Structures and Their Analysis - Springer978-3-319-31081-7/1.pdf · students (See, for instance, D.L. Schodek and M. Bechthold, Structures, 6th edition, Pearson Prentice Hall 2008.)

Structures and Their Analysis

Maurice Bernard Fuchs

Structures and Their Analysis

123

Maurice Bernard FuchsSchool of Mechanical EngineeringTel Aviv UniversityTel AvivIsrael

ISBN 978-3-319-31079-4 ISBN 978-3-319-31081-7 (eBook)DOI 10.1007/978-3-319-31081-7

Library of Congress Control Number: 2016934026

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ForRinna

Preface

There are two main methods for analyzing a structure: the force method and thedisplacement method. Some will argue the former is passé. We do not follow thatargument. The force method may have lost its prominence as a structural analysistool, but it is still conducive to the way engineers think about a structure. Forces andequilibrium are intuitive from infancy, for example, when we tried out our first solosteps. There is also the early childhood sense of triumph when an eighth woodenblock is positioned correctly on a stack of seven, or the feeling of disappointmentwhen the last cube makes the tower tumble.

This innate feeling of forces and equilibrium has remained with us. We usuallythink of structures in terms of forces, although in practice our computers will per-form the analysis by way of displacements. This duality is reflected in this treatise.

My first book on structures was a text written by André Paduart of the FreeUniversity of Brussels (ULB). It was called M; N; T ; R; δ. The title started outwith the internal loads: M for bending Moment, N for Normal force, T for shearforce (effort Tranchant—the text was in French), R for Reactions I guess, and lastlyδ for displacements. At that time (1963) displacements were indeed a nuisance andyou computed them only when you absolutely needed them. In those days theanalysis method was the force method.

Things have changed. Nowadays the analysis method of choice is indisputablythe displacement method, where the basic variables are the displacements of thestructure. Once these have been computed the internal forces and reactions easilyfollow suit. The shift in paradigm was heralded by the advent of tremendous, readilyavailable computing power. Arguably, a basic tenet in the classical days of structuralanalysis was never to have more than three linear equations with three unknowns atany time, no matter how complex the structure. This led to some ingenious modelingand clever assumptions in order to avoid the three-equations obstacle.

The displacement method, on the other hand, albeit deceptively simple andrelatively boring, because it involves tedious matrix calculations, requires thesolution of many linear equations with as many unknowns. With the finite-elementmethod, an offshoot of the displacement method, it is not unusual to solve tens of

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thousands of linear equations when, for instance, considering the analysis of anaircraft wing or a cable-stayed bridge, a task perfectly suited to computerizedcalculations. Moreover, the writing of the equations, what we call assembly, is alsoautomated in the displacement method; again something computers do well.

This is also how we present these two aspects of analysis: performing an analysisby the force method is a human endeavor. It can be applied to small andmedium-sized structures. It requires an understanding of the behavior of thestructure. Its results, the internal loads, can be understood and related to and, as weshall see, even qualitatively predicted in some cases. The displacement method, onthe other hand, is for computers. It is algorithmic and can easily be programmed, itmanipulates vast arrays of numbers, and does not require too deep an understandingof the process.

This book is intended for a graduate level of instruction although undergraduatematerial is embedded in the text. Like most structural courses it assumes that thereader has some basic knowledge of the field. Courses in Statics and SolidMechanics (Strength of Materials) are prerequisites, and essentials of these coursesare the first topics of this book.

Part I: Preliminaries

We start with principles of equilibrium (Chap. 1) from which we need to postulatethe existence of internal loads. This is followed immediately by a probably toolengthy chapter (Chap. 2) on how to compute and display these internal forces bymeans of the nsm-diagrams. We next discuss the principle of virtual work (Chap. 3)without having introduced Hooke’s law. This is very important from a didacticviewpoint. It shows that the principle, in its basic form, is a mathematical conceptand not really connected to a structure (a structure is something that deforms underloading). We also show that for the principle to work we need to define quantitiesthat turn out to be deformations (Chap. 4). This way the principle is not some hattrick but simply integration by parts, a mathematical tool. In the next chapter(Chap. 5) we introduce elasticity, that is, the linear relation between the internalforces and the deformations. When external loads are applied to a structure, itdeforms so as to engender internal forces which balance the applied ones. Wediscuss the Timoshenko beam and its relation to the Bernoulli–Euler beam used inengineering beam theory, a graduate course topic. Finally, in the last chapter ofPart I (Chap. 6) we introduce a technique for calculating a displacement or arotation at any location along a structure. The method is closely connected to virtualwork and is often taught in introductory courses to structures.

viii Preface

Part II: The Structures

The first chapter in Part II (Chap. 7) describes the types of structures dealt with inthis book: beams, plane frames and, to a lesser extent, plane trusses. Such structuresare composed of line elements fully or partially connected to one another or to theground, as described in this chapter. The following chapter (Chap. 8) is an intro-duction to and a comparison of the two main methods of analysis. This is done byanalyzing a stylized structure, a rigid beam on several elastic supports, by both theforce method and the displacement method. The last chapter of Part II (Chap. 9) isusually taught to architectural students but unfortunately less so to engineeringstudents (See, for instance, D.L. Schodek and M. Bechthold, Structures, 6th edition,Pearson Prentice Hall 2008.) We consider this material extremely important. Itoften gives a hint of the expected results before actually performing the analysis.

Part III: Flexibility

The flexibility matrix (Chap. 10) lies at the heart of the force method. The chapterstarts with the Betti–Maxwell reciprocal theorems, the notion of degree of freedom,leading to the definition of influence or flexibility coefficients, which are theconstituents of the flexibility matrix. Several illustrated examples close the chapter,but not before discussing why normal deformations in the presence of bendingdeformation are usually neglected. Redundancy is treated in the followingchapter (Chap. 11). The force method is applicable to redundant structures only.Determining the degree of redundancy is therefore the first step of an analysis bythe force method. We also consider redundancy in the context of safety and stiffnessof structures. The next chapter is the core of the force method (Chap. 12). We learnthe important notions of releasing a structure and writing the compatibility equa-tions. Thereafter, the task is reduced to the analysis of a statically determinatestructure where all the applied loads are known. The chapter closes with severalillustrated problems. This is followed by a chapter dealing with the effect of heat onstructures (Chap. 13). It includes prestressing and manufacturing deformations ormisalignments. The chapter emphasizes that all this is applicable to redundantstructures only. Determinate structures without applied forces are guaranteed to befree of internal forces.

Part IV: Stiffness

We introduce the stiffness method by way of trusses (Chap. 14). We find theanalysis of trusses by the force method too tedious. The stiffness method, which isessentially a numerical method to be solved algorithmically, is ideally suited to

Preface ix

trusses as an analysis tool. The following chapter (Chap. 15) introduces the stiffnessmatrix of an element. This potent concept allows us to analyze with the same easeany structure if we know the stiffness matrices of all its elements. An intermediatechapter (Chap. 16) essentially indicates a systematic way (congruent transforma-tions) to obtain the element stiffness matrix in oblique coordinates. The followingchapter (Chap. 17) is the core of the displacement (stiffness) method. We show themeaning of the equilibrium equations which compute the unknown nodal dis-placements. Next, we indicate the concept of ‘assembly’, an efficient algorithm towrite the equilibrium equations, essentially, the system stiffness matrix. Everythingwe have done until now assumed that the structure was submitted to point loadsapplied to the nodes. This excludes distributed forces and temperature effects(applied strain). The final chapter of Part IV (Chap. 18) addresses this subject. Bymeans of fixed-end reactions the problem is circumvented in a most elegant manner.This concludes the analysis methods presented in this text.

Part V: Four Additional Topics

The next part of the book touches on a few aspects of structural analysis and designwhich complement well the subject matter of this text.

A nice theorem which may be of practical use is presented next (Chap. 19).It states that a roller support of a continuous beam is optimally positioned (from astiffness perspective) if the slope of the beam over the support is horizontal.A simple numerical technique to find the optimal location is also indicated.

A redundant truss cannot, in general, be fully stressed. In the following chapter(Chap. 20) we show a class of redundant trusses which can be fully stressed.

The next chapter (Chap. 21) indicates a way to cast the analysis equations of aframe in the form of truss equations by means of unimodal normal, shear, andmoment elements. Consequently, a unified approach to structural analysis is pos-sible by considering any frame as a generalized truss.

In the penultimate chapter of this book (Chap. 22) we present a very elegantformula for the internal forces in trusses explicitly in terms of the stiffnesses of thebars. Unfortunately, the formula is plagued by the so-called curse of dimensionality.

Part VI: Epilogue

Finally, in the concluding chapter (Chap. 23) we summarize the essentials of all thatwas said by means of the simplest structure of them all: an assembly of linearsprings.

x Preface

One of our guidelines when writing the book was to keep everything as simpleas possible so as to convey the basic ideas without unnecessary burdens. We useplane structures, straight-linear uniform elements, uniformly distributed forces,beams with lateral forces only (which precludes normal internal forces in beams).Using planar structures ignores torsional effects, which is a drawback. Students willhave ample leisure to include all this when taking a course in finite elements orwhen performing a finite-element analysis. We have also avoided talking ofenergy-related concepts (Castigliano, Clapeyron, etc.). Instead, we emphasizeVirtual Work. The latter has a large enough gangplank to take all these methods onboard. This was also the reason for avoiding, as much as possible, analyzing trussesmanually. In our view trusses are computer territory. All this made it possible tomake space for understanding structural behavior which should after all be the mainobjective of a book on structures.

I have read and browsed through many books on structures. A few stand out. KHGerstle’s Basic Structural Analysis, Prentice-Hall, 1974, is probably my firstchoice. I have benefitted a lot from Schodek’s Structures, mentioned earlier. Thenthere is, of course, SO Asplund (Structural Mechanics: Classical and MatrixMethods, Prentice-Hall, 1966), an amazing book. I have seen it referred to as thebible of modern structural theory. I fully agree with this description.

I am indebted to Professor Z. Hashin (TAU) and to the late Professors M.A.Brull (TAU) and A. Libai (Technion), who in their own way have been instru-mental in making my career in structures possible. I am grateful to my graduate andPhD students who were an inspiration. Many thanks go to Tim Love of CambridgeUniversity, a rare altruist, who was so helpful during a 2-months’ summer stay atthe Engineering Department. Thanks to Miriam Hercberg for her help in checkingthe manuscript and to Guy Shiber for verifying and correcting the equations, cal-culations, and figures.

Finally, I wish to express my gratitude to the Faculty of Engineering at Tel AvivUniversity, a haven of serenity and genuine academic spirit, for providing thematerial conditions to do research, support students, finance for sabbaticals, andparticipation in scientific meetings and conferences.

December 2015

Preface xi

Acknowledgments

Figure 19.5 reprinted from Computers & Structures, M.B. Fuchs and M.A. Brull,A New Strain Energy Theorem and Its Use in the Optimum Design of ContinuousBeams, Vol. 10, pp. 647-657, 1979, with permission from Elsevier.

Figure 20.2, 20.3 and the figure on the cover, reprinted from AIAA Journal,M.B. Fuchs and L.P. Felton, On a Class of Fully Stressed Trusses, Vol. 12, No. 11,pp.1597-1599, 1974, with permission from AIAA.

Figures 21.1, 21.2, 21.3, 21.4, Problem 21.5 reprinted from Computers &Structures, M.B. Fuchs, Unimodal Formulation of the Analysis and Design ofFramed Structures, Vol. 63, No. 4, pp. 739-747, 1997, with permission fromElsevier.

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Contents

Part I Preliminaries

1 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 In the Beginning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Resultant Force and Resultant Moment . . . . . . . . . . . . . . . . 51.3 Equilibrium of a Solid. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Three Scalar Conditions. . . . . . . . . . . . . . . . . . . . 71.4 Statically Equivalent Forces . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Sliding Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.2 Couples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.3 Reduction to One Force and One Couple . . . . . . . . 101.4.4 Uniformly Distributed Forces . . . . . . . . . . . . . . . . 111.4.5 Three Conditions Out of Five . . . . . . . . . . . . . . . . 12

1.5 Illustrated Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.5.1 Moments with Respect to 3 Collinear Points . . . . . 131.5.2 Auto-Equilibrated 2-Force and 3-Force Systems . . . 14

1.6 The Internal Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.6.1 Free-Body Diagrams . . . . . . . . . . . . . . . . . . . . . . 141.6.2 Equivalent System of Internal Forces. . . . . . . . . . . 161.6.3 Normal Force, Shear Force and Bending

Moment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.7 Equilibrium Conditions Along the Element . . . . . . . . . . . . . 18

1.7.1 Equilibrium of a Lamella dx of a PrismaticElement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.7.2 Equilibrium at a Concentrated Load . . . . . . . . . . . 201.8 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2 The nsmðxÞ Diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.1 Line Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Positive, Negative nsm . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 Calculating nsm at a Section . . . . . . . . . . . . . . . . . . . . . . . 25

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2.4 The nsm Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.1 Basic Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.2 Practical Guidelines for Positioning

the s and m Diagrams . . . . . . . . . . . . . . . . . . . . . 282.4.3 Relations Between the s and m Diagrams. . . . . . . . 29

2.5 Typical Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.6 Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.6.1 Point Force on a Symmetric Framewith Inclined Columns. . . . . . . . . . . . . . . . . . . . . 36

2.6.2 Discontinuities at Point Loads . . . . . . . . . . . . . . . 372.7 Inverse Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.8.1 You Cannot Split a Point . . . . . . . . . . . . . . . . . . . 402.8.2 What Is in a Name? . . . . . . . . . . . . . . . . . . . . . . 402.8.3 Why Tensile Side for Bending Moments? . . . . . . . 412.8.4 How Are the Applied Forces Annulled?. . . . . . . . . 422.8.5 Noise on the Way to Equilibrium . . . . . . . . . . . . . 43

2.9 Illustrated Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.9.1 Frame with Point Loads. . . . . . . . . . . . . . . . . . . . 442.9.2 Distributed Forces Between Point Loads . . . . . . . . 452.9.3 Inverse Analysis of a Frame . . . . . . . . . . . . . . . . . 482.9.4 Kinked Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.9.5 Broken Line with End-Couples. . . . . . . . . . . . . . . 50

2.10 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.1 What Is Virtual Work? . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 Virtual Work for Particles in a Plane. . . . . . . . . . . . . . . . . . 563.3 Virtual Work for a Frame Element . . . . . . . . . . . . . . . . . . . 573.4 Equilibrium and Compatible Systems . . . . . . . . . . . . . . . . . 60

3.4.1 Equilibrium Systems . . . . . . . . . . . . . . . . . . . . . . 603.4.2 Compatible Systems . . . . . . . . . . . . . . . . . . . . . . 60

3.5 Point Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.6.1 Cθ and ðmκdx) Are Virtual WorkExpressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.7 Illustrated Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.7.1 An Example with Axial Forces . . . . . . . . . . . . . . . 643.7.2 Beam with Arc Shape Deformation . . . . . . . . . . . . 65

3.8 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4 Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.1 The Bernoulli Hypothesis of Plane Sections . . . . . . . . . . . . . 674.2 Study of Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

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4.3 Geometric Interpretation of the Deformations . . . . . . . . . . . . 714.3.1 Extensional Deformation Mode εðxÞ . . . . . . . . . . . 714.3.2 Shear Deformation Mode γðxÞ . . . . . . . . . . . . . . . 724.3.3 Bending Deformation Mode κðxÞ . . . . . . . . . . . . . 72

4.4 Zero Deformation—Rigid-Body Displacement . . . . . . . . . . . 734.5 In Vector Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.6 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.1 Elastic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2 Hooke’s Law at Location y, z on the Cross-Section . . . . . . . 765.3 The Timoshenko Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.4 Euler–Bernoulli Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.4.1 Virtual Work for Euler–Bernoulli Deformations . . . 825.4.2 A Note on Curvature. . . . . . . . . . . . . . . . . . . . . . 835.4.3 Rods and Beams. . . . . . . . . . . . . . . . . . . . . . . . . 84

5.5 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6 The Unit-Load Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.1 The Equilibrium Configuration . . . . . . . . . . . . . . . . . . . . . . 85

6.1.1 The Unit-Load Method for Beams . . . . . . . . . . . . 866.1.2 Redundant Beams . . . . . . . . . . . . . . . . . . . . . . . . 88

6.2 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.2.1 Computing a Displacement . . . . . . . . . . . . . . . . . 906.2.2 Computing a Rotation . . . . . . . . . . . . . . . . . . . . . 916.2.3 Frames, Beams and Trusses: Numerical

Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.2.4 Formula for Calculating the Integrals. . . . . . . . . . . 93

6.3 Trusses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.4 Illustrated Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.4.1 A Kinked Bar or Shifting from Normalto Bending Mode . . . . . . . . . . . . . . . . . . . . . . . . 96

6.4.2 Simple Beam with Distributed Forces . . . . . . . . . . 996.4.3 A Simple Frame . . . . . . . . . . . . . . . . . . . . . . . . . 1006.4.4 Ineffective Equilibrium Systems . . . . . . . . . . . . . . 104

6.5 Statically Redundant Examples. . . . . . . . . . . . . . . . . . . . . . 1046.5.1 The Propped Cantilever . . . . . . . . . . . . . . . . . . . . 1056.5.2 A Redundant Truss . . . . . . . . . . . . . . . . . . . . . . . 1076.5.3 A Portal Frame. . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.6 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Part II The Structures

7 Types of Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.1 The Frame, the Beam and the Truss . . . . . . . . . . . . . . . . . . 1137.2 Full Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

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7.3 Partial Connections Between Elements (Releases) . . . . . . . . . 1167.4 Partial Connections Between Elements and the Ground

(Supports) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197.5 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8 Structural Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1218.1 The Two Main Methods of Analysis . . . . . . . . . . . . . . . . . . 1218.2 Analysis by the Force Method . . . . . . . . . . . . . . . . . . . . . . 122

8.2.1 Determining the Statical Redundancy . . . . . . . . . . 1228.2.2 Releasing the Structure . . . . . . . . . . . . . . . . . . . . 1238.2.3 The Compatibility Equations . . . . . . . . . . . . . . . . 124

8.3 Analysis by the Displacement Method. . . . . . . . . . . . . . . . . 1258.3.1 Determining the Kinematical Degrees

of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1258.3.2 The Equilibrium Equations. . . . . . . . . . . . . . . . . . 126

8.4 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

9 Qualitative Analysis and Design . . . . . . . . . . . . . . . . . . . . . . . . . 1299.1 An Introductory Example . . . . . . . . . . . . . . . . . . . . . . . . . 1299.2 The Basic Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1319.3 Qualitative Analysis of Portal Frame . . . . . . . . . . . . . . . . . . 1329.4 Qualitative Design of Truss-Beams . . . . . . . . . . . . . . . . . . . 1329.5 Illustrated Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

9.5.1 Continuous Beam . . . . . . . . . . . . . . . . . . . . . . . . 1349.5.2 Continuous Beam on Five Supports . . . . . . . . . . . 1359.5.3 Fixed Portal Frame with Distributed Forces . . . . . . 1369.5.4 Four-Cell-Truss-Beam on Three Supports. . . . . . . . 1379.5.5 Simple Truss-Beam . . . . . . . . . . . . . . . . . . . . . . . 1399.5.6 Simply Supported Portal Frame . . . . . . . . . . . . . . 139

9.6 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Part III Flexibility

10 Flexibility Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14310.1 Virtual Work with Related ε and n . . . . . . . . . . . . . . . . . . . 14310.2 Betti and Maxwell Theorems . . . . . . . . . . . . . . . . . . . . . . . 145

10.2.1 Betti’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 14510.2.2 Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . 14610.2.3 Maxwell’s Theorem . . . . . . . . . . . . . . . . . . . . . . 147

10.3 Computing Influence or Flexibility Coefficients . . . . . . . . . . 14810.4 Flexibility Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14910.5 Properties of Flexibility Matrices . . . . . . . . . . . . . . . . . . . . 15110.6 Forces and Displacements at the Degrees of Freedom . . . . . . 15410.7 Generalized Degrees of Freedom . . . . . . . . . . . . . . . . . . . . 155

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10.8 Illustrated Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15510.8.1 Propped Cantilever with Hinge . . . . . . . . . . . . . . . 15510.8.2 Portal Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . 15710.8.3 Simple Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . 15810.8.4 Beam on Elastic Supports . . . . . . . . . . . . . . . . . . 15910.8.5 Generalized Flexibility Matrix . . . . . . . . . . . . . . . 16010.8.6 Truss-Beam on Elastic Supports . . . . . . . . . . . . . . 16110.8.7 Comparing the Beam and the Truss. . . . . . . . . . . . 16110.8.8 Propped Cantilever . . . . . . . . . . . . . . . . . . . . . . . 164

10.9 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

11 Redundancy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16711.1 Typical Redundant Structures. . . . . . . . . . . . . . . . . . . . . . . 16711.2 Definition of Statical Redundancy. . . . . . . . . . . . . . . . . . . . 16811.3 Counting the Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . 169

11.3.1 Elements and Nodes . . . . . . . . . . . . . . . . . . . . . . 16911.3.2 Other Decompositions . . . . . . . . . . . . . . . . . . . . . 17111.3.3 Internal Releases. . . . . . . . . . . . . . . . . . . . . . . . . 17211.3.4 Frames with Truss Elements. . . . . . . . . . . . . . . . . 173

11.4 Redundancy by Inspection . . . . . . . . . . . . . . . . . . . . . . . . . 17511.4.1 The Simple Tree . . . . . . . . . . . . . . . . . . . . . . . . . 17511.4.2 Recognizing the Redundancy by Releasing

the Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17611.5 Beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

11.5.1 Counting Unknowns and Equations. . . . . . . . . . . . 17711.5.2 Redundancy by Inspection . . . . . . . . . . . . . . . . . . 178

11.6 Trusses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18011.6.1 Counting Unknowns and Equations. . . . . . . . . . . . 18011.6.2 Redundancy by Inspection . . . . . . . . . . . . . . . . . . 180

11.7 Mechanisms and Conditionally Stable Structures . . . . . . . . . 18211.7.1 Mechanism Versus Structure . . . . . . . . . . . . . . . . 18211.7.2 Conditionally Stable Structures . . . . . . . . . . . . . . . 18411.7.3 R� 0 is not always sufficient for stability. . . . . . . . 18511.7.4 Large Internal Forces in Quasi Non-stable

Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18611.8 Properties of Statically Redundant structures . . . . . . . . . . . . 187

11.8.1 Redundant Structures Are Safer . . . . . . . . . . . . . . 18711.8.2 Redundant Structures Are Stiffer . . . . . . . . . . . . . . 18811.8.3 Flow of Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 189

11.9 Properties of Statically Determinate Structures . . . . . . . . . . . 18911.9.1 No External Forces Means No Internal Forces . . . . 19011.9.2 Heating Will Not Produce Internal Forces . . . . . . . 191

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11.9.3 No Prestressing Possible . . . . . . . . . . . . . . . . . . . 19211.9.4 Non-sensitivity to the Exact Dimensions

of the Elements . . . . . . . . . . . . . . . . . . . . . . . . . 19311.10 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

12 The Force Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19512.1 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19512.2 Released Structures Subjected to the Released Loads . . . . . . 196

12.2.1 R ¼ 1 Propped Cantilever . . . . . . . . . . . . . . . . . . 19612.2.2 R ¼ 2 Continuous Beam . . . . . . . . . . . . . . . . . . . 197

12.3 The Essence of the Force Method. . . . . . . . . . . . . . . . . . . . 19812.3.1 R ¼ 1 Cantilever Beam with Elastic Spring . . . . . . 19912.3.2 R ¼ 2 Continuous Beam . . . . . . . . . . . . . . . . . . . 200

12.4 Non-valid Releases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20212.5 Formalizing the Compatibility Equations . . . . . . . . . . . . . . . 20312.6 Implementation of the Compatibility Equations . . . . . . . . . . 20512.7 Solved Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

12.7.1 R ¼ 1 Propped Cantilever . . . . . . . . . . . . . . . . . . 20612.7.2 R ¼ 2 Beam on Mixed Supports . . . . . . . . . . . . . . 20812.7.3 R ¼ 2 Beam: Releasing the Moments over

Two Supports. . . . . . . . . . . . . . . . . . . . . . . . . . . 21012.7.4 Same R ¼ 2 Beam: Disconnecting

Two Supports. . . . . . . . . . . . . . . . . . . . . . . . . . . 21112.7.5 Beam on Many Supports—Three Moments

Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21112.7.6 Beam Stiffened by Underlying Truss . . . . . . . . . . . 21612.7.7 R ¼ 2 Portal Frame . . . . . . . . . . . . . . . . . . . . . . . 21812.7.8 R ¼ 2 Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . 22212.7.9 R ¼ 1 Cable-Braced Frame . . . . . . . . . . . . . . . . . 22412.7.10 Fixed Circular Arch with Applied Couple . . . . . . . 22712.7.11 Grid of Beams Under Lateral Force. . . . . . . . . . . . 22912.7.12 R ¼ 2 Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

12.8 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

13 Applied Strains and Initial Stresses . . . . . . . . . . . . . . . . . . . . . . . 23513.1 Initial Internal Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23513.2 Mechanical Strain Versus Applied Strain . . . . . . . . . . . . . . . 23513.3 Applied Strains in Axial and Bending Elements . . . . . . . . . . 236

13.3.1 The Case of Statically Determinate Structures. . . . . 23813.3.2 Displacements of Determinate Structures

Under Applied Strain . . . . . . . . . . . . . . . . . . . . . 23913.3.3 Extending the Scope of Applied Strain . . . . . . . . . 240

13.4 Applied Strain in Redundant Structures . . . . . . . . . . . . . . . . 24013.4.1 Internal Forces Due to Applied Strain . . . . . . . . . . 24013.4.2 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . 24113.4.3 The General Case . . . . . . . . . . . . . . . . . . . . . . . . 243

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13.5 Illustrated Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24313.5.1 Cantilever Propped on Two Supports . . . . . . . . . . 24313.5.2 Determinate and Redundant Trusses . . . . . . . . . . . 24613.5.3 Fixed Elements. . . . . . . . . . . . . . . . . . . . . . . . . . 24813.5.4 Partially Heated Beam . . . . . . . . . . . . . . . . . . . . . 25013.5.5 A Heated Roof . . . . . . . . . . . . . . . . . . . . . . . . . . 25113.5.6 Rectangular Box with Applied Curvature

on Top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25213.6 Summary of the Technique . . . . . . . . . . . . . . . . . . . . . . . . 25313.7 Lack-of-Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

13.7.1 Lack-of-Fit as Applied Gaps . . . . . . . . . . . . . . . . 25413.7.2 The Compatibility Equations . . . . . . . . . . . . . . . . 255

13.8 Prestressing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25513.8.1 Why Prestressing?. . . . . . . . . . . . . . . . . . . . . . . . 25613.8.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

13.9 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

Part IV Stiffness

14 Introducing the Stiffness Method . . . . . . . . . . . . . . . . . . . . . . . . 26114.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26114.2 Pin-Jointed Ideal Trusses . . . . . . . . . . . . . . . . . . . . . . . . . . 26214.3 Rigid-Jointed Trusses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26414.4 The Fundamental Analysis Equations for Trusses . . . . . . . . . 26514.5 Virtual Work, R ¼ QT . . . . . . . . . . . . . . . . . . . . . . . . . . . 26614.6 Solving the Equations by the Stiffness Method . . . . . . . . . . . 26714.7 Element Stiffness Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 26814.8 Solved Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26914.9 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

15 Element Stiffness Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27315.1 Typical Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27315.2 Element Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 27415.3 Some Properties of K . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

15.3.1 Physical Interpretation of the Coefficients of K . . . . 27515.3.2 The Stiffness Matrix Is Symmetric . . . . . . . . . . . . 27615.3.3 Relations Between Rows/Columns . . . . . . . . . . . . 27715.3.4 An Element Stiffness Matrix Is Singular . . . . . . . . 27915.3.5 Orthogonality of the Flexural and Extensional

Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27915.3.6 How Many Independent Coefficients

Are There?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28115.4 Formula for a Stiffness Coefficient . . . . . . . . . . . . . . . . . . . 282

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15.5 Computing the Stiffness Matrix of Uniform Elements . . . . . . 28315.5.1 Uniform Beam (Flexural) Element Matrix . . . . . . . 28415.5.2 Uniform Rod (Extensional) Element Matrix . . . . . . 28615.5.3 Extensional and Rotational Spring Element

Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28615.6 Illustrated Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28715.7 Element Stiffness Matrices by Shape Functions . . . . . . . . . . 289

15.7.1 The Flexural Matrix . . . . . . . . . . . . . . . . . . . . . . 28915.7.2 The Extensional Matrix . . . . . . . . . . . . . . . . . . . . 291

15.8 Sample Rod and Beam Element Stiffness Matrices . . . . . . . . 29315.8.1 Stepped Rod Element . . . . . . . . . . . . . . . . . . . . . 29315.8.2 Tapered Rod Element . . . . . . . . . . . . . . . . . . . . . 29415.8.3 Beam Element with Central Hinge . . . . . . . . . . . . 29515.8.4 Beam Element with Central Lateral Guide . . . . . . . 297

15.9 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29815.10 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

16 Change of Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29916.1 The Uniform Frame Element . . . . . . . . . . . . . . . . . . . . . . . 29916.2 Congruent Transformations . . . . . . . . . . . . . . . . . . . . . . . . 300

16.2.1 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . 30216.2.2 Stiffness Matrix of an Oblique Element . . . . . . . . . 302

16.3 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

17 Assembling the System Stiffness Matrix. . . . . . . . . . . . . . . . . . . . 30717.1 Modeling the Structure as the Sum of Its Elements . . . . . . . . 307

17.1.1 Elements and Nodes . . . . . . . . . . . . . . . . . . . . . . 30717.1.2 Nodal or Global Coordinates, Degrees

of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30817.1.3 Element or Local Coordinates. . . . . . . . . . . . . . . . 309

17.2 The Nodal Equilibrium Equations . . . . . . . . . . . . . . . . . . . . 31017.2.1 If the End-Displacements Are Known ... . . . . . . . . 31017.2.2 If the End-Displacements Are Wrong ... . . . . . . . . 31217.2.3 Equilibrium at the Unknown Degrees

of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31417.3 Assembling the System Stiffness Matrix . . . . . . . . . . . . . . . 316

17.3.1 An Example for Assembling K. . . . . . . . . . . . . . . 31717.3.2 Equilibrium and Assembly Are Equivalent . . . . . . . 319

17.4 More on Degrees of Freedom. . . . . . . . . . . . . . . . . . . . . . . 32317.4.1 Degrees of Freedom in the Force

and Displacement Methods . . . . . . . . . . . . . . . . . 32417.4.2 Singular Stiffness Matrices . . . . . . . . . . . . . . . . . . 324

17.5 Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32617.5.1 Inextensible Elements . . . . . . . . . . . . . . . . . . . . . 32617.5.2 Congruent Transformation . . . . . . . . . . . . . . . . . . 326

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17.5.3 Direct Assembly . . . . . . . . . . . . . . . . . . . . . . . . . 32817.5.4 Axial Elements Combined with Frame

Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33117.5.5 Trusses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

17.6 Illustrated Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33517.6.1 Fixed Beam with Elastic Supports. . . . . . . . . . . . . 33517.6.2 Beam with Guides and a Roller . . . . . . . . . . . . . . 337

17.7 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

18 Loads on Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34118.1 Element Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34118.2 Fixed-End Reactions � f.e.r. . . . . . . . . . . . . . . . . . . . . . . . 342

18.2.1 Typical f.e.r.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34318.2.2 The Method in Three Steps . . . . . . . . . . . . . . . . . 34418.2.3 Equivalent Nodal Loads. . . . . . . . . . . . . . . . . . . . 346

18.3 Illustrated Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34618.3.1 Fixed Beam with Intermediate Support . . . . . . . . . 34618.3.2 Frame with Rigid Element . . . . . . . . . . . . . . . . . . 34818.3.3 Fixed Haunched Beam . . . . . . . . . . . . . . . . . . . . 35018.3.4 Fixed Beam with Central ‘Haunch’ . . . . . . . . . . . . 35118.3.5 Applied Strain Yielding Zero Equivalent Loads . . . 35218.3.6 Continuous Beam with Element Loads . . . . . . . . . 35318.3.7 Frame with Element Loads. . . . . . . . . . . . . . . . . . 35518.3.8 Fixed Beam with Central Guide . . . . . . . . . . . . . . 359

18.4 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

Part V Four Additional Topics

19 A Strain Energy Theorem—Moving Supports . . . . . . . . . . . . . . . 36519.1 Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36519.2 Minimum Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 36619.3 Designing for Zero Slope . . . . . . . . . . . . . . . . . . . . . . . . . 36719.4 Illustrated Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

19.4.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36819.4.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36819.4.3 Example 3: Several Moving Supports . . . . . . . . . . 368

19.5 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

20 Fully-Stressed Trusses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37120.1 Fully Stressed Trusses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37120.2 General Redundant Truss. . . . . . . . . . . . . . . . . . . . . . . . . . 37220.3 Circle-Chord Trusses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37320.4 Extended Circle-Chord Trusses. . . . . . . . . . . . . . . . . . . . . . 37420.5 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

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21 Frames Viewed as Generalized Trusses . . . . . . . . . . . . . . . . . . . . 37721.1 Element Stiffness Matrices in Modal Coordinates . . . . . . . . . 377

21.1.1 Axial Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . 37821.1.2 Bending Stiffness Matrix . . . . . . . . . . . . . . . . . . . 379

21.2 Coordinate Transformation. . . . . . . . . . . . . . . . . . . . . . . . . 37921.3 Unimodal Normal, Moment and Shear Elements. . . . . . . . . . 38021.4 Frames Viewed as Trusses. . . . . . . . . . . . . . . . . . . . . . . . . 38121.5 Explicit Analysis of a Propped Cantilever . . . . . . . . . . . . . . 38221.6 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

22 Explicit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38522.1 The Explicit Expression. . . . . . . . . . . . . . . . . . . . . . . . . . . 38522.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38822.3 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

Part VI Epilogue

23 Plus ça Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39323.1 The Three Pillars of Structures . . . . . . . . . . . . . . . . . . . . . . 393

23.1.1 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39323.1.2 Deformation (Virtual Work) . . . . . . . . . . . . . . . . . 39423.1.3 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

23.2 Methods of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39623.2.1 The Basic Method. . . . . . . . . . . . . . . . . . . . . . . . 39623.2.2 The Displacement Method . . . . . . . . . . . . . . . . . . 39823.2.3 The Force Method . . . . . . . . . . . . . . . . . . . . . . . 399

23.3 Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40223.3.1 Columns and Rows of Q . . . . . . . . . . . . . . . . . . . 40223.3.2 On the Equilibrium and Compatibility

Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40323.4 Plus ça Change ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

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About the Author

Maurice Bernard Fuchs was brought up in Antwerp, Belgium, and graduatedfrom the Ecole Polytechnique of the Free University of Brussels (ULB) in civil andaeronautical engineering, specializing in structures. He earned an ScD inTechnology at the Faculty of Aeronautical Engineering at the Israel Institute ofTechnology (Technion) and joined the department of Solid Mechanics, Materialsand Structures of Tel Aviv University in 1977, where he has been a faculty memberever since. He was the first head of the then newly established School ofMechanical Engineering and is currently Professor Emeritus at that school.

The author has taught basic and advanced undergraduate courses in structuralanalysis and aircraft analysis and design, and graduate courses in finite-elementanalysis and structural optimization. He has also taught structures courses toarchitectural students at the Faculty of Architecture at the Technion and at theSchool of Architecture of Tel Aviv University. The author was a visiting assistantprofessor at the School of Engineering at UCLA, a visiting scholar at theEngineering Department of Cambridge University, and a visiting professor atthe Joseph Fourier Université de Grenoble. He is a founding member of theInternational Society of Structural and Multidisciplinary Optimization (ISSMO) andwas a regular contributor to its conferences and meetings. His main research field isthe design of optimal structures on which he has published over 50 papers.

The author has two married daughters living in Israel. Sossy Fuchs-Aroch, aclinical psychologist, raises her family in Binyamina, a sleepy village some 70 kmnorth of Tel Aviv, not far from a beautiful Mediterranean beach. Dr. Judith Weiss, apostdoctoral fellow at the Hebrew University, specializes in Christian Kabbala andother mystic stuff. She lives in Jerusalem with her family in a calm oasis in anotherwise bustling capital city.

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