Murari L. Gambhir

Stability Analysis and Design of Structures

Murari L. Gambhir

Stability Analysis and Design of Structures

With 159 Figures

~Springer

Dr. Murari LaI Gambhir Thapar Institute Engineering & Technology Department of Civil Engineering 147001 Patiala, India mlg_nrg@rediffmail.com

ISBN 978-3-642-05866-0 ISBN 978-3-662-09996-4 (eBook) DOI 10.1007/978-3-662-09996-4

Library of Congress Control Number: 2004106461

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Preface

The stability considerations are extremely important and inevitable in the design of many engineering structures compassing aeronautical engineering, civil engineering, mechanical engineering, naval architecture and applied mechanics wherein a designer is confronted by numerous stability problems. Most of the national standards have based their design codal provisions on the stability criteria, especially in design of steel structures. In view of making the engineers appreciative of limitations associated with many structural design codal provisions, most of the engineering colleges and universities offer the course on the subject as a part of curriculum.

A number of books on the subject are available in the market, which had been writ­ten before mid-eighties and treated the problems normally encountered in engineering mainly by classical techniques. In view of rapid advancements and improvements in the methods of analysis and in the computing environment, stiffness methods sup­ported by numerical techniques are being extensively applied to relatively complex real-life problems. The later approach is emphasized in the present book.

The text is specially designed to cater to the classroom or self-study needs of stu­dents at advanced undergraduate and graduate level in structural engineering, applied mechanics, aeronautical engineering, mechanical engineering and naval architecture. Although the special problems pertaining to these disciplines differ philosophically but analytical and design principles discussed in the text are generally applicable to all of them. The emphasis is on fundamental theory rather than specific applications.

The text addresses to the stability of key structural elements: rigid-body assem­blage, column, beam-column, beam, rigid frame, thin plate, arch, ring and shell. The text begins with introduction to general basic principles of mechanics. This is followed by a detailed discussion on stability analysis of rigid-body assemblage, column, beam-column, beam, rigid frame, plates, arch and shell arranged in different chapters from 1 to 9. In Chap. 10, the elastic theories of buckling have been extended to the inelastic range. Where as in Chap. 11 on the design for structural stability, the American national standard, Australian standard AS: 1250-1981, British code BS: 5940-1985 (Part-I) and Indian code of practice IS: 800-1984 have been compared for the provisions related to stability considerations and number of design illustra­tions have also been given. Each chapter contains numerous worked-out problems

VI Preface

to clarify the discussion of practical applications that will facilitate comprehension of basic principles from the field of stability theory. Wherever possible alternate ap­proaches to the solution of important problems have been given. Tables and formulae are devised in the form suitable for the use in the design office. Thus the book would also prove useful to the practicing engineers engaged in actual design. In addition exercise problems designed to support and extend the treatment are given at the end of each chapter. For more important ones answers have also been given. The illustration problems have been treated by the practical methods, which are best suited. There is conscious effort to present results in non-dimensional form to render the subject mat­ter independent of system of units. These non-dimensional parameters facilitate the application of results to different materials and structural configurations encountered in practice. A large amount of practical data in tabular form and simplified formulae are given to make them suitable for the use in the design of various components.

It is the opinion of the author that the undergraduate students should study first six chapters as a part of their required program of study. The remaining chapters can be studied at the graduate level. To make the fundamentals of stability analysis more understandable and meaningful, this text should be used at the level when the student has attained the basic knowledge of statics, solid mechanics or strength of materials and calculus. Only a minimum knowledge of calculus, Fourier series and Bessel functions is assumed on the part of reader. However, for reference necessary background information needed to deal with problems involving differential equations and Bessel functions is given in the appendix. The subject matter and its presentation sequence has been class tested over the past two decades. In the process students have made valuable suggestions for which author is grateful.

The author wishes to express his sincere gratitude to the authors of various books on the subject who have been an inspiration to developing this text. The author thanks all those who have assisted in various ways in preparation of this text. Particularly, he wishes to acknowledge the assistance rendered by Dr. Puneet Gambhir, Er. Mohit Gambhir and Er. Neha Gambhir in preparation of manuscript. The author is extremely grateful to his wife Ms Saroj Gambhir for the patience she has shown while he was busy completing this job. The assistance and advice received from Dr. Thomas Ditzinger and Ms. Gaby Maas, the Editor, of Springer-Verlag is gratefully acknowledged. The author welcomes suggestions from the readers for improvement in the subject matter in any manner.

Patiala, India May 12,2004

M. L. Gambhir

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Definitions of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Structural Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Methods for Stability Ananlysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Idealization of Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Equations of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Free-Body Diagrams........................................ 13 2.5 Work of Externally Applied Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5.1 Eigenwork and Displacement Works . . . . . . . . . . . . . . . . . . . . 17 2.5.2 Linear Springs....................................... 18 2.5.3 Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5.4 The Principle of Superposition of Mechanical Work . . . . . . . 22 2.5.5 Non-Linearities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6 Work of Internal Forces: Strain Energy. . . . . . . . . . . . . . . . . . . . . . . . . 25 2.7 The Work Equation......................................... 29 2.8 Energy Theorems of Elastic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.9 Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.9.1 Total Potential Energy of a Deformable Body. . . . . . . . . . . . . 44 2.9.2 Principle of Stationary Potential Energy . . . . . . . . . . . . . . . . . 45 2.9.3 Applications of Total Potential Energy Principles . . . . . . . . . 46

2.10 Methods of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.1 0.1 Method of Trial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.10.2 Galerkin Method..................................... 60 2.10.3 Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.10.4 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.11 Orthogonality of Buckling Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

VIII Contents

3 Rigid-Body Assemblages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.2 Methods of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.2.1 Equilibrium Approach................................ 87 3.2.2 Energy Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.3 Single-Degree-of-Freedom Rigid-Bar Assemblages . . . . . . . . . . . . . . 92 3.3.1 Modeling of Elastically Deformable Elements

by Equivalent Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.4 Two-Degree-of-Freedom Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.5 Discrete Element Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.6 Problems ................................................. 114

4 Buckling of Axially Loaded Members (Columns) .................. 119 4.1 Introduction ............................................... 119 4.2 Buckling Loads for Members with Different End Conditions . . . . . . 119

4.2.1 Hinged-Hinged Strut ................................. 120 4.2.2 Fixed-Free Cantilever Strut . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.2.3 Fixed-Hinged Strut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.2.4 Fixed-Fixed Strut .................................... 124 4.2.5 Struts with Elastic Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.2.6 Framed Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.3 Concept of Effective Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.4 Approximate Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.5 Large Deflection Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

5 Stability Analysis of Beam-Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.1 Introduction ............................................... 171 5.2 Derivation of Basic Equations ................................ 171 5.3 Analysis of Beam-Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

5.3.1 Beam-Column with Concentrated Loads . . . . . . . . . . . . . . . . . 173 5.3.2 Beam-Column with an Interior Moment . . . . . . . . . . . . . . . . . 176 5.3.3 Beam-Column Subjected to End Moments ............... 177 5.3.4 Beam-Columns Subjected to Distributed Loads ........... 181 5.3.5 Rotationally Restrained Beam-Columns ................. 184

5.4 Beam-Column with Elastic Supports .......................... 185 5.4.1 Differential Equation Method .......................... 185 5.4.2 Numerical Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

5.5 Strut with Initial Eccentricity ................................. 199 5.6 Interaction Equation ........................................ 201 5.7 Problems ................................................. 205

Contents IX

6 Stability Analysis of Frames ................................... 213 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.2 Classical Approach ......................................... 213

6.2.1 Continuous Columns and Beam-Columns ................ 213 6.2.2 Rigid-Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

6.3 Semi-Geometrical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 6.4 Stiffness Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

6.4.1 Criterion for Determination of Critical Load. . . . . . . . . . . . . . 233 6.4.2 Stiffness Matrix Including Axial Force Effects . . . . . . . . . . . . 235

6.5 Stability Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 6.5.1 Member with No Lateral Displacement .................. 240 6.5.2 Member Subjected to a Relative End Displacement .4. . . . . . 242

6.6 Frames with Sidesway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 6.6.1 Single-Bay Multi-Storey Frames ....................... 257 6.6.2 Multi-Bay Rigid Frames .............................. 265 6.6.3 Substitute Frame Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

6.7 Rigidly Connected Trusses ................................... 272 6.8 Moment Distribution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 6.9 Problems ................................................. 283

7 Buckling of Members Having Open Sections ..................... 291 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 7.2 Torsional Buckling ......................................... 291

7.2.1 Member Subjected to Torque .......................... 291 7 .2.2 Member Subjected to Axial Force . . . . . . . . . . . . . . . . . . . . . . 299

7.3 Lateral Buckling of Beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 7.3.1 Torsional Buckling due to Flexure ...................... 315 7.3.2 Torsional Buckling due to Flexure and Axial Force ........ 323

7.4 Lateral Buckling of Beams with Transverse Loads . . . . . . . . . . . . . . . 324 7.4.1 Lateral Buckling of a Cantilever Beam .................. 324 7 .4.2 Lateral Buckling of a Simply Supported Beam. . . . . . . . . . . . 328

7.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

8 Elastic Buckling of Thin Flat Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 8.2 Governing Differential Equations of Bending . . . . . . . . . . . . . . . . . . . 336

8.2.1 Boundary Conditions ................................. 341 8.3 Energy Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

8.3.1 Strain Energy of Plates ............................... 342 8.3.2 Potential Energy due to In-Plane Forces ................. 343

8.4 Buckling Analysis of Rectangular Plates . . . . . . . . . . . . . . . . . . . . . . . 344 8.4.1 Governing Differential Equation Solution . . . . . . . . . . . . . . . . 344 8.4.2 Stationary Potential Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 362

8.5 Buckling of Web Plates of Girders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 8.5.1 Buckling of Rectangular Plate in Shear . . . . . . . . . . . . . . . . . . 368

X Contents

8.5.2 Buckling of Rectangular Plate due to Non-Uniform Longitudinal Stresses ................................. 371

8.5.3 Buckling of Stiffened Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 8.6 Strength of Thin Plates in Compression . . . . . . . . . . . . . . . . . . . . . . . . 383 8.7 Plates Under Longitudinal Compression and Normal Loading ..... 387

8.7.1 Governing Differential Equation Method ................ 387 8.7.2 Energy Approach .................................... 390

8.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

9 Stability Analysis of Arches, Rings and Shells ..................... 397 9.1 Introduction ............................................... 397 9.2 Arches .................................................... 398

9.2.1 Flat Arches ......................................... 398 9.2.2 Circular Arches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

9.3 Stability of rings and tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 9.4 Elastic Instability of Thin Shells .............................. 410

9.4.1 Governing Differential Equation ........................ 411 9.4.2 Energy Approach .................................... 415

9.5 Problems ................................................. 419

10 Inelastic Buckling of Structures ................................ 425 10.1 Introduction ............................................... 425 10.2 Inelastic Buckling of Straight Columns ........................ 425

10.2.1 Stress-Strain Relationship ............................. 427 10.3 Theories of Inelastic Buckling ................................ 430

10.3.1 Reduced Modulus Theory ............................. 430 10.3.2 Tangent Modulus Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

10.4 Eccentrically Loaded Columns ............................... 436 10.4.1 Analysis of Short Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

10.5 Inelastic Buckling by Torsion and Flexure ...................... 443 10.6 Lateral Buckling of Beams in the Inelastic Range . . . . . . . . . . . . . . . . 443 10.7 Inelastic Buckling of Plates .................................. 444

10.7.1 Plates Subjected to Uniaxial Loading .................... 445 10.7.2 Plate Subjected to In plane Biaxial Loading .............. 448

10.8 Inelastic Buckling of the Shells ............................... 449 10.9 Problems ................................................. 451

11 Structural Design For Stability Of Members ..................... 453 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 11.2 Column Design Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 11.3 Local Plate Buckling of Structural Members . . . . . . . . . . . . . . . . . . . . 457

11.3 .1 Average Shear Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 11.3.2 Flexural Buckling of Webs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 11.3.3 Built-up Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

11.4 Beam Design Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

Contents XI

11.4.1 Lateral Buckling of Beams ............................ 468 11.4.2 Effective Length of Compression Flange . . . . . . . . . . . . . . . . . 469 11.4.3 Codal Provisions ..................................... 471 11.4.4 Bearing Compressive Stress ........................... 476

11.5 Stiffeners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 11.5.1 Vertical Stiffeners .................................... 478 11.5.2 Horizontal Stiffeners ................................. 479

11.6 Beam-Column Design Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 11.6.1 Codal Provisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 11.6.2 Design of a Beam-Column Member. . . . . . . . . . . . . . . . . . . . . 484

11.7 Optimum Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 11.8 Problems ................................................. 491

Appendix A: Stability Functions .................................... 493 A.l Stability Functions for Compression Members . . . . . . . . . . . . . . . . . . 493 A.2 Stability Functions for Tension Members . . . . . . . . . . . . . . . . . . . . . . . 497 A.3 Stability Magnification Factors for Members with Lateral Load . . . . 499

Appendix B: Effective Length of Stepped and Multiple Level Load Columns .............................. 503

Appendix C: Mathematical Essentials ............................... 515 C.1 Linear Differential Equations ................................. 515 C.2 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 C.3 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

Appendix D: General References ................................... 529

Subject Index .................................................... 531

The information contained in this text has been either generated or obtained from the sources believed to be reliable and much care has been taken by the author and the publishers to make the book error (factual or printing) free. However, nei­ther the Springer-Verlag nor its author guarantees the accuracy or completeness of any information published herein, and neither the author nor the publisher shall be responsible for any errors, omissions, or damages arising out of the use of this information. This work is published with an intention of making the fundamental principles of stability analysis clear, and not to render engineering or other pro­fessional services. For such services, the assistance of an appropriate professional should be sought.

About the Author

M.L. GAMBIDR is currently Director of Rayat Institute of Engineering and Infor­mation Technology, Railmajra, Punjab, India. Previously he was Professor & Head of Civil Engineering Department, and Dean Planning & Resource Generation at the Thapar Institute of Engineering & Technology, Patiala. He obtained his Bache­lor's and Master's degrees from University of Roorkee (presently Indian Institute of Technology, Roorkee), and his Ph.D. from Queen's University, Kingston, Canada.

His major research interests have been in the areas of structural Engineering particularly in structural failures and rehabilitation of structures, structural reliability; structural stability and dynamics; and High Performance Concrete. He has supervised 40 Masters and 6 Doctoral theses. He has wide experience in structural design of diverse types of structures in reinforced concrete and structural steel. Dr. Gambhir has published over sixty technical papers in reputed journals and has authored: Concrete Technology, Tata McGraw-Hill Publishing Company, 3rd Edition; and Reinforced Concrete Design, Macmillan (I) Ltd., 1st Edition.

He has been recipient of several awards including Agra University Bursary, National Scholarship, Research Fellowship, Roorkee University (now liT Roorkee) Gold Medal, Canadian Commonwealth Scholarship and merit scholarships. He is a member of Indian Society for Technical Education and the Indian Society for Earthquake Technology. He has been member of numerous committees.