Structural Analysis with Applications to Aerospace Structures PDF

Structural Analysis SOLID MECHANICS AND ITS APPLICATIONS Volume 163 Series Editor: G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI Aims and Scope of the Series Th...

Structural Analysis SOLID MECHANICS AND ITS APPLICATIONS Volume 163 Series Editor: G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative research- ers giving vision and insight in answering these questions on the subject of mech- anics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it in- cludes the foundation of mechanics; variational formulations; computational mech- anics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plas- ticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. For other titles published in this series, go to www.springer.com/series/6557 O.A. Bauchau J.I. Craig Structural Analysis With Applications to Aerospace Structures O.A. Bauchau J.I. Craig School of Aerospace Engineering School of Aerospace Engineering Georgia Institute of Technology Georgia Institute of Technology Atlanta, Georgia Atlanta, Georgia USA USA ISBN 978-90-481-2515-9 e-ISBN 978-90-481-2516-6 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009932893 © Springer Science + Business Media B.V. 2009 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written per- mission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To our wives, Yi-Ling and Nancy, and our families Preface Engineered structures are almost as old as human civilization and undoubtedly began with rudimentary tools and the first dwellings outside caves. Great progress has been made over thousands of years, and our world is now filled with engineered struc- tures from nano-scale machines to soaring buildings. Aerospace structures ranging from fragile human-powered aircraft to sleek jets and thundering rockets are, in our opinion, among the most challenging and creative examples of these efforts. The study of mechanics and structural analysis has been an important area of en- gineering over the past 300 years, and some of the greatest minds have contributed to its development. Newton formulated the most basic principles of equilibrium in the 17th century, but fundamental contributions have continued well into the 20th century. Today, structural analysis is generally considered to be a mature field with well-established principles and practical tools for analysis and design. A key rea- son for this is, without doubt, the emergence of the finite element method and its widespread application in all areas of structural engineering. As a result, much of today’s emphasis in the field is no longer on structural analysis, but instead is on the use of new materials and design synthesis. The field of aerospace structural analysis began with the first attempts to build flying machines, but even today, it is a much smaller and narrower field treated in far fewer textbooks as compared to the fields of structural analysis in civil and mechan- ical engineering. Engineering students have access to several excellent texts such as those by Donaldson and Megson , but many other notable textbooks are now out of print. This textbook has emerged over the past two decades from our efforts to teach core courses in advanced structural analysis to undergraduate and graduate students in aerospace engineering. By the time students enroll in the undergraduate course, they have studied statics and covered introductory mechanics of deformable bodies dealing primarily with beam bending. These introductory courses are taught using texts devoted largely to applications in civil and mechanical engineering, leaving our students with little appreciation for some of the unique and challenging features of aerospace structures, which often involve thin-walled structures made of fiber- reinforced composite materials. In addition, while in widespread use in industry and VIII Preface the subject of numerous specialized textbooks, the finite element method is only slowly finding its way into general structural analysis texts as older applied methods and special analysis techniques are phased out. The book is divided into four parts. The first part deals with basic tools and concepts that provide the foundation for the other three parts. It begins with an intro- duction to the equations of linear elasticity, which underlie all of structural analysis. A second chapter presents the constitutive laws for homogeneous, isotropic and lin- early elastic material but also includes an introduction to anisotropic materials and particularly to transversely isotropic materials that are typical of layered composites. The first part concludes with chapter 4, which defines isostatic and hyperstatic prob- lems and introduces the fundamental solution procedures of structural analysis: the displacement method and the force method. Part 2 develops Euler-Bernoulli beam theory with emphasis on the treatment of beams presenting general cross-sectional configurations. Torsion of circular cross- sections is discussed next, along with Saint-Venant torsion theory for bars of arbitrary shape. A lengthy chapter is devoted to thin-walled beams typical of those used in aerospace structures. Coupled bending-twisting and nonuniform torsion problems are also addressed. Part 3 introduces the two fundamental principles of virtual work that are the ba- sis for the powerful and versatile energy methods. They provide tools to treat more realistic and complex problems in an efficient manner. A key topic in Part 3 is the de- velopment of methods to obtain approximate solution for complex problems. First, the Rayleigh-Ritz method is introduced in a cursory manner; next, applications of the weak statement of equilibrium and of energy principles are presented in a more formal manner; finally, the finite element method applied to trusses and beams is presented. Part 3 concludes with a formal introduction of variational methods and general statements of the energy principles introduced earlier in more applied con- texts. Part 4 covers a selection of advanced topics of particular relevance to aerospace structural analysis. These include introductions to plasticity and thermal stresses, buckling of beams, shear deformations in beams and Kirchhoff plate theory. In our experience, engineering students generally grasp concepts more quickly when presented first with practical examples, which then lead to broader generaliza- tions. Consequently, most concepts are first introduced by means of simple examples; more formal and abstract statements are presented later, when the student has a better grasp of the significance of the concepts. Furthermore, each chapter provides numer- ous examples to demonstrate the application of the theory to practical problems. Some of the examples are re-examined in successive chapters to illustrate alternative or more versatile solution methods. Step-by-step descriptions of important solution procedures are provided. As often as possible, the analysis of structural problems is approached in a unified manner. First, kinematic assumptions are presented that describe the structure’s dis- placement field in an approximate manner; next, the strain field is evaluated based on the strain-displacement relationships; finally, the constitutive laws lead to the stress field for which equilibrium equations are then established. In our experience, this ap-  Preface IX proach reduces the confusion that students often face when presented with develop- ments that don’t seem to follow any obvious direction or strategy but yet, inevitably lead to the expected solution. The topics covered in parts 1 and 2 along with chapters 9 and 10 from part 3 form the basis for a four semester-hour course in advanced aerospace structural analysis taught to junior and senior undergraduate students. An introductory graduate level course covers part 2 and selected chapters in parts 3 and 4, but only after a brief review of the material in part 1. A second graduate level course focusing on varia- tional end energy methods covers part 3 and selected chapters in part 4. A number of homework problems are included throughout these chapters. Some are straightfor- ward applications of simple concepts, others are small projects that require the use of computers and mathematical software, and others involve conceptual questions that are more appropriate for quizzes and exams. A thorough study of differential calculus including a basic treatment of ordinary and partial differential equations is a prerequisite. Additional topics from linear al- gebra and differential geometry are needed, and these are reviewed in an appendix. Notation is a challenging issue in structural analysis. Given the limitations of the Latin and Greek alphabets, the same symbols are sometimes used for different purposes, but mostly in different contexts. Consequently, no attempt has been made to provide a comprehensive list of symbols, which would lead to even more confu- sion. Also, in mechanics and structural analysis, sign conventions present a major hurdle for all students. To ease this problem, easy to remember sign conventions are used systematically. Stresses and force resultants are positive on positive faces when acting along positive coordinate directions. Moments and torques are positive on positive faces when acting about positive coordinate directions using the right-hand rule. In a few instances, new or less familiar terms have been chosen because of their importance in aerospace structural analysis. For instance, the terms “isostatic” and “hyperstatic” structures are used to describe statically determinate and indetermi- nate structures, respectively, because these terms concisely define concepts that often puzzle and confuse students. Beam bending stiffnesses are indicated with the symbol “H” rather than the more common “EI.” When dealing exclusively with homoge- neous material, notation “EI” is easy to understand, but in presence of heteroge- neous composite materials, encapsulating the spatially varying elasticity modulus in the definition of the bending stiffness is a more rational approach. It is traditional to use a bold typeface to represent vectors, arrays, and matri- ces, but this is very difficult to reproduce in handwriting, whether in a lecture or in personal notes. Instead, we have adopted a notation that is more suitable for hand- written notes. Vectors and arrays are denoted using an underline, such as u or F. Unit vectors are used frequently and are assigned a special notation using a single overbar, such as ı̄1 , which denotes the first Cartesian coordinate axis. We also use the over- bar to denote non-dimensional scalar quantities, i.e., k̄ is a non-dimensional stiffness coefficient. This is inconsistent, but the two uses are in such different contexts that it should not lead to confusion. Matrices are indicated using a double-underline, i.e., C indicates a matrix of M rows and N columns. X Preface Finally, we are indebted to the many students at Georgia Tech who have given us helpful and constructive feedback over the past decade as we developed the course notes that are the predecessor of this book. We have tried to constructively utilize their initial confusion and probing questions to clarify and refine the treatment of important but confusing topics. We are also grateful for the many discussions and valuable feedback from our colleagues, Profs. Erian Armanios, Sathya Hanagud, Dewey Hodges, George Kardomateas, Massimo Ruzzene, and Virgil Smith, several of whom have used our notes for teaching advanced aerospace structural analysis here at Georgia Tech. Atlanta, Georgia, Olivier Bauchau July 2009 James Craig Contents Part I Basic tools and concepts 1 Basic equations of linear elasticity................................ 3 1.1 The concept of stress........................................ 3 1.1.1 The state of stress at a point........................... 3 1.1.2 Volume equilibrium equations......................... 7 1.1.3 Surface equilibrium equations.......................... 10 1.2 Analysis of the state of stress at a point........................ 11 1.2.1 Stress components acting on an arbitrary face............ 11 1.2.2 Principal stresses.................................... 13 1.2.3 Rotation of stresses.................................. 14 1.2.4 Problems........................................... 19 1.3 The state of plane stress..................................... 20 1.3.1 Equilibrium equations................................ 20 1.3.2 Stresses acting on an arbitrary face within the sheet........ 21 1.3.3 Principal stresses.................................... 22 1.3.4 Rotation of stresses.................................. 24 1.3.5 Special states of stress................................ 26 1.3.6 Mohr’s circle for plane stress.......................... 27 1.3.7 Lamé’s ellipse....................................... 30 1.3.8 Problems........................................... 31 1.4 The concept of strain........................................ 33 1.4.1 The state of strain at a point........................... 34 1.4.2 The volumetric strain................................. 37 1.5 Analysis of the state of strain at a point........................ 38 1.5.1 Rotation of strains................................... 38 1.5.2 Principal strains..................................... 40 1.6 The state of plane strain..................................... 41 1.6.1 Strain-displacement relations for plane strain............. 41 1.6.2 Rotation of strains................................... 42 XII Contents 1.6.3 Principal strains..................................... 43 1.6.4 Mohr’s circle for plane strain.......................... 44 1.7 Measurement of strains...................................... 45 1.7.1 Problems........................................... 49 1.8 Strain compatibility equations................................ 50 2 Constitutive behavior of materials............................... 53 2.1 Constitutive laws for isotropic materials........................ 55 2.1.1 Homogeneous, isotropic, linearly elastic materials......... 55 2.1.2 Thermal effects...................................... 59 2.1.3 Problems........................................... 61 2.1.4 Ductile materials..................................... 63 2.1.5 Brittle materials..................................... 65 2.2 Allowable stress............................................ 66 2.3 Yielding under combined loading............................. 68 2.3.1 Tresca’s criterion.................................... 68 2.3.2 Von Mises’ criterion.................................. 70 2.3.3 Comparing Tresca’s and von Mises’ criteria.............. 71 2.3.4 Problems........................................... 73 2.4 Material selection for structural performance.................... 73 2.4.1 Strength design...................................... 74 2.4.2 Stiffness design...................................... 74 2.4.3 Buckling design..................................... 75 2.5 Composite materials........................................ 76 2.5.1 Basic characteristics.................................. 76 2.5.2 Stress diffusion in composites.......................... 78 2.6 Constitutive laws for anisotropic materials...................... 82 2.6.1 Constitutive laws for a lamina in the fiber aligned triad..... 85 2.6.2 Constitutive laws for a lamina in an arbitrary triad......... 87 2.7 Strength of a transversely isotropic lamina...................... 94 2.7.1 Strength of a lamina under simple loading conditions...... 94 2.7.2 Strength of a lamina under combined loading conditions... 95 2.7.3 The Tsai-Wu failure criterion.......................... 96 2.7.4 The reserve factor.................................... 98 3 Linear elasticity solutions....................................... 101 3.1 Solution procedures......................................... 102 3.1.1 Displacement formulation............................. 103 3.1.2 Stress formulation................................... 103 3.1.3 Solutions to elasticity problems........................ 104 3.2 Plane strain problems....................................... 110 3.3 Plane stress problems....................................... 111 3.4 Plane strain and plane stress in polar coordinates................ 113 3.5 Problem featuring cylindrical symmetry........................ 116 3.5.1 Problems........................................... 133  Contents XIII 4 Engineering structural analysis.................................. 137 4.1 Solution approaches........................................ 137 4.2 Bar under constant axial force................................ 138 4.3 Hyperstatic systems......................................... 144 4.3.1 Solution procedures.................................. 145 4.3.2 The displacement or stiffness method................... 146 4.3.3 The force or flexibility method......................... 151 4.3.4 Problems........................................... 156 4.3.5 Thermal effects in hyperstatic system................... 157 4.3.6 Manufacturing imperfection effects in hyperstatic system... 161 4.3.7 Problems........................................... 164 4.4 Pressure vessels............................................ 165 4.4.1 Rings under internal pressure.......................... 165 4.4.2 Cylindrical pressure vessels........................... 166 4.4.3 Spherical pressure vessels............................. 167 4.4.4 Problems........................................... 168 4.5 Saint-Venant’s principle..................................... 169 Part II Beams and thin-wall structures 5 Euler-Bernoulli beam theory.................................... 173 5.1 The Euler-Bernoulli assumptions............................. 174 5.2 Implications of the Euler-Bernoulli assumptions................. 175 5.3 Stress resultants............................................ 177 5.4 Beams subjected to axial loads............................... 178 5.4.1 Kinematic description................................ 179 5.4.2 Sectional constitutive law............................. 179 5.4.3 Equilibrium equations................................ 180 5.4.4 Governing equations................................. 181 5.4.5 The sectional axial stiffness............................ 182 5.4.6 The axial stress distribution............................ 182 5.4.7 Problems........................................... 185 5.5 Beams subjected to transverse loads........................... 186 5.5.1 Kinematic description................................ 186 5.5.2 Sectional constitutive law............................. 187 5.5.3 Equilibrium equations................................ 188 5.5.4 Governing equations................................. 189 5.5.5 The sectional bending stiffness......................... 191 5.5.6 The axial stress distribution............................ 193 5.5.7 Rational design of beams under bending................. 194 5.5.8 Problems........................................... 211 5.6 Beams subjected to combined axial and transverse loads.......... 217 5.6.1 Kinematic description................................ 217 5.6.2 Sectional constitutive law............................. 217 XIV Contents 5.6.3 Equilibrium equations................................ 218 5.6.4 Governing equations................................. 219 6 Three-dimensional beam theory................................. 223 6.1 Kinematic description....................................... 224 6.2 Sectional constitutive law.................................... 225 6.3 Sectional equilibrium equations............................... 227 6.4 Governing equations........................................ 229 6.5 Decoupling the three-dimensional problem..................... 230 6.5.1 Definition of the principal axes of bending............... 231 6.5.2 Decoupled governing equations........................ 232 6.6 The principal centroidal axes of bending....................... 233 6.6.1 The bending stiffness ellipse........................... 235 6.7 The neutral axis............................................ 236 6.8 Evaluation of sectional stiffnesses............................. 240 6.8.1 The parallel axis theorem............................. 240 6.8.2 Thin-walled sections................................. 242 6.8.3 Triangular area equivalence method..................... 243 6.8.4 Useful results....................................... 244 6.8.5 Problems........................................... 247 6.9 Summary of three-dimensional beam theory.................... 248 6.9.1 Discussion of the results.............................. 255 6.10 Problems................................................. 255 7 Torsion....................................................... 261 7.1 Torsion of circular cylinders.................................. 261 7.1.1 Kinematic description................................ 262 7.1.2 The stress field...................................... 264 7.1.3 Sectional constitutive law............................. 265 7.1.4 Equilibrium equations................................ 266 7.1.5 Governing equations................................. 266 7.1.6 The torsional stiffness................................ 267 7.1.7 Measuring the torsional stiffness....................... 267 7.1.8 The shear stress distribution........................... 268 7.1.9 Rational design of cylinders under torsion............... 270 7.1.10 Problems........................................... 271 7.2 Torsion combined with axial force and bending moments......... 271 7.2.1 Problems........................................... 274 7.3 Torsion of bars with arbitrary cross-sections.................... 275 7.3.1 Introduction......................................... 275 7.3.2 Saint-Venant’s solution............................... 276 7.3.3 Saint-Venant’s solution for a rectangular cross-section..... 284 7.3.4 Problems........................................... 289 7.4 Torsion of a thin rectangular cross-section...................... 290 7.5 Torsion of thin-walled open sections........................... 292  Contents XV 7.5.1 Problems........................................... 294 8 Thin-walled beams............................................. 297 8.1 Basic equations for thin-walled beams.......................... 297 8.1.1 The thin wall assumption.............................. 297 8.1.2 Stress flows......................................... 298 8.1.3 Stress resultants..................................... 299 8.1.4 Sign conventions..................................... 301 8.1.5 Local equilibrium equation............................ 302 8.2 Bending of thin-walled beams................................ 303 8.2.1 Problems........................................... 304 8.3 Shearing of thin-walled beams................................ 307 8.3.1 Shearing of open sections............................. 308 8.3.2 Evaluation of stiffness static moments................... 308 8.3.3 Shear flow distributions in open sections................. 309 8.3.4 Problems........................................... 317 8.3.5 Shear center for open sections.......................... 318 8.3.6 Problems........................................... 324 8.3.7 Shearing of closed sections............................ 325 8.3.8 Shearing of multi-cellular sections...................... 329 8.3.9 Problems........................................... 332 8.4 The shear center............................................ 334 8.4.1 Calculation of the shear center location.................. 334 8.4.2 Problems........................................... 341 8.5 Torsion of thin-walled beams................................. 343 8.5.1 Torsion of open sections.............................. 343 8.5.2 Torsion of closed section.............................. 343 8.5.3 Comparison of open and closed sections................. 345 8.5.4 Torsion of combined open and closed sections............ 346 8.5.5 Torsion of multi-cellular sections....................... 347 8.5.6 Problems........................................... 351 8.6 Coupled bending-torsion problems............................ 354 8.6.1 Problems........................................... 361 8.7 Warping of thin-walled beams under torsion.................... 362 8.7.1 Kinematic description................................ 362 8.7.2 Stress-strain relations................................. 363 8.7.3 Warping of open sections.............................. 364 8.7.4 Problems........................................... 369 8.7.5 Warping of closed sections............................ 369 8.7.6 Warping of multi-cellular sections...................... 371 8.8 Equivalence of the shear and twist centers...................... 371 8.9 Non-uniform torsion........................................ 372 8.9.1 Problems........................................... 376 8.10 Structural idealization....................................... 377 8.10.1 Sheet-stringer approximation of a thin-walled section...... 378 XVI Contents 8.10.2 Axial stress in the stringers............................ 381 8.10.3 Shear flow in the sheet components..................... 381 8.10.4 Torsion of sheet-stringer sections....................... 384 8.10.5 Problems........................................... 390 Part III Energy and variational methods 9 Virtual work principles......................................... 395 9.1 Introduction............................................... 395 9.2 Equilibrium and work fundamentals........................... 396 9.2.1 Static equilibrium conditions.......................... 396 9.2.2 Concept of mechanical work........................... 399 9.3 Principle of virtual work..................................... 400 9.3.1 Principle of virtual work for a single particle............. 400 9.3.2 Kinematically admissible virtual displacements........... 405 9.3.3 Use of infinitesimal displacements as virtual displacements. 410 9.3.4 Principle of virtual work for a system of particles......... 412 9.4 Principle of virtual work applied to mechanical systems.......... 415 9.4.1 Generalized coordinates and forces..................... 420 9.4.2 Problems........................................... 425 9.5 Principle of virtual work applied to truss structures.............. 428 9.5.1 Truss structures...................................... 428 9.5.2 Solution using Newton’s law........................... 431 9.5.3 Solution using kinematically admissible virtual displacements432 9.5.4 Solution using arbitrary virtual displacements............ 433 9.6 Principle of complementary virtual work....................... 437 9.6.1 Compatibility equations for a planar truss................ 438 9.6.2 Principle of complementary virtual work for trusses....... 441 9.6.3 Complementary virtual work.......................... 445 9.6.4 Applications to trusses................................ 446 9.6.5 Problems........................................... 449 9.6.6 Unit load method for trusses........................... 449 9.6.7 Problems........................................... 455 9.7 Internal virtual work in beams and solids....................... 456 9.7.1 Beam bending....................................... 457 9.7.2 Beam twisting....................................... 458 9.7.3 Three-dimensional solid.............................. 459 9.7.4 Euler-Bernoulli beam................................. 461 9.7.5 Problems........................................... 462 9.7.6 Unit load method for beams........................... 462 9.7.7 Problems........................................... 470 9.8 Application of the unit load method to hyperstatic problems....... 472 9.8.1 Force method for trusses.............................. 473 9.8.2 Force method for beams.............................. 480  Contents XVII 9.8.3 Combined truss and beam problems..................... 485 9.8.4 Multiple redundancies................................ 487 9.8.5 Problems........................................... 489 10 Energy methods................................................ 493 10.1 Conservative forces......................................... 494 10.1.1 Potential for internal and external forces................. 497 10.1.2 Calculation of the potential functions.................... 498 10.2 Principle of minimum total potential energy.................... 500 10.2.1 Non-conservative external forces....................... 503 10.3 Strain energy in springs..................................... 504 10.3.1 Rectilinear springs................................... 504 10.3.2 Torsional springs.................................... 508 10.3.3 Bars............................................... 509 10.3.4 Problems........................................... 513 10.4 Strain energy in beams...................................... 514 10.4.1 Beam under axial loads............................... 514 10.4.2 Beam under transverse loads........................... 515 10.4.3 Beam under torsional loads............................ 516 10.4.4 Relationship with virtual work......................... 517 10.5 Strain energy in solids....................................... 519 10.5.1 Three-dimensional solid.............................. 519 10.5.2 Three-dimensional beams............................. 520 10.6 Applications to trusses and beams............................. 521 10.6.1 Applications to trusses................................ 521 10.6.2 Problems........................................... 525 10.6.3 Applications to beams................................ 527 10.7 Development of a finite element formulation for trusses.......... 529 10.7.1 General description of the problem..................... 530 10.7.2 Kinematics of an element............................. 531 10.7.3 Element elongation and force.......................... 533 10.7.4 Element strain energy and stiffness matrix............... 533 10.7.5 Element external potential and load array................ 535 10.7.6 Assembly procedure.................................. 535 10.7.7 Alternative description of the assembly procedure......... 538 10.7.8 Derivation of the governing equations................... 539 10.7.9 Solution procedure................................... 540 10.7.10Solution procedure using partitioning................... 542 10.7.11Post-processing...................................... 544 10.7.12Problems........................................... 547 10.8 Principle of minimum complementary energy................... 548 10.8.1 The potential of the prescribed displacements............. 548 10.8.2 Constitutive laws for elastic materials................... 550 10.8.3 The principle of minimum complementary energy......... 551 10.8.4 The principle of least work............................ 553 XVIII Contents 10.8.5 Problems........................................... 559 10.9 Energy theorems........................................... 560 10.9.1 Clapeyron’s theorem................................. 561 10.9.2 Castigliano’s first theorem............................. 563 10.9.3 Crotti-Engesser theorem.............................. 564 10.9.4 Castigliano’s second theorem.......................... 565 10.9.5 Applications of energy theorems....................... 565 10.9.6 The dummy load method.............................. 570 10.9.7 Unit load method revisited............................ 572 10.9.8 Problems........................................... 576 10.10Reciprocity theorems....................................... 577 10.10.1Betti’s theorem...................................... 577 10.10.2Maxwell’s theorem................................... 579 10.10.3Problems........................................... 581 11 Variational and approximate solutions............................ 583 11.1 Approach................................................. 583 11.2 Rayleigh-Ritz method for beam bending....................... 585 11.2.1 Statement of the problem.............................. 585 11.2.2 Description of the Rayleigh-Ritz method................ 586 11.2.3 Discussion of the Rayleigh-Ritz method................. 587 11.2.4 Problems........................................... 602 11.3 The strong and weak statements of equilibrium.................. 603 11.3.1 The weak form for beams under axial loads.............. 604 11.3.2 Approximate solutions for beams under axial loads........ 608 11.3.3 Problems........................................... 616 11.3.4 The weak form for beams under transverse loads.......... 617 11.3.5 Approximate solutions for beams under transverse loads... 621 11.3.6 Problems........................................... 624 11.3.7 Equivalence with energy principles..................... 625 11.3.8 The principle of minimum total potential energy.......... 627 11.3.9 Treatment of the boundary conditions................... 629 11.3.10Summary........................................... 638 11.4 Formal procedures for the derivation of approximate solutions..... 638 11.4.1 Basic approximations................................. 639 11.4.2 Principle of virtual work.............................. 640 11.4.3 The principle of minimum total potential energy.......... 643 11.4.4 Problems........................................... 650 11.5 A finite element formulation for beams........................ 654 11.5.1 General description of the problem..................... 655 11.5.2 Kinematics of an element............................. 657 11.5.3 Element displacement field............................ 658 11.5.4 Element curvature field............................... 660 11.5.5 Element strain energy and stiffness matrix............... 660 11.5.6 Element external potential and load array................ 661  Contents XIX 11.5.7 Assembly procedure.................................. 662 11.5.8 Alternative description of the assembly procedure......... 664 11.5.9 Derivation of the governing equations................... 666 11.5.10Solution procedure................................... 666 11.5.11Summary........................................... 670 11.5.12Problems........................................... 671 12 Variational and energy principles................................ 673 12.1 Mathematical preliminaries.................................. 673 12.1.1 Stationary point of a function.......................... 674 12.1.2 Lagrange multiplier method........................... 675 12.1.3 Stationary point of a definite integral.................... 677 12.2 Variational and energy principles............................. 679 12.2.1 Review of the equations of linear elasticity............... 680 12.2.2 The principle of virtual work.......................... 682 12.2.3 The principle of complementary virtual work............. 683 12.2.4 Strain and complementary strain energy density functions.. 685 12.2.5 The principle of minimum total potential energy.......... 686 12.2.6 The principle of minimum complementary energy......... 688 12.2.7 Energy theorems..................................... 690 12.2.8 Hu-Washizu’s principle............................... 690 12.2.9 Hellinger-Reissner’s principle.......................... 694 12.3 Applications of variational and energy principles................ 695 12.3.1 The shear lag problem................................ 697 12.3.2 The Saint-Venant torsion problem...................... 701 12.3.3 The Saint-Venant torsion problem using the Prandtl stress function............................................ 703 12.3.4 The non-uniform torsion problem....................... 707 12.3.5 The non-uniform torsion problem (closed sections)........ 709 12.3.6 The non-uniform torsion problem (open sections)......... 712 12.3.7 Problems........................................... 713 Part IV Advanced topics 13 Introduction to plasticity and thermal stresses..................... 721 13.1 Yielding under combined loading............................. 721 13.1.1 Introduction to yield criteria........................... 722 13.1.2 Tresca’s criterion.................................... 724 13.1.3 Von Mises’ criterion.................................. 724 13.1.4 Problems........................................... 725 13.2 Applications of yield criteria to structural problems.............. 725 13.2.1 Problems........................................... 731 13.2.2 Plastic bending...................................... 732 13.2.3 Problems........................................... 737 XX Contents 13.2.4 Plastic torsion....................................... 737 13.3 Thermal stresses in structures................................ 741 13.3.1 The direct method.................................... 742 13.3.2 Problems........................................... 746 13.3.3 The constraint method................................ 746 13.4 Application to bars, trusses and beams......................... 748 13.4.1 Applications to bars and trusses........................ 748 13.4.2 Problems........................................... 753 13.4.3 Application to beams................................. 753 13.4.4 Problems........................................... 760 14 Buckling of beams.............................................. 763 14.1 Rigid bar with root torsional spring............................ 763 14.1.1 Analysis of a perfect system........................... 763 14.1.2 Analysis of an imperfect system........................ 765 14.2 Buckling of beams.......................................... 767 14.2.1 Equilibrium equations................................ 767 14.2.2 Buckling of a simply-supported beam (equilibrium approach) 769 14.2.3 Buckling of a simply-supported beam (imperfection approach)........................................... 771 14.2.4 Work done by the axial force.......................... 774 14.2.5 Buckling of a simply-supported beam (energy approach)... 776 14.2.6 Applications to beam buckling......................... 780 14.2.7 Buckling of beams with various end conditions........... 784 14.2.8 Problems........................................... 784 14.3 Buckling of sandwich beams................................. 789 15 Shearing deformations in beams................................. 793 15.1 Introduction............................................... 793 15.1.1 A simplified approach................................ 794 15.1.2 An equilibrium approach.............................. 795 15.1.3 Problems........................................... 801 15.2 Shear deformable beams: an energy approach................... 801 15.2.1 Shearing effects on beam deflections.................... 805 15.2.2 Shearing effects on buckling........................... 812 15.2.3 Problems........................................... 814 16 Kirchhoff plate theory.......................................... 819 16.1 Governing equations of Kirchhoff plate theory.................. 820 16.1.1 Kirchhoff assumptions................................ 820 16.1.2 Stress resultants..................................... 824 16.1.3 Equilibrium equations................................ 826 16.1.4 Constitutive laws.................................... 828 16.1.5 Stresses due to in-plane forces and bending moments...... 830 16.1.6 Summary of Kirchhoff plate theory..................... 830  Contents XXI 16.2 The bending problem....................................... 831 16.2.1 Typical boundary conditions........................... 832 16.2.2 Simple plate bending solutions......................... 836 16.2.3 Problems........................................... 839 16.3 Anisotropic plates.......................................... 840 16.3.1 Laminated composite plates........................... 840 16.3.2 Constitutive laws for laminated composite plates.......... 841 16.3.3 The in-plane stiffness matrix........................... 842 16.3.4 Problems........................................... 844 16.3.5 The bending stiffness matrix........................... 845 16.3.6 The coupling stiffness matrix.......................... 847 16.3.7 Problems........................................... 849 16.3.8 Directionally stiffened plates........................... 849 16.3.9 Governing equations for anisotropic plates............... 851 16.4 Solution techniques for rectangular plates...................... 853 16.4.1 Navier’s solution for simply supported plates............. 853 16.4.2 Problems........................................... 858 16.4.3 Lévy’s solution...................................... 859 16.4.4 Problems........................................... 864 16.5 Circular plates............................................. 865 16.5.1 Governing equations for the bending of circular plates..... 865 16.5.2 Circular plates subjected to loading presenting circular symmetry........................................... 868 16.5.3 Problems........................................... 871 16.5.4 Circular plates subjected to arbitrary loading............. 872 16.5.5 Problems........................................... 874 16.6 Energy formulation of Kirchhoff plate theory................... 875 16.6.1 The virtual work done by the internal stresses............ 876 16.6.2 The virtual work done by the applied loads............... 880 16.6.3 The principle of virtual work for Kirchhoff plates......... 881 16.6.4 The principle of minimum total potential energy for Kirchhoff plates..................................... 882 16.6.5 Approximate solutions for Kirchhoff plates.............. 884 16.6.6 Solutions based on partial approximation................ 891 16.6.7 Problems........................................... 895 16.7 Buckling of plates.......................................... 896 16.7.1 Equilibrium formulation.............................. 897 16.7.2 Energy formulation.................................. 904 16.7.3 Problems........................................... 913 A Appendix: mathematical tools................................... 915 A.1 Notation.................................................. 915 A.2 Vectors, arrays, matrices and linear algebra..................... 916 A.2.1 Vectors, arrays and matrices........................... 916 A.2.2 Vector, array and matrix operations..................... 918 1 Basic equations of linear elasticity Structural analysis is concerned with the evaluation of deformations and stresses aris- ing within a solid object under the action of applied loads. If time is not explicitly considered as an independent variable, the analysis is said to be static; otherwise it is referred to as structural dynamic analysis, or simply structural dynamics. Under the assumption of small deformations and linearly elastic material behavior, three- dimensional formulations result in a set of fifteen linear first order partial differential equations involving the displacement field (three components), the stress field (six components) and the strain field (six components). This chapter presents the deriva- tion of these governing equations. In many applications, this complex problem can be reduced to simpler, two-dimensional formulations called plane stress and plane strain problems. For most situations, it is not possible to develop analytical solutions of these equations. Consequently, structural analysis is concerned with the analysis of struc- tural components, such as bars, beams, plates, or shells, which will be addressed in subsequent chapters. In each case, assumptions are made about the behavior of these structural components, which considerably simplify the analysis process. For instance, given a suitable set of assumptions, the analysis of bar and beam problems reduces to the solution of one-dimensional equations for which analytical solutions are easily obtained. 1.1 The concept of stress 1.1.1 The state of stress at a point The state of stress in a solid body is a measure of the intensity of forces acting within the solid. It can be visualized by cutting the solid by a plane normal to unit vector, n̄, to create two free bodies which reveal the forces acting on the exposed surfaces. From basic statics, it is well-known that the distribution of forces and moments that will appear on the surface of the cut can be represented by an equipollent force, F , acting at a point of the surface and a couple, M. Newton’s 3rd law also requires 4 1 Basic equations of linear elasticity a force and couple of equal magnitudes and opposite directions to act on the two surfaces created by the cut through the solid, as depicted in fig. 1.1. (See appendix A for a description of the vector, array and matrix notations used in this text.) Plane of M the cut F F Free Body Solid Body n diagram M Applied loads Small surface An on the cut F P Fn n Mn M Fig. 1.1. A solid body cut by a plane to isolate a free body. Consider now a small surface of area An located at point P on the surface gen- erated by the cut in the solid. The forces and moments acting on this surface are equipollent to a force, F n , and couple, M n ; note that these resultants are, in gen- eral, different, in both magnitude and orientation, from the corresponding resultants acting on the entire surface of the cut, as shown in fig. 1.1. Let the small surface be smaller and smaller until it becomes an element of infinitesimal area dAn → 0. As the surface shrinks to a differential size, the force and couple acting on the element keep decreasing in magnitude and changing in orientation whereas the normal to the surface remains the unit vector n̄ of constant direction in space. This limiting process gives rise to the concept of stress vector, which is defined as µ ¶ Fn τ n = lim. (1.1) dAn →0 dAn The existence of the stress vector, i.e., the existence of the limit in eq. (1.1), is a fundamental assumption of continuum mechanics. In this limiting process, it is as- sumed that the couple, M n , becomes smaller and smaller and, in the limit, M n → 0 as dAn → 0; this is also an assumption of continuum mechanics which seems to be reasonable because in the limiting process, both forces and moment arms become increasingly small. Forces decrease because the area they act on decreases and mo- ment arms decrease because the dimensions of the surface decrease. At the limit, the couple is the product of a differential element of force by a differential element of moment arm, giving rise to a negligible, second order differential quantity. In conclusion, whereas an equipollent couple might act on the entire surface of the cut, the equipollent couple is assumed to vanish on a differential element of area of the same cut. The total force acting on a differential element of area, dAn , is  1.1 The concept of stress 5 F n = dAn τ n. (1.2) Clearly, the stress vector has units of force per unit area. In the SI system, this is measured in Newton per square meters, or Pascals (Pa). During the limiting process de- i3 Cut normal t23 to axis i scribed in the previous paragraph, 2 s3 t2 the surface orientation, as defined t3 by the normal to the surface, is kept P t32 constant in space. Had a different dA3 s2 normal been selected, a different P dA2 t31 t21 stress vector would have been ob- tained. i2 t13 To illustrate this point, con- Cut normal sider a solid body and a coordinate to axis i3 system, I, consisting of three mu- dA1 t1 tually orthogonal unit vectors, I = i1 Cut normal s1 P t12 (ı̄1 , ı̄2 , ı̄3 ), as shown in fig. 1.2. to axis i1 First, the solid is cut at point P by a plane normal to axis ı̄1 ; on the sur- Fig. 1.2. A rigid body cut at point P by three planes face of the cut, at point P, a differ- orthogonal to the Cartesian axes. ential element of surface with an area dA1 is defined and let τ 1 be the stress vector acting on this face. Next, the solid is cut at the same point by a plane normal to axis ı̄2 ; at point P, let τ 2 be the stress vector acting on the differ- ential element of surface with an area dA2. Finally, the process is repeated a third time for a plane normal to axis ı̄3 ; at point P, the stress vector τ 3 is acting on the differential element of surface with an area dA3. Clearly, three stress vectors, τ 1 , τ 2 , and τ 3 are acting at the same point P, but on three mutually orthogonal faces normal to axes ı̄1 , ı̄2 , and ı̄3 , respectively. Because these three stress vectors are acting on three faces with different orientations, there is no reason to believe that those stress vectors should be identical. To further understand the state of stress at point P, the components of each stress vectors acting on the three faces are defined τ 1 = σ1 ı̄1 + τ12 ı̄2 + τ13 ı̄3 , (1.3a) τ 2 = τ21 ı̄1 + σ2 ı̄2 + τ23 ı̄3 , (1.3b) τ 3 = τ31 ı̄1 + τ32 ı̄2 + σ3 ı̄3. (1.3c) The stress components σ1 , σ2 , and σ3 are called direct, or normal stresses; they act on faces normal to axes ı̄1 , ı̄2 , and ı̄3 , respectively, in directions along axes ı̄1 , ı̄2 , and ı̄3 , respectively. The stress components τ12 and τ13 are called shearing or shear stresses; both act on the face normal to axis ı̄1 , in directions of axes ı̄2 and ı̄3 , respectively. Similarly, stress components τ21 and τ23 both act on the face normal to axis ı̄2 , in directions of axes ı̄1 and ı̄3 , respectively. Finally, stress components τ31 and τ32 both act on the face normal to axis ı̄3 , in directions along axes ı̄1 and ı̄2 , 6 1 Basic equations of linear elasticity respectively. The various stress components appearing in eq. (1.3) are referred to as the engineering stress components. The units of stress components are identical to those of the stress vector, force per unit area, or Pascal. The stress components represented i3 in fig. 1.2 are all defined as positive. s3 Furthermore, the three faces depicted in this figure are positive faces. A face t32 s1 is positive when the outward normal to t21t31 t12 t23 the face, i.e., the normal pointing away t13 s2 t13 s2 from the body, is in the same direction as the axis to which the face is normal; t23 t21 i2 t12 a face is negative when its outward nor- t31 mal is pointing in the direction opposite s1 t32 to the axis to which the face is normal. i1 s3 The positive directions of stress com- ponents acting on negative faces are the Fig. 1.3. Sign conventions for the stress com- opposite of those for stress components ponents acting on a differential volume ele- acting on positive faces. This sign con- ment. All stress components shown here are vention is illustrated in fig. 1.3, which positive. shows positive stress components acting on the six faces of a cube of differential size. Positive stress components are shown in solid lines on the three positive faces of the cube; positive stress components are shown in dotted lines on the three nega- tive (hidden) faces of the cube. Taken together, the direct stress components σ1 , σ2 , and σ3 and the shear stress components, τ12 and τ13 , τ21 and τ23 , and τ31 and τ32 , fully characterize the state of stress at point P. It will be shown in a later section that if the stress components acting on three orthogonal faces are known, it is possible to compute the stress com- ponents acting at the same point, on a face of arbitrary orientation. This discussion underlines the fact that the state of stress at a point is a complex concept: its complete definition requires the knowledge of nine stress components acting on three mutually orthogonal faces. This should be contrasted with the concept of force. A force is vector quantity that is characterized by its magnitude and orientation. Alternatively, a force can be defined by the three components of the force vector in a given coordinate system. The definition of a force thus requires three quantities, whereas the definition of the stress state requires nine quantities. A force is a vector, which is referred to as a first order tensor, whereas a state of stress is a second order tensor. Several quantities commonly used in solid mechanics are also second order tensors: the strain tensor, the bending stiffnesses of a beam, and the mass moments of inertia of a solid object. The first two of these quantities will be introduced in later sections and chapters. Much like the case for vectors, all second order tensors will be shown to possess certain common characteristics.  1.1 The concept of stress 7 1.1.2 Volume equilibrium equations In general, the state of stress varies throughout a solid body, and hence, stresses acting on two parallel faces located a small distance apart are not equal. Consider, for instance, the two opposite faces of a differential volume element that are normal to axis ı̄2 , as shown in fig. 1.4. The axial stress component on the negative face at coordinate x2 is σ2 , but the stress components on the positive face at coordinate x2 + dx2 will be slightly different and written as σ2 (x2 + dx2 ). If σ2 (x2 ) is an analytic function, it is then possible to express σ2 (x2 + dx2 ) in terms of σ2 (x2 ) using a Taylor series expansion to find ¯ ∂σ2 ¯¯ σ2 (x2 + dx2 ) = σ2 (x2 ) + dx2 +... higher order terms in dx2. ∂x2 ¯x2 This expansion is a fundamental step in the derivation of the differential equa- tions governing the behavior of a continuum. The stress component on the positive face at coordinate x2 + dx2 can be written as σ2 (x2 + dx2 ) ≈ σ2 + (∂σ2 /∂x2 )dx2. The same Taylor series expansion technique can be applied to all other direct and shear stress components. s3 + (ds3/dx3) dx3 t32 + (dt32/dx3) dx3 t31 + (dt31/dx3) dx3 t23 + (dt23/dx2) dx2 i3 t21 s2 + (ds2/dx2) dx2 s2 t23 t21 + (dt21/dx2) dx2 t31 i2 t32 i1 s3 Fig. 1.4. Stress components acting on a differential element of volume. For clarity of the figure, the stress components acting on the faces normal to ı̄1 are not shown. Consider now the differential element of volume depicted in fig. 1.4. It is sub- jected to stress components acting on its six external faces and to body forces per unit volume, represented by a vector b acting at its centroid. These body forces could be gravity forces, inertial forces, or forces of an electric or magnetic origin; the com- ponents of this body force vector resolved in coordinate system I = (ı̄1 , ı̄2 , ı̄3 ) as b = b1 ı̄1 + b2 ı̄2 + b3 ı̄3. The units of the force vector are force per unit volume or Newton per cubic meter. Force equilibrium According to Newton’s law, static equilibrium requires the sum of all the forces acting on this differential element to vanish. Considering all the forces acting along 8 1 Basic equations of linear elasticity the direction of axis ı̄1 , the equilibrium condition is µ ¶ ∂σ1 − σ1 dx2 dx3 + σ1 + dx1 dx2 dx3 ∂x1 µ ¶ ∂τ21 − τ21 dx1 dx3 + τ21 + dx2 dx1 dx3 ∂x2 µ ¶ ∂τ31 − τ31 dx1 dx2 + τ31 + dx3 dx1 dx2 + b1 dx1 dx2 dx3 = 0. ∂x3 This equation states an equilibrium of forces, and therefore the stress components must be multiplied by the area of the surface on which they act to yield the corre- sponding force. Similarly, the component of the body force per unit volume of the body is multiplied by the volume of the differential element, dx1 dx2 dx3 , to give the body force acting on the element. After simplification, this equilibrium condition becomes · ¸ ∂σ1 ∂τ21 ∂τ31 + + + b1 dx1 dx2 dx3 = 0. ∂x1 ∂x2 ∂x3 This equation is satisfied when the expression in brackets vanishes, and this yields the equilibrium equation in the direction of axis ı̄1 ∂σ1 ∂τ21 ∂τ31 + + + b1 = 0. ∂x1 ∂x2 ∂x3 For the same reasons, forces along axes ı̄2 and ı̄3 must vanish as well, and a similar reasoning yields the following three equilibrium equations ∂σ1 ∂τ21 ∂τ31 + + + b1 = 0, (1.4a) ∂x1 ∂x2 ∂x3 ∂τ12 ∂σ2 ∂τ32 + + + b2 = 0, (1.4b) ∂x1 ∂x2 ∂x3 ∂τ13 ∂τ23 ∂σ3 + + + b3 = 0, (1.4c) ∂x1 ∂x2 ∂x3 which must be satisfied at all points inside the body. The equilibrium conditions implied by Newton’s law, eqs. (1.4), have been writ- ten by considering an differential element of the undeformed body. Of course, when forces are applied, the body deforms and so does every single differential element. Strictly speaking, equilibrium should be enforced on the deformed configuration of the body, rather than its undeformed configuration. Indeed, stresses are only present when external forces are applied and the body is deformed. When no forces are ap- plied, the body is undeformed, but stresses all vanish. Unfortunately, it is difficult to write equilibrium conditions on the deformed con- figuration of the body because this configuration is unknown; indeed, the goal of the theory of elasticity is to predict the deformation of elastic bodies under load. It is a basic assumption of the linear theory of elasticity developed here that the displacements of the body under the applied loads are very small, and hence, the  1.1 The concept of stress 9 difference between the deformed and undeformed configurations of the body is very small. Under this assumption, it is justified to impose equilibrium conditions to the undeformed configuration of the body, because it is nearly identical to its deformed configuration. Moment equilibrium To satisfy all equilibrium requirements, the sum of all the moments acting on the differential element of volume depicted in fig. 1.4 must also vanish. Consider first the moment equilibrium about axis ı̄1. The contributions of the direct stresses and of the body forces can be eliminated by choosing an axis passing through the center of the differential element. The resulting moment equilibrium equation is µ ¶ dx2 ∂τ23 dx2 τ23 dx1 dx3 + τ23 + dx2 dx1 dx3 2 ∂x2 2 µ ¶ dx3 ∂τ32 dx3 −τ32 dx1 dx2 − τ32 + dx3 dx1 dx2 2 ∂x3 2 · ¸ ∂τ23 dx2 ∂τ32 dx3 = τ23 − τ32 + − dx1 dx2 dx3 = 0. ∂x2 2 ∂x3 2 The bracketed expression must vanish and after neglecting higher order terms, this reduces to the following equilibrium condition τ23 − τ32 = 0. Enforcing the vanishing of the sum of t t the moments about axes ı̄2 and ı̄3 leads to similar equations, τ23 = τ32 , τ13 = τ31 , τ12 = τ21. (1.5) 90o 90o t t The implication of these equalities is sum- marized by the principle of reciprocity Fig. 1.5. Reciprocity of the shearing of shear stresses, which is illustrated in stresses acting on two orthogonal faces. fig. 1.5. Principle 1 (Principle of reciprocity of shear stresses) Shear stresses acting in the direction normal to the common edge of two orthogonal faces must be equal in magnitude and be simultaneously oriented toward or away from the common edge. Another implication of the reciprocity of the shear stresses is that of the nine components of stresses, six only are independent. It is common practice to arrange the stress tensor components in a 3×3 matrix format   σ1 τ12 τ13  τ12 σ2 τ23 . (1.6) τ13 τ23 σ3 The principle of reciprocity implies the symmetry of the stress tensor. 10 1 Basic equations of linear elasticity 1.1.3 Surface equilibrium equations At the outer surface of the body, the stresses acting inside the body must be in equi- librium with the externally applied surface tractions. Surface tractions are repre- sented by a stress vector, t, that can be resolved in reference frame I = (ı̄1 , ı̄2 , ı̄3 ) as t = t1 ı̄1 +t2 ı̄2 +t3 ı̄3. Figure 1.6 shows a free body in the form of a differential tetra- hedron bounded by three negative faces cut through the body in directions normal to axes ı̄1 , ı̄2 , and ı̄3 , and by a fourth face, ABC, of area dAn , which is a differential element of the outer surface of the body. The unit normal to this element of area is denoted n̄, and its components in coordinate system I are n̄ = n1 ı̄1 + n2 ı̄2 + n3 ı̄3. Note that n1 , n2 , and n3 are the cosines of the angle between n̄ and ı̄1 , n̄ and ı̄2 , and n̄ and ı̄3 , respectively, also called the direction cosines of n̄: n1 = n̄·ı̄1 = cos(n̄, ı̄1 ), n2 = n̄ · ı̄2 = cos(n̄, ı̄2 ), and n3 = n̄ · ı̄3 = cos(n̄, ı̄3 ). Fig. 1.6. A tetrahedron with one face along the outer surface of the body. Equilibrium of forces acting along axis ı̄1 implies dx1 dx2 dx3 t1 dAn = σ1 dA1 + τ21 dA2 + τ31 dA3 − b1 , (1.7) 6 where dA1 , dA2 , and dA3 are the areas of triangles OBC, OAC and OAB, respec- tively, and the last term represents the body force times the volume of the tetrahe- dron. The areas of the three faces normal to the axes are found by projecting face ABC onto planes normal to the axes using the direction cosines to find dA1 = n1 dAn , dA2 = n2 dAn , and dA3 = n3 dAn. (1.8) Dividing eq. (1.7) by dAn then yields the first component of the surface traction vector t1 = σ1 n1 + τ21 n2 + τ31 n3 , where the body force term vanishes because it is a higher order differential term. The same procedure can be followed to express equilibrium conditions along the  1.2 Analysis of the state of stress at a point 11 directions of axes ı̄2 and ı̄3. The three components of the surface traction vector then become t1 = σ1 n1 + τ12 n2 + τ13 n3 , (1.9a) t2 = τ12 n1 + σ2 n2 + τ23 n3 , (1.9b) t3 = τ31 n1 + τ32 n2 + σ3 n3. (1.9c) A body is said to be in equilibrium if eqs. (1.4) are satisfied at all points inside the body, and eqs. (1.9) are satisfied at all points of its external surface. 1.2 Analysis of the state of stress at a point The state of stress at a point is characterized in the previous section by the normal and shear stress components acting on the faces of a differential element of volume cut from the solid. The faces of this cube are cut normal to the axes of a Cartesian ref- erence frame I = (ı̄1 , ı̄2 , ı̄3 ), and the stress vector acting on these faces are resolved along the same axes. Clearly, another face at an arbitrary orientation with respect to these axes can be selected. In section 1.2.1, it will be shown that the stresses acting on this face can be related to the stresses acting on the faces normal to axes ı̄1 , ı̄2 , and ı̄3. This important result implies that once the stress components are known on three mutually orthogonal faces at a point, they are known on any face passing through that point. Hence, the state of stress at a point is fully defined once the stress components acting on three mutually orthogonal faces at a point are known. 1.2.1 Stress components acting on an arbitrary face To establish relationships between stresses, it is necessary to consider force or mo- ment equilibrium due to these stresses, and this must be done with reference to a specific free body diagram. Figure 1.7 shows a specific free body constructed from a tetrahedron defined by three faces cut normal to axes ı̄1 , ı̄2 , and ı̄3 , and a fourth face normal to unit vector n̄ = n1 ı̄1 + n2 ı̄1 + n3 ı̄3 , of arbitrary orientation. This tetra- hedron is known as Cauchy’s tetrahedron. The components, n1 , n2 , and n3 , of this unit vector are the direction cosines of unit vector n̄, i.e., the cosines of the angles between n̄ and ı̄1 , n̄ and ı̄2 , and n̄ and ı̄3 , respectively. Figure 1.7 shows the stress components acting on faces COB, AOC and AOB, of area dA1 , dA2 , and dA3 , respectively; the stress vector, τn , acts on face ABC of area dAn. The body force vector, b, is also acting on this tetrahedron. Equilibrium of forces acting on tetrahedron OABC requires τ 1 dA1 + τ 2 dA2 + τ 3 dA3 = τ n dAn + b dV, where τ 1 , τ 2 and τ 3 are the stress vectors acting on the faces normal to axes ı̄1 , ı̄2 , and ı̄3 , respectively, and dV is the volume of the tetrahedron. Dividing this equilibrium equation by dAn and using eq. (1.8) gives the stress vector acting of the inclined face as 12 1 Basic equations of linear elasticity n i3 i3 C tn sn n s1 s tns t21 t12 t13 i2 i2 s2 dA2 dA1 t23 O t31 B t32 dA3 s3 i1 A dAn i1 Stresses on three Stress on the face normal faces normal to n Fig. 1.7. Differential tetrahedron element with one face, ABC, normal to unit vector n and the other three faces normal to axes ı̄1 , ı̄2 , and ı̄3 , respectively. τ n = τ 1 n1 + τ 2 n2 + τ 3 n3 − b dV/dAn The body force term is multiplied by a higher order term, dV/dAn , which can be neglected in the equilibrium condition. Expanding the three stress vectors in terms on the stress components then yields τ n = (σ1 ı̄1 +τ12 ı̄2 +τ13 ı̄3 ) n1 +(τ21 ı̄1 +σ2 ı̄2 +τ23 ı̄3 ) n2 +(τ31 ı̄1 +τ32 ı̄2 +σ3 ı̄3 ) n3. (1.10) To determine the direct stress, σn , acting on face ABC, it is necessary to project this vector equation in the direction of unit vector n̄. This can be achieved by taking the dot product of the stress vector by unit vector n̄ to find n̄ · τ n = n̄· [(σ1 ı̄1 + τ12 ı̄2 + τ13 ı̄3 ) n1 + (τ21 ı̄1 + σ2 ı̄2 + τ23 ı̄3 ) n2 + (τ31 ı̄1 + τ32 ı̄2 + σ3 ı̄3 ) n3 ]. Because n̄ = n1 ı̄1 + n2 ı̄1 + n3 ı̄3 , this yields σn = (σ1 n1 + τ12 n2 + τ13 n3 ) n1 + (τ21 n1 + σ2 n2 + τ23 n3 ) n2 + (τ31 n1 + τ32 n2 + σ3 n3 ) n3 , and finally, after minor a rearrangement of terms, σn = σ1 n21 + σ2 n22 + σ3 n23 + 2τ23 n2 n3 + 2τ13 n1 n3 + 2τ12 n1 n2. (1.11) The stress components acting in the plane of face ABC can be evaluated in a similar manner by projecting eq. (1.10) along a unit vector in the plane of face ABC. Consider a unit vector, s̄ = s1 ı̄1 + s2 ı̄1 + s3 ı̄3 , normal to n̄, i.e., such that n̄ · s̄ = 0. The shear stress component acting on face ABC in the direction of unit vector s̄ is denoted τns and is obtained by projecting eq. (1.10) along vector s̄ to find τns =(σ1 s1 + τ12 s2 + τ13 s3 ) n1 + (τ21 s1 + σ2 s2 + τ23 s3 ) n2 + (τ31 s1 + τ32 s2 + σ3 s3 ) n3 ,  1.2 Analysis of the state of stress at a point 13 and finally, after minor a rearrangement of terms, τns = σ1 n1 s1 + σ2 n2 s2 + σ3 n3 s3 + τ12 (n2 s1 + n1 s2 ) (1.12) + τ13 (n1 s3 + n3 s1 ) + τ23 (n2 s3 + n3 s2 ). Equations. (1.11) and (1.12) express an important result of continuum mechanics. They imply that once the stress components acting on three mutually orthogonal faces are known, the stress components on a face of arbitrary orientation can be readily computed. To evaluate the direct stress component acting on an arbitrary face, all that is required are the direction cosines of the normal to the face. Evaluation the shear stress component acting on the same face requires, in addition, the direction cosines of the direction of the shear stress component in that face. Consider the following question: how much information is required to fully de- fine the state of stress at point P of a solid? Clearly, the body can be cut at this point by a plane of arbitrary orientation. The stress vector acting on this face gives information about the state of stress at point P. The stress vector acting on a face with another orientation would give additional information about the state of stress at the same point. If additional faces are considered, each new stress vector pro- vides additional information. This reasoning would seem to imply that the complete knowledge of the state of stress at a point requires an infinite amount of information, specifically, the stress vectors acting on all the possible faces passing through point P. Equations. (1.11) and (1.12), however, demonstrate the fallacy of this reasoning: once the stress vectors acting on three mutually orthogonal faces are known, the stress vector acting on any other face can be readily predicted. In conclusion, com- plete definition of the state of stress at a point only requires knowledge of the stress vectors, or equivalently of the stress tensor components, acting on three mutually orthogonal faces. 1.2.2 Principal stresses As discussed in the previous section, eqs. (1.11) and (1.12) enable the computa- tion of the stress components acting on a face of arbitrary orientation, based on the knowledge of the stress components acting on three mutually orthogonal faces. As illustrated in fig. 1.7, the stress vector acting on a face of arbitrary orientation has, in general, a component σn n̄, acting in the direction normal to the face, and a compo- nent τns s̄, acting within the plane of the face. This discussion raises the following question: is there a face orientation for which the stress vector is exactly normal to the face? In other words, does a particular orientation, n̄, exist for which the stress vector acting on this face consists solely of τ n = σp n̄, where σp is the yet unknown magnitude of this direct stress component? Introducing this expression into eq. (1.10) results in σp n̄ = (σ1 ı̄1 +τ12 ı̄2 +τ13 ı̄3 ) n1 +(τ21 ı̄1 +σ2 ı̄2 +τ23 ı̄3 ) n2 +(τ31 ı̄1 +τ32 ı̄2 +σ3 ı̄3 ) n3. This equation alone does not allow the determination of both σp and of unit vector n̄. Projecting this vector relationship along axes ı̄1 , ı̄2 , and ı̄3 leads to the following three scalar equations 14 1 Basic equations of linear elasticity (σ1 − σp ) n1 + τ12 n2 + τ13 n3 = 0, τ12 n1 + (σ2 − σp ) n2 + τ23 n3 = 0, τ13 n1 + τ23 n2 + (σ3 − σp ) n3 = 0, respectively. The unknowns of the problem are the direction cosines, n1 , n2 , and n3 that define the orientation of the face on which shear stresses vanish, and the magnitude, σp , of the direct stress component acting on this face. These equations are recast as a homogeneous system of linear equations for the unknown direction cosines    σ1 − σp τ12 τ13 n1   τ12 σ2 − σp τ23  n2 = 0. (1.13)   τ13 τ23 σ3 − σp n3 Since this is a homogeneous system of equations, the trivial solution, n1 = n2 = n3 = 0, is, in general, the solution of this system. When the determinant of the system vanishes, however, non-trivial solutions will exist. The vanishing of the de- terminant of the system leads to the cubic equation for the magnitude of the direct stress σp3 − I1 σp2 + I2 σp − I3 = 0, (1.14) where the quantities I1 , I2 , and I3 are defined as I1 = σ1 + σ2 + σ3 , (1.15a) 2 2 2 I2 = σ1 σ2 + σ2 σ3 + σ3 σ1 − τ12 − τ13 − τ23 , (1.15b) 2 2 2 I3 = σ1 σ2 σ3 − σ1 τ23 − σ2 τ13 − σ3 τ12 + 2τ12 τ13 τ23 , (1.15c) are called the three stress invariants. The solutions of eq. (1.14) are called the principal stresses. Since this is a cubic equation, three solutions exist, denoted σp1 , σp2 , and σp3. For each of these three solutions, the matrix of the system of equations defined by eq. (1.13) has a zero de- terminant, and a non-trivial solution exists for the directions cosines that now define the direction of a face on which the shear stresses vanish. This direction is called a principal stress direction. Because the equations to be solved are homogeneous, their solution will include an arbitrary constant, which can be determined by enforcing the normality condition for unit vector n̄, n21 + n22 + n23 = 1. This solution process can be repeated for each of the three principal stresses. This will result in three different principal stress directions. It can be shown that these three directions are mutually orthogonal. 1.2.3 Rotation of stresses In the previous sections, free body diagrams are formed with faces cut in directions normal to axes of the orthonormal basis I = (ı̄1 , ı̄2 , ı̄3 ), and the stress vectors are resolved into stress components along the same directions. The orientation of this basis is entirely arbitrary: basis I ∗ = (ı̄∗1 , ı̄∗2 , ı̄∗3 ) could also have been selected, and  1.2 Analysis of the state of stress at a point 15 an analysis identical to that of the previous sections would have led to the definition of normal stresses σ1∗ , σ2∗ , σ3∗ , and shear stresses τ23 ∗ ∗ , τ13 ∗ , τ12. A typical equilibrium equation at a point of the body would be written as ∂σ1∗ ∂τ ∗ ∂τ ∗ ∗ + 21∗ + 31 + b∗1 = 0, (1.16) ∂x1 ∂x2 ∂x∗3 where the notation (.)∗ is used to indicate the components of the corresponding quan- tity resolved in basis I ∗. A typical surface traction would be defined as t∗1 = n∗1 σ1∗ + n∗2 τ21 ∗ + n∗3 τ31 ∗. (1.17) Although expressed in different reference frames, eqs. (1.4) and (1.16), or (1.9) and (1.17) express the same equilibrium conditions for the body. Two orthonormal bases, I and I ∗ , are involved in this problem. The orientation of basis I ∗ relative to basis I is discussed in section A.3.1 and leads to the definition of the matrix of direction cosines, or rotation matrix, R, given by eq. (A.36). Consider the stress component σ1∗ : it represents the magnitude of the direct stress component acting on the face normal to axis ı̄∗1. Equation (1.11) can now be used to express this stress component in terms of the stress components resolved in axis system I to find σ1∗ = σ1 `21 + σ2 `22 + σ3 `23 + 2τ23 `2 `3 + 2τ13 `1 `3 + 2τ12 `1 `2 , (1.18) where `1 , `2 , and `3 , are the direction cosines of unit vector ı̄∗1. Similar equations can be derived to express the stress components σ2∗ and σ3∗ in terms of the stress components resolved in axis system I. For σ2∗ , the direction cosines `1 , `2 , and `3 appearing in eq. (1.18) are replaced by direction cosines m1 , m2 , and m3 , respec- tively, whereas direction cosines n1 , n2 , and n3 will appear in the expression for σ3∗. Coordinate rotations are defined in appendix A.3. The shear stress components follow from eq. (1.12) as ∗ τ12 = σ1 `1 m1 + σ2 `2 m2 + σ3 `3 m3 + τ12 (`2 m1 + `1 m2 ) (1.19) + τ13 (`1 m3 + `3 m1 ) + τ23 (`2 m3 + `3 m2 ). Here again, similar relationships can be derived for the remaining shear stress com- ∗ ∗ ponents, τ13 and τ23 , through appropriate cyclic permutation of the indices. All these relationships can be combined into the following compact matrix equa- tion  ∗ ∗ ∗   σ1 τ12 τ13 σ1 τ12 τ13 τ21 ∗ ∗  σ2∗ τ23 = RT τ12 σ2 τ23  R, (1.20) ∗ ∗ ∗ τ31 τ32 σ3 τ13 τ23 σ3 where R is the rotation matrix defined by eq. (A.36). This equation concisely en- capsulates the relationship between the stress components resolved in two different coordinate systems, and it can be used to compute the stress components resolved in basis I ∗ in terms of the stress components resolved in basis I. 16 1 Basic equations of linear elasticity Finally, since the principal stresses at a point are independent of the particular coordinate system used to define the stress state, the coefficients of the cubic equation that determines the principal stresses, eq. (1.14), must be invariant with respect to reference frames. This is the very reason why quantities I1 , I2 , and I3 defined by eq. (1.15) are called the stress invariants. The word “invariant” refers to the fact that these quantities are invariant with respect to a change of coordinate system. Let I ∗ and I be two different orthonormal bases, I1 = σ1∗ + σ2∗ + σ3∗ = σ1 + σ2 + σ3 , (1.21a) I2 = σ1∗ σ2∗+ σ2∗ σ3∗ + σ3∗ σ1∗ ∗2 − τ12 ∗2 − τ13 ∗2 − τ23 2 2 2 = σ1 σ2 + σ2 σ3 + σ3 σ1 − τ12 − τ13 − τ23 , (1.21b) ∗ ∗ ∗ ∗ ∗2 ∗ ∗2 ∗ ∗2 ∗ ∗ ∗ I3 = σ1 σ2 σ3 − σ1 τ23 − σ2 τ13 − σ3 τ12 + 2τ12 τ13 τ23 2 2 2 = σ1 σ2 σ3 − σ1 τ23 − σ2 τ13 − σ3 τ12 + 2τ12 τ13 τ23. (1.21c) Tedious algebra using eqs. (1.20) to write the stress components resolved in basis I ∗ in terms of the stresses components resolved in basis I will reveal that the above relationships are correct. Example 1.1. Computing principal stresses Consider the following stress tensor   −5 −4 0 S = −4 1 0. 0 1 Compute the principal stresses and the principal stress directions. The stress invari- ants defined by eq. (1.15) are computed as I1 = −3, I2 = −25 and I3 = −21. The principal stress equation, eq. (1.14), now becomes σp3 + 3σp2 − 25σp + 21 = (σp − 1)(σp2 + 4σp − 21) = 0, The solutions of this cubic equations yield the principal stresses as σp1 = 3, σp2 = 1 and σp3 = −7. Next, the principal direction associated with σp1 = 3 is computed. The homoge- neous system defined by eq. (1.13) becomes    −8 −4 0 n1  −4 −2 0 n2 = 0.   0 0 −2 n3 The determinant of this system vanishes because the first two equations are a multiple of each other. The first equation yields n1 = α and n2 = −2α, where α is an arbitrary constant, whereas the third equation gives n3 = 0. Since the principal √ 2 2 2 2 √ be unit vector, n1 + n2 + n3 = 1, or 5α = 1; finally n1 = 1/ 5, direction must n2 = −2/ 5 and n3 = 0. Proceeding in a similar manner for the other two principal stresses, the three principal directions are found to be  1.2 Analysis of the state of stress at a point 17       1  0 −2 1  