value increases asymptotically to its limit value: FA.I V.a = 2π 2 .

222

6 Optimization of Axially Compressed Rods with Mixed Boundary …

Exact solution

V V

F F

→

1 √2

Approximate solution

=

1 √2

Fig. 6.9 Relative mass, Mixed eigenvalue problem (M), bimodal case, α = 2. Optimization problem in Lagrange sense

The plots on Fig. 6.11 shows the similar behavior of the invariant stiffness parameter F B.I V . With the increasing parameter μ and vanishing parameter q, the parameter F B.I V monotonically increases towards its upper limit: FB.I V.a =

9π 2 . 5

For the optimization problem in Lagrange sense, the isoperimetric inequality could be formulated as the strict inequality [51]. This isoperimetric inequality is sharp. The optimal solution could not be attained for any positive value of parameter q. The analogous consideration, but using completely different arguments, was deliberated by [52]. In other words, the original Lagrange problem possesses no attainable solution from the both points of view of the optimal control theory and variational calculus. Nevertheless, the reasonable solution could be established for the practical applications. Namely, the certain cross-section possesses the smallest area. The additional restriction to the minimal area of the cross-section presumes: μ ≥ A(x) ≥ q ≥

a

min .

(6.54)

6.10 Isoperimetric Inequalities

223 Approximate solution

Exact solution

Fig. 6.10 Factor A, Mixed eigenvalue problem (M), bimodal case, α = 2. Optimization problem in Lagrange sense

As mentioned above, the formulation of the optimization problem with the additional restrictions to the cross-sections of the column (6.53) is referred to as the problem in Nikolai sense. The solution of optimization of the problem in Nikolai sense is appropriate for the engineering applications, because in this instance the stress in the narrowest cross-section remains limited. The formula (6.37) delivers the attainable solution of the optimization problem in Nikolai sense. The setting q=

a

min ,

a

2 3

min

= Pmin .

(6.55)

in Eq. (6.37) presents the unique solution of the optimization problem for the mixedtype of boundary conditions. Thus, for the optimization problem in Nikolai sense, the isoperimetric inequality formulates as the not-strict, sharp inequality.

224

6 Optimization of Axially Compressed Rods with Mixed Boundary …

Approximate solution

=

Exact solution

Fig. 6.11 Factor B, Mixed eigenvalue problem (M), bimodal case, α = 2. Optimization problem in Lagrange sense

6.11 Conclusions The optimization of a column, compressed by axial forces was solved in closed form. The alternative designs were characterized by positive cross-sectional area functions. The critical values of buckling were equal for all alternative designs of the columns. The optimization problem in Lagrange sense searches the minimal mass of the column. The method of dimensionless factors was used for the optimization analysis. The applied method for integration of the optimization criteria delivers different length and volumes of the optimal columns. Instead of the seeking for the columns of the fixed length and volume, the columns with the different lengths and cross-sections are compared using the invariant factors. The isoperimetric inequalities were rigorously justified by means of the Hölder inequality about the mean values. The authors demonstrated that Euler’s column with boundary conditions of mixed type, which is governed by necessary optimality conditions in the bimodal case, possesses the largest buckling load among all columns with the same weight. The optimal singlesize column with one fully clamped end and other moving clamped end has the shape

6.12 Summary of Principal Results

225

of hourglass, as shown on Chap. 5, Fig. 5.2, case V. The optimal full-length column with both clamped ends comprises two single-size columns (Chap. 5, Fig. 5.2, case VI). The closed form solution contains one auxiliary parameterμ. For each positive value of the parameter μ the necessary optimality conditions are resolved in closed form. Using the method of Hölder inequalities, for each given positive value of parameter μ the isoperimetric inequality for the buckling eigenvalue is uniquely specified (ibid., [39]). The statement of the isoperimetric inequality is based on the customary optimality conditions for multiple eigenvalues. Accordingly, the existence of the optimal solution is guaranteed for each positive value of the auxiliary parameter. The mass of the established optimal design is the monotonically decreasing function of the auxiliary parameter μ. Because there is no smallest positive real number, there is also no attainable solution of the optimization problem in Lagrange sense. In this problem, the authors search the solution among the positive functions. With the disappearing parameter μ, the optimal cross-sections asymptotically adjust to its limit shape. The solution of the alternative, regularized Nikolai problem with the additional restrictions to the cross-sections is appropriate for the engineering applications, because the stress in the narrowest cross-section remains limited. The solution of optimization problem in Nikolai sense leads to the not-strict, sharp isoperimetric inequality.

6.12 Summary of Principal Results • The closed form solution of the optimization problem with mixed boundary conditions in Lagrange sense; • The necessary optimality conditions guarantee the additional grade of symmetry of the solution; • The necessary optimality conditions are the first integrals of the differential equations; • The necessary optimality conditions similar to the first integral of energy of differential equations of dynamics; • For each given positive value of the auxiliary parameter there exits the isoperimetric inequality between the length, critical buckling load and the material volume of the compressed strut. • The existence of the optimal solution is stated for each positive value of the auxiliary parameter; • The mass of the established optimal design is the monotonically decreasing function of the auxiliary parameter; • The optimal cross-sections asymptotically transform with the vanishing auxiliary parameter to the limit shape.

226

6 Optimization of Axially Compressed Rods with Mixed Boundary …

6.13 List of Symbols

m m1, m2 m M = |℘|

a

Φi (P), i = 0, . . . , 4 F ℘ = m 1 + im 2 = M · exp(i θ ) ℵ I1 , I2 F A , FB Ω = m/ν Δ C ⟨ϕ⟩ =

1 2l

∫l −l

ν, C FA.c , FB.c 2L L l La ls lm la

a

min

A

ϕ(ξ )dξ

Actual buckling moment of an arbitrary singlelength column Actual moments of bimodal problem Admissible buckling moment of an arbitrary single-length column Amplitude of the complex moment Arbitrary cross-sectional area Auxiliary functions, Eq. (6.37) Axial force in the column Complex bimodal moment Complex differential Eq. (6.24) Cross-section’s principal moments of the cross-second order Dimensionless factors Dimensionless parameter Eigenvalue of the dimensionless eigenvalue problem Shape factor Integrals of functions ϕ(x) Integration constants of Eq. (6.25) Invariant factors for the columns with the constant cross-section Length of an arbitrary double-length column Length of an arbitrary single-length column (L = 2l) Length of t an arbitrary symmetry element (half-length column) Length of the optimal single-length column in the asymptotic case (A) Length of the optimal symmetry element (halflength column, S) Length of the optimal symmetry element (halflength column, M) Length of the optimal symmetry element (halflength column, A), the asymptotic case Minimal area of the cross-section in Nikolai 2 3 problem, min = Pmin Optimal cross-sectional area

a

References

R, P, S R = 1−α ,P = 1+α

227

2 , α+1

S = −2

θ ℵ R = Reℵ, ℵ I = Imℵ E˜ i , i = 1, .., 4 E i , i = 1, .., 4, E 1 = μ2/3 , E 2 = 2 2 q 3 ≡ P, M 3 = M ε ε εs εm εa εa

V v Va vs vm va

Parameters of Emden–Fowler equation Parameters of Emden–Fowler equation for the unimodal column Phase of the complex moment Real and imaginary parts of the equation ℵ Roots of the Eq. (6.34) in the asymptotic case Roots of the equation F(s) Eq. (6.34) Total stiffness of an arbitrary single-length column Total stiffness of an arbitrary symmetry element (half-length column) Total stiffness of the optimal symmetry element (half-length column, S) Total stiffness of the optimal symmetry element (half-length column, M) Total stiffness of the optimal symmetry element (half-length column, A), the asymptotic case Total stiffness of the optimal symmetry element (single-length column, A), asymptotic case Volume of an arbitrary single-length column Volume of an arbitrary symmetry element (half-length column) Volume of the optimal single-length column in the asymptotic case (A) Volume of the optimal symmetry element (half-length column, S) Volume of the optimal symmetry element (half-length column, M) Volume of the optimal symmetry element (half-length column, A) in the asymptotic case

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27. Egorov, Y. V., & Karaa, S. (1996). Sur La Forme Optimale D’une Colonne En Compression. C. R. Academy of Science Paris, 322, 519–524. 28. Egorov, Y. V., & Kondrat’ev, V. A. (1996). Estimates for the first eigenvalue in some SturmLiouville problems. Russian Mathematical Surveys, 51(3), 439–508; Uspekhi Mat. Nauk, 51(3), 73–144. 29. Egorov, Yu. V., & Kodratiev, V. A. (1996). On the optimal column shape. Doklady Math, 54(2), 748–750. 30. Egorov, Y., & Kodratiev, V. (1996). On spectral theory of elliptic operators. Birkhauser Verlag. 31. Seyranian, A. P., & Privalova, O. G. (2003). The Lagrange problem on an optimal column: Old and new results. Structural Multidisciplinary Optimization, 25, 393–410. 32. Atanackovic, T. M., & Seyranian, A. P. (2008). Application of Pontryagin’s principle to bimodal optimization problems. Structural and Multidisciplinary Optimization, 37, 1–12. 33. Young, L. C. (1937). Generalized curves and the existence of an attained absolute minimum in the calculus of variations. Comptes Rendus de la Société des Sciences et des Lettres de Varsovie, 30, 211–234. 34. Athans, M., & Falb, P. L. (1966). Optimal control: An introduction to the theory and its applications (p. 879). McGraw-Hill. 35. Hazewinkel, M. (Ed.). (2001). Sturm-Liouville theory. Springer-Verlag. 36. Zettl, A. (2005). Sturm-Liouville theory. American Mathematical Society. 37. Agarwal, R. P., & O’Regan, D. (2008). An introduction to ordinary differential equations. Springer. 38. Karaa, S. (2003). Properties of the first eigenfunctions of the clamped column equation. Theoretical Applied Mechanics, 30(4), 265–276. 39. Kobelev, V. (2016). Some exact analytical solution in structural optimization. Mechanics Based Design of Structures and Machines. https://doi.org/10.1080/15397734.2016.1143374 40. Timoshenko, S., & Gere, J. M. (1961). Theory of elastic stability. McGraw-Hill. 41. Jerath, S. (2021). Structural stability theory and practice. John Wiley & Sons. 42. Pachpatte, B. G. (2005). Mathematical inequalities. North-Holland Mathematical Library, 67, Elsevier. 43. Zwillinger, D., & Dobrushkin, V. (2022). Handbook of differential equations (4rd ed.). Taylor & Francis Group, LLC, 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742. 44. Berkovic, L. M. (1997). Generalized Emden-Fowler equation. Symmetry in Nonlinear Mathematical Physics, 1, 155–163. 45. Olver, P. J. (2012). Applications of lie groups to differential equations. Band 107 von Graduate Texts in Mathematics, Springer Science & Business Media, 497 p. ISBN: 1468402749, 9781468402742. 46. Oliveri, F. (2010). Lie symmetries of differential equations: Classical results and recent contributions. Symmetry, 2, 658–706. https://doi.org/10.3390/sym2020658 47. Gradshteyn, I. S., Ryzhik, I. M. (2014) Table of integrals, series, and products (8th ed.). Academic Press. https://doi.org/10.1016/C2010-0-64839-5. ISBN 978-0-12-384933-5. 48. Abramowitz, M., & Stegun I. A. (Eds.). (1983). Handbook of mathematical functions with formulas, graphs, and mathematical tables. Applied Mathematics Series 55. United States Department of Commerce, National Bureau of Standards; Dover Publications. 49. Olver, F. W. J., et al. (2010). NIST handbook of mathematical functions (p. 967). Cambridge University Press. 50. Maple. (2022). Maplesoft. Une Division De Waterloo Maple Inc. 2021. 51. Chavel, I. (2001). Isoperimetric inequalities differential geometric and analytic perspectives. Cambridge University Press. 52. Cox, S. J., & McCarthy, C. M. (1998). The shape of the tallest column. SIAM Journal of Mathematical Analysis, 29(3), 547–554.

Chapter 7

Stability Optimization of Twisted Rods

Abstract The stability problems are illustrated in this Chapter. In the context of twisted rods, the counterpart for Euler’s buckling problem is Greenhill’s problem, which studies the forming of a loop in an elastic bar under pure torsion. The authors search the optimal shape of the rod along its axis. A priori form of the cross-section remains unknown. For the solution of the actual problem the stability equations take into account all possible convex, simply connected shapes of the cross-section. Thus, the authors drop the assumption about the equality of principle moments of inertia for the cross-section. The cross-sections are similar geometric figures related by a homothetic transformation with respect to a homothetic center on the axis of the rod and vary along its axis. The distribution of material along the length of a twisted rod is optimized so that the rod is of the constant volume and will support the maximal moment without spatial buckling. The cross section that delivers the maximum or the minimum for the critical eigenvalue must be determined among all convex, simply connected domains. The authors demonstrate at the beginning the validity of static Euler’s approach for simply supported rod (hinged), twisted by the conservative moment. The applied method for integration of the optimization criteria delivers different length and volumes of the optimal twisted rods. Instead of the seeking for the twisted rods of the fixed length and volume, the authors directly compare the twisted rods with the different lengths and cross-sections using the invariant factors. The solution of optimization problem for twisted rod is stated in closed form in terms of the higher transcendental functions. In the torsion stability problem, the optimal shape of cross-section is the equilateral triangle. Keywords Stability optimization · Torsional buckling

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Kobelev, Fundamentals of Structural Optimization, Mathematical Engineering, https://doi.org/10.1007/978-3-031-34632-3_7

231

232

7 Stability Optimization of Twisted Rods

7.1 Isoperimetric Inequality for Twisted Rod with Arbitrary Convex, Simply-Connected Cross-Section Consider a thin elastic rod with isotropic cross-section, twisted by couples applied at its ends alone (Greenhill’s problem [1]). A long rod subjected to the action of torsional couples. If the torque increases, the rod may lose its straight form. For the certain torque moment, the rod axis forms a spatial, helical curve. Suppose that the ends of the rod are fixed pivotally and that the pairs whose moments are equal to M are applied at the ends of the rod, and the vector-moments of the pairs keep direction along the initial axis of the rod. The case of the constant moments of inertia of the rod section relative to all central axes was considered in Timoshenko and Gere [2, Chap. 5], Jerath [3, Chap. 6]. The system under consideration is nonconservative, because in different variants of the rod transition from the initial position to the curved position including the rotation of the end sections around the x-axis, the nares can produce different work. Generally saying, such problems require a dynamic approach [4]. As an exception, the static method leads for the problem under consideration to correct results. According to Euler’s theory, the magnitude of the critical moment is determined by the smallest positive eigenvalue M, Sect. 2.11 in Bolotin [5]. The static Euler’s approach was shown to be valid for the simply supported rod (hinged), twisted by the conservative moment [6]. The numerical solution of problems of optimizing stability of the simply supported twisted rod were studied in Banichuk and Barsuk [7]. The cross-sectional area variation that presents the largest value to the critical moment causing a loss of stability was searched. In course of optimization, the constraints on the volume of material and on the admissible thickness of the rod were gratified. The distribution of material along the length of a twisted rod is optimized so that the rod is of the constant volume V and will support the maximal moment without spatial buckling. The cross section that delivers the maximum or the minimum for the critical eigenvalue must be determined among all convex, simply connected domains. The closed-form analytical solutions was found for the optimal shape of the rod along its axis with the application of the variational methods [8]. The stability equations are valid for all possible convex, simply connected shapes of the cross-section. The beam possesses the similarly shaped cross-sections with the variable cross-sectional area along the axis. The cross-sections in different points along the axis are considered as the similar geometric figures. These figures are related by a homothetic transformation with respect to a homothetic center on the axis of the rod. The size of the cross-section varies along the axis of the rod:

a

a(x) > 0.

(7.1)

The function (x) is the locally integrable function over the length of the rod (x ∈ G L , G L = [−L , L]).

7.1 Isoperimetric Inequality for Twisted Rod with Arbitrary Convex …

a

def

where ⟨ f ⟩ =

⟨ ⟩ = V,

233

∫L f (x)d x.

(7.2)

−L

With respect to lines passing through a point on the neutral axis of bending and parallel to the axes y and z, the second moments of area are I11 , I12 , I22 . [ I=

] I11 I12 . I12 I22

I11 and I22 signify the bending moments of inertia around the y- and z-axes, respectively. The centrifugal moment of inertia is denoted as I12 [9]. The principal moments of inertia for the cross-section I I , I I I are the eigenvalues of the matrix I and must not be a-priory equal. The second moments of area are the powers of the cross-sectional area : I11 = C11

a, α

I12 = C12

a, α

I22 = C22

a

a. α

The constant α is the shape exponent. The values C11 , C12, C22 are the components of tensor of the second rank. Certain axes Y, Z coincide with the directions of principal axes of the second moment of area through its center of gravity. The angle between the principal axis of the second moment and y axis is Θ. This angle remains constant for all cross-sections along the axis of the rod. The moments of inertia are the functions of the angle Θ: I11 = II cos2 Θ + III sin2 Θ, I12 = (II − III ) sin Θ cos Θ, I22 = II sin2 Θ + III cos2 Θ, II = C I

a, α

III = C I I

a. α

(7.3)

Using the Bernoulli–Euler model, in this and the next Chapter we study the midline stability of an axially twisted rod. The rod is twisted by conservative couples M. The displacements in directions, transversal to the axis of the rod, are and The curvatures of the axis k z , k y in course of buckling are assumed to be small. Thus, the curvatures of the rod central line are:

y

kz =

y, ''

ky =

z

z. ''

With the curvatures of the rod line, the bending moments read: Mz = E I11 k z + E I12 k y ,

M y = E I12 k z + E I22 k y .

(7.4)

234

7 Stability Optimization of Twisted Rods

The torque remains constant along the axis of the rod. If the midline of the rod gradually deviates from being straight for some or all of its length, the axial torque gives rise of bending moments of the rod axis. The geometrically linear equations are applicable for the treatment of the stability problem. Initially the axis of the rod is straight and there are no bending moments M y , Mz . If the initially small couple M gradually increases, there exists a critical value of bending instability of the rod. The differential equations for stability will be: ( (

E I11 E I12

)''

y ) y '' ''

( + E I12 ( '' + E I22

)''

z ) z '' ''

''

z

= −μ

y

=μ

'''

'''

,

.

(7.5)

Consider the beam hinged on two supports x = L and x = −L: |

y||| y||

x=L

y y

x=−L E I11 ''

E I11

|

z||| z||

= 0,

''

= 0, + E I12 + E I12

x=L

= 0,

= 0, x=−L | '' = 0, E I12 + E I22 | |x=L | '' = 0, E I12 '' + E I22 |

z z

'' |

x=−L

y y

|

z ||| z || '' ''

x=L

= 0,

x=−L

(7.6)

= 0.

The bending moments vanish at both endpoints of the rod x = ±L. For constant cross-section ac (x) = const, the solution of (7.5) reduces to: (

( ) ) μc x μc x = C2 + C3 sin √ + C4 cos √ , II III II III ) ( cos(Θ) sin(Θ) μc x C2 + (C3 d1 + C4 d2 ) cos √ = C1 + (II − I I I ) cos2 (Θ) − II II III ) ( μc x + (C3 d2 − C4 d1 ) sin √ , II III √ II III cos(Θ) sin(Θ)(II − I I I ) d1 = , d2 = , d0 = (II − I I I ) cos2 (Θ) − II . d0 d0 (7.7)

y z

The last four integration constants C1 , C2 , C3 , C4 follow from the absence of the lateral displacements at both end points. The non-trivial solution appears, if the determinant of the characteristic equation vanishes: √ ) ( μc L 4 II III π μc L =0→ √ = . sin √ d0 2 II III II III Thus, the critical moment will be equal for the constant cross-section to [10]:

7.1 Isoperimetric Inequality for Twisted Rod with Arbitrary Convex …

μc =

235

π √ II III . 2L

We study only the optimization problems, for which both moments of inertia are proportional to one function j (x): II = ηI j,

III = ηII j,

√

√ II III = j η I η I I .

The values ηI , ηII remain√constant along the rod length. The square root of the α is proportional to the quantity α with product √ of area moments: II III = C C = C I C I I . This value is the isotropic shape factor. The shape exponent α takes the values of 1, 2 and 3. The case α = 2 corresponds to a proportional scaling of the cross-section. The Euler characteristic or, alternatively, the topological genus of a surface represents the number of the “holes”. By determining the moment projections on the axes, we arrive at the following differential equations:

a

( √ E II III

y

''

)''

z

= −μ

'''

a

(

,

EJ

√

II III

z

''

)''

y

=μ

'''

.

(7.8)

The problem (7.8) is neither self-adjoint nor conservative in the classical sense. However, it is a conservative system of the second kind [6], since: [( √ ⟨ E II III

y

''

)''

]

[( √ E II III

z y ⟩=⟨

+M

'''

''

z

''

)''

]

y z ⟩.

−M

'''

''

(7.9)

As shown in Sect. 4.2 of Leipholz [6], the system is self-adjoint in a generalized sense with respect to operator: [ ] T

y =y . ''

If moments of the second order are not equal (η1 /= η2 ), critical eigenvalue M will be proportional to the geometric mean of the principal moments of the second order: M=

√ ηI · ηII · M.

The critical eigenvalue of Eqs. (7.7) and (7.8) is M = M. In the simplest, isotropic case, both equal principal moments the second order are equal: ηI = ηII = 1.

236

7 Stability Optimization of Twisted Rods

Among others, this setting is valid for the rod with the cross-section in form of an equilateral triangle and for the rod with a circular cross-section. Among the simply connected cross-sections, worth be mentioned the triangle and circular sections. The optimal convex shape of the simply connected cross-section was determined in Ting [11]. Of all convex domains with the area Δ , the equilateral triangle yields the maximum of the product I I I I I :

a

√

√

3 18

II II I ≤

a

2 Δ

= jΔ .

Thus, the rod with the cross-section in form of an equilateral triangle delivers the maximum for critical eigenvalue for all convex domains of the same cross-sectional area √Δ . For the cross-section in form of the equilateral triangle the shape factor C = 3/18 ≈ 0.9622. For the circular cross-section the shape factor is C = (4π )−1 ≈ 0.07957... The shaft with the circular, simply connected cross-section Θ delivers correspondingly the minimum for critical eigenvalue:

a

a

√

II II I ≥

1 4π

a

2 Θ

= jΘ .

Two other cases describe the situations in which the form of transverse crosssection endures the transformation such that one of dimensions of the cross-section changes. The case α = 1 corresponds the variable wall thickness (as the design variable) and constant mean diameter of the tube. The case α = 3 corresponds the changeable mean diameter of the tube (as the design variable) and constant wall thickness.

7.2 Optimization Problem and Isoperimetric Inequality for Stability Based on the above solution, the distribution of material along the length of a twisted rod will be optimized. The principal moments of the second order will be equal. All admissible shafts are of the same volume V. The distribution of material along the length of a twisted shaft is optimized so that the rod is of constant volume and provides the maximal moment without spatial buckling. The material volume V[ ] of the rod with the cross-sectional area > 0 is equal to:

a

a

⟨ ⟩ = ⟨A⟩ = V.

a

(7.10)

7.2 Optimization Problem and Isoperimetric Inequality for Stability

237

The formal statement of the optimization problem is the following with Eq. (7.10): ] [ [ M∗∗ = M Aα , , = max μ

yz a

(x)

a , y, z α

]

= M∗ .

(7.11)

The constancy of volume among all competitive designs plays the role of the isoperimetric condition.Therefore, the optimal solution for boundary conditions produces the optimum function A, for which the critical moment M[A] is greater, than the critical moment M[ ] for any arbitrary thickness distribution with the same volume of material of the compressed ring:

a

A = optimum

a

A ≥ 0, ⟨A⟩ = ⟨ ⟩ = V, M∗∗ = M[A] ≥ μ[ ] = M∗ .

a⇔

a

a

The augmented functional for the optimization problem (7.11) reads: [( L[ ] = V[ ] + λ⟨ E j

a

a

y

''

)''

]

z y

+μ

'''

''

−

[( Ej

z

''

)''

]

y z⟩

−μ

'''

''

(7.12)

The necessary optimality condition results from first variation of the augmented Lagrangian (7.12). With an auxiliary constant c = (ECα)−1 λ, the optimality condition reads: −c + Aα−1 K2 = 0, K2 = κ y2 + κz2 . The spatial curvature is signed in Eq. (7.13) as K = variables: κy =

z, ''

κz =

√

(7.13)

κ y2 + κz2 , where in the new

y. ''

Let optimal distribution of the cross-sectional area along the span of the rod is A( ). The application of the Fermat’s principle for the optimization problems leads to the necessary optimality condition:

x

A = (K/c)2/(1−α) ,

J = CAα .

(7.14)

238

7 Stability Optimization of Twisted Rods

The necessary optimality condition (7.14) is the requirement that the augmented Lagrangian has a stationary value. The optimal second moment of the cross-section follows from (7.14) as: J = EC · (K/c)2α/(1−α) =

(

y

''2

z

''2

+

)α/(1−α)

.

(7.15)

The applied method of scaling allows the arbitrary selection of the constant c. For briefness of the governing equations, the authors set the constant c as the positive solution of the equation: EαC(1/c)2α/(1−α) = 1. With this choice the bending stiffness reduces to: EJ =

(

y

''2

+

z

''2

)α/(1−α)

≡ K2α/(1−α) .

(7.16)

7.3 Closed-Form Solution of Optimization Problem The order of both Eqs. (7.8) could be reduced by one using the boundary value conditions (7.7). The buckling equations transform with Eq. (7.16) to: [ ] ( e1 κ y , κz = [ ] ( e2 κ y , κz =

y y

''2 ''2

z +z +

''2 ''2

)α/(1−α) )α/(1−α)

y z

z

''

= −M ' ,

''

= M '.

y

(7.17)

For the solution of two simultaneous Eqs. (7.17), new functions K(x), θ (x) are introduced: κ y = K cos θ, κz = K sin θ.

(7.18)

Equation (7.17) are equivalent to two simultaneous equations for the new unknowns K, θ : [ ] [ ] e~1 [K, θ] = e1 κ y , κz sin(θ ) − e2 κ y , κz cos(θ ) = 0, [ ] [ ] e~2 [K, θ] = e1 κ y , κz cos(θ ) + e2 κ y , κz sin(θ ) = 0, K(L) = K(−L) = 0, θ (0) = 0, A(0) = 1. The equations e~1 [K, θ] and e~2 [K, θ] in (7.19) reduce to:

(7.19)

7.3 Closed-Form Solution of Optimization Problem

239

[ ] − (α − 1)K2 θ '' + K' 2(α + 1)Kθ ' + (α − 1)MK(3α−1)/(α−1) = 0, ) ( 1 − α 2 K4 K'' − K5 (1 − α)2 θ '2 + α(1 + α)K3 K'2 + (1 − α)2 Mθ ' K(7α−5)/(α−1) = 0.

(7.20)

It is possible to solve Eq. (7.18) for K, θ . It is preferably to find the solution for the functions A, θ , using Eq. (7.12) to replace K by A: ] 1 [ Aα+1 θ '' + A' 2(α + 1)Aα θ ' + (α − 1)MA' = 0, 2 ( ) 2(1 + α)A'' − 1 − α 2 A−1 A'2 − 4Aθ '2 + 4A1−α Mθ ' = 0.

(7.21)

The integration constant could be put to zero from the symmetry considerations: ϑ(0) = 0. The solution ϑ(x) of the second Eq. (7.21) with respect to θ (x) reads: α−1 ϑ= Λ 2

∫x

A−α (z)dz.

(7.22)

0

Substitution of ϑ(x) instead of θ (x) into the first Eq. (7.21) leads to an equation in terms of A only: (1 − α)A'2 − 2AA'' + (α − 1)M2 A2−2α = 0.

(7.23)

For solution the dependent and independent variables in (7.23) must be exchanged: ] [ ( )2 dx d2x 1−α 2 2−2α d x . = −1 M A dA2 2A dA dA

(7.24)

Equation (7.24) is the equation of the second order with missing x(A) and allows the order reduction to an equation of the first order. The reduction of order follows the substitution: Ξ=

dx . dA

Equation (7.24) transforms to Emden–Fowler equation [12, 13]: ) 1 − α ( 2 2−2α 2 dΞ = · M A Ξ − 1 · Ξ. dA 2A The solution of Emden–Fowler equation of the first order reads: ∫A x =− 0

tα √ dt + C2 for 0 < A ≤ 1, right half, C1 t α+1 − M2 t 2

240

7 Stability Optimization of Twisted Rods

∫A x= 0

tα √ dt − C2 C1 t α+1 − M2 t 2

for 0 < A ≤ 1, left half.

(7.25)

The solution A(x) is an even function of x. According to the symmetry of the solution, dA/d x must vanish in the point x = 0. From this condition follows, that C2 = L. The integral Eq. (7.25) is summable, if C1 = M2 . The resulting equation of the first order is solvable in quadrature: 1 x=L− M

∫A √ 0

x=

1 M

∫A 0

t α−1 1 − t α−1

dt for 0 < A ≤ 1, right half,

t α−1 dt − L for 0 < A ≤ 1, left half. √ 1 − t α−1

(7.26)

The function A(x), as parametrically specified by Eqs. (7.25) and (7.26), is an even function of variable x. The singular integrals in (7.25) and (7.26) express in terms of the hypergeometric and beta functions [14, 15]: ∫A 0

∫A 0

Aa+1 · F2 dt = √ a+1 2 1 − tb ta

([

) ] [ ] 1 a+1 a+b+1 b , for 0 < A ≤ 1; , ,A 2 b b

√ [ )] (√ π dt = √ 1 − erf −ln(A) for 0 < A ≤ 1, √ ε 1 − tε tε

∫1 0

a

√t dt 1−t b

(7.27)

) ( for a > 0, b > 0; = b1 B 21 , a+1 b (7.28)

∫1 0

ε

√t dt 1−t ε

=

√ π √ , ε

for ε → 0.

The expression erf x is the Gauss error function. With the formulas (7.27) the integrals (7.26) for different shape exponents α express as: ([ ) ] [ ] 1 α 2α − 1 Aα α−1 for α > 1; , ,A · F2 , x =1− αM 2 2 α−1 α−1 √ (√ ) 1 π x= erf −ln(A) + o(α − 1) for α → 1 and α > 1. (7.29) M α−1 ) √ √ 2 ( 2 − 2 1 − A − A 1 − A + o(α − 2) for α → 2; x =1− 3M

7.3 Closed-Form Solution of Optimization Problem

x =1−

) √ 1 ( arcsin(A) − A 1 − A2 + o(α − 3) 2M

241

for α → 3.

(7.30)

The functions in Eqs. (7.27) to (7.30) are the real functions of 0 < A < 1. The limit in Eq. (7.28) is the right-hand limit α → 1+. The area of the optimal cross-sections vanishes in the end points of the rod, where the bending moment disappears: x(A = 0) = L . Substitution of the condition A(0) = 1 into (7.26) leads to: 1 L= M

∫1 0

t α−1 dt. √ 1 − t α−1

(7.31)

The integral (7.31) evaluates with Eq. (7.28) in terms of Beta-function: L=

( ) 1 α 1 1 B , . Mα−1 2 α−1

(7.32)

For volume evaluation the integrand in Eq. (7.31) must be multiplied by area t: V=

1 M

∫1 0

( ) 1 α+1 tα 1 1 B , . dt = √ Mα−1 2 α−1 1 − t α−1

(7.33)

For estimation of elastic energy E the integrand in Eq. (7.28) must be multiplied by area t α [14]: 1 E= M

∫1 0

( ) 1 2α 1 1 B , . dt = √ Mα−1 2 α−1 1 − t α−1 t 2α−1

(7.34)

For briefness, the normalized axial coordinate will be used: X = x/L, −1 ≤ X ≤ 1. Finally, the dependence of the optimal shape A upon the coordinate X is given implicitly in terms of hypergeometric function: Aα α−1 ( 1 α ) 2 F2 X =1− α B 2 , α−1

([

) ] [ ] 1 α 2α − 1 α−1 . , , ,A 2 α−1 α−1

(7.35)

The expression for volume of the half of the optimal rod reads:

v

( ) B 21 , α+1 V α−1 = = (1 α ). L B 2 , α−1

(7.36)

242 Table 7.1 Normalized and normalized elastic energy for different shape exponents α = 1, 2, 3

7 Stability Optimization of Twisted Rods

Normalized volume

α 1 2 3

v= v= v= v=

V

L √1 2 4 5 8 3π

Normalized elastic energy ε= ε= ε= ε=

E

L. √1 2 24 35 32 15π

For elastic energy E of the half of the optimal rod, the following formula is valid: ( 2α ) B 21 , α−1 E ε = = (1 α ). L B 2 , α−1

(7.37)

For practically interesting cases the shape reduces to the elementary functions, as shown in Table 7.1. The buckling shape of the twisted column with the simply supported (hinged) ends is demonstrated on Fig. 7.1 for α = 2. For the other values of the shape exponent α the buckling shapes are similar. Figure 7.2 displays the areas of cross-sections of the optimal twisted columns for possible values of shape exponent α = 1, 2, 3. Second moments of inertia of the cross-sections of the optimal twisted columns are shown for α = 1, 2, 3 on Fig. 7.3.

7.4 Effectiveness of Optimal Designs The influence of the shape exponent α influences the estimations for the optimization effects. Another argument for the introduction of the invariant optimization factors is methodical. In the variational calculus is common to get one factor as the optimization objective and the others as the a-priori given constraints. To convert it into an unconstrained problem the method of Lagrange multipliers is commonly used. For the twisted rod the optimality expresses with the isoperimetric inequalities. The resulting unconstrained problem with Lagrange multiplies increases number of variables. The new number of unknown variables is the original number of variables plus the original number of constraints. The constraints are usually solved for some of the variables in terms of the others, and the former can be substituted out of the objective function, leaving an unconstrained problem in a smaller number of variables. This method of solution of leads to the nonlinear algebraic equations for Lagrange multiplies. These nonlinear equations in the most cases do not possess the closed analytical solutions and are solvable only numerically. The authors use systematically the method of dimensionless factors for the optimization analysis. The applied method for integration of the optimization criteria delivers different length and volumes of the optimal twisted rods. Instead of the seeking for the twisted rods of the fixed length and volume, the authors directly

7.4 Effectiveness of Optimal Designs

243

y z

Fig. 7.1 The shaft with the simply supported (hinged) ends and its buckling curve. The projections (x), (x) of the spatial buckling curve in Cartesian coordinates are shown in the graphs below

compare the twisted rods with the different lengths and cross-sections using the invariant factors. Instead of dealing with the Lagrange multipliers, the authors introduce the certain invariant factors. Consider the twisted rods with the same form of cross-sections, fixing the exponent α. For each fixed value of α the authors introduce the two dimensionless factors:

244

7 Stability Optimization of Twisted Rods

Fig. 7.2 Areas of cross-sections A of the optimal twisted rods for α = 1, 2, 3

FV = M

L p1 , V p2

FE = 2M

V p3 . E p4

(7.38)

For some arbitrary powers p1 , p2 , p3 , p4 , the factors alter for any affine transformation of the rod. The affine transformation of the rod is the product of two elementary transformations, namely homothetic axial and transverse scaling. The homothety ratio ζ multiplies lengths by ζ . In other words, ζ is the ratio of magnification or dilation factor or scale factor or similitude ratio. The cross-section function A(ξ ) scales by another factor q, such that for the affine transformed twisted rod the cross-section function will be qA(ξ ). Apparently, the eigenvalue M alters in course of the affine transformation of the twisted rod. The authors use the factors FV , FE for the comparisons of different designs. The critical buckling moment ECM inherits the factor C and is proportional to this value. Evidently, that the ratios of the buckling loads for different designs with the same form of the cross-sections do not depend on the constants C. For different cross-sections the actual value of C have to be used. With the methods of dimensional analysis, the authors can immediately determine the distinctive choice of powers p1 = 1 + α,

p2 = α,

p3 = 2 + α,

p4 = 1 + α.

(7.39)

7.4 Effectiveness of Optimal Designs

245

Fig. 7.3 Second moments of inertia of the cross-sections of the optimal twisted rods for α = 1, 2, 3

Using the expressions (7.38) and (7.39), the factors result to: FV =

L 1+α , Vα

FE = 2

V 2+α . E 1+α

(7.40)

The factors FV , FE do not alter for any affine transformation of the twisted rod. Thus, the factors FV , FE are the invariants to the affine transformation of the twisted rod and provide a natural basis for the comparison of different designs. With the above factor, the estimation of the effect of mass optimization turns out to be trivial. For this purpose, the authors consider the reference design with the constant cross-section along the span. The invariant factor for the reference design is F˜V . The factor equals for all exponents α and for the boundary conditions with both hinged ends reads: F˜V = 1.

(7.41)

The greater the factor is, the higher the buckling moment for the given length and volume of the twisted rod. For example, the buckling force of the reference clamped twisted rod is four times the buckling force of the reference twisted rod with the hinged ends. The dual formulations are typical the optimization of buckling twisted rod as well. For the dual formulations, the masses of the twisted rods for the fixed lengths and

246

7 Stability Optimization of Twisted Rods

fixed buckling forces are compared. The volumes and masses of the optimal and reference twisted rods relate to each other as the inverse roots of the order α of the factors FV : V = ˜ Vr e f

√ α

F˜V , FV

(7.42)

Specifically, the twisted rod with the higher value of the factor FV possesses the lower mass. The results of the evaluation of the dimensionless volume factors and volumes for the fixed critical eigenvalue M = π/2 and the half-length L = 1 are presented in Table 7.2. The volumes of the optimal columns v and the volume optimization factor FV are shown as the functions α of on Fig. 7.4. Figure 7.5 demonstrates the volumes v and elastic energy ε of the optimal columns. With the above deliberations, the optimal shapes of the torqued shaft could be displayed for another standard end conditions: (a) Upper end pinned; lower end translation free clamped. Total length of the shaft is L. (b) Upper end pinned; lower end pinned. Total length of the shaft is 2L. (c) Upper end clamped (fixed); lower end translation free clamped. Total length of the shaft is 2L. (d) Upper end clamped (fixed); lower end clamped (fixed). Total length of the shaft is 4L. If the cross-sections in each of the repetitive elements match, the critical torque of these shafts remains M. Figure 7.6 demonstrates the optimal shapes of the torqued shaft for different end conditions.

Table 7.2 Dimensionless volume factor and volume for the fixed critical eigenvalue M = π/2 and the half-length L = 1 Shape exponent

FV

V (Δ = π/2, L = 1)/2

Constant cross section, reference α=0

1

1

√ 2

√ 2/2

α=1 α=2

25/16

4/5

α=3

27π 3 /512

8/3π

7.5 Conclusions

Fig. 7.4 Volumes of the optimal twisted rods

247

v and the volume optimization factor F

V

7.5 Conclusions In this Chapter, the authors study a thin elastic rod with isotropic cross-section, twisted by couples applied at its ends alone (Greenhill’s problem). According to Euler’s theory, the magnitude of the critical moment was determined by the smallest positive eigenvalue M. The solution of optimization problem for twisted rod was stated in closed form. The solution expresses in terms of the higher transcendental functions. The final formulas involved the length of the rod, its volume and critical torque. Notable, that in the torsion stability problem the optimal shape of the rod is roughly parabolic along the axis of the rod. As usual, the optimal shape of crosssection is the equilateral triangle.

248

7 Stability Optimization of Twisted Rods

Fig. 7.5 Volumes

v and elastic energy ε of the optimal twisted rods

7.6 Summary of Principal Results • the validity of static Euler’s approach was demonstrated for simply supported rod (hinged), twisted by the conservative moment; • the applied method for integration of the optimization criteria delivers different length and volumes of the optimal twisted rods; • the solution of optimization problem for twisted rod is stated in closed form in terms of the higher transcendental functions; • the optimal shapes of the torqued shaft could be displayed for another standard end conditions: – – – –

Upper end pinned; lower end translation free clamped; Upper end pinned; lower end pinned; Upper end clamped (fixed); lower end translation free clamped; Upper end clamped (fixed); lower end clamped (fixed).

7.6 Summary of Principal Results

249

Fig. 7.6 Optimal shapes of the torqued shaft for different end conditions: a Upper end pinned; lower end translation free clamped. b Upper end pinned; lower end pinned. c Upper end clamped (fixed); lower end translation free clamped. d Upper end clamped (fixed); lower end clamped (fixed)

250

7 Stability Optimization of Twisted Rods

7.7 List of Symbols A

a V

[

I= M C L[ ] c

a

I11 I12

Optimal area of cross-section Arbitrary area of cross-section ] Volume of the twisted shaft I12 Matrix of the bending moments I22 eigenvalue of the dimensionless eigenvalue problem shape factor of cross-section augmented functional for the optimization problem Auxiliary constant (Lagrange multiplier)

References 1. Greenhill, A. G. (1883). On the strength of shafting when exposed both to torsion and to end thrust. In Institution of Mechanical Engineers, Proceedings (pp. 182–225). 2. Timoshenko, S., & Gere, J. M. (1961). Theory of elastic stability. McGraw-Hill. 3. Jerath, S. (2021). Structural stability theory and practice. Wiley. 4. Kirillov, O. N. (2013). Nonconservative stability problems of modern physics. Walter de Gruyter GmbH. 5. Bolotin, V. V. (1963). Nonconservative problems of the theory of elastic stability. Pergamon Press. 6. Leipholz, H. H. E. (1974). On conservative elastic systems of the first and second kind. Ingenieur-Archiv, 43, 255–271. 7. Banichuk, N. V., & Barsuk, A. A. (1982). On stability of torsioned elastic rods. Izvestiya Akademii Nauk SSSR, Mekhanika Tverdogo Tela (MTT ), 7(6), 148–154. 8. Kobelev, V. (2023). Closed form solution in the buckling optimization problem of twisted shafts. Applied Mechanics, 4(1), 317–333. https://doi.org/10.3390/applmech4010018 9. Andersen, L., & Nielsen, S. R. K. (2008). Elastic beams in three dimensions. Structural Mechanics, Lecture Notes (Vol. 23). Aalborg University, Department of Civil Engineering. http://homes.civil.aau.dk/jc/FemteSemester/Beams3D.pdf. ISSN 1901-7286. 10. Biezeno, C. B., & Grammel, R. (1956). Engineering dynamics (Vol. 1955). Blackie. 11. Ting, T. W. (1963). Isoperimetric inequality for moments of inertia of plane convex sets. Transactions of the American Mathematical Society, 107(3), 421–431. 12. Zwillinger, D., & Dobrushkin, V. (2022). Handbook of differential equations (4rd Ed.). Taylor & Francis Group, LLC. 13. Berkovic, L. M. (1997). Generalized Emden-Fowler equation. Symmetry in Nonlinear Mathematical Physics, 1, 155–163. 14. Gradshteyn, I. S., & Ryzhik, I. M. (2014). Table of integrals, series, and products (8th Ed.). Academic Press. https://doi.org/10.1016/C2010-0-64839-5, ISBN 978-0-12-384933-5. 15. Abramowitz, M., & Stegun, I. A. (Eds.). (1983). Handbook of mathematical functions with formulas, graphs, and mathematical tables. Applied Mathematics Series, 55. United States Department of Commerce, National Bureau of Standards; Dover Publications.

Chapter 8

Periodic Greenhill’s Problem for Twisted Elastic Rod

Abstract In this chapter the authors demonstrate the isoperimetric inequality arising in exactly solvable structural optimization problem of stability under torque load. The periodic Greenhill problem describes the forming of a loop in an elastic bar under torsion. The inequality for infinite rod with periodical cross-section with two types of supports is rigorously verified. The optimal shape of the twisted rod is constant along its length and the optimal shape of cross-section is the equilateral triangle. The technique to demonstrate of isoperimetric inequalities exploits the variational method and the Hölder inequality. Keywords Stability optimization · Torsional buckling · Periodic Greenhill’s problem

8.1 Periodic Greenhill’s Problem The minor counterpart for the celebrated Euler buckling problem is the Greenhill’s problem, which studies the forming of a loop in an elastic bar under torsion [1]. In the paper [2] the problem of the optimal design of columns under combined compression and torsion was investigate and the cross-sectional area varying along the axis of the column which leads to the maximal critical loading was sought. The varying crosssection was approximated either by a function with free parameters or is determined using Pontriagin’s maximum principle. According to Euler’s stability theory, the magnitude of the critical moment is determined by the smallest positive eigenvalue M = μ[ ]. The authors demonstrate the validity of static Euler approach for periodically supported infinite rod, twisted by the conservative moment. At first, consider an infinite thin elastic rod with an isotropic cross-section. The second moment of inertia of the cross-sectional area with respect to lines passing through a point on the neutral axis of bending and parallel to the axes y and z is the positive function j = C α with the period L > 0:

a

a

j (x) = j(x + L) > 0.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Kobelev, Fundamentals of Structural Optimization, Mathematical Engineering, https://doi.org/10.1007/978-3-031-34632-3_8

251

252

8 Periodic Greenhill’s Problem for Twisted Elastic Rod

The function j −1 (x) is locally integrable. The rod is twisted by conservative couples, such that the torque M is constant over the axis of the rod. Using the Bernoulli–Euler model, the differential equation of twisted rod [3], Eq. (7.5) in terms of lateral deflections y(x), z(x) read: ( (

E j y '' E jz

)''

) '' ''

= −Mz ''' , = My ''' .

The elastic modulus of material is E. The corresponding equations in terms of bending moments: m z = E j y '' , m y = E j z '' . are (

m ''z = −M ·

my Ej

)'

, m ''y = M ·

(

mz Ej

)'

.

(8.1)

x = k L for k = . . . , −2, −1, 0, 1, 2, . . .

(8.2)

The coordinates of the supports are:

8.2 Periodic Conditions Two periodic conditions at the points (8.2) are considered. (A) The elastic rod is periodically simply supported with fixed edges. The lateral forces '

Q z (k L) = m y (k L),

'

Q y (k L) = −m z (k L).

satisfy the conditions: '

'

'

'

m y (k L) = −m y (k L + L), m z (k L) = −m z (k L + L)

(8.3)

The conditions (8.3) express the periodical directional alteration of lateral reaction forces. Assume the initial conditions for k = 0: m y (0) = c1 , m z (0) = c2 . with arbitrary constants c1 , c2 . The solution of the Eq. (8.1) reads:

(8.4)

8.2 Periodic Conditions

253

m y (x) = c1 cos F(x) + c2 sin F(x), m z (x) = c2 cos F(x) − c1 sin F(x).

(8.5)

where ∫x F(x) = M 0

dξ . E j (ξ )

(8.6)

The periodicity condition (8.3) for the functions (8.5) requires, that '

'

↔ −b = −c1 sin F(L) + b cos F(L),

'

'

↔ c1 = −b sin F(L) − c1 cos F(L).

m y (0) = −m y (L) m z (0) = −m z (L)

(8.7)

The condition (8.3) is satisfied for arbitrary constants a, b, only if the determinant of (8.7) vanishes 2 + 2 cos F(L) = 0. This equation represents the stability equation for the case (A). Its first positive solution is F(L) = π. With this condition the critical eigenvalue of the Greenhill problem with a periodically variable cross-section reads: μ= ∫L 0

πE j −1 d x

.

(8.8)

(B) The elastic rod is clamped on free movable edges: m y (k L) = −m y (k L + L),

m z (k L) = −m z (k L + L).

(8.9)

The conditions (8.9) express the sign change of reaction moments at the clamped supports. Consider again the solution (8.5). The periodicity condition (8.9) for the functions (8.5) requires, that m y (0) = −m y (L) ↔ c1 = −c1 cos F(L) − c2 sin F(L), m z (0) = −m z (L) ↔ c2 = −c2 cos F(L) + c1 sin F(L). The condition (8.9) is satisfied, only if the determinant of (8.10) vanishes: 2 + 2 cos F(L) = 0.

(8.10)

254

8 Periodic Greenhill’s Problem for Twisted Elastic Rod

Its first positive solution is: F(L) = π. Equation (8.8) provides the stability equation also for the case (B).

8.3 Stability of Twisted, Periodically Supported Rod with Varying Stiffness Functions j and j −1 are locally integrable and periodic with the period L: j(x) = j (x + L). The moment of inertia of congruent figures could be represented in terms of their cross-sectional areas: ( ( ) ) L L −x = + x , for 0 ≤ x ≤ L . j = C α (x), (x) = (x + L), 2 2

a

a

a

a

a

The shape exponent α assumes depending on the form of the cross-section the values of 1, 2 and 3. The cases α = 1 and α = 3 describe the situations in which the form of transverse cross-section undergoes an affine transformation such that one of the geometrical dimensions of the cross-section changes. In this article only the case α = 2 is studied. This case corresponds to a congruent change in the form of the cross-section. C is the shape factor. For the circular cross-section the shape factor is C = (4π )−1 . For the cross-section in the form of an equilateral triangle the value for √ the shape factor is C = 3/18. The following notation is used for brevity 1 ⟨f⟩ = L

∫L f (x)d x. 0

With this notation the critical torque of the conservative Greenhill problems with an arbitrary, variable cross-section reads:

a

a

M ∗ = μ[ ] =

π E . ⟨ j −1 ⟩ L

(8.11)

8.4 Optimization Problem for Periodically Supported Twisted Rod

255

8.4 Optimization Problem for Periodically Supported Twisted Rod Based on this solution, the distribution of material along the length of a twisted rod is optimized:

a

a

M ∗∗ = μ[A] = max μ[ ]. (t)

(8.12)

The volume of one section the rod is given

a

V = ⟨ ⟩ = ⟨A⟩

(8.13)

Consequently, the optimal solution for boundary conditions produces the optimum function A, for which the critical load M is greater, than the critical compression load for any arbitrary thickness distribution a with the same volume of material of the compressed ring: A = optimum

a

A ≥ 0,

a

⟨A⟩ = ⟨ ⟩ = V

a

μ[A] ≡ M∗∗ ≥ μ[ ] ≡ M∗

The constancy of volume among all competitive designs plays the role of the isoperimetric condition. The inequality about the mean values states, [4], [5] for α ≥ −1 reads: ⟨

⟩1/ p ⟨ q ⟩1/q ⟨ ⟩−1 jp ≤ j if p ≤ q ⇒ j −1 ≤ ⟨ j⟩ = ⟨J ⟩, p = −1, q = 1, α α with j = C , J = CA , A = const. (8.14)

a

where the equality is attained only for: A=

a for 0 ≤ x ≤ L .

(8.15)

Substitution (8.15) into (8.11) states the upper boundary M∗∗ for critical buckling moment in form of isoperimetric inequality

a

M∗ = μ[ ] =

πEJ πE ≤ = μ[A] ≡ M∗∗ . −1 ⟨ j ⟩L L

(8.16)

a

The equality in (8.16) attains only for the rod with constant cross-section ≡ A . The distribution of material along the length of a twisted rod is optimized so that the rod is of constant volume and provides the maximal moment without spatial buckling. The periodically supported rod with constant cross-section A0 provides the maximal buckling moment among all rods with the same section volumes.

256

8 Periodic Greenhill’s Problem for Twisted Elastic Rod

8.5 Isoperimetric Inequality for Periodically Supported Twisted Rod The authors consider now a rod with an unknown a priori form of the cross-section. The authors search once again the optimal shape of the rod along its axis, but now the cross-section is not assumed to be isotropic. Namely, the cross section that delivers the maximum or the minimum for the critical eigenvalue must be determined among all convex, simply connected domains. For the solution of the actual problem the stability equations must be generalized, taking into account all possible convex, simply connected shapes of the cross-section. Let I1 and I2 denote the moments of inertia of Ω about the principal axes of inertia through the center of gravity. The cross-sections are similar geometric figures related by a homothetic transformation with respect to a homothetic center on the axis of the rod and vary along its axis: II = CI

a (x), I α

II

= CI I

a (x), j = √C C a , J = √C C α

I

II

α

I

IIA

α

.

(8.17)

The shape factors C I , C I I in Eq. (8.17) are the moments of inertia of the reference cross-section with = 1. The bending moments in terms of curvatures of the rod are:

a

m z = E I I I y '' , m y = E I I z '' .

(8.18)

The equations an infinite periodic uniform beam resting on an arbitrary number of simple supports in terms of lateral displacements y(x), z(x) read: m ''z = −Mz ''' , m ''y = My ''' .

(8.19)

The solution of the Eqs. (8.18) and (8.19) with the initial conditions (8.4) is: √ m y (x) = c1 sin G(x) + c2 √ m z (x) = −c1

CI cos G(x), CI I

CI I cos G(x) + c2 sin G(x). CI

(8.20)

In Eq. (8.20), the function is: μ G(x) = √ E II II I

∫x 0

dξ . j(ξ )

(8.21)

The corresponding periodicity conditions (A) or (B) for the functions (8.20) is satisfied for arbitrary constants c1 , c2 , only if.

8.6 Conclusions

257

2 + 2 cos G(L) = 0. This equation delivers the stability equation with the first positive solution: G(L) = π. The critical eigenvalue is:

a

M∗ = μ[ ] =

π E , ⟨ j −1 ⟩ L

j=

√ CI CI I

a. α

(8.22)

The distribution of material along the length of a twisted rod is optimized based on the solution (8.22) using the same arguments, as in the Sect. 8.2. For all rods of the given section length and given section volume the critical eigenvalue of the Greenhill problem attains its maximum for the rod with constant cross-section A0 over the span length: √ CI CI I α A = M∗∗ . M = μ[ ] ≤ μ[A] = π E L

a

∗

(8.23)

The last step is to determine the optimal convex, simply connected shape of the cross-section. In this case the shape exponent is equal: α = 2. The solution grounds on the isoperimetric equation [6], Eq. (5.2). Namely, of all convex domains the equilateral triangle yields the maximum of II III , such that: √

II II I

√ √ 3 3 2 √ . A , ≤ CI CI I ≤ 18 18

(8.24)

Thus, the rod with the cross-section in form of an equilateral triangle delivers the maximum for critical eigenvalue for all convex, simply connected domains of the same cross-sectional area A. The isoperimetric inequality is stated: ∗

a

M = μ[ ] ≤ μ[A] = π E

√

3 A2 ≡ M∗∗ . 18 L

(8.25)

The shaft with the circular cross-section delivers correspondingly the minimum for critical eigenvalue.

8.6 Conclusions In this Chapter the optimization problem for twisted rod of an infinite length, which rested on periodically located moment-free bearings, was examined. Remarkable, that in the torsion problem the optimal shape of the optimal rod is constant along its

258

8 Periodic Greenhill’s Problem for Twisted Elastic Rod

length and the shape of cross-section is the equilateral triangle. The solution of the optimization problem expresses in form of isoperimetric inequality, involving the length of the rod, its volume and critical torque. This isoperimetric inequality was rigorously justified by means of Hölder inequality about the mean values.

8.7 Summary of Principal Results • The optimal distribution of material along the length of a twisted rod, which has constant volume and provides the maximal moment without spatial buckling, is investigated; • the periodically supported rod with constant cross-section provides the maximal buckling moment among all rods with the same section volumes; • the twisted rod with the cross-section in form of an equilateral triangle delivers the maximum for critical eigenvalue for all convex, simply connected domains of the same cross-sectional area; • the twisted rod with the circular cross-section delivers correspondingly the minimum for critical eigenvalue.

8.8 List of Symbols A

Area of the optimal cross-section (x) = (x + L) periodic cross-section y(x), z(x) Lateral displacements m y (x), m z (x) Bending moments V Reference volume of the twisted rod M∗ ≡ μ[ ]. Critical eigenvalue of the dimensionless eigenvalue problem M ∗∗ ≡ μ[A]. Extreme eigenvalue √ of the dimensionless eigenvalue problem C Shape factor, CΔ = 183 , CΘ = (4π )−1 I1 , I2 Moments of inertia of some reference cross-section

a

a

a

a

a

References 1. Greenhill, A. G. (1883). On the strength of shafting when exposed both to torsion and to end thrust. Institution of Mechanical Engineers, Proceedings, 182–225. 2. Kruzelecki, J., & Ortwein, R. (2012). Optimal design of clamped columns for stability under combined axial compression and torsion. Structural and Multidisciplinary Optimization, 45, 729–737. 3. Ziegler, H. (1977). Principles of structural stability, 2. ed. Birkhäuser. 4. Pachpatte, B. G. (2005). Mathematical inequalities. North-Holland Mathematical Library, 67. Elsevier.

References

259

5. Bullen, P. S. (2003). Handbook of means and their inequalities mathematics and its applications. Kluwer Academic Publishers. 6. Ting T. W. (1963). Isoperimetric inequality for moments of inertia of plane convex sets. Transactions of the American Mathematical Society, 107(3), 421–431.

Chapter 9

Optimization of Concurrently Compressed and Torqued Rod

Abstract This chapter considers the stability problems in the context of twisted and compressed rods are demonstrated. The complement for Euler’s buckling problem is Greenhill’s problem, which studies the forming of a loop in an elastic bar under simultaneous torsion and compression (Greenhill in On the strength of shafting when exposed both to torsion and to end thrust. Inst Mech Eng Proc: 182–225, 1883 [1]). The authors search the optimal distribution of bending flexure along the axis of the rod. For the solution of the actual problem the stability equations take into account all possible convex, simply connected shapes of the cross-section. The authors study the cross-sections with equal principal moments of inertia. The cross-sections are similar geometric figures related by a homothetic transformation with respect to a homothetic center on the axis of the rod and vary along its axis. The cross section that delivers the maximum or the minimum for the critical eigenvalue must be determined among all convex, simply connected domains. The optimal form of the cross-section is known to be an equilateral triangle. The distribution of material along the length of a twisted and compressed rod is optimized so that the rod must support the maximal moment without spatial buckling, presuming its volume remains constant among all admissible rods. The static Euler’s approach is applicable for simply supported rod (hinged), twisted by the conservative moment and axial compressing force. For determining the optimal solution, the authors directly compare the twisted rods with the different lengths and cross-sections using the invariant factors. The solution of optimization problem for simultaneously twisted and compressed rod is stated in closed form in terms of the higher transcendental functions. Keywords Stability optimization · Simultaneous compression · Torsional buckling

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Kobelev, Fundamentals of Structural Optimization, Mathematical Engineering, https://doi.org/10.1007/978-3-031-34632-3_9

261

262

9 Optimization of Concurrently Compressed and Torqued Rod

9.1 Twisted and Axially Compressed Shafts with Convex Simply-Connected Cross-Sections Consider a thin elastic rod with isotropic cross-section, compressed by axial forces and twisted by couples applied at its ends (Generalized Greenhill’s problem [1, 2]. The rod possessed the similarly shaped cross-sections with the varying crosssectional area. The optimization problem of instantly twisted and compressed rod was studied numerically in [3]. The optimization problem consisted in finding the distribution of cross-sectional areas that assigns the largest value to the critical moment causing a loss of stability and such that the constraint on the volume of material and the constraints on the admissible thickness of the rod were satisfied. Pastrone [4] deliberated stability of an elastic rod subjected to a twist and compression and f non-uniqueness of equilibriums. Several solutions, which generalize the Greenhill theory, were derived for straight rods and helixes. Spasic [5] studied the problem of determining the compressed and twisted column of minimal mass. For a given combination of torque and compression, the shape of the minimal mass column was determined numerically. A cross-sectional area varying along the axis of the column was sought. This area function leads to the greatest critical loading of buckling. The varying cross-section was approximated parametrically. A function with free parameters represented the varying cross-section area. Otherwise, the varying cross-section was determined using Pontriagin’s maximum principle, and the corresponding control function was found also numerically. Following the structural optimization theory, the shape of the column of minimal was determined numerically for a given combination of compression and torque. Atanackovi´c [6, 7] studied the stability regions for an elastic rod under axial force and torque. The regions of stability follow from the bifurcation points of an equation, which results from the first integrals of the constitutive equations. Glavardanov et al. [8] examined the regions of stability for an elastic beam supported by Cardan joints at both ends. The beam is loaded by a compressive force and a couple. The compressibility of the beam axis was considered. The optimal design of compressed rods under simultaneous compression and torsion was examined in [9]. A cross-sectional area varying along the axis of the column which leads to the maximal critical loading was sought. The varying crosssection was approximated by a function with free parameters. Alternatively, the varying cross-section was determined numerically using Pontriagin’s maximum principle. Consider a thin elastic rod with isotropic cross-section, twisted by couples and compressed by axial forces applied at its ends. The distribution of material along the length of a twisted rod is optimized so that the rod is of the constant volume 2V and will support the maximal moment without spatial buckling. The cross section that delivers the maximum or the minimum for the critical eigenvalue must be determined among all convex, simply connected domains. A priori form of the cross-section remains unknown. For the solution of the actual problem the stability equations take into account all possible convex, simply connected shapes of the cross-section. Thus,

9.2 Optimization Problem and Isoperimetric Inequality for Stability

263

the authors drop the assumption about the equality of principle moments of inertia for the cross-section. The cross-sections are similar geometric figures related by a homothetic transformation with respect to a homothetic center on the axis of the rod and vary along its axis. The area of an arbitrary cross-section with the coordinate x is (x). The area of the optimal cross-section with the coordinate x is A(x). The cross-sectional area (x) is positive, locally integrable, even control function of the independent coordinate. Its integral over the half-length of the rod pronounces the half-volume of the rod:

a

a

a

∫L

V[ ] =

a(x)d x.

(9.1)

0

In this Chapter, two principal moments of inertia in each cross-section are assumed to be equal: II = III = j,

j =C

a, α

here C is the shape factor, and α is the shape exponent. The shape exponent α takes the values of 1, 2 and 3. Two cases describe the situations in which the form of transverse cross-section undergoes the transformation one of the geometrical dimensions of the cross-section alternates proportionally to a design parameter. The optimal shape of cross-section was stated by [10]. The technically important case is the thin-walled tubes with the variable thickness of wall and the mean diameter of tube. The case α = 1 corresponds the changeable wall thickness and constant mean diameter of the tube. The design variable in this case is the thickness of the material. The case α = 3 corresponds the variable mean diameter of the tube and constant wall thickness. The design variable in this case is the mean diameter of the tube. The case α = 2 corresponds to a similar variation of the form of the cross-section. The above equality is valid for the cross-sections in form of equilateral triangles and circles. The circles lead to the minimal principal moments of inertia for the given area of cross-section. The equilateral triangles lead to the maximal principal moments of inertia.

9.2 Optimization Problem and Isoperimetric Inequality for Stability Based on the above solution, the distribution of material along the length of a compressed and torqued rod will be optimized. The distribution of material along the length of a compressed and torqued rod is optimized so that the rod is of constant volume and provides the maximal moment without spatial buckling. The material volume of the rod is given by 2V. The half-volume of the rod is V. We search the

264

9 Optimization of Concurrently Compressed and Torqued Rod

rod of the fixed volume 2V and fixed torque M, which supports the maximal axial force Δ without spatial buckling. The formulation of the optimization problem is the following: ] [ ] [ Δ∗∗ = Δ A, , ≥ Δ , , = Δ∗ ,

yz

ayz

a

def

2V = ⟨ ⟩ = ⟨A⟩, ⟨ f ⟩ =

∫L f (x)d x.

(9.2)

−L

The constancy of volume among all competitive designs plays the role of the isoperimetric condition. Accordingly, the optimal solution for boundary conditions produces the optimum function A, for which the critical compression load Δ is greater, than the critical compression load for any arbitrary thickness distribution with the same volume of material of the compressed ring:

a

A = optimum

a,

A ≥ 0,

a = Δ[A] ≥ Δ[a] = Δ .

M∗∗ = M[A] = M[ ] = M∗ ,

a

⟨A⟩ = ⟨ ⟩, Δ∗∗

∗

Consider the beam hinged on two supports x = L and = −L: |

y||| y||

x=L

|

= 0, = 0,

x=−L | | = 0, E j '' | |x=L | E j '' | = 0, x=−L

z y

z||| z||

x=L

= 0, = 0,

x=−L | | E j '' | = 0, |x=L | E j '' | = 0. x=−L

z z

(9.3)

The differential equations of simultaneously compressed and twisted rod are based on the Bernoulli–Euler model. The geometrically linear equilibrium equations describe the buckling of the rod [11]: ( Ej ( Ej

y

''

z

''

)''

)''

z

=M

= −M

'''

y

y, ''

+Δ

'''

z

+Δ

'''

(9.4) .

[ The Rayleigh’s quotient for the generalized Greenhill problem is R C The augmented functional for the optimization problem (9.1)–(9.5) reads:

(9.5)

a , y˜ , z˜]. α

9.2 Optimization Problem and Isoperimetric Inequality for Stability

265

) ( ) ( ] def ⟨E j ˜ ''2 + ˜ ''2 + 2M ˜ ' ˜ '' − ˜ '' ˜ ' ⟩ [ , R , ˜, ˜ = ⟨ ˜ '2 + ˜ '2 ⟩ [ ] ] [ Δ , , = min R , ˜ , ˜ ,

y z

ayz

zy z y

y z a y z yz a y z ] [ L[a] = ⟨a⟩ + λΔ a, y, z . ˜ ,˜

(9.6) The first variation of the augmented Lagrangian from Eq. (9.6) reads: (

)

y +z ⟩ δL[a] = ⟨δ a⟩ + λ , ⟨y + z ⟩ δ(E j ) = ECδ(a ) = αC E a δ a. ⟨δ(E j) ·

''2

'2

''2

'2

α

α−1

The necessary optimality condition follows from the vanishing of the first variation of the augmented Lagrangian with an auxiliary constant:

a

δL[ ] = ⟨1 −

a

α−1

K2 /c⟩, c = (ECα)−1 λ.

(9.7)

The spatial curvature in Eq. (9.7) reads: K2 = κ y2 + κz2 =

y +z ''2

''2

.

The application of the Fermat’s principle for the optimization problems leads to: ( )1/(1−α) A = K2 /c ,

J = CAα .

(9.8)

The optimal second moment of the cross-section follows from Eq. (9.8) as: ( )α/(1−α) ( J = EC · K2 /c =

y +z ''2

''2

)α/(1−α)

.

(9.9)

The applied method of scaling allows an arbitrary selection of the constant c. For briefness of the governing equations, the authors set the constant c as: c = (EC)

1−α α

.

With this choice the bending stiffness reduces to: EJ =

(

y +z ''2

''2

)α/(1−α)

≡ K2α/(1−α) .

(9.10)

266

9 Optimization of Concurrently Compressed and Torqued Rod

9.3 Closed-Form Solution of Optimization Task Due to the mirror symmetry of the problem with respect to the point x = 0, one half of the rod 0 ≤ x ≤ L will be studied. Using Eq. (9.10), the buckling Eqs. (9.4) and (9.5) convert to: [

z ,y ''

''

z ,y

''

e1 [ e2

''

]

]

= =

( (

y +z ''2

y +z ''2

)α/(1−α)

y

−Λ + Mz' ,

)α/(1−α)

z

−Λ − M

''2

''2

''

''

y

z

y. '

(9.11) (9.12)

For the handling of the governing equations, the authors introduce the new dimensionless parameter Ω: ( ) Δ 2 2 Ω= 3 M

with Δ∗∗ = Δ, M∗∗ = M.

(9.13)

If solely an axial force compresses the rod, the Euler’s eigenvalue Δ determines the buckling compression. Otherwise, if only a moment twists the rod, the buckling state results from the eigenvalue of the pure Greenhill problem M. If both loads are simultaneously applied, the parameter Ω relates two critical buckling states. The eigenvalue of the pure Greenhill problem M follows from the Euler’s eigenvalue Δ with this parameter Ω: √ M=2

Δ . 3Ω

The analysis of the solution clarifies with the parameter Ω. Two mentioned above pure solutions result as the opposite limit cases from the general formulas. One asymptotic case will be Ω → 0, Δ → 0. The axial compression disappears. In this case, the torsional eigenvalue M remains bonded and corresponds to the critical pure torque in the Greenhill problem. The other asymptotic case is Ω → ∞, M → 0. The compression eigenvalue Δ is bonded, this case corresponds to the solution of the Euler problem of the pure compressed rod. The torque is absent and the instability is caused only by the axial compression. The authors proceed with the solution of optimization problem. For the solution of two simultaneous Eqs. (9.12) and (9.13), two auxiliary functions K(x), θ (x) will be selected:

z

''

= K cos θ,

y

''

= K sin θ.

(9.14)

After the substitution (9.14) into (9.11) and (9.12), appear two differential equations for the auxiliary functions K, θ . It is possible to solve these equations for

9.3 Closed-Form Solution of Optimization Task

267

K, θ , but the solution for the functions A, θ is preferrable. With this choice of the unknowns, the solution immediately results to the optimal cross-section area A. Using Eq. (9.9), the authors replace K by A : √ ''

'

'

Aθ + (α + 1)θ A + (α − 1) α − 1 '2 2A '2 4 A − θ + A + 2A α+1 1+α

√

''

Λ ' −α A A = 0, 3Ω

Δ 1−α ' 2ΔA1−α A θ + = 0. 3Ω 1+α

(9.15)

(9.16)

Equation (9.16) leads to the solution for function θ . The integration constant could be put to zero from the symmetry considerations: θ (0) = 0. The corresponding solution of Eq. (9.16) reads: √

Δ θ = (1 − α) 3Ω

∫x

A−α (z)dz.

(9.17)

0

Substitution of this solution into the Eq. (9.15) leads to the nonlinear ordinary differential equation only for the function A: A'' +

α − 1 '2 2Δ 1−α 2Δ(1 − α) 1−2α A − 2A + A A + = 0. 2A α+1 3Ω

(9.18)

The Eq. (9.18) contains three dimensionless parameters: α, Ω, Δ. The critical value for M results with Eq. (9.14). For the closed form solution, the dependent and independent variables in Eq. (9.18) must be exchanged: ) ( ( ) 1 − α 2 A1−2α + 3ΩA1−α d x 3 d2x α − 1 dx + 2Δ = . dA2 2A dA 3Ω(1 + α) dA

(9.19)

Equation (9.19) is of the second order with missing dependent and independent variables. Interchanging the independent and dependent variables, the Eq. (9.19) transforms to Emden–Fowler equation [12, 13]. The reduction of order follows the substitution: Ξ=

dx . dA

With the new dependent variable Ξ, the nonlinear ordinary differential equation of the first order is: ) ( 1 − α 2 A1−2α + 3ΩA1−α 3 α−1 dΞ = Ξ + 2Δ Ξ. (9.20) dA 2A 3Ω(1 + α)

268

9 Optimization of Concurrently Compressed and Torqued Rod

The differential Eq. (9.20) is solvable by quadrature: √ Ξ(A) =

√ 3Ω(1 + α) 4A

] [ −4Δ(1 + α)A2+2α + 3Ω C1 (1 + α)A3α+1 − 4ΔA3α+2 ) ( , 3ΔΩAα+1 + (1 + α) AΔ − 43 C1 ΩAα (9.21) ∫A x=

Ξ(t)dt + C2 .

(9.22)

0

The integrals contain two integration constants: C1 , C2 . The integral for the Eq. (9.22) is summable, if the first integration constant is equal to: C1 =

4Δ 3Ω + 1 + α . 3Ω 1+α

(9.23)

With this value, the solution (9.19) reads: 1 x= √ 2 Δ

∫A √ 0

t α−1 t α−1 (1 + α + 3Ω) − 3t α Ω − 1 − α

dt + C2 , for 0 < A < 1, right half;

(9.24) 1 x =− √ 2 Δ

∫A √ 0

t α−1 t α−1 (1 + α + 3Ω) − 3t α Ω − 1 − α

dt + C2 , for 0 < A < 1, left half.

(9.25) The function A(x), which is defined implicitly with Eqs. (9.24) and (9.25), is an even function of x according the symmetry conditions. Consequently, according to the symmetry of the solution, dA/d x must vanish in the point x = 0. The resulting equation of the first order is solvable in quadrature: 1 x= √ 2 Δ

∫A 0

t α−1 √ dt, for 0 < A < 1, right half; t α−1 (1 + α + 3Ω) − 3t α Ω − 1 − α (9.26)

1 x =− √ 2 Δ

∫A √ 0

t α−1 t α−1 (1 + α + 3Ω) − 3t α Ω − 1 − α

dt, for 0 < A < 1, left half. (9.27)

9.4 Special Cases

269

The functions in Eqs. (9.26) and (9.27) are the real functions of 0 < A < 1. The limit is the right-hand limit α → 1+. The area of the optimal cross-sections vanishes in the end points of the rod, where the bending moment disappears: x(A = 0) = L .

(9.28)

The formulas (9.26) and (9.27) provide the most general expression for the solution of the declared optimization problem for all admissible values of the design parameters: √ α, Δ, M = 2

Δ . 3Ω

For some value of parameter α, the integrals (9.26)–(9.27) reduce to the higher transcendental functions [14]. For the significant for application cases, the solutions of the optimal rods express of the standard special functions.

9.4 Special Cases 9.4.1 Optimal Rod for Greenhill Torsion At first, the mentioned above Euler and pure Greenhill load buckling states result from the general expressions (9.26) and (9.27) as the special cases. The solution reduces in the asymptotic case Δ → 0 to the solution of the optimization problem for the pure twisted rod with the vanishing axial compression. The closed form solution for the optimal Greenhill rod for all admissible values of α reads: 1 x= 3MΩ

∫A 0

α−1

t 2 dt for 0 < A < 1, right half. √ 1−t

(9.29)

The singular integral in (9.29) expresses in terms of the beta B and hypergeometric F 2 2 functions [15]: √ ([ ] [ ] ) 1 α+1 3+α 2 Aα+1 , , , A for α > 1, F x= 2 3MΩ(α + 1) 2 2 2 2 ( ) ∫1 α−1 1 α+1 1 t 2 1 B , , L= dt = √ 3MΩ 2 2 3MΩ 1−t 0 √ ([ ] [ 3+α ] ) , ,A x 2 Aα+1 2 F2 21 , α+1 2 ( 1 α+1 )2 X= = . L α+1 B 2, 2

(9.30)

270

9 Optimization of Concurrently Compressed and Torqued Rod

The solution (9.30) is the solution of the optimization problem for the twisted rod with the vanishing axial compression.

9.4.2 Optimal Strut for Euler Compression Similarly, the other asymptotic case M → 0 delivers the solution of the optimization problem for the solely compressed rod with the zero torque. For the optimal Euler rod, the closed form solution evaluates for all admissible values of α to: 1 x= √ 2 (1 + α)Δ

∫A 0

t α−1 dt for 0 < A ≤ 1, right half. √ t α−1 − 1

(9.31)

The singular integral in Eq. (9.31) expresses again in the higher functions. The integral in Eq. (9.30) can be found with the hypergeometric function (ibid., [15]): x=

iAα F2 √ 2α (1 + α)Δ 2

([

) ] [ ] 1 α 2α − 1 for α > 1. , , , Aα−1 2 α−1 α−1

(9.32)

The half-length of the rod follows from (9.32) as the limit value A = 1: 1 L= √ 2 (1 + α)Δ

∫1 0

) ( √ [ 2α−1 i π α−1 ( 3α−1 ) . dt = √ √ 2α (1 + α)Δ [ 2α−2 t α−1 − 1 t α−1

(9.33)

From Eqs. (9.32) and (9.33) follows the normalized coordinate (0 ≤ X ≤ 1): ( 3α−1 ) ) ([ ] [ ] [ 2α−2 1 α 2α − 1 x Aα α−1 . , , ,A X = = ( 2α−1 ) √ 2 F2 L 2 α−1 α−1 π [ α−1

(9.34)

9.5 Arbitrary Relation Compression to Torque 9.5.1 Optimal Rod for Shape Exponent α = 1 In this section the solutions of optimization problems are studied for an arbitrary relation between the compression force and the torque moment. The closed form solution of the principal Eqs. (9.26) and (9.27) could be obtained for the practically interesting cases of shape exponent α = 1, 2, 3 (Fig. 9.1). This figure shows the typical buckling mode of the optimal columns.

9.5 Arbitrary Relation Compression to Torque

271

Fig. 9.1 The buckling shape of the simultaneously twisted and compressed rod with the simply supported (hinged) ends. The compression force is F and the torque is M

In the case α = 1 the solution reads (Fig. 9.2): 1

x= √ 2 3ΩΔ

∫A √ 0

) √ 1 ( 1 − 1 − A for 0 < A < 1, right half. dt = √ 1−t 3ΩΔ 1

(9.35) The half-length of the rod follows from (9.35) as: 1

L= √ 2 3ΩΔ

∫1 √ 0

1

1 dt = √ . 1−t 3ΩΔ

(9.36)

The relation of the normalized coordinate to the optimal cross-section is a quadratic function: X≡

√ x = 1 − 1 − A, A = 2X − X2 . L

(9.37)

272

9 Optimization of Concurrently Compressed and Torqued Rod

Fig. 9.2 Areas of cross-sections of the optimal twisted rods for α = 1 for all values of parameter Ω

The half-volume of the optimal rod evaluates trivially to: 1

Vα=1 = √ 2 3ΩΔ

vα=1

∫1 0

√ 2 3 dt = √ , √ 1−t 9 ΩΔ t

2 Vα=1 = . = L 3

(9.38)

Notably, √ that the shape in the case α = 1 depends implicitly on the parameter Δ M = 2 3Ω .

9.5.2 Optimal Rod for Shape Exponent α = 2 In the case α = 2, the solution reads (Fig. 9.3): √ ∫A 1 3 dt x= √ √ − t)(Ωt − 1) (1 6 Δ 0 √ i 3 ϒ(A, Ω) − ϒ(0, Ω) =− for 0 < A < 1, right half, √ 12 Ω3 Δ

(9.39)

9.5 Arbitrary Relation Compression to Torque

273

Fig. 9.3 Three-dimensional representation for cross-sections of the optimal twisted rods for α = 2 for all values of parameter Ω

√ i 3 ϒ(1, Ω) − ϒ(0, Ω) L=− . √ 12 Ω3 Δ

(9.40)

The auxiliary function in Eq. (9.39) is: ) √ (√ √ ϒ(A, Ω) = 2 Ω 1 − A 1 − ΩA ( ) √ √ √ − (1 + Ω) ln 1 + Ω − 2AΩ + 2 Ω 1 − A 1 − ΩA .

(9.41)

The shape of the optimal rod depends on the parameter Ω. If the Ω parameter is less than the critical value, Ω2 = 1, the functions of the cross-sectional area are roughly parabolic. Otherwise, the functions possess the saddle points. The shapes for the values of the parameter, less than critical Ω < Ω2 = 1, are shown on upper half of Fig. 9.4. If parameter is higher than critical value Ω > Ω2 = 1, the shapes are plotted on the lower half of this figure. For comparison, both cases are shown overlapped on Fig. 9.5. With the Eqs. (9.36) and (9.37), the normalized axial coordinate 0 < x < 1 of the cross-section A reads: X(A, Ω) ≡

ϒ(A, Ω) − ϒ(0, Ω) x . = L ϒ(1, Ω) − ϒ(0, Ω)

(9.42)

The volume of the rod is easily calculated with the integration of Eq. (9.3) by parts. Remembering that A(0) = 1, X (0, Ω) = 1, the authors get the expression for the half-volume of the rod Vα=2 . Dividing the half-volume of the rod by half-length, the normalized half-volume results:

v

α=2

Vα=2 =1− = L

∫1 0

x(t, Ω)dt.

(9.43)

274

9 Optimization of Concurrently Compressed and Torqued Rod

Fig. 9.4 Areas of cross-sections of the optimal twisted rods for α = 2 for values of parameter Ω, which are less than critical (above) and higher than critical (below)

The integral (9.43) allows the expression in closed form: ( ) ( )] [ ( ) ] [ √ 4 Ω(Ω + 1) 3Ω2 + 2Ω + 3 χ2 − 4Ω 3Ω2 + 4Ω + 3 χ1 + 8Ω 3Ω2 + 4Ω + 3 χ2 − 12Ω3/2 (Ω + 1) [ ] , α=2 = [ ] 16Ω3/2 (Ω + 1)2 χ2 − 16Ω2 (Ω + 1) χ1 + 32Ω2 χ2 − 16Ω5/2 ( √ ) χ1 = ln 1 + Ω − 2 Ω + ln(1 − Ω),

v

(9.44)

χ2 = ln(1 − Ω).

The asymptotic case Ω → 0 delivers the solution of the optimization problem for the solely compressed rod with the zero torque. This solution is the limit case of (9.43) as Ω → 0: √ X˜ = lim X = 1 − 1 − A − Ω→0 ( ) A˜ = A ≈ 1 − σ˜ + σ1˜ , Ω→0 ∫1

X˜ α=2 =

0

˜ AdX = 45 .

A 2

√

1 − A,

√ σ˜ =

3

X˜ − 1 +

√

˜ X˜ 2 − 2X,

(9.45)

9.5 Arbitrary Relation Compression to Torque

275

Fig. 9.5 Areas of cross-sections of the optimal twisted rods for α = 2 for all values of parameter Ω. For comparison, the values of the parameter above and below the critical value are shown on one plot

9.5.3 Optimal Rod for Shape Exponent α = 3 Finally, the authors study the last practically interesting case = 3. For this case, the optimal shape is solvable in closed form. The governing Eqs. (9.26) and (9.27) reduce in this case to the following integrals: √ ∫A 3 t2 √ x= √ ( 6 Δ (1 − t) Ωt 2 − 0

8 dt for 0 < A < 1, right half, Ω3 = . ) 3 Ω3 − 1) 2 (t (9.46)

For the brevity of the formulas, the authors introduce the auxiliary variable Z=

√

3Ω + 1.

With the variable Z , the axial coordinate (9.46) rewrites as an integral with the variable upper limit:

276

9 Optimization of Concurrently Compressed and Torqued Rod

1 x(A, Ω) = √ 2 Δ

∫A √ 0

t2 dt (1 − t)(t Z − t − 2)(t Z + t + 2)

for 0 < A < 1, right half.

(9.47)

Analogously, the half-length and the half-volume reduce to the similar integrals with the fixed limit of integration: 1 L(Ω) = √ 2 Δ 1 vα=3 (Ω) = √ 2 Δ

∫1 √ 0

∫1 √ 0

t2 dt, (1 − t)(a Z − t − 2)(t Z + t + 2)

t3 dt, (1 − t)(a Z − t − 2)(t Z + t + 2)

(9.48)

(9.49)

The integrals (9.48) and (9.49) express in closed form: x(A, Ω) = c1 · [Φ(A) − Φ(0)] + c2 · [ψ(A) − ψ(0)],

(9.50)

L(Ω) = c1 · [Φ(1) − Φ(0)] + c2 · [ψ(1) − ψ(0)],

(9.51)

v

α=3 (Ω)

= c3 · [Φ(1) − Φ(0)] + c4 · [ψ(1) − ψ(0)].

(9.52)

The coefficients in Eqs. (9.50)–(9.52) are:

z z z z z z z z

(

) √ +3 +3 2 √ ( )3/2 ( )2 , 3 Δ 1− +1 ( )3 √ 2 +3 +3 8 √ ( )5/2 ( )3 , 15 Δ 1− +1

c1 = c3 =

z z z z z z z z z (

2

2

c2 = c4 =

) +3 )3/2 (

( ), +3 1− +1 ( ) 4 +3 2 +12 8 √ )5/2 ( ( )2 . 15 Δ √ +3 1− +1 √2 3 Δ √

(9.53)

The special functions ψ(A), Φ(A) in Eqs. (9.50)–(9.52) are the elliptic integrals of the first and second kinds correspondingly [14]: ) ( ) ( 1 1 , Φ(A) = E Q(A), , ψ(A) = F Q(A), Q(1) Q(1) √ (Z − 1)(2 + (Z + 1)A) Q(A) = . √ 2 Z

(9.54)

9.5 Arbitrary Relation Compression to Torque

277

( )2 The shape of the optimal rod depends on the parameter (9.13), Ω = Δ3 M2 . If the Ω parameter is less than the critical value Ω3 = 8/3, the functions of the crosssectional area are roughly parabolic. Otherwise, the functions possess the saddle points. The shapes for the values of the parameter, less than critical Ω < Ω3 , are shown on upper half of Fig. 9.6. If parameter is higher than critical value Ω > Ω3 , the shapes are plotted on the lower half of this figure. The combination of both cases is displayed on Fig. 9.7.

Fig. 9.6 Areas of cross-sections of the optimal twisted rods for α = 3 for values of parameter Ω. Above are shown the plots for parameter Ω, which are less than critical value. The plots for parameter Ω, which are higher than critical are displayed below

278

9 Optimization of Concurrently Compressed and Torqued Rod

Fig. 9.7 Areas of cross-sections of the optimal twisted rods for α = 3 for all values of parameter Ω. The values of the parameter above and below the critical value are shown overlapping on one plot

9.6 Mass Comparisons of Optimal Shafts to Constant-Cross-Section Shafts Evidently, that the exponent α influences the estimations for the optimization effects. For comparison, the authors need the expression of the volume of the loaded rods with different length. The rods possess the constant cross-section c along their axes. The area moment of the second order is j = C αc . The rods with the definite form of cross-section are compressed by the axial force CΔc and twisted by moment CMc . The stability criterion for the simultaneously compressed and twisted rods with the constant cross-section along its length reads [16, 17]:

a

(

C Mc L E jπ

)2 +

2CΛc L ≤ 1. E jπ2

a

(9.55)

9.6 Mass Comparisons of Optimal Shafts to Constant-Cross-Section Shafts

279

a

If the cross-sectional area if the constant rod is c , the volume of the half of the rod will be Vc = c L. The authors substitute into Eq. (9.55) the critical values:

a

√ Mc =

4Δc . 3Ω

From the resulting equation results the expression of the stability criteria in terms of the compressional eigenvalue Δ and the dimensionless parameter Ω: 4ΔL 2 2ΔL + α 2 ≤ 1. 2 3Ω 2α π c cπ

a

a

(9.56)

Using the expression (9.56), the critical compression load of the rod with the constant cross-section of the reference rod r e f reads:

a

Δc =

a

3Ωπ 2 2α c ). ( 2 c 3 αc Ω + 2L

a a

(9.57)

The cross-section of the uniform rod with the critical eigenvalue Δc results from Eq. (9.57) after the solution this equation for c . For comparison, the authors need the cross-section for the rods with Δc = 1. The uniform cross-section of the rod with the half-length L and critical compression eigenvalue Δc = 1 results from (9.57) finally as:

a

√ √ 4π 2 Ω + 3Ω2 + 3Ω α , L c = 3π 2 Ω Vc Vc = L c , r e f = . L

a

a v

v

(9.58)

The half-volumes of the uniform rods r e f with the half-length L = 1 and various values Ω and α are shown on Fig. 9.8. On the other hand, the volumes of the optimal rods α (Ω) for the same critical compression load Δ = 1 were already determined above and are given by Eqs. (9.38), (9.45) and (9.52). With this information the authors can relate the volumes of the rods with the optimal and with the uniform cross-sections. Subsequently, the masses of the twisted rods for the same half-length L = 1 and equal buckling forces are compared. The volumes and masses of the optimal and reference twisted rods relate to each other as: χ (α, Ω) =

v

v (Ω) . v α

ref

The values χ are shown as the functions of α and Ω on Fig. 9.9.

(9.59)

280

9 Optimization of Concurrently Compressed and Torqued Rod

Fig. 9.8 Volumes of the reference rods with the constant cross-sections for different α and Ω for the constant length L = 1

Fig. 9.9 Relations χ (α, Ω) of masses of optimal rods to the masses of the rods with the constant cross-sections for α = 1, 2, 3

9.7 Conclusions

281

Fig. 9.10 Relations χ of masses of optimal rods to the masses of the rods with the constant cross√ sections as functions of M/ Δ and α = 1, 2, 3

Alternatively, the authors can represent the values R as the functions of the ratios torque to compression loads √MΔ . For this purpose, the abscissa axis on Fig. 9.10 is scaled using the formula: M √ = Δ

√

4 . 3Ω

Remarkably, that the optimal rods possess less mass, than the rods with the constant cross-sections only for definite values of the ratio Ω.

9.7 Conclusions In this Chapter, the solution of optimization problem for simultaneously twisted and compressed strut was investigated. The strut can bend arbitrary in space and its buckling form resembles the three-dimensional spiral. The optimal distribution

282

9 Optimization of Concurrently Compressed and Torqued Rod

of bending stiffness along the axis of the strut is searched. The stability equations account all possible convex, simply connected shapes of the cross-section. Among these forms of the cross-sections, the equilateral triangle sections possess the equal principal moments of inertia for all lateral directions. The sections in form of the equilateral triangles are the optimal for the investigated problem. The application of the variational methods leads to the necessary optimality conditions. The optimality equation is of the Emden–Fowler type. Its solution expresses in terms of the higher transcendental functions. The final formulas involve the length of the rod, its volume and critical torque and axial compression force. Remarkable, that in the torsion stability problem the optimal shape of the rod is roughly parabolic along its length and the optimal shape of cross-section is the equilateral triangle.

9.8 Summary of Principal Results • The optimal distribution of material along the length of a twisted and compressed rod, which supports the maximal moment without spatial buckling, presuming its volume remains constant, is found • The static Euler’s approach is applicable for simply supported rod (hinged), twisted by the conservative moment and axial compressing force. • the twisted rods with the different lengths and cross-sections using the invariant factors are compared. • The solution of optimization problem for simultaneously twisted and compressed rod is stated in closed form in terms of the higher transcendental functions.

9.9 List of Symbols A(x)

The optimal distribution of material along the length of a twisted rod is An arbitrary distribution of material along the length of a rod (x) Lateral displacements (x), (x) m y (x), m z (x) Bending moments V Reference volume of the twisted rod M Critical dimensionless torque eigenvalue Δ Critical dimensionless compression eigenvalue ( ) Δ 2 2 Ratio critical compression to critical torque Ω= 3 M

a y z

References

283

References 1. Greenhill A.G. (1883) On the strength of shafting when exposed both to torsion and to end thrust. Inst Mech Eng Proc: 182–225 2. Timoshenko, S., & Gere, J. M. (1961). Theory of elastic stability. McGraw-Hill. 3. Banichuk N.V., Barsuk A.A. (1982) Optimization of stability of elastic beams against simultaneous compression and torsion. In: Applied problems of strength and plasticity. Automation and algorithms for solving problems in elastoplastic state (collection). Gorkii, Gorkii State University, pp 122–126 4. Pastrone, F. (1992). Torsional instabilities of Greenhill type in elastic rods. Rendiconti del Seminario Matematico della Università di Padova, 87(1992), 77–91. 5. Spasic D.T. (1993) Stability of a compressed and twisted linearly-elastic rod, Ph.D. thesis, University of Novi Sady 6. Atanackovic T.M. (2002) Stability bounds and optimal shape of elastic rods. In: Seyranian A.P., Elishakoff I. (eds) Modern problems of structural stability. International Centre for Mechanical Sciences, vol 436. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2560-1_1 7. Glavardanov V.B., Atanackovic T.M. (2001) Optimal shape of a twisted and compressed rod. Euro J Mechan A/Sol 20(5):795–809 8. Glavardanov V.B., Maretic R.B., Grahovac N.M. (2009) Buckling of a twisted and compressed rod supported by Cardan joints. Euro J Mechan A/Solids 28(1):131–140 9. Kruzelecki, J., & Ortwein, R. (2012). Optimal design of clamped columns for stability under combined axial compression and torsion. Struct Multidisc Optim, 45, 729–737. 10. Ting T.W. (1963) Isoperimetric inequality for moments of inertia of plane convex sets. Trans Am Math Soc 107(3):421–431 11. Jerath, S. (2021). Structural stability theory and practice. Wiley. 12. Zwillinger D., Dobrushkin V. (2022) Handbook of differential equations, 4rd edn. Taylor & Francis Group, LLC, 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 13. Berkovic, L. M. (1997). Generalized Emden-Fowler equation. Symmetry Nonlin Math Phys, 1, 155–163. 14. Abramowitz M., Stegun I.A. (eds) (1983) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Applied mathematics series, 55. United States Department of Commerce, National Bureau of Standards, Dover Publications, Washington D.C., New York 15. Gradshteyn I.S., Ryzhik I.M. (2014) Table of integrals, series, and products, 8th edn. Academic Press. https://doi.org/10.1016/C2010-0-64839-5, ISBN 978-0-12-384933-5 16. Grammel R. (1923) Das kritische Drillungsmoment von Wellen. Zeitschrift für angewandte Mathematik und Mechanik, Heft 4, Band 3 17. Beck M. (1955) Knickung gerader Stäbe durch Druck und konservative Torsion. IngenieurArchiv, Band XXILI, Heft 4

Chapter 10

Optimization for Buckling of Conservative Systems of Second Kind

Abstract In this Chapter the authors investigate the optimization problem for conservative systems of the second kind. The known example of a conservative system of the second kind is the generalized Pfluger column. The conservative systems of the second kind are truly non-self-adjoint and non-conservative systems in a classical sense, but self-adjoint in a generalized sense. These systems buckle by divergence and possess a generalized conservation theorem and a generalized Rayleigh quotient. Optimization problems for the generalized Pfluger column with pinned–pinned or sliding ends, subjected to distributed compressive follower forces are considered. It is shown that by means of a special transformation of independent variables, the problem is reduced to a classical conservative bifurcation problem for the column loaded by the axially distributed load. The optimal solutions for some load distributions are found in closed form. Keywords Stability optimization · Non-self-adjoint system · Pfluger column · Conservative system of the second kind

10.1 Pfluger Column The shape of columns for which the buckling load is the largest among all columns of given length and volume, for various boundary conditions and a “dead” conservative load and the corresponding isoperimetric inequalities were discussed above. The stability equations are reduced to second order self-adjoint equations with homogeneous, but generally mixed boundary conditions. Due to the conservative character of the load, the stability problem was analyzed by means of the static Euler method. The general nonconservative systems are described by non-self-adjoint equations of motion and lose stability by bifurcation (divergence) and/or flutter [1]. The stability of such systems is studied by the dynamic method. The optimization of non-conservative systems has been surveyed by Weisshaar and Plaut [2], Seiranyan and Scharanuk [3], Gajewski and Zyczkowski [4]. However, certain non-self-adjoint systems have only a bifurcation type instability, despite the presence of nonconservative forces [5, 6]. Such a system is called a © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Kobelev, Fundamentals of Structural Optimization, Mathematical Engineering, https://doi.org/10.1007/978-3-031-34632-3_10

285

286

10 Optimization for Buckling of Conservative Systems of Second Kind

conservative system of the kind. This system is self-adjoint in a generalized sense with respect to an assigned operator; there exist a Lyapunov functional and a generalized Rayleigh quotient. An elastic rod, loaded by the distributed follower force F F(x), is called a generalized Pfluger column . The dimensionless function F = F(x), (a ≤ x ≤ b) describes the distribution of the axial force along the axis of the column. The proportionality constant F is a load parameter with the dimension of force. The column is simplysupported or guided on its ends. For the small positive values F, the column remains stable, but for a definite load parameter the column eventually begins to oscillate and with an increasing load parameter it buckles. The differential equations of simultaneously compressed rod are based on the Bernoulli–Euler model. Stability of the generalized Pfluger column is described by the eigenvalue problem governed by the differential equation: Lw = −ρω2 w + (E j wx x )x x + FF(x)wx x = 0.

(10.1)

The boundary conditions for the pinned end are: w = 0,

E j wx x = 0.

(10.2)

Correspondingly, the boundary conditions for the guided end are: wx = 0, (E jwx x )x = 0.

(10.3)

The values in Eqs. (10.1–10.3) are: w = w(x) ω ρ j =C α α C >0

a

a

the lateral deflection of the column, the frequency of free vibration, the linear density of the column, second moment of area, the shape exponent, the shape factor, cross-sectional area.

The column with guided ends ) admits a translatory isoperimetric mode. ( The operator F(x) d 2 /d x 2 is not selfadjoint (in the classical sense) with respect to the prescribed boundary conditions. However, the operator L = −ρω2 +

( ) d2 d2 d2 E j + FF(x) 2 . 2 2 dx dx dx

(10.4)

is selfadjoint in a generalized sense [5], with respect to the operator: M = Ej

d2 . dx2

(10.5)

10.1 Pfluger Column

287

Generalized selfadjointness implies: def

⟨Lu · Mv⟩ = ⟨Lv · Mu⟩, where ⟨ f ⟩ =

∫b f d x.

(10.6)

a

The equality (10.6) holds for all admissible functions u(x) and v(z), satisfying the same boundary conditions, which are prescribed for w(z). Because the operator L is selfadjoint in a generalized sense, the generalized Pfluger column can only have a bifurcation, static mode of instability (divergence). Consequently, the Euler method can be applied to study the stability. This exceptional property eases the mathematical handling of the optimization problem. From the stability theory, there exists a generalized Rayleigh quotient for approximating the lowest buckling load. The lowest buckling load is the minimum of the generalized Rayleigh quotient in the space of admissible functions w: F = min Rw [w], w )]2 [ ( 2 d ⟨ d x E j dd xw2 ⟩ Rw [w] = [ 2 ]2 . ⟨F(x) dd xw2 ⟩

(10.7)

In terms of bending moment m(x) = E j

d 2w , dx2

the stability Eq. (10.1) assumes the form: Ej

d 2m + FFm = 0. dx2

(10.8)

The boundary conditions for (10.8) in the case of guided ends are written in the form: dm = 0. dx The conditions for pinned ends in term of moments read: m = 0. Equation (10.8) and the corresponding Rayleigh quotient (10.7) could be written in a dimensionless form. The dimensionless critical parameter Δ is directly proportional to the critical load parameter F:

288

10 Optimization for Buckling of Conservative Systems of Second Kind

Δ = F /EC. For briefness of equations, the authors will put in the Chapter: C = 1, E = 1. The parameter Δ is a principal positive eigenvalue for the differential equation with the certain set of the above discussed boundary conditions:

a

d 2m + ΔF dx2

−α

m = 0.

(10.9)

The critical dimensionless parameter Δ is the lower bound of the generalized Rayleigh quotient among all admissible functions:

a

a

Δ[ ] = min R[ , m], m [ [2 ⟩ ⟨ dm dx . R[ , m] = −α ⟨F(x) m2⟩

a

a

(10.10)

The boundary conditions will be considered thereafter in one of the following forms: m(a) = 0, m(b) = 0, for both pinned ends; m x (a) = 0, m(b) = 0, for right pinned end and left guided end; m(a) = 0, m x (b) = 0, for left pinned end and right guided end.

(10.11)

The admissible functions in (10.10) are the differentiable functions, which satisfy the corresponding boundary conditions. The boundary conditions (10.11) are of Sturm type, so the principal eigenvalue is simple. We demonstrate the solution of the Eq. (10.8) for the constant cross-section = 1, a = 0, b = 1 and

a

F = (1 − x)k , for k = 0, 1, 2. The column with both pinned ends will be studied, id est the first case of Eq. (10.11). The solutions are shown in the ( √ ) ( √ ) (√ ) (√ ) k = 0 : m = sin Δ cos x Δ − sin x Δ cos Δ , Δ = π 2 ; ( ) √ √ √ 3/2 · 1 − x , Δ = 18.9562; k = 1 : m = 2 3iΔ1/6 · I1/3 − 2i Δ(1−x) (√ ) ( √3 ) √ Δ(1−x)2 i Δ k = 2 : m = −π exp 2 · J1/4 − · 1 − x , Δ = 30.933. 2 (10.12) The functions Iυ (x), Jυ (x) are the Bessel functions. The critical values of the dimensionless critical parameter Δ of the columns with the constant cross-section are used as the references for the estimation of optimization effect.

10.2 Stability Optimization for “Pfluger Column”

289

10.2 Stability Optimization for “Pfluger Column” Consider now the problem of the maximization of the critical buckling load. The design variable is the dimensionless cross-sectional area (x), and the shape of the cross-section is prescribed. For each special case, the distribution of the compression force along the length of the rod is assumed to be a priori defined and does not alter in course of the optimization [7]. The constancy of volume among all competitive designs plays the role of the isoperimetric condition. Accordingly, the optimal solution for boundary conditions produces the optimum function A, for which the critical compression load is greater, than the critical compression load for any arbitrary thickness distribution with the same volume of material of the compressed column.:

a

a

A = optimum

a,

A ≥ 0,

a

⟨A⟩ = ⟨ ⟩ = V ,

a

Δ∗∗ = Δ[A] ≥ Δ[ ] = Δ∗ .

a a = Δ[A] = min R[A, m] = max Δ[a], a

The design objective is the critical buckling load Δ∗∗ = max Δ∗ [ ]: (x)

Δ∗∗

(x)

m

a

a

Δ∗ = Δ[ ] = min R[ , m]. m

(10.13) (10.14)

The function A exposes the required solution of the optimization problem. The admissible function, which delivers the minimum of the generalized Rayleigh quotient for the optimal cross-sectional area A(x), is M(x). The necessary optimality condition could be obtained using the method of Lagrange multipliers [8]: FM 2 = c L Aα+1 .

(10.15)

The symbol c L > 0 in Eq. (10.15) is a Lagrange multiplier. The similar optimization tasks were also studied in [9, 10].

290

10 Optimization for Buckling of Conservative Systems of Second Kind

10.3 Auxiliary Conservative System of First Kind: “Generalized Euler Column” There exists an auxiliary conservative system of the first kind, which serves for the resolution of the above nonconservative system. This conservative system is called the generalized Euler column and is characterized by the fact that the distributed loading G(x) remains parallel to the undeformed axis. Assuming that the end x = −1 is guided: m x (a) = 0, w(b) = 0.

(10.16)

the authors can write the differential equation of the generalized Euler column as (−1 ≤ x ≤ 1): Kw = −ρω2 w + ( j wx x )x x + F[G(x)wx ]x = 0.

(10.17)

The end x = 1 is either pinned or guided. The differential operator, analogous to operator (10.4), is selfadjoint in the classical sense: K = −ρω2 +

d2 dx2

( j

d2 dx2

) +F

( ) d d G =0 dx dx

(10.18)

The stability of the column is studied by the Euler bifurcation method. The Rayleigh quotient for this problem is written as:

Δ[ j, G] = min

a

⟨

w

α

(

d2w dx2

)2

⟩

( )2 . ⟨G dw ⟩ dx

(10.19)

The Euler–Lagrange equation for functional (10.19) is: d2 dx2

(

a

w dx2

αd

2

)

( ) d dw +Δ G = 0. dx dx

(10.20)

In the static case, Eq. (10.18) reduces to Eq. (10.20), multiplied by kα . Integrating the differential Eq. (10.20) once, the authors get the equation: m x + ΔGwx = C1

(10.21)

Applying the boundary conditions (10.16) for the guided end, one can find that C1 = 0. Dividing by G, differentiating and writing in a dimensionless form, the authors can deduce the equivalent equation in terms of the bending moment m,

a

( −1 ) G mx x + Δ

−α

m = 0.

(10.22)

10.4 Isoperimetric Inequality

291

Notable, that the equations of the Pfluger column allow the transformation to the equations of the auxiliary conservative system of the first kind. For this purpose, the new independent variable y introduces as: ∫x y(x) =

F(ξ )dξ.

(10.23)

a

The lower bound in the integral (10.23) is insignificant. In terms of the new variable (10.23), the equation of the Pfluger column Eq. (10.9) converts to:

a

( ) Fm y y + Δ

−α

m = 0.

(10.24)

With the following substitution, the Eq. (10.24) matches to Eq. (10.22): G = 1/F.

(10.25)

Consequently, the authors can conclude that for certain boundary conditions, by means of transformation of (10.23) and (10.25), the optimization problem for the generalized Pfluger column turns into the optimization problem for a column, loaded by distributed conservative forces.

10.4 Isoperimetric Inequality The isoperimetric inequality could be stated for the maximal buckling load of the Pfluger column. There exist the functions M and A, such that: FM 2 = c L Aα+1 ,

d2 M + Δ∗∗ FA−α M = 0. dx2

(10.26)

Function A satisfies the isoperimetric condition, while M satisfies the boundary conditions and the auxiliary normalization condition: c L = 1.

(10.27)

As noted above, the column with the thickness distribution A is optimal. As already pointed in (10.11), the function M represents the bending moments in the state of buckling for the optimal column:

292

10 Optimization for Buckling of Conservative Systems of Second Kind

Δ∗∗ [A] = R[A, M] = min R[A, m]. m

The admissible functions m must satisfy the certain boundary conditions, which will be specified below. The isoperimetric inequality can be stated for the column with the optimal thickness distribution A. For this purpose, consider a non-optimal column with an arbitrary distribution of thickness (x) > 0. Thus, and A are different. To find the principal eigenvalue for the non-optimal column, the variational principle will be applied. The variational principle assures that:

a

a

a

a

a

Δ∗ = Δ[ ] = min R[ , m] ≤ R[ , M]. m

This inequality is trivial, because the function M is not the fundamental form for an arbitrary (x). Namely, the function M is the fundamental form only for the optimal cross-section function A, which can differ from . From the other side, the values M, Δ∗∗ , satisfy the equation with the optimal function A:

a

a

d2 M + Δ∗∗ FA−α M = 0, dx2

a

but do not satisfy the analogous equation with an arbitrary function ,

a

d2 M + Δ∗∗ F dx2

−α

M /= 0.

The following inequality could be immediately approved:

a

Δ∗ = R[ , M] ≤ R[A, M] = Δ∗∗ .

(10.28)

In Eq. (10.28), the denominators of fractions to the left and right of the third inequality sign are identical. To approve the inequality (10.28), the denominators must be compared. The pursued inequality (10.28) is equivalent to another desired inequality: ∫b

FA−α M 2 d x ≤

a

Substitution of the optimality condition

∫b

a

F a

−α

M 2 d x.

(10.29)

10.5 Optimal Shapes of “Generalized Pfluger Columns”

293

FM 2 = Aα+1 reduces the examined inequality (10.29) into: ∫b FA

−α

∫b M dx =

A

2

a

A

α+1

∫b dx ≡

a

∫B (

∫B dy ≤ A

−α

A

A

a

∫b Ad x ≤

a

)α dy with

a

−α

Aα+1 d x,

a

dy = A. dx

(10.30)

The concluding step is the check the validity of the inequality in Eq. (10.30). Opportunely, the inequality (10.30) follows immediately from the generalized mean values inequality, [11, 12]. √

Ap

/1/ p

a √ p

1≤

√ ≤

Aα

a

/1/α

Aq

a

q

/1/q if p ≤ q; (10.31)

if p → 0, q = α > p > 0.

α

With the Eq. (10.31), the inequality in (10.29) is proved to be valid. Finally, the desired inequality is stated:

a

a

Δ∗ = Δ[ ] = min R[ , m] ≤ R[A, M] ≡ Δ[A] = Δ∗∗ . m

(10.32)

The inequality (10.32) expresses, that the optimal generalized Pfluger column has the largest buckling load among all generalized Pfluger columns with the same weight.

10.5 Optimal Shapes of “Generalized Pfluger Columns” 1. The methods of finding the optimal shapes of the Euler columns were demonstrated above in Chap. 2. The only difference to the previously discussed results is the given variable axial force distribution F(x) along the axis of the column. From the optimality condition (10.26) follows, that for the optimal column the moment relates to the cross-section area and axial force as: FM 2 = Aα+1 . Substitution of the optimality condition (10.26) into the governing equation leads to the ordinary differential equation for the unknown function M:

294

10 Optimization for Buckling of Conservative Systems of Second Kind

d2 M 1−α 1 + ΔF 1+α M 1+α = 0. dx2

(10.33)

One of the boundary conditions (10.12), (10.13) or (10.14) must specify the values of the function M on both ends x = a, x = b. The simplest task is to determine the force distributions, which cause the certain buckling modes. If the moment M along the axis of the rod is given, the function F for distribution of the axial follower force results from the inversion of Eq. (10.33): )1+α ( 1 d2 M M α−1 . F= − Δ dx2

(10.34)

Another, challenging task is to determine the moment M for a given function F. The closed form solution of the nonlinear eigenvalue problem could be expressed in 1 closed form only for some exceptional cases. Furthermore, the coefficient F 1+α of the nonlinear ordinary differential Eq. (10.33) is variable. This feature complicates the closed-form solution of the nonlinear eigenvalue problem. For an arbitrary function F(x) and α /= 1 the eigenvalue problem allows no closed form solution and requires the applications of the numerical methods. 2. For α = 1, the eigenvalue problem for the Eq. (10.33) with boundary conditions Eq. (10.12) reduces in this case to: √ d2 M + Δ F = 0, 2 dx

M(a) = M(b) = 0.

(10.35)

The solution of (10.35) is: ⎤ ⎡ ∫ x ∫θ √ ∫ b ∫θ √ Δ⎣ F(η)dηdθ − 2 F(η)dηdθ ⎦, M(x) = (1 + x) · 2 a

a

a

(10.36)

a

Once the solution of the eigenvalue problem M is found, the optimal cross-section A follows from the optimality condition: √ FM 2 = A2 → A = M F. For certain functions F the integrals in Eq. (10.36) could be expressed in terms of higher transcendental functions. The closed form solution could be arrived for the load distribution function )1/k ( F(x) = 1 − x n .

(10.37)

with positive integer numbers n, k. In this case, the integrals in Eq. (10.36) reduce to the hypergeometric function. As the resulting equations are somewhat bulky, the authors demonstrate only some particular results.

10.5 Optimal Shapes of “Generalized Pfluger Columns”

295

In the simplest case (n = 2, k = 1) the distribution (10.37) of the load reduces to: F = 1 − x 2. The solution in this case is: ] [ √ )3 Δπ Δ 1( 2 2 2 , − x arcsin x + 1 − x − 1 − x M(x) = 4 2 3 √ ] [ √ √ )3 Δ Δπ 1( 2 A(x) = 1 − x − x arcsin x + 1 − x 2 − 1 − x 2 2 . 4 2 3

(10.38)

The volume of the optimal column for the values n = 2, m = 1 evaluates with (10.38) to: ∫1 −1

( ) 128 Δ 3π 2 − . Ad x = 2 2 15

(10.39)

For somewhat more general case of an arbitrary integer m and n = 2, the distribution of the load (10.37) shortens to the function: F=

√ k

1 − x 2.

(10.40)

The integral (10.36) in this case could be compactly presented as well: M(x) = ϑ1 (k) − ϑ2 (k), ( √ [ 1+2k Δ π ( 2k ) , for k /= 1, 2 [ ( 1+3k ϑ1 (k) = 2k ) Δπ , for k = 1; 4 [ [ 1 [ 2) ([ 1 ( Δ 2k+1 , 2 ,x , for k /= 1, 2 F2 − 2 , − 2k 2k+1 √ ( ) 23 ] ϑ2 (k) = Δ [ 1 2 , for k = 1. x arcsin x + 1 − x 2 − 3 1 − x 2

(10.41)

The case k = 1 was already shown in Eq. (10.38). Analogous solutions could be established in closed forms also for different boundary conditions. Namely, the actual method is applicable for the boundary conditions (10.13) and (10.14). For the boundary conditions (10.13), the solution of (10.35) reads: ⎡ M(x) = Δ⎣(1 − x) ·

∫1 √ −1

F(η)dη −

∫ x ∫θ √ 1

1

⎤ F(η)dηdθ ⎦.

(10.42)

296

10 Optimization for Buckling of Conservative Systems of Second Kind

√ For n = 2 and an arbitrary integer k, the substitution of function F = k 1 − x 2 from Eq. (10.40) into Eq. (10.42) delivers the moment solution of the generalized Pfluger column: M(x) = (2 − x)ϑ1 (k) − ϑ2 (k).

(10.43)

Similarly, for the boundary conditions (10.14) the solution of (10.35) reads: ⎡ M(x) = Δ⎣(1 + x) ·

∫1 √ −1

F(η)dη −

∫ x ∫θ √

⎤ F(η)dηdθ ⎦.

(10.44)

−1 −1

For m√ = 2 and an arbitrary integer k, the replacement of the distribution function F = k 1 − x 2 from Eq. (10.40) in Eq. (10.44) expresses the moment of the optimal rod to: M(x) = (2 + x)ϑ1 (k) − ϑ2 (k).

(10.45)

The optimal cross-sections follow with Eqs. (10.41), (10.43) and (10.45) from the optimality condition (10.26) and the expression (10.40). The shapes of the optimal columns are shown on Fig. 10.1. 3. For an arbitrary value of shape exponent α, the closed form solution is conceivable for F = 1. In this case, the Eq. (10.33) simplifies to [7]: d2 M 1−α + ΔM 1+α = 0. 2 dx

Fig. 10.1 1 The shapes A of optimal columns, loaded by the distributed load F(η) = (1 − x n )1/k for k = 1, 2, 3 and n = 2.

(10.46)

10.5 Optimal Shapes of “Generalized Pfluger Columns”

297

Interchanging the independent and dependent variables, the Eq. (10.46) transforms to Emden–Fowler equation [13, 14]: ) ( dx 3 d2x 1−α 1+α = Δ M . d M2 dM

(10.47)

The order of the Eq. (10.47) could be reduced by one. The solution of (10.47) with the boundary condition x(M = 1) = 0 reads: ∫M √

x =±

dt c − Δ · (1 + α) · t

1

2 1+α

.

(10.48)

The signs plus and minus point to the left and right half of the optimal shape. The optimal solution must be symmetric for the mirror symmetry of the boundary conditions. The optimal shape in the even function of the independent coordinate x, if c = Δ · (1 + α). The substitution of this value into (10.48) leads to the closed form solution in terms of hypergeometric function [15]: ) ([[ ] [ ]] 1 1+α 3+α M 2 , , , M 1+α x = ±√ 2 F2 2 2 2 Δ · (1 + α) ( 3+α ) √ [ 2 π ( ). −√ Δ · (1 + α) [ 1 + α2

(10.49)

For the interesting cases of shape exponent α = 1, 2, 3, the expression (10.49) expresses over the elementary functions: √

√ x =±

1−M for α = 1; x =± √ Δ/2 3 √

2M 2/3

[

√ ( )) M 2/3 ( · π − 2 arcsin M 1/3 + M 1 − M 2/3 2 Δ

(10.50) ] for α = 2; (10.51)

√ ( √ ) √ 2 x =± √ 1 − M · 2 + M for α = 3. 3 Δ

(10.52)

The moment distributions (10.50)–(10.52) of the optimal columns are shown on Fig. 10.2. The shapes result from the moment distributions with the optimality condition: M 2 = Aα+1 . For λ = 1, it follows:

298

10 Optimization for Buckling of Conservative Systems of Second Kind

Fig. 10.2 Moment distributions M(x) for the optimal columns, loaded by the constant load = 1, for α = 1, 2, 3

([[ ] [ ]] ) 1+α 1 1+α 3+α A 2 , ,A , F x = ±√ 2 2 2 2 Δ · (1 + α) 2 ) ( √ [ 3+α π 2 ( ). −√ Δ · (1 + α) [ 1 + α2

(10.53)

The shapes of the optimal columns are shown on Fig. 10.3. The critical dimensionless load parameter Δ and Lagrange multiplier for all shown solutions were set to one (Fig. 10.4). 4. The evaluation of the volume v of the one half of optimal column is displayed on Fig. 10.4 . The evaluation of the half-volume of the optimal columns is based on integration by parts of the volume. This integration leads to the formula: ∫b v=

ad x = a

a

∫1 x|x=b x=a

−

xd 0

a=

√

) ( √ 1 + α π [ 3+α 2 ). ( √ 2 Δ· [ 2 + α2

(10.54)

10.6 Conclusions This Chapter deals with the problem of the optimal tapering of a conservative system of the second kind, namely the generalized Pfluger column. Well known, that the conservative systems of the second kind are non-self-adjoint and nonconservative systems in a classical sense, but are certain self-adjoint systems. The self-adjointness for such systems is stated in slightly extended sense. This nonconservative systems

10.6 Conclusions

299

Fig. 10.3 The shapes A(x) of optimal columns, loaded by the constant load F = 1, for α = 1, 2, 3 Fig. 10.4 The half-volume v of optimal columns, loaded by the constant load F = 1 for α = 1, 2, 3

300

10 Optimization for Buckling of Conservative Systems of Second Kind

are selfadjoint in a generalized sense, with respect to the certain operator. Therefore, these systems buckle by divergence and possess a generalized conservation theorem and a generalized Rayleigh quotient. For the generalized Pfluger column, subjected to distributed compressive follower forces, the optimization problems were studied. Optimization problems were considered for columns with pinned–pinned or sliding ends. It was shown by means of a special transformation of independent variables, that the Pfluger column is reduced to an auxiliary, ordinary conservative bifurcation problem. The auxiliary problem is the Euler column, which is loaded by the definite axial distributed load. The optimal solutions of the auxiliary are found in closed form problems for some load distributions. Moreover, the isoperimetric inequalities are established for these optimization problems.

10.7 Summary of Principal Results • The optimization problem for a conservative system of the second kind (generalized Pfluger column) is solved. • Optimization problems for the generalized Pfluger column with pinned–pinned or sliding ends, subjected to distributed compressive follower forces are considered. • The problem is deduced to a classical conservative bifurcation problem for the column loaded by the axially distributed load. • The optimal solutions for some load distributions are found in closed form. • For the optimization problems analytical solutions are obtained and the corresponding isoperimetric inequalities are approved.

10.8 List of Symbols FF(x) F(x) L M Δ

a

A

Distributed follower force Distribution of the axial force along the axis of the column Differential operator for Pfluger column Adjoint operator Dimensionless critical parameter Δ = F /EC Area an arbitrary cross-sectional distribution Optimal area cross-sectional distribution

References

301

References 1. Kirillov ON (2013) Nonconservative stability problems of modern physics. Walter de Gruyter GmbH, Berlin, Boston 2. Weisshaar TA, Plaut RH (1981) Structural optimization under nonconservative loading. In: Haug EJ, Cea J, Sijthoff and Noordhoff, Rijn AAD (eds) Optimization of distributed parameter structures, pp 843–864 3. Seiranyan, A. P., & Scharanuk, A. W. (1983). Sensitivity analysis and optimization of critical parameters in problems of dynamic stability. Mech Solids (MTT), 18, 173–182. 4. Gajewski, A., & Zyczkowski, M. (1988). Optimal structural design under stability constraints. Kluwer Academic Publishers. 5. Leipholz HHE (1974) On conservative elastic systems of the first and second kind. Ing-Archiv: 255–271 6. Leipholz HHE (1974) On a generalization of the concept of self-adjointness and of Rayleigh’s quotient. Mech Res Commun: 67–72 7. Kobelev VV (1993) Isoperimetric inequalities in stability problems. In: Rozvany GIN (ed) Optimization of large structural systems. NATO ASI series, vol 231. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-9577-8_60 8. Kobelev, V. V. (1993). Isoperimetric inequalities in optimal structural design. Struct Optim, 6, 38–51. https://doi.org/10.1007/BF01743173 9. Atanackovic TA, Simic SS (1999) On the optimal shape of a Pflüger column. Euro J Mech A/ Sol 18(5):903–913. https://doi.org/10.1016/S0997-7538(99)00128-X 10. Trung TD, Le BH, Huong CQ (2012) Analyzing and optimizing of a Pfluger column. Vietnam J Sci Technol 48(5). https://doi.org/10.15625/0866-708X/48/5/1184 11. Pachpatte BG (2005) Mathematical inequalities. North-Holland Mathematical Library, 67. Elsevier, Amsterdam 12. Beckenbach EF, Bellman R (1971) Inequalities. Springer-Verlag, Berlin, Heidelberg, N.-Y. 13. Zwillinger D, Dobrushkin V (2022) Handbook of differential equations, 4rd edn. Taylor & Francis Group, LLC, 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 14. Berkovic, L. M. (1997). Generalized Emden-Fowler equation. Symmetry Nonlin Math Phys, 1, 155–163. 15. Abramowitz M, Stegun IA (eds) (1983) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Applied mathematics series, 55. United States Department of Commerce, National Bureau of Standards, Dover Publications, Washington D.C., New York

Chapter 11

Structural Optimization for Stability of Circular Rings

Abstract The isoperimetric inequalities for stability problems of the circular ring are illustrated in this chapter. The problem of maximizing the critical load causing a loss of stability in an elastic inextensible circular ring under hydrostatic pressure is studied. The undeformed ring has the form of a circle of unit radius, and its thickness, and hence the flexural rigidity, varies along the arc. The thickness distribution must be determined from the condition of the maximum critical load causing the loss of stability, under the condition that the mass of the ring remains constant. It is shown that of all circular rings of the same mass, the ring of constant thickness can bear the greatest load before losing stability. Keywords Optimization of circular ring · Stability of arc

11.1 Stability of Circular Rings and Arches Of considerable practical significance is the problem of stability of a closed circular ring under the action of radial load, which is uniformly distributed along the circumference of the ring. Such rings usually serve to absorb radial forces transmitted to the cylindrical shell attached to them, or as ribs supporting the shell (Fig. 11.1). Consider the conditions for the loss of stability of a circular ring acted upon by a uniformly distributed, compressive hydrostatic load. Let us denote by F the force per unit length of the circumference. Under the action of hydrostatic pressure, the elementary load vectors remain normal to the curved axis of the ring, and the work done by this load is equal to the product of the pressure and the difference in the areas bounded by the ring in its deformed and undeformed states. Therefore, the external load is conservative, and the phenomenon of loss of stability can be studied using static methods. The authors shall assume that the bending occurs in the plane of curvature of the ring. It is also assumed that one of the principal central axes of symmetry of the transverse cross-section lies in the plane of curvature of the ring. The differential equations of the circular elastic line are based on the Bernoulli–Euler model. When the compressive load reaches its critical value, the initial circular form of the ring becomes unstable and a perturbed state of equilibrium appears. Let us © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Kobelev, Fundamentals of Structural Optimization, Mathematical Engineering, https://doi.org/10.1007/978-3-031-34632-3_11

303

304

11 Structural Optimization for Stability of Circular Rings

Fig. 11.1 Forces and moments acting on the radially hydrostatically compressed circular ring

denote by M, Q and N the bending moment, transverse and axial forces in the ring cross-section, force N being positive in compression. The equations of equilibrium of an element of length dξ , determining the projections of forces in the directions of tangent and normal, and also the moments relative to the center, are [1]: Q dN + + qξ = 0, dξ r N dQ + − qr = 0, dξ r dN dM +r − rqξ = 0. − dξ dξ

−

(11.1)

The coordinate ξ is counted along the arc, r to the center. The value r denotes the radius of the circle, passing through the center of gravity of the section, qξ and qr are the projections of the external load on the corresponding directions. It is assumed that the load F is always directed along the normal, furthermore F = qr . Comparing the above equations, we obtain: Q=

dM , dξ

d3 M qξ dqr 1 dM + − = 0. + 2 3 dξ r dξ r dξ

(11.2)

Up to the moment of ring bulging, the axial force is N0 = Fr . The expression w(ξ ) has the meaning of deflection in the radial direction in plane of the ring. The change in curvature is expressed in terms of deflection w as follows [2]:

11.1 Stability of Circular Rings and Arches

305

k = ddξw + rw . 2

2

(11.3)

2

The first term in this expression is the same as in the case of a straight beam, the second term is found by taking into account the change in radius of curvature: 1 w 1 1 ≈ 2. − = r1 r r −w r The bending moment in the cross-section is equal: M = EJ

k = EJ

(

) d 2w w . + dξ 2 r2

(11.4)

Substitution of the bending moment in the above differential equation for M, results the differential equation for w ( EJ

d 5w 2 d 3w 1 dw + 2 + 4 5 dξ r dξ 3 r dξ

) =

qξ d qr − . r dξ

(11.5)

In stability problems, the radial component of the forces N arising from ring bulging have to be considered: ( qr = N0

d 2w w + 2 2 dξ r

)

( =

) d 2w w Fr. + dξ 2 r2

(11.6)

Considering qξ = 0, we find the sought differential equation: ( EJ

2 d 3w 1 dw d 5w + + 4 5 2 3 dξ r dξ r dξ

)

( + Fr

d 3w 1 dw + 2 3 dξ r dξ

) =0

(11.7)

The radial deflection is assumed as a sinus function: w ∼ sin

ξn , r

where n is the number of full waves generated along the circumference. If = 1, we obtain the displacement of the ring as a solid body. Excluding this case, we will take n ≥ 2. Substituting expression for the radial deflection into (11.7), we arrive at the following dependence: ( )EJ F = n2 − 1 3 . r The minimum of the above critical load corresponds to two waves on the circumference n = 2 and is equal to [3]:

306

11 Structural Optimization for Stability of Circular Rings

Fig. 11.2 Circular arch, spanning the central corner 2γ and hinged at the ends

F=

3E J . r3

(11.8)

We have obtained the well-known formula for critical pressure, often used in practical calculations Let’s pass to the case of circular arch, spanning the central corner 2γ and hinged at its ends (Fig. 11.2). The expression for radial deflection to the ring is taken as: w ∼ sin

π nξ , l

where l = 2r γ is the length of the midline. Substituting the assumed radial deflection into Eq. (11.7), then we obtain: Fr π2 1 = n2 2 − 2 . EJ l r

(11.9)

The number of half-waves n has to be determined from the condition that the total length of the centerline remains unchanged. The extensional deformation of the midline is [4]: ε=

w dv − , dξ r

where v is the displacement along the arc. The second term in this expression accounts for the shortening of the arc. Assuming ε = 0, we find: w dv = . dξ r Both ends of the arch will remain stationary under the condition: ∫l wdξ = 0 0

11.2 Optimization for Stability of Circular Rings and Arches

307

The smallest number n, at which this equality holds is n = 2. With the length of the midline l = 2r γ , the critical load form (11.9) yields to: ( F=

) π2 EJ −1 . 2 γ r3

(11.10)

From this we can obtain the formula (11.10) for the ring if we put γ = π/2; this angle corresponds to the arc located between the inflection points of the elastic line of the ring. As we have seen, an articulated arch with a non-stretch axis bulge along two half-waves. However, for a hollow arch it may turn out that the bulging will occur at a lower load if we consider the midline to be deformable.

11.2 Optimization for Stability of Circular Rings and Arches The optimization problem for stability of arches was studied in several papers, starting from [5]. The paper of [6] considers the problem of optimal shaping of arches subjected to general loading. The examination includes the search for the optimum center line form, as well as the optimum area distribution along the arch. The optimization problem is studied with objective function and constraints. As the design functionals, the definite combinations of arch thrust, material volume, total arc length, and enclosed area are treated. In the paper [7], the model of funicular design for arches was revised. In the cited paper the equations determining the buckling load and transitional bending moments were derived. In [8], the problem of determining the optimal cross-sectional area function of funicular, transverse vibrating arch under external load and geometrical constraints was solved by the use of Pontryagin maximum principle. An elastic, plane and funicular circular arch loaded by uniformly distributed radial pressure was investigated in [9]. Fundamental buckling modes corresponding to two lowest critical loads both for out-of-plane and in-plane buckling of the arch were studied. Depth and width of a rectangular cross-section were treated as independent control functions. The optimization problem determines the cross-sectional dimensions as the design variables and consists in minimization the total volume of the arch under given external pressure and geometrical constraints. The paper [10] examined the numerical problem of the optimal arch as a uniformly compressed arch subjected to static loads. An analytical solution for the optimal shape of a plane-statically determined arch subjected to uniform vertical loads was demonstrated in [11]. The problem of a catenary subjected to the self-weight is extended to an inverted catenary subjected to the self-weight and to a constant vertical load distribution.

308

11 Structural Optimization for Stability of Circular Rings

The analytical solutions for the cable (or arch) of uniform self-weight and uniform vertical load were presented in [12].

11.3 Basic Equations and Formulation of Optimization Then the critical load F will be equal to the product of elasticity modulus E on the minimum value λ of the Rayleigh’s quotient R[ , w]:

a

)2 ( ⟨ j (ξ ) · w '' + w ⟩ . F = EΔ , Δ = λ[ ] = min R[ , w], R[ , w] = w ⟨w '2 − w 2 ⟩ (11.11) ∗

∗

a

a

a

where the following notations are used for brevity: ∫2π

1 ⟨f⟩= 2π def

f (ξ )dξ, r = 1. 0

a

a

The authors denote by = (ξ ) a dimensionless, 2π -periodic function proportional to the area of transverse cross-section of the ring, such that

a

V=⟨ ⟩

a

for (ξ ) > 0.

(11.12)

In the expression (11.11), the second moment of area reads: j =C

a (ξ ), α

0 ≤ ξ < 2π.

The shape exponent α assumes depending on the form of the cross-section the values of 1, 2 and 3; C is the shape factor. For brevity we put C = 1. The condition of (11.12) expresses the requirement that the volume of the ring material must be constant for all rings under consideration. The constancy of volume among all competitive designs plays the role of the isoperimetric condition. The minimum in (11.11) is found from the set of all 2π -periodic, twice differentiable functions, such that ⟨w⟩ = ⟨w sin ξ ⟩ = ⟨w cos ξ ⟩ = 0.

(11.13)

The problem of optimization is formulated in the following manner: find the area of the cross-section A satisfying the condition (11.13) such, that the critical load F = EΔ∗ causing the loss of stability has its extreme value λ[a] = λ[A] = Δ amax a

∗∗

⟨ ⟩=V, >0

.

(11.14)

11.4 Transforming of Variational Formulation and General Properties …

309

Accordingly, the optimal solution for boundary conditions produces the optimum function A, for which the critical compression load is greater, than the critical compression load for any arbitrary thickness distribution with the same volume of material of the compressed ring:

a

A = optimum

a,

A ≥ 0,

a

⟨ A⟩ = ⟨ ⟩ = V , Δ∗∗ ≥ Δ∗ .

It will be shown that a ring of constant thickness A = 1 delivers the maximum in (11.14), and that the following inequality holds:

a

Δ∗ = λ[ ] ≤ λ[A] = Δ∗∗ ,

for A = 1, V = 1.

(11.15)

Equality in (11.15) is attained from (11.8), when the ring is of constant thickness A = 1 and Δ∗∗ = 3..

11.4 Transforming of Variational Formulation and General Properties of Boundary Value Problem Before proving inequality (11.15), the authors shall mention some facts, which are essential for further discussion. The authors note that Δ[a] is the smallest positive eigenvalue of the self-conjugate boundary value problem with periodic boundary conditions m '' + m + λ

a

−α

m = 0,

m(0) = m(2π ), m ' (0) = m ' (2π ).

(11.16)

The authors can confirm this by writing out Euler’s equations for the functional (11.11) and putting m=

a (w α

''

) +w .

The authors know ([13], Ch. VI, Sect. 3) that the eigenvalues of the boundary value problem (11.16) are simple or double, and that they can be arranged to form a non-decreasing sequence λ0 , λ1 , . . . , λi , λi+1 , . . . , (λi ≤ λi+1 ) so that the fundamental function corresponding to the eigenvalue λi has, in the semi-interval [0, 2π ) exactly i zeros if i is even, and i + 1 zeros if i is odd. Furthermore,

310

11 Structural Optimization for Stability of Circular Rings

λ0 ≤ λ1 = λ2 = 0. The fundamental functions cos ξ and sin ξ correspond to the doubly degenerate eigenvalue. λ1 = λ2 , which is equal to zero. The third and fourth eigenvalues have a physical meaning. These eigenvalues define the dimensionless critical load. The value Δ[ ] is equal to the third eigenvalue:

a

a

Δ∗ = λ[ ] = λ3 . The critical load causing loss of stability may be double:

a

λ[ ] = λ3 = λ4 . Let us denote by u(ξ ) and v(ξ ) the fundamental functions of the boundary value problem (11.16). The fundamental function u(ξ ) corresponds to the eigenvalue λ3 . The fundamental function v(ξ ) corresponds to the eigenvalue λ4 . These functions change their sign in the half open interval [0, 2π ) exactly four times. The authors shall assume that the functions are orthogonal and normalized

a

⟨

a

a

⟩ ⟨ u = 1,

−α 2

a

⟩ ⟨ v = 1,

−α 2

a

−α

⟩ uv = 0

(11.17)

For any given , the functional space ∏( ) is introduced:

a

def

∏[ ] =

(

) m 1 |m 1 ∈ C 2 [0, 2π ), m 1 (ξ ) = m 1 (ξ + 2π ), . ⟨m 1 ⟩ = ⟨m 1 sin ξ ⟩ = ⟨m 1 cos ξ ⟩ = 0, ⟨ −α m 21 ⟩ = 1

a

a

Based on the definition of the functional space ∏( ), another functional space T ( ) is defined:

a

a

a

a a

| ⟨ def { T [ ] = m 2 |m 1 ∈ ∏( ), m 2 ∈ ∏( ),

−α

⟩ } m1m2 = 0

The eigenvalues of the boundary value problem (11.16) can be regarded as extreme values of the following variational problem:

a

λ3 = min R[m, ], m∈∏[a]

λ4 =

amin

m, ∈T[a]

a

R[m, ].

(11.18)

11.5 The Proof of Isoperimetric Inequality

311

As demonstrated above, for the ring is of constant thickness the number of full waves is two and the fundamental functions are: √ 2 cos 2ξ, U ∈ ∏[A], √ V = 2 sin 2ξ, V ∈ T[A].

U=

(11.19)

The substitution of the fundamental functions (11.19) into (11.18) delivers the dimensionless eigenvalues for the buckling of the ring of constant thickness (11.8): min R[m, A] = ⟨U '2 − U 2 ⟩ = Δ∗∗ ≡ 3,

m∈∏[A]

min R[m, A] = ⟨V '2 − V 2 ⟩ = Δ∗∗ ≡ 3.

m∈T [A]

11.5 The Proof of Isoperimetric Inequality For any function a > 0, satisfying conditions (11.13), holds the inequality:

a

Δ∗ = λ[ ] ≤ λ[A] = Δ∗∗

for A = 1.

(11.20)

The equality (11.20) is attained only for the ring is of constant thickness: A ≡ 1, for 0 ≤ ξ < 2π.

a

For an arbitrary function and admissible periodical functions U, V the following inequality is valid:

a

a

Δ∗ ≡ λ3 [ ] ≤ λ4 [ ], where

a a Using the definition of ∏(a) and T(a) ⟨ a u ⟩ = 1, ⟨a v ⟩ = 1, ⟨a a

λ3 [ ] =

−α 2

⟨v '2 − v 2 ⟩ ⟨u '2 − u 2 ⟩ , λ . [ ] = 4 ⟨ −α · u 2 ⟩ ⟨a −α · v 2 ⟩

−α 2

the authors have the following relation

−α

⟩ uv = 0,

(11.21)

312

11 Structural Optimization for Stability of Circular Rings

a

λ3 [ ] ≤

a

a

⟨u '2 − u 2 ⟩ ⟨v '2 − v 2 ⟩ ⟨u '2 + v '2 − v 2 − v 2 ⟩ λ3 [ ] + λ4 [ ] ) ( = + = 2 2⟨ −α · u 2 ⟩ 2⟨ −α · v 2 ⟩ ⟨ −α · u 2 + v 2 ⟩ (11.22)

a

a

a

for any congruence functions

a

a

u ∈ ∏[ ], v ∈ T [ ]. In particular, the authors can put in (11.22): −U sin θ + V cos θ U cos θ + V sin θ , v= . √ √ ς ς

u=

(11.23)

Here

a

ς =⟨

−α

( 2 ) U + V 2 ⟩/2 > 0

and θ is the solution of the trigonometric equation:

a

⟨

−α

a

( 2 ) U − V 2 ⟩ cos 2θ + ⟨

−α

U V ⟩ sin 2θ = 0.

With such defined constants ς and θ , the functions (11.23) satisfy the requirements (11.21). Substituting (11.23) in the last term of the inequality (11.22), the authors can continue the sequence of inequalities:

a

λ3 [ ] ≤

⟨u '2 + v '2 − u 2 − v 2 ⟩ ⟨U '2 + V '2 − U 2 − V 2 ⟩ ( ( ) ) = ⟨ −α · u 2 + v 2 ⟩ ⟨ −α · U 2 + V 2 ⟩

a

a

(11.24)

U 2 + V 2 = 2.

(11.25)

From (11.9) it follows that

The denominator Eq. (11.24) reduces with Eq. (11.25) to:

a

⟨

−α

a

( 2 )⟩ ⟨ U + V2 = 2

−α

⟩

.

(11.26)

With Eq. (11.26), the right side of (11.24) reduces to the expression: ⟩ ⟨ '2 U + V '2 − U 2 − V 2 ⟨U '2 + V '2 − U 2 − V 2 ⟩ ⟨ )⟩ ( λ3 [ ] ≤ . = −α · U 2 + V 2 2⟨ −α ⟩

a

a

a

(11.27)

The denominator in Eq. (11.27) could be evaluated using the inequality about the mean values states [14, 15]. The Jensen’s inequality states for α ≥ −1 states, that:

11.6 Conclusions

313

a⟩ ⟨ a ⟩

⟨

p 1/ p

≤

−α −1/α

a ⟩ if p ≤ q; ≤ ⟨a⟩ = 1 for p = −α ≤ −1, q = 1.

⟨

q 1/q

(11.28)

Important, that from the Jensen’s inequlity (11.28) follows the decisive estimation:

a

⟨

−α

⟩

≥ 1,

a

for ⟨ ⟩ = 1,

a > 0,

α ≥ −1.

(11.29)

⟨ ⟩ The equality a −α = 1 in Eq. (11.29) attained only for the function, which is identically equal to unity: A ≡ 1.

(11.30)

Consequently, (11.27) could be extended to ⟩ ⟩ ⟨ '2 ⟨ '2 U + V '2 − U 2 − V 2 U − U 2 + V '2 − V 2 Δ∗∗ + Δ∗∗ ⟩ ⟨ Δ ≡ λ3 [ ] ≤ ≤ ≡ ≡ Δ∗∗ . 2 2 2 −α ∗

a

a

(11.31)

Finally, with (11.31) the inequality (11.5) is proved. The inequality (11.31) demonstrates that a ring of constant thickness delivers the maximum of critical load causing the loss of stability, presuming the volume of the material of the ring remains constant.

11.6 Conclusions In this chapter the optimization of the circular ring under the action of the hydrostatic compression was investigated. The optimization aim was the critical pressure. It was assumed, that the mass of the ring is fixed. The from the literature known solutions for the circular arcs show the optimal thickness of the arc must be variable. There arises the question, if the optimal thickness of the complete ring is variable or constant. The solution of the optimization problem for the complete ring considerably differs from the solution or the corresponding problem for the arc. The eigenvalue problem for a simply supported arc has only one critical fundamental solution, which leads to the variable thickness of the optimal arc. Contrarily, the eigenvalue problem for the ring allows multiple, namely at most two, eigenvalues. Accordingly, the solution of the optimization problem demonstrates, that the optimal thickness distribution for the complete ring reduces to the constant thickness. The isoperimetric inequalities were rigorously justified by means of the Hölder inequality about the mean values. Consequently, the optimal one is the ring of constant thickness acted upon by a uniformly distributed, compressive hydrostatic load.

314

11 Structural Optimization for Stability of Circular Rings

11.7 Summary of Principal Results • The optimization of the critical load, which causes a buckling of an elastic inextensible circular ring under hydrostatic pressure is solved in form of isoperimetric inequality; • The ring of the constant thickness attains the maximum of critical load causing the radial instability, among the all rings of the same volume of material.

11.8 List of Symbols F = ECΔ Critical load of a circular ring with radius r = 1, acted by uniformly distributed, compressive hydrostatic load λ[ ] Dimensionless critical parameter for an arbitrary cross-sectional distribution Δ[A] Extreme value of critical load R[ , w] Rayleigh’s quotient M Adjoint operator F[E J, F] Generalized Rayleigh quotient Δ Dimensionless critical parameter Δ = F /EC Area an arbitrary cross-sectional distribution A Optimal area cross-sectional distribution M Bending moment, transverse in the ring cross-section Q Transverse forces in the ring cross-section N Axial forces in the ring cross-section v The displacement along the arc ξ Coordinate counted along the arc qξ Projection of the external load on the arc direction qr Projection of the external load on the radial direction w Deflection in the radial direction in plane of the ring κ Change in curvature of the ring ε Extensional deformation of the midline of the ring α Shape exponent C Shape factor

a

a

a

References

315

References 1. Timoshenko, S., & Woinowsky-Krieger, S. (1957). Theory of plate and shell (2nd ed.). McGraw Hill. 2. Ventsel, E., & Krauthammer, T. (2001). Thin plates and shells, theory, analysis, and applications. Marcel Dekker AG. 3. Timoshenko, S., & Gere, J. M. (1961). Theory of elastic stability. McGraw-Hill. 4. Libai, A., & Simmonds, J. G. (1988). The nonlinear theory of elastic shells. Academic. 5. Budiansky, B., Frauenthal, J. C., & Hutchinson, J. W. (1969). An optimal arches. Transactions of ASME, Series E: Journal of Applied Mechanics, 29(1), 880–882. 6. Farshad, M. (1976). On optimal form of arches. Journal of the Franklin Institute, 302(2), 187–194. 7. Tadjbakhsh, I. G. (1981). Stability and optimum design of arch-type structures. International Journal of Solids and Structures, 17(6), 565–574. 8. Błachut, J., & Gajewski, A. (1981). On unimodal and bimodal optimal design of funicular arches. International Journal of Solids and Structures, 17(7), 653–667. 9. Bochenek, B., & Gajewski, A. (1989). Multimodal optimization of arches under stability constraints with two independent design functions. International Journal of Solids and Structures, 25(1), 67–74. 10. Serra, M. (1994). Optimal arch: Approximate analytical and numerical solutions. Computers & Structures, 52(6), 1213–1220. 11. Marano, G. C., Trentadue, F., Petrone, F. (2014). Optimal arch shape solution under static vertical loads. Acta Mechanica, 225(3), 679–686. 12. Wang, C. Y., & Wang, C. M. (2015). Closed-form solutions for funicular cables and arches. Acta Mechanica, 226(5), 1641–1645. 13. Sansone, S. (1941). Equazioni differenziali nel campo reale. Zanichelli. 14. Pachpatte, B. G. (2005). Mathematical inequalities. North-Holland Mathematical Library (Vol. 67). Elsevier. 15. Bullen, P. S. (2003). Handbook of means and their inequalities mathematics and its applications. Kluwer Academic Publishers.

Chapter 12

Stability Optimization of Axially Compressed Rods on Elastic Foundations

Abstract In this Chapter, the optimization problem for an axially compressed rod or strut on an elastic foundation is studied. The axis of the rod coincides with the boundary of the elastic foundation. Common models of foundations are deliberated: Winkler, Filonenko-Borodich, Pasternak, and Reissner-Kerr. The direction of the applied forces coincides with the axis of the undeformed rod. The critical values of buckling remain equal among all competitive designs of the rods. The principal results are the closed form solutions and the exact bounds for the masses of the optimal struts. Keywords Optimization of buckling · Contact problem · Winkler foundation · Filonenko-Borodich foundation · Pasternak foundation · Reissner foundation · Kerr foundation

12.1 Stability for Axially Compressed Rods on Elastic Foundation The axis of the unloaded beam is straight and its length is 2l. The strut is placed horizontally (along x axis) and contacts with the foundation in the region Gl = [−l, l] (Fig. 12.1). The length of the contact region could be reflected as infinite. It is presumed, that no friction is present at the beam-foundation interface and that the weight of the beam is negligible in comparison with the reaction force. The deflection of the bent axis of the rod in the moment of buckling is (x), x ∈ Gl . The deflection (x) is four times continuously differentiable on Gl . Bending of the beam in the presence of axial load is considered. It is compressed by forces F in x direction applied to its ends. The bending moment reads:

w

w

m = EJ

w. ''

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Kobelev, Fundamentals of Structural Optimization, Mathematical Engineering, https://doi.org/10.1007/978-3-031-34632-3_12

317

318

12 Stability Optimization of Axially Compressed Rods on Elastic Foundations

Fig. 12.1 Models of the elastic foundations for struts of infinite length (left) and finite length (right)

Using the Bernoulli-Euler model, the differential equation of the elastic line reads: (

Ej

w)

'' ''

+F

w + s (x) = 0, ''

i

x ∈ Gl , Gl = [−l, l].

(12.1)

s

In Eq. (12.1), E is the Young’s modulus of the beam, i , i = 1, 2, 3 is the contact pressure for both models of elastic foundations; x is distance measured along the beam. In general case, the equilibrium Eq. (12.1) will be a singular integral–differential of the fourth order with the variable coefficient j (x). The cross-sectional area is the positive and integrable control function. The second moment of area per unit

a

12.1 Stability for Axially Compressed Rods on Elastic Foundation

a

319

width of the plate is j = C α , the shape exponent α = 1, 2, 3; C is the shape factor .The bending moment, the area of cross-section and the second moment of area per unit width of the support plate j are two times continuously differentiable functions on Gl . There are several common models of the elastic foundations (Table 12.1, [1–3]). The comprehensive account of the elastic and viscoelastic models of foundation was presented in [4]. The case i = 1 designates the layer of the finite thickness h. Thus, a weightless beam rests on an infinite in horizontal direction elastic layer. The dimensionless parameter ε = l/ h characterizes the length to thickness ratio. The values E f , G f , are the Young and shear moduli of the elastic medium. Its Poisson coefficient is ν f . The deformation of the layer occurs under conditions of planar deformation. The case i = 2 describes the unbounded elastic medium. The foundation is treated as a two-dimensional, frictionless, elastic half space. The unbounded elastic medium results from the elastic layer in limit case of an infinitely thick layer ε → 0. The model of Winkler foundation has the index i = 3. The Winkler foundation is asymptotic limit of the elastic layer for fading thickness ε → ∞. The local reaction per unit length of the rod be proportional to the local deflection (Table 12.1). In this case the medium is treated as the continuously distributed set of linear springs, which stiffness corresponds to the modulus of foundation κ. We mention the two-parameter Reissner-Pasternak model [5–7]. Other types of elastic foundations were developed by Kerr [8, 9]. The Kerr foundation model was

a

Table. 12.1 Models of the elastic foundations Equations of the elastic foundations Hypotheses of the models

s

3

=κ

w

Winkler foundation [12] Narrowly spaced, independent linear springs; limit case of an elastic layer with an infinitesimally small thickness

s = κ (w − μ w ) s − λ s = κw 2 d2 κ dx2

3

3

s

3

Springs are coupled to a stretched elastic membrane

2 d2 3 κ dx2

s −λ s 3

Filonenko-Borodich foundation [13]

2 d2 3 κ dx2

=κ

(

=κ

(

Pastenak foundation [14]

w − μ w)

w + μ w) 4 d4 κ dx4

2 d2 κ dx2

Spring elements embedded in an elastic compressible medium Shear interactions between the spring elements Reissner foundation [15] Spring elements embedded in an elastic compressible medium Springs are coupled to a stretched elastic membrane as well Hetenyi foundation [16, 17] Spring elements embedded in an elastic beam Interaction between the single spring elements due to bending

320

12 Stability Optimization of Axially Compressed Rods on Elastic Foundations

Table. 12.2 Coefficients of the elastic foundations for Reissner or Kerr model [23]

Bounded layer (I) w I,I I κ

wI =

1−2ν f 2(1−ν f )

1−ν f

(1−2ν f )(1+ν f )

2

Ef h

Unbounded layer (II) wI I =

1 2

1 Ef 1−ν 2f h

Θ wI h

Θ wI I h

λ2κ

3−4ν f h 2 1−2ν f 3

h2 3

μ2κ

h2 1−ν f

h2 3

shown to be identical to a Vlasov foundation model [10, 11]. Some models do not explicitly result from the hypotheses of the homogeneous isotropic elastic layer of small thickness. The presented below optimization routine is applicable for these models of elastic foundations as well. The stability of a compressed elastic strut with the common models of elastic foundation was considered in [18–22]. The Kerr model of an elastic foundation was used for analysis of buckling problems in [23]. In the citing paper, the coefficients of Reissner or Kerr models were derived from the equations of elasticity theory (Table 12.2). The coefficients λκ and μκ are of the dimensions of length. For the unbounded layer λκ = μκ , and the Reissner model reduces back to Winkler model. The following elastic constant of the elastic layer is used in Table 12.2: Ef ). Θ= ( 2 1 − ν 2f

12.2 Deformation of an Infinite Elastic Layer 12.2.1 Elastic Layer Under the Surface Load 1. In this section the normal loading of the elastic layer on a solid base is outlined. Consider the problem of equilibrium of an infinite, linearly elastic, isotropic and homogeneous strip or layer (−∞ < x < ∞ :, 0 ≤ z ≤ h). Along its bottom surface z = h, the elastic layer rests upon the solid base. There are two possibilities for the contact conditions along the bottom of the layer z = h, −∞ < x < ∞ (Chap. 1, [24]). The first possibility is marked with the Latin index I. The layer is bounded, or fastened to the base (on the bottom of the strip);

12.2 Deformation of an Infinite Elastic Layer

321

u x (x, z = h) = 0, u z (x, z = h) = 0.

(12.2)

The second possibility is the free gliding of the elastic layer over the surface of the solid base z = h. In this case, the layer glides without friction, but without any gap upon the solid foundation below it. This possibility for an unbounded layer is marked with the Latin index II: τx z (x, z = h) = 0, u z (x, z = h) = 0.

(12.3)

Boundary conditions on the upper surface depend upon the position of the point on it. On free part of the upper surface two components of stress vanish: τx z (x, z = 0) = 0, σz (x, z = 0) = 0, x∈ / Gl = [−l, l].

(12.4)

There are two options for load conditions on the other part of the upper surface. In the first option, the downward translation along the interval is prescribed: τx z (x, z = 0) = 0, u z (x, z = 0) = (x),

w

x ∈ Gl = [−l, l].

(12.5)

In the second option, the layer under the action of the given distributed normal load on the upper surface z = 0. The pressure on the part of surface is: τx z (x, z = 0) = 0, σz (x, z = 0) = i (x),

s

x ∈ Gl = [−l, l].

(12.6)

2. The deformation of an isotropic elastic plate under the surface loads was studied in [25]. The method of integral Fourier transform was traditionally used to solve the above problem [26, 27]. The contact pressure S and normal translation are liked to each other via the integral equation [28]:

w

1 π

∫l −l

( t−x dt = Θ (x), (t)K h

s

(

w

∫∞ K (x) = 0

L i (ξ ) cos(ξ · x)dξ. (12.7) ξ

The approximate solutions of integral Eq. (12.7) can be found in the above cited monograph [29]. As shown later, the formulas for half-infinite elastic medium will result from the expressions (12.7) as the limit case of an infinitely high thickness h, or

322

12 Stability Optimization of Axially Compressed Rods on Elastic Foundations

ε → 0. Oppositely, the formulas for Winkler foundation result from the expressions (12.7) in the limit case of an infinitesimally small thickness h, or ε → ∞. For the bounded, or fastened layer at the base the following relations are valid: L I (u) = L I (u) → w I u, u→0

x

2 sinh(2u) − 4u , K = 3 − 4ν f , 2 K cosh(2u) + 1 + K2 + 4u 2 1 − 2ν f 4(K − 1) lim L I (u) = 1, w I = ( . )2 = (1 + K)2 2 1 − νf

u→∞

(12.8) (12.9)

The second possibility is the unbounded, freely sliding layer. In this case the function in the expression for the function L i (ξ ) in Eq. (12.7) will be: L I I (u) =

cosh(2u) − 1 , sinh(2u) + 2u

L I I (u) → w I I u, lim L I I (u) = 1, w I I = u→0

u→∞

1 = lim w I . 2 ν f →0

(12.10) (12.11)

w

The formulas (12.7) determine the displacement (x) under the action of normal force (x) on its boundary are essential for formulation of the governing Eq. (12.1).

s

12.2.2 Elastic Layer of Intermediate Thickness 1. The bounded and unbounded case of moderate thickness of the layer must be studied separately. In both cases, there is a known approximation of the functions L i (ξ ), i = I, I I : L˜ i (ξ ) = tanh(wi ξ ),

L˜ i (ξ ) → wi ξ, lim L˜ i (ξ ) = 1. ξ →0

ξ →∞

(12.12)

It is important to note that the approximation L˜ i (ξ ) (12.12) fully matches asymptotic behavior of L i (ξ ). The deviation between the original function L i (ξ ) and its approximation L˜ i (ξ ) in each case is about 10% for all 0 < ξ < ∞. For briefness of formulas, we omit the index i. ˜ ) into expression (12.7) leads The substitution of Eq. (12.12) for the function L(ξ to the approximate kernel K˜ (x), x /= 0: ∫∞ ˜ ∫∞ tanh(w ξ ) L(ξ ) cos(ξ x)dξ = cos(ξ x)dξ K˜ (x) = ξ ξ 0

0

( πx ) ] [ ( πx ) ]} 1 {[ = ln cosh + 1 / cosh −1 . 2 2w 2w

(12.13)

12.2 Deformation of an Infinite Elastic Layer

323

The substitution of (12.13) in (12.7) simplifies it to: 1 π

∫l

s(t) K˜

(

−l

( t−x dt = Θ (x), h

w

| π x || | K˜ (x) = − ln|tanh |. 4w

(12.14)

For moderate thickness of the layer, Eq. (12.14) will be used further as the approximation for the original integral Eq. (12.7). For the solution of (12.14) the dimensionless variables are introduced: S=

s,

W=

Θ

w, l

ω=

πl t x , t= , x= . 2w h l l

(12.15)

In the dimensionless variables, the integral Eq. (12.14) reduces to: 1 − π

∫1 −1

| ( (| | ωt ω x || | dt = W(x), x ∈ G1 = [−1, 1]. − S(t) ln|tanh 2 2 |

(12.16)

Equation (12.16) admits closed form solution [30]. The known solution of Eq. (12.16) with the singularities on the ends of the interval G1 , reads:

f

∫1 (t) cosh2 (ω x) r(t) C cosh(ω) ω − dt, S(x) = 2 π r(x) π cosh (ωt) r(x) sinh(ω(t − x)) −1 √ d . r(x) = cosh2 (ω) − cosh2 (ωx), = dt

f

w

(12.17)

The constant C have to be determined from the resolvability condition: −

1 π

∫1 −1

| ( (| | ωt || dt = W(0). S(t) ln||tanh 2 |

The solution (12.17) was applied for the analysis of the rigid punch with the sharp ends. In this case, the displacement exists on both ends. Contrarily, the displacement and contact stress vanish on both pivot points x = 1 and x = −1 in the considered task. Therefore, we search another solution of Eq. (12.16) for the optimization problem. The required solution vanishes on the ends of the interval G1 . The expression for the contact stress reads:

s

1

=Θ·S

(x ) l

ω ≡ Θ · S(x), S(x) = − π

∫1 −1

f

(t) r(x) dt. r(t) sinh(ω(t − x))

(12.18)

324

12 Stability Optimization of Axially Compressed Rods on Elastic Foundations

Equations (12.17) and (12.18) are the looked-for expression to contact force, which depends upon the derivative of the certain function W(x), x ∈ G1 . The function 1 (x) is required for the solution of the optimization problem.

s

2. As will be shown later, for a special setting α = 1 of optimization problem the displacement (x) on the strut will be known. In this case, the normal displacement for this setting follows from the necessary optimality condition. Remarkable, that the normal displacement of the optimal strut matches the normal displacement of the rigid cylindrical indenter. Thus, the boundary reaction on the contact surface (x) depends upon its normal displacement. Accordingly, the inverse formulas for (12.7) will be necessary. The inverse formulas deliver the contact stresses under the known indenter, which penetrates in the free boundary of the layer. The inverse formulas will be applied for calculation of the reaction force (x) of the compressed elastic strut on the surface of layer in the moment of its buckling. In all cases, the elastic fiber slides freely along the contact surface on the upper surface of the elastic layer.

w s

s

The contact problem for a rigid cylinder, which is pressed without friction between cylinder and layer on a bounded elastic layer was solved in [31]. The cylinder is long enough to ensure a plane deformation. Asymptotic solutions are presented when the ratio of the half width l of the contact area to the thickness h of the layer is small. The contact stress on the cylindrical surface according to citing above paper [31] reads:

s (x) =

∞ ( (2k ∑ l

1

· pk

(x ) , l

lim

s

1

def

=

s

1∞

≡

s

h→∞ h √ p0 (t) = 2lΘ 1 − t 2 , −l ≤ x ≤ l, −1 ≤ t ≤ 1, k=0

2

√ = 2Θ l 2 − x 2 , (12.19)

√ 1 α1lΘ 1 − t 2 , 4 ] √ 1[ p2 (t) = 4α2 t 2 + α12 + 5α2 lΘ 1 − t 2 . (12.20) 32 ( ) The functions αi ν f , i = 1, 2, 3 in Eq. (12.20) were calculated by the interpolation of the coefficients from [31]: p1 (t) =

( ) ( ) ( ) α1 ν f = −6.834893 + 8.4099062 exp ν f − 3.80877946 exp 2ν f , ( ) ( ) ( ) α2 ν f = 12.213291 − 15.5615088 exp ν f + 7.939283 exp 2ν f , ( ) ( ) ( ) α3 ν f = −19.420652 + 28.5944616 exp ν f − 14.7696433 exp 2ν f . (12.21) The interpolation formulas (12.21) together with the points form (Table 12.1, [31]) are displayed in Fig. 12.2. This result is essential to the solution of the optimization problem. The total force normal to the boundary of layer reads:

12.2 Deformation of an Infinite Elastic Layer

325

Fig. 12.2 Calculated values and their approximations by functions α0 (ν), α1 (ν), α2 (ν)

∫l P1 =

s (x)d x = Θπl 64h 1

−l

2[

4

] α12 l 4 + 8α1 h 2 l 2 + 6α3l 4 + 64h 4 .

(12.22)

With the length-to-depth ratio ε = l/ h, Eq. (12.22) turns into: P1 =

) ] Θπl 2 [( 6α3 + α12 ε4 + 8α1 ε2 + 64 . 64

(12.23)

For the infinite thickness of the layer: P2 = lim P1 = Θπl 2 . ε→0

(12.24)

The indentation of a cylinder in an elastic layer resting on the rigid base was systematically refined in [32, 33]. 3. The formulas (12.19) and (12.20) are valid for the length-to-depth ratio ε = hl < 0.4 [34]. The thickness of the layer is higher, than the length of contact region. Alternatively, for ε → ∞ the layer is very thin. The adequate approximation is the Winkler foundation.

326

12 Stability Optimization of Axially Compressed Rods on Elastic Foundations

For the practical application the interpolation between two extreme cases could be used. For interpolation, we use the sum of the reaction forces in both states of a semiinfinite elastic medium for an infinite thickness of layer 3 and for an infinitesimal thin layer 1∞ . Accordingly, the reaction force in contact reads:

s

s

s ≈s 1

1∞

+

s ≡s +s 3

2

3

√ l2 − x2 = 2Θ l 2 − x 2 + Θ . wi h

(12.25)

In Eq. (12.25), the Winkler modulus for a thin elastic layer is given in Table 12.2. The formula (12.25) displays correctly the behavior of the normal stress in the contact region.

12.2.3 Limit Case of Half-Infinite Elastic Medium Another limit case corresponds to the very thick layer. The formulas for half-infinite elastic medium originate from the expressions (12.7) as the limit case of an infinitely high thickness h → ∞. The function K (x) for the thick layer behaves as [35]: K (x) → − ln|x|.

(12.26)

h→∞

The substitution (12.21) into (12.7) leas to: ∫l

w

Θ (x) =

s (t) ln|t − x|dt.

(12.27)

2

−l

The infinitely thick layer behaves as the half-infinite plate. The differentiation (12.27) leads to the common expression for the gradient of downwards translation of elastic half-space under the surface load:

Θ

w

'

1 (x) = π

∫l

s (t) dt, s (x) = − √l − x t−x π 2

2

2

∫l

2

−l

−l

Θ

w√(t)dt '

(t − x) l 2 − t 2

.

(12.28)

Equation (12.28) corresponds to Eq. (1.20). For an infinite length of the strut, Eq. (12.28) shows that Θ ' (x) is the Hilbert transform of 2 (t) (Chap. 15, Vol. 2, [36]). The common relation for the inversion of Hilbert transform is (ibid, Table 15.1):

w

s

Θ

w (x) = π1 '

∫∞

s (t) dt, s (t) = − 1 ∫ Θ w (t) dt. t−x π t−x ∞

2

2

−∞

−∞

'

(12.29)

12.2 Deformation of an Infinite Elastic Layer

327

12.2.4 Limit Case of Thin Elastic Layer In this section the deformation of the thin elastic layer on a solid base is briefly displayed. This subject was comprehensively studied in [37–39]. In the limit case of an infinitesimally thin layer, there exists the direct solution. At first, the asymptotic behavior of the functions L i for small arguments h → 0, ε → ∞ contribute to the integrals with the variable upper boundary X < ∞: ∫X K (x, X ) = 0

L i (ξ ) cos(ξ · x)dξ → wi h→0 ξ

∫X cos(ξ · x)dξ = wi

sin(X · x) , x

0

(12.30) where according to (12.9) and (12.11): L i (ξ ) → wi . ξ h→0

(12.31)

For the estimation of the pointwise normal loading, we use the Dirac delta function for the stress: (x) = δ(x − x0 ), ϕ0 ∈ Gl :

s

w

1 Θ (x) = π

∫l −l

( ( ( ∫l 1 t−x t−x , X dt = , X dt. δ(t − x0 )K (t)K h π h

s

(

−l

(12.32) The substitution (12.20) into (12.32) delivers: ( ) 0 Θ 1 sin X x−x h . (x) = wi h π x − x0

w

w

(12.33)

This, Θ (x)/wi h turns at X → ∞ to the Dirac delta function, as shown in Arfken et al. [40, Sect. 1.11]. Thus, the displacement under the load (x) = δ(x − x0 ) is directly proportional to the concentrated load in the point x0 :

s

w(x) = wΘh δ(x − x ). i

0

(12.34)

If the normal displacement is given, the reaction at a point on the foundation is proportional to the deflection at that point only:

s (x) → s (x) = wΘh w(x). 1

h→0

3

i

(12.35)

328

12 Stability Optimization of Axially Compressed Rods on Elastic Foundations

Consequently, the thin layer behaves as the local foundation with the Winkler modulus. The Winkler moduli correspond to the values from Table 12.2: κ=

( )2 Ef 1−νf 2Θ 1 − ν f Θ )( ) = , bounded layer (fixed on the solid basis) = ( wI h h 1 − 2ν f 1 − 2ν f 1 + ν f h

(12.36) Ef 2Θ 1 Θ = = , unbounded layer (glides without friction over solid basis) κ= 2 wI I h h 1−νf h

(12.37) With Eqs. (12.36) and (12.37), the right side of the equation for stability of the elastic strut Eq. (12.1) reads:

s

3

w

=κ .

(12.38)

The resulting equations demonstrates that the thin elastic layer is statically equivalent to Winkler foundation. The latter depicts a series of independent and closely spaced linear springs. Although it is capable of modelling the foundation behavior, the Winkler approach is unable to represent the continuous nature of strut deformation for the layer of moderate thickness. This happens because due to the linear one-parameter assumption involved in its formulation. Considering solitary stiffness in the pressure-deflection relation, the Winkler model fails to describe a continuous response of the moderately thick layers.

12.3 Stability of Infinitely Long, Homogeneous Struts 12.3.1 Stability of Homogeneous Infinite Strut on Winkler Foundation In this section, the boundary conditions for elastic strut in the state of compression will be briefly discoursed. The following boundary conditions are usually formulated [41, 42]. I. The boundary conditions for the rod which is clamped at x = −l and free at x = l are the following

w|

x=−l

= 0,

w || '

x=−l

= 0,

Ej

w || ''

x=l

= 0,

(

Ej

|

w ) || '' '

x=l

= 0. (12.39)

II. If the rod is clamped x = −l and hinged at x = l, the boundary conditions assume the form

w|

x=−l

= 0,

w || '

x=−l

= 0,

Ej

w || ''

x=l

= 0,

w|

x=l

= 0.

(12.40)

12.3 Stability of Infinitely Long, Homogeneous Struts

329

III. If the rod is hinged at x = l and x = −l the boundary conditions are: Ej

w || ''

x=−l

= 0,

w|

x=−l

= 0,

Ej

w || ''

x=l

w|

= 0,

x=l

= 0.

(12.41)

IV. The end x = L is clumped and its transversal movement is restricted. The other end x = −l can freely move in the direction, orthogonal to the axis of the rod. However, the direction of this end remains parallel to the axis of undeformed rod in course of deformation: | | ( )| '| '' ' | '| |x=l = 0, = 0, E j = 0, = 0. (12.42) | x=l x=−l

w

w

w

x=−l

w

If a rod is subjected to a gradually increasing load, when the load reaches a critical level, the rod may suddenly change shape and the component is said to have buckled. The sudden change in shape under load occurs, which look like as the bowing of the initially straight axis of the rod. The x axis passes through the point of the rod’s support. The stability of the homogeneous beam on the Winkler foundation was studied in Timoshenko and Gere [43]. The linear dependence of the local displacement upon the local applied force is typical for the infinitely thin elastic foundation layer (Sect. 12.1.3) The study of stability of an infinitely long, homogeneous elastic beam with the constant cross-section (x) = ac , j = jc = C c α is modest. Using the Bernoulli–Euler model, the differential equation with the boundary conditions I for the clamped homogeneous rod will be:

a

(

E jc

w)

'' ''

+F

w

''

+κ

a

w = 0, w(l) = w (l) = w(−l) = w (−l) = 0. ''

''

(12.43)

The Winkler modulus for a thin elastic layer is given by Eqs. (12.18) and (12.19). The solution of the differential equation assumes the form:

w(x) = exp(iΩx).

(12.44)

For an infinite strut, the buckling mode leads to the downward deflection 3 = κ cos(Ωx). The corresponding characteristic equation has the form:

w = cos(Ωx), such that the normal contact pressure will be s Ω4 − PΩ2 + Q = 0,

P=

F κ , Q= . E jc E jc

(12.45)

The roots of characteristic Eq. (12.45) are: [ [ √( ( √( ( | | 2 |P |P P P 2 √ √ + − − Q, Ω2 = −Ω4 = − Q. Ω1 = −Ω3 = 2 2 2 2 The solution of Eq. (12.1) takes the form:

330

12 Stability Optimization of Axially Compressed Rods on Elastic Foundations

w(x) = A sin(Ω x) + B sin(Ω x) + C cos(Ω x) + D cos(Ω x). 1

2

1

2

With the given boundary conditions (12.43), we obtain the following characteristic determinant: | | | | 0 0 1 1 | | 2 2 | | 0 0 −Ω1 −Ω2 | = 0. | | sin(Ω1l) sin(Ω2 l) cos(Ω1l) cos(Ω2 l) || | | Ω2 sin(Ω l) Ω2 sin(Ω l) Ω2 cos(Ω l) Ω2 cos(Ω l) | 1 2 1 2 1 2 1 2

(12.46)

Calculation of the determinant (12.46) gives: ( 2 )2 Ω1 − Ω22 sin(Ω1l) sin(Ω2 l) = 0. Equating sin(Ω1l) and sin(Ω2 l) to zero, we obtain: Ω1l = n π, Ω2 l = n π, n = 1, 2, 3. With the expressions p1 , p2 for we find: F n2π 2 κ l2 = 2 + . E jc l E jc n 2 π 2 The curved line consists of n half-waves of a sine wave: ( nπ ) x . (x) ∼ sin l

w

In contrast to a freely deflecting rod, here the number of half-waves n /= 1. The number of half-waves must be determined from the condition of minimum load: κ π4 = n 2 (n + 1)2 4 . E jc l The corresponding critical values of load are: ]π2 [ F = n 2 + (n + 1)2 2 . E jc l If the number of half-waves n is large enough, we can write: ]π2 [ F π2 = n 2 + (n + 1)2 2 ≈ 2n 2 2 , E jc l l

π4 π4 κ = n 2 (n + 1)2 4 ≈ n 4 4 . E jc l l

The critical load in the model of Winkler foundation for the constant cross-section of the beam turns out to be:

12.3 Stability of Infinitely Long, Homogeneous Struts

F(Ω) = E jc Ω2 +

331

κ . Ω2

(12.47)

Its minimal value corresponds to a critical parameter: ( ) F Ω∗ = min F(Ω). Ω

Accordingly, the critical parameter will be: Ω∗ =

√ 4

κ . E jc

With the critical parameter Ω∗ , the critical compression force in the case of Winkler foundation (12.47) results: (

√

F3 = 2 E jc κ, κ =

Θ , wI h Θ , wI I h

layer fixed; layer glides without friction.

(12.48)

The evaluation of critical compression force for Reissner foundation model (Table 12.1) uses the similar technique. For a continuous sinusoidal wave from, the equation of the contact stress is:

s

3

−

λ2κ

s

d2 3 =κ dx2

(

z−

μ2κ

d2

z)

dx2

) ( = κ 1 + Ω2 μκ .

The periodical solution of the above equation delivers the contact pressure along the length of the strut:

s

3

= κ cos(Ω x)

1 + Ω 2 μκ . 1 + Ω 2 λκ

The substitution of contact reaction leads to the characteristic equation: ( )( ) ( ) 1 + Ω2 λκ E jc Ω4 − FΩ2 + 1 + Ω2 μκ κ = 0. The solution of characteristic equation delivers the critical compression force for Reissner foundation: ( ( ) ) E jc 1 + Ω2 λκ Ω2 + 1 + Ω2 μκ κ ( ) . F (Ω) = 1 + Ω2 λκ Ω2 Again, its minimal value corresponds to a critical parameter: ( ) F Ω∗ = min F(Ω). Ω

332

12 Stability Optimization of Axially Compressed Rods on Elastic Foundations

The derivative of the critical compression force by Ω leads to the algebraic polynomial equation of the eighth order. For small λκ and μκ this equation reduces to: ( ) E jc Ω4 − κ dF(Ω) =2 + 2Ωκλ2κ λ2κ − μ2κ = 0. 3 dΩ Ω The solution of the above equation brings the critical parameter: √ ∗

Ω =

4

E jc +

κ ( ). λ2κ − μ2κ

κλ2κ

Consequently, the critical compression force for Reissner foundation results as F3 = F (Ω∗ ): ( ( √ √ ) ( κ 2 F3 = 2 E jc κ + κ μ2κ − λ2κ 1 − λκ . E jc

(12.49)

The formula (12.49) is valid for the relatively small values of λκ and μκ . For λk = 0, Eq. (12.49) reduces to Filonenko-Borodich foundation, for which the critical compression force in the strut is equal to: √ F3 = 2 E jc κ + κμ2κ . For μk = 0, Eq. (12.49) results the critical compression force of Pasternak foundation: ( ( √ √ √ κ 2 2 F3 = 2 E jc κ + κλκ 1 − λκ ≈ 2 E jc κ + κλ2κ . E jc In the cases of the unbounded struts both parameters are equal: μκ = λκ (Table 12.2). Thus, the critical compression force matches the critical compression force of the strut on the Winkler foundation, Eq. (12.48): √ F3 = 2 E jc κ

12.3.2 Stability of Homogeneous Strut on Semi-infinite Elastic Foundation The stability of the infinite elastic, homogeneous strut on the semi-infinite elastic foundation was studied in [44, 45] using the Bernoulli–Euler model. This situation could be treated as the limit case of the infinitely thick elastic layer (Sect. 12.2.3).

12.3 Stability of Infinitely Long, Homogeneous Struts

333

Substitution of the contact reaction Eq. (12.29) into governing Eq. (12.1) leads: (

E jc

w

) '' ''

+F

w

''

Θ − π

w dt

∫∞

'

t−x

−∞

= 0.

(12.50)

z

The solution of the differential equation assumes the form (x) = cos(Ωx). The substitution of this function into (12.50) leads to the: E jc Ω4 − FΩ2 + ΩΘ = 0.

(12.51)

The solution of Eq. (12.51) gives the critical compression force F (Ω) = E jc Ω2 +

Θ . Ω

The minimal value of the critical compression force will be attained for the critical parameter: (

∗

Ω =

Θ 2E jc

(1/3 .

With this value, the critical compression force in the case of elastic foundation delivers: F2 =

3 2/3 Θ (2E jc )1/3 . 2

(12.52)

12.3.3 Stability of Homogeneous Strut on Elastic Layer For the layer of moderately high thickness the solution could be determined by the asymptotic methods. For the solution of the optimization problem, we apply the contact pressure from the first two terms of the governing Eq. (12.1) with = cos(Ω x). Using the Bernoulli–Euler model„ the equation of beam reads:

z

s (x) = −( E j w ) 1

c

'' ''

−F

w

''

( ) = −E jc Ω4 + FΩ2 cos(Ωx).

(12.53)

Equation (12.53) is substitutes into Eq. (12.7): 1 π

∫∞

s (t)K 1

−∞

(

( t−x dt = Θ (x), h

w

∫∞ K (x) = 0

L i (ξ ) cos(ξ · x)dξ. ξ

(12.54)

334

12 Stability Optimization of Axially Compressed Rods on Elastic Foundations

1 π

∫∞

⎤ ⎡∞ ( ( ∫ t − x L (ξ ) i ⎣ cos ξ · dξ ⎦dt = Θ (x). 1 (t) ξ h

s

−∞

w

(12.55)

0

We change the order of integration in Eq. (12.55): ∫∞ 0

⎡ ∞ ∫ L i (ξ ) 1 ⎣ ξ π −∞

⎤ ( ( t−x dt ⎦dξ = Θ (x). 1 (t) cos ξ · h

s

w

(12.56)

The integral over ϕ calculates: 1 π

∫∞

(

s (t) cos ξ t −h x 1

−∞

( ( ) ( x) dt = − cos ξ hΩ 2 −E jc Ω 2 + F [δ(hΩ − ξ ) + δ(hΩ + ξ )], h

(12.57) In (12.57), δ(x) is the Dirac function. The substitution of (12.53) into (12.57) leads to the characteristic equation: (

) E jc Ω3 − FΩ L(hΩ) + Θ = 0.

(12.58)

From Eq. (12.58) the critical value for compression determines immediately as: F = E jc Ω2 +

Θ . ΩL(hΩ)

(12.59)

The wave parameter Ω is have to be defined. This parameter must be chosen from the condition of the minimum load. The closed from solution for the functions L I (u) and L I I (u) from (12.9) and (12.11) is evidently impossible. We convert Eq. (12.57) into the Taylor series with respect to h. For the function L I (u) it reads: F = E jc Ω2 +

(K + 1)2 Θ K3 − 4K2 + K + 6 − Θh, K = 3 − 4ν f . 4Ω2 (K − 1) h 6(K − 1) (12.60)

The condition of the minimum of F with respect to Ω reads: 2E jc Ω =

(K + 1)2 Θ . (K − 1) 2Ω 3 h

(12.61)

The solution of Eq. (12.61) delivers the optimal value of the critical wave parameter ω :

12.4 Optimal Strut on Elastic Foundations

√ ∗

Ω =

4

335

(K + 1)2 Θ . 4(K − 1) E jc h

(12.62)

The substitution of Ω∗ into Eq. (12.60) delivers the critical compression load F2 for the struts on the elastic foundations of moderate thickness, which is ideally connected to the solid base: √ √ 2 E jc Θ E jc Θ F2 = (K+1) ≈ 3.26 . . . , for ν f = 13 . K−1 h h (12.63) F2 → ∞, for ν f = 21 For the elastic foundations of moderate thickness, which could glide without friction over the solid base, the function L I I (u) should be used. In this case, the critical compression load for the struts is: √ F2 =

E jc Θ ≈ 2.82 . . . 8 h

√

E jc Θ . h

(12.64)

If the material is stretched from one end, the Poisson effect describes compression of the material in the perpendicular direction of the applied stress. Both formulas (12.63) and (12.64) result for ν f = 0 the same value, because the Poisson effect in elastic layer in this situation disappears.

12.4 Optimal Strut on Elastic Foundations 12.4.1 Formulation of Optimization Problem The optimal solution for boundary conditions produces the optimum function, for which the critical compression force for any arbitrary thickness distribution are equal, and the volume of material of the optimal compressed strut on the elastic foundation is minimal. The constancy of critical compression forces for all competitive designs plays the role of the isoperimetric condition:

a

A = optimum , such that

a a

A ≥ 0, j = C α , J = CAα , V ∗∗ = ⟨A⟩ ≤ ⟨ ⟩ = V ∗ , F[A] = F[ ].

a

(12.65)

336

12 Stability Optimization of Axially Compressed Rods on Elastic Foundations

12.4.2 Optimality Conditions The strut with the thickness distribution A satisfies the necessary optimality condition of (12.63) Aα−1

z

'' 2

= cL .

(12.66)

The boundary conditions were discussed in the Sect. 12.2. Here c L > 0 is the Lagrange multiplier for the isoperimetric optimization problem:

a

a

def

V ∗∗ ≤ V ∗ for all , V ∗ = ⟨ ⟩ =

∫

ad x,

l

−l

def

V ∗∗ = ⟨A⟩ =

∫

l

−l

a

Ad x, F[A] = F[ ].

In general case, the solution could be arrived with the numerical methods [46–48]. Remarkably, in an exceptional case a closed form solution could be established for shape exponent α = 1. Consider the hinged rod on its ends x = l and = −l; thus, the boundary conditions are III. Using the Bernoulli–Euler model„ the static equations for the Winkler foundation with the boundary conditions are: (

EJ

w || ''

x=−l

EJ

w)

'' ''

w|

= 0,

+F

x=−l

w

''

+κ

= 0,

w = 0,

EJ

w ||

J = CAα ,

''

x=l

= 0,

w|

x=l

(12.67) = 0.

(12.68)

Equations (12.66) and (12.67) together with the corresponding boundary conditions (12.68) are nonlinear, singular differential equations for two functions A(x), (x). The optimality condition leads in the case α = 1 to the following boundary value problem:

w

w = c , w| w=l −x . '' 2

L

2

2

x=−l

= 0,

w|

x=l

= 0, (12.69)

Because of the linearity of the eigenvalue problem, for the solution we can set an arbitrary value of the Lagrange multiplier c L . For briefness, the value c L = 4 will be used. According to (12.69), the deformed shape of the optimal strut must be cylindrical. Notably, that the normal displacement of the optimal strut (12.69) matches the normal displacement of the rigid cylindrical indenter. The problem rigid cylindrical indenter was broadly studied in the previously cited documents. The elastic foundation supports the strut and advantages the reduction of the mass.

12.4 Optimal Strut on Elastic Foundations

337

12.4.3 Optimal Strut on Winkler Foundation

w

Using the solution of the boundary value problem (12.69) for the function (x), we obtain:

s

3

) ( = κ l 2 − x 2 , P3 =

∫l

s d x = 43 l κ. 3

−l

2

(12.70)

The solution A3 of the ordinary differential Eq. (12.67) for A, reads: ) ( 2ECA'' + 2F − κ l 2 − x 2 = 0, A|x=l = A|x=−l = 0, )( ) (2 l − x 2 −5l 2 + x 2 , A3 = A0 + B3 , B3 = κ 24EC l2 − x2 A0 = F. 2EC

(12.71)

The function A3 vanishes on the both hinged ends due to Eq. (12.68). The function A0 represents the optimal cross-section distribution of the free strut without the external support. If the elastic foundation disappears (κ = 0), the optimal volume of the strut will be: V0 = ⟨A0 ⟩ =

2Fl 3 . 3EC

From (12.71), the volume of the optimal rod on the Winkler foundation is equal to: V ∗∗ = ⟨A3 ⟩ = V 3 ≡ V0 + V30 , V30 = ⟨B3 ⟩ = −

4κl 5 . 15EC

(12.72)

For a given axial compression, the volume of the optimal strut on the elastic foundation V3 is less than the volume of the free strut V0 . The volume of the optimal strut V3 will be positive, if the required force F is greater than: F3.crit =

2 2 κl . 5

This means, that the infinitely thin strut on the elastic foundation has the critical compression force F3.crit , such that no additional elastic stiffening is necessary.

338

12 Stability Optimization of Axially Compressed Rods on Elastic Foundations

12.4.4 Optimal Strut on Reissner Foundation Instead of Winkler foundation, the Reissner foundation is supposed for the calculations. The vertical displacement results from optimality conditions (12.69). Resolving the model of Reissner foundation for 3 , we obtain the contact stress:

s

3

s

3

s

s

s

s

( ) d2 3 = κ l 2 − x 2 + 2μ2κ , 3 (l) = 3 (−l) = 0, (12.73) dx2 ( ) ( )⎤ ⎡ exp l−x + exp l+x ) ( ) (2 λ λκ κ ⎦. ( ) = κ l − x 2 + 2κ μ2κ − λ2κ ⎣1 − 2l 1 + exp λκ

− λ2κ

The function A3 nullifies on the both hinged ends. The solution A3 of the ordinary differential equation could be easily determined from the governing Eq. (12.1): 2ECA'' + 2F −

s

A|x=l = A|x=−l = 0, ) ( 2 ( ( 2 ) 2 ) 2 κ l −x + o μ2κ , λ2κ . A3 = A0 + B3 + λκ − μκ 2EC 3

= 0,

(12.74)

Integration of the expression A3 from (12.74) results the volume of the optimal rod on the Reissner foundation: ] [ ) 5 ( 2 2 ∗∗ V = ⟨A3 ⟩ = V3 ≡ V0 + 1 − 3 λκ − μκ V30 . (12.75) 2l

12.4.5 Optimal Strut on Half-Infinite Elastic Space For the elastic strut contacted with an infinite medium, the static equations of the Bernoulli-Euler model, read: (

ECAα

Θ

w)

'' ''

w (x) = '

1 π

∫l −l

w +s ''

+F

s

2 (t)

t−x

dt,

2

= 0,

∫l −l

w

'

√ (t)dt l 2 −t 2

= 0,

s (x) = − 2

z

√ ∫l Θ ' (t)dt l 2 −x 2 √ . π (t−x) l 2 −t 2 −l

(12.76)

Equations (12.76) and (12.66) together with the corresponding boundary conditions (12.68) are nonlinear, singular integral–differential equations for two functions A(x), (x).

w

12.4 Optimal Strut on Elastic Foundations

339

The authors proceed with the solution of the optimization problem for α = 1. The deflection of the strut follows once again from the optimality conditions (12.67):

z=l The contact pressure (12.76) as:

s

2

2

− x 2.

and the total force to the foundation P2 evaluate from

s

2

√ = 2Θ l 2 − x 2 , ∫l

P2 =

(12.77)

s d x = π Θl . 2

2

(12.78)

−l

The substitution of (12.77) into the governing equation results in the differential equation for the unknown optimization variable A. The boundary conditions follow from Eq. (12.13): √ ECA'' + F − Θ l 2 − x 2 = 0, A(−l) = A(l) = 0.

(12.79)

From the boundary value problem (12.79) follows the expected solution of the optimization problem:

B2 =

A2 = A0 + B2 ,

(12.80)

( x )] ( ) Θ [ √2 2 l − x 2 2l 2 + x 2 − 3πl 3 + 6xl 2 arcsin . 12EC l

(12.81)

The material volume of the optimal rod reads finally as: V ∗∗ = ⟨A2 ⟩ = V2 ≡ V0 + V20 , V0 =

2 Fl 3 , V20 = 3 EC

∫l B2 d x = − −l

3π Θl 4 . 16 EC

(12.82)

For a given axial compression, the volume of the optimal strut on half-infinite elastic medium V ∗∗ ≡ V 2 is less than the volume of the free strut V0 as well. The elastic medium supports the strut and advantages the reduction of the mass. The volume of the optimal strut V2 will be positive, if the required force F is greater than: F2.crit =

9 Θlπ. 32

340

12 Stability Optimization of Axially Compressed Rods on Elastic Foundations

This means, that the infinitely thin strut on the half-infinite elastic medium has the critical compression force F2.crit , such that no additional elastic stiffening is required.

12.5 Optimal Strut on an Elastic Layer 12.5.1 Optimization of Compressed Strut on Elastic Layer The authors proceed with the solution of the optimization problem. The equilibrium equation for the elastic strut follows from (12.1): (

ECAα

w

w)

'' ''

+F

w +s ''

1

= 0.

(12.83)

The expression (t) symbolizes the deformation of the strut and upward displacement of the contact surface. Equations (12.83) and (12.66) together with the corresponding boundary conditions (12.11) are nonlinear, singular integral–differential equations for two functions A(x), (x). For an exceptional case α = 1, the deformation of the strut results immediately from the necessary optimality condition (12.66). The displacement of the optimal strut is given by Eq. (12.69). Notably, that the normal displacement of the optimal strut matches the normal displacement of the rigid cylindrical indenter. The formulas (12.19) and (12.25) deliver the contact stresses under the cylindrical indenter, which penetrates in the free boundary of the layer. The inverse formulas will be applied for calculation of the reaction force S(ϕ) of the compressed elastic strut on the surface of layer in the moment of its buckling. In all cases, the elastic fiber slides freely along the contact surface on the upper surface of the elastic layer. After the substitution of the upwards displacement of the optimal strut, results the ordinary differential equation for A(x):

w

ECA'' + F −

1 2

s

1

= 0. A(l) = A(−l) = 0.

(12.84)

1. At first, the expression for the normal reaction according to Meijers [31], Eq. (12.10) is used. The solution of Eq. (12.84) expresses in closed form. This solution reads in the order of ε2 = l 2 / h 2 as: ( ( x )]( ( 2 ) α1 l 2 Θ [ √2 2 3 2 2 , 2 l − x 2l + x − 3πl + 6xl arcsin 1+ A1 = A0 + 12EC l 8 h2 ) ( α A1 = A0 + 1 + 1 ε2 B2 . (12.85) 8 The volume of the optimal strut is: V ∗∗ = ⟨A1 ⟩ ≡ V1 ,

12.5 Optimal Strut on an Elastic Layer

341

Fig. 12.3 The interpolation function for the contact pressure in the elastic layer of moderate thickness ( √ ) 1= 2 2 2 l 2 − t 2 + l w−t Θ as ih function of h and t

s

[ ] 4 ) l4 l2 Θl π ( 2 3α1 + 17α2 4 + 24α1 2 + 192 . V1 = V0 − 1024 h h EC

(12.86)

For the visualization of the results, the optimal area of the cross-section A1 from Eq. (12.85) is plotted on Fig. 12.5. For the calculations with the Meijers model, the following values of the mechanical constant and thicknesses of layer were used for calculations (Figs. 12.3 and 12.4): 1 , E = 1, E f = 1, F = 1, l = 1, C = 1, 3 h = 10; 5; 2; 1, 0.75.

νf =

(12.87)

The contact forces from the Meijers model are displayed by dashed lines. Each line corresponds to the different thickness of the elastic layer. 2. At second, the expression for the normal reaction with the interpolation model Eq. (12.25) is used. For the interpolation model, the Winkler modulus of the thin elastic layer results from Eq. (12.36): κ=

Θ w I,I I h

.

With this setting, the solution of Eq. (12.84) reads: ( x )] ( ) Θ [ √2 2 l − x 2 2l 2 + x 2 − 3πl 3 + 6xl 2 arcsin 12EC l )( ) (2 2 2 2 −5l l − x + x Θ + ≡ A0 + B1 + B2 . (12.88) w I,I I h 24EC

A1 = A0 +

342

12 Stability Optimization of Axially Compressed Rods on Elastic Foundations

Fig. 12.4 Total reaction forces as function of the layer thickness h and Poisson ratio ν

The volume of the optimal strut for the interpolation model (12.89) is: V ∗∗ = ⟨A1 ⟩ = V 1 ≡ V0 + V20 + V30 , ( V1 = V0 + 1 +

( ( 64 α ) ε V20 ≈ V0 + 1 + 1 ε2 V20 45π w I,I I 8

(12.89) (12.90)

Figure 12.6 displays the optimal area of the cross-section A1 from Eq. (12.88). For the calculations with the interpolation model, the above values of the mechanical constants (12.87) were used. The contact forces from the interpolation model are displayed by dot-dashed lines. For comparison, Fig. 12.7 shows both optimal solutions for Meijers and interpolation models of contact stresses.

12.6 Appendix. Direct Calculation of Hilbert Integrals

343

Fig. 12.5 Optimal shapes of struts. Calculations for the strut on an unbounded medium and for the elastic struts of different thicknesses. Calculation of contact pressure with the Meijers formulas

12.6 Appendix. Direct Calculation of Hilbert Integrals The normal contact pressure calculates as the principal value of the Hilbert integral 49:

s

2

s

Θ = π

∫l −l

Ef Gf ω sin(ωt)dt )= , Θ= ( , t−x 1 − νf 2 1 − ν 2f

ωΘ {ψ1 (x, ω) · cos(ωx) + ψ2 (x, ω) · sin(ωx)}, 2 = π ψ1 (x, ω) = Si(ω · (l − x)) + Si(ω · (l + x)), ψ2 (x, ω) = Ci(ω · (l − x)) − Ci(ω · (l + x)). The values E f , G f , are the Young and shear moduli of the elastic medium. Its Poisson coefficient is ν f .The asymptotic expansions of the functions ψi (x, ω), i = 1, 2 over l reads: ( ( 1 2 cos(ωl) cos(ωx) +O 2 , ψ1 (x, ω) = π − l→∞ ωl l

344

12 Stability Optimization of Axially Compressed Rods on Elastic Foundations

Fig. 12.6 Optimal shapes of struts. Calculations for the optimal struts on different elastic foundations: an unbounded medium and the elastic layers of different thicknesses. Calculation of contact pressure with the interpolation formulas

( ( 1 2 cos(ωl) cos(ωx) +O 2 , l→∞ ωl l ( ( 1 = . Θω cos(ωx) + O 2 l→∞ l

ψ2 (x, ω) = −

s 12.7 Conclusions

The optimization problem for an axially compressed elastic beams or struts on an elastic foundation is solved for the regular models of foundations. The following models of elastic foundations were applied for the solution of optimization problems: Winkler, Filonenko-Borodich, Pasternak, and Reissner-Kerr. In the optimization formulations, the critical values of buckling remain equal among all competitive designs of the rods. The optimization aim was represented by the mass of the struts pro its unit length and width. The closed-form solutions for the simply-supported conditions on the ends of the rod were derived. The solutions were expressed in terms of the elementary or the higher transcendental functions.

12.8 Summary of Principal Results

345

Fig. 12.7 Optimal shapes of struts. Calculations for a free column, for the strut on an unbounded medium and for the elastic struts of different thicknesses. Calculation of contact pressure with the interpolation and Meijers formulas

12.8 Summary of Principal Results • Stability of the axially compressed strut on the elastic foundation is studied • For thin layers the models of elastic foundations of Winkler, Filonenko-Borodich, Pasternak, Reissner applied • For thick layers the Meijers and interpolation model are used • The stability of infinite struts is studied for reference • The shape optimization problem for maximal buckling load is studied for all foundation models • The closed-form solutions are found for the linear dependence of bending stiffness of strut upon the cross-sectional area.

346

12 Stability Optimization of Axially Compressed Rods on Elastic Foundations

12.9 List of Symbols νf Ef E κ λk , μk

Poisson coefficient of an elastic plate or layer Young modulus of an elastic plate or layer Young modulus of an elastic beam Modulus of Winkler foundation Characteristic lengths of Kerr-Reissner foundation model Contact pressure for Winkler and Kerr-Reissner foundations Contact pressure for semi-infinite elastic plate Contact pressure for an elastic layer or strip Optimal area of strut for Winkler foundation Optimal area of strut for semi-infinite elastic plate Optimal area of strut for elastic layer Optimal volume for Winkler foundation Optimal volume for semi-infinite elastic plate Optimal volume for elastic layer Elastic constant for plane strain

s s s

1

2 3

A1 A2 A3 V ∗∗ ≡ V 1 V ∗∗ ≡ V 2 V ∗∗ ≡ V 3 K = 3 − 4ν f Θ=

E ( f ) 2 1−ν 2f 1−2ν f

wI =

=

2(1−ν f ) w I I = 21

2

ε = l/h 2l h Ω∗ F

Gf 1−ν f

=

4(K−1) (1+K)2

Elastic constant for foundation Elastic constant for bounded layer Elastic constant for unbounded layer Length-to-depth ratio Length of the strut Thickness of the elastic layer Critical wave parameter Compression force in the strut

References 1. Kerr, A. D. (1964). Elastic and viscoelastic foundation models. ASME Journal of Applied Mechanics, 31, 491–498. 2. Teodoru, I.-B. (2009). Beams on elastic foundation. The simplified continuum approach. Buletinul Institutului Politehnic din Ia¸si Publicat de Universitatea Tehnic˘a „Gheorghe Asachi” din ˘ Ia¸si Tomul LV (LIX), Fasc. 4, 2009 Sec¸tia Construc¸tii. Arhitectur˘ a. 3. Vlasov, V. Z. (1964). Selected works (Vol. III, p. 508). USSR Academy of Sciences. 4. Younesian, D., Hosseinkhani, A., Askari, H., et al. (2019). Elastic and viscoelastic foundations: a review on linear and nonlinear vibration modeling and applications. Nonlinear Dynamics, 97, 853–895. https://doi.org/10.1007/s11071-019-04977-9 5. Kiani, Y., Bagherizadeh, E., & Eslami, M. R. (2011). Thermal buckling of clamped thin rectangular FGM plates resting on Pasternak elastic foundation (three approximate analytical solutions). Zeitschrift für Angewandte Mathematik und Mechanik, 91, 581. 6. Eisenberger, M., & Clastornik, J. (1987). Beams on variable two-parameter elastic foundation. Journal of Engineering Mechanics, 113, 1454.

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Index

A Airy stress function, 8 Axisymmetric contact problem averaged displacement, 54 concentric indenters, 81, 82 design parameters, 73, 84–86 energy capacity, 58, 92 form factor, 54, 58, 60, 65, 73, 76 integral equation, 48, 55 magnification factor, 56 normal displacement, 50 normal force, 50 partial contact energy, 56 ring-shaped indenter, 76 series for displacement, 54 series for stress, 53 shape function, 54 single indenter, 76 spring rate, 58, 83, 84 stiffness, 58, 83, 84 stiffness maximization, 73, 74, 84, 85 stiffness minimization, 74, 84–86 stored elastic energy, 56, 57 stored elastic energy, ring-shaped indenter, 78 stress minimization, 67 total contact energy, 56 total contact force, 56, 70 total energy maximization, 73, 74, 85 total energy minimization, 74, 84, 86 total force maximization, 67

B Base functions first type, 13

second type, 16 third type, 21 Bernoulli-Euler model axial compression and lateral contact, 318, 329, 332, 333, 336, 338 circular ring, 303 compressed rod, 166, 286 twisted and compressed rod, 264 twisted rod, 233, 252 Boundary conditions mixed type, 199 periodic, 309 Sturm type, 178 Boussinesq-Papkovich solution, 42 Buckling moment actual, 174 admissible, 174

C Charged disk electric field, 90 electrostatic potential, 90 surface density of charge, 89 Compressed and torqued rod, 262 augmented Lagrangian, 264 Fermat’s principle, 265 necessary optimality condition, 265 Condition isoperimetric, 175, 199, 200, 237, 255, 264, 289, 308, 335 orthogonality, 200 Conducting disk magnetic field, 92 magnetostatic potential, 92 surface density of current, 92

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Kobelev, Fundamentals of Structural Optimization, Mathematical Engineering, https://doi.org/10.1007/978-3-031-34632-3

349

350 Cross-section circular, 167, 236 triangle, 167, 236 Cross-sectional area, 101, 122, 167, 199, 233, 254, 263, 286, 318

D Disk elastostatics, 47 electrostatics, 89 magnetostatics, 92 Double periodic inclusions lattice, 141 optimal,approximate, 150

E Eigenvalues mixed type boundary conditions, 199 multiple, 199 simple, 178, 288 Sturm type boundary conditions, 178, 288 Elastic layer, 28 Elastostatic displacement problem, 4, 50 mixed problem, 4, 50, 103, 112, 140, 141 traction problem, 4, 48, 53 Emden-Fowler equation, 180, 205, 239, 267, 297 generalized, 206 one-parameter Lie group, 180, 205 Euler column, 168, 199, 200 generalized, 290

F Falling factorial, 54, 57 Flamant solution, 25 periodic, 34 Foundation elastic layer, 319, 320, 322 Kerr model, 319 Pasternak model, 319 Reissner model, 319 semi-infinite medium, 319 Winkler foundation, 319, 328 Function Bessel, 54, 288 beta, 240, 269 digamma, 33 dilogarithm, 18, 36

Index elliptic integral,complete, 49, 50, 53, 55, 67, 76 elliptic integral,incomplete, 209, 276 Gauss error, 240 hypergeometric, 47, 49, 54, 58, 60, 65, 91, 95, 181, 182, 240, 241, 269, 270, 294, 297 integral cosine, 21 integral sine, 21 Jacobi ϑ, 144 Jacobi elliptic, 144, 147, 150 Jacobi elliptic inverse, 152 Lerch, 27, 28 Meijer G , 51 Natanson, 144 Weierstrass ζ, 144 Weierstrass σ, 144 G Greenhill’s problem, 232 generalized, 262 generalized, Emden-Fowler equation, 267 periodic, 251 periodic torsion, 252 torsion, 233 torsion and compression, 263 torsion, Emden-Fowler equation, 239 H Hankel inverses, 46, 47 Hölder conjugates, 202 I Inequality buckling of rod,isoperimetric, 203 Hölder, 202, 203 mean values, 203, 255, 293, 312 Integral equation axisymmetric contact problem, 50 Carleman type, 53 Cauchy kernel, 132 Hilbert kernel, 133, 135 Melan’s problem, 123 Melan’s problem,double-periodic, 143 Melan’s problem,periodic, 133, 135 plane contact problem, 12 singular, 12 J Jacobi

Index elliptic function, 144, 150 inverse elliptic function, 152 lattice parameter τ, 144 nome q, 144 polynomials, 54, 183 ϑ-function, 144 K Kernel Cauchy, 11, 12, 104, 127, 132 Dolgikh-Fil’shtinskii- Natanson, 143 elliptic integral,complete, 49 generalized Hilbert, 123, 127 Hilbert, 124, 125 logarithmic, 12 ζ -Weierstraß, 149 L Lattice Bravais, 144 chess-board, 148 equiharmonic case, 144 fundamental period parallelogram, 142 generators, 142 invariants, 144 lemniscatic case, 148, 158 primitive translation vectors, 142 ratio of periods, 146 rectangular, 146 square, 148 unit cell, 142 Legendre polynomials, 55 Lie symmetry analysis, 179

M Melan’s problem, 117 double-periodic, 141 first, 102 first,periodic, 122 second, 101–103 second,periodic, 122

N Necessary optimality condition, 109, 127, 178, 199, 237, 265, 289, 336 bimoldal case, 203

P Periodic inclusions

351 centers of the inclusions, 119 coaxial, 156 inclination angle, 119 maximum of stress in the stiffener, 132, 136 optimization problem, 126 parallel, 154, 157 strain due to the action of periodic forces, 121 Pfluger column, 286 closed-form solution, 294, 297 Emden-Fowler equation, 297 generalized selfadjointness, 287 isoperimetric inequality, 293 optimality conditions, 289 optimization problem, 289 Rayleigh quotient,generalized, 287, 288 volume of optimal column, 298 Pochhammer symbol, 54, 57 Polynomials Chebyshev, 13, 106, 129 Jacobi, 54, 183 Legendre, 55 Weierstrass, 144

R Rayleigh’s quotient buckling of circular ring, 308 buckling of rod, 174, 201 Greenhill problem,generalized, 264

S Second moments of area, 233, 286, 318 principal, 233, 263 Shape exponent, 167, 208, 233, 235, 254, 257, 263, 270, 272, 275, 286, 308, 319, 336 Shape factor, 167, 254, 263, 286, 308, 319 anisotropic, 233 isotropic, 235 Sonine and Schafheitlin formula, 49, 91, 95 Stability boundary conditions, 169, 328 circular ring, 303 elastic layer, 340 Rayleigh-Ritz method, 172 Reissner model, 331 semi-infinite medium, 332 variation principle, 172, 173 Winkler foundation, 329, 330

352 W Weierstrass p-function, 144

Index polynomial, 144 σ-function, 144 ζ-function, 144