Structural Mechanics Analytical and Numerical Approaches for Structural Analysis Lingyi Lu† School of Civil Engineering Southeast University, Nanjing, China Junbo Jia Aker Solutions, Bergen, Norway Zhuo Tang National Wind Institute Texas Tech University, Lubbock, Texas, USA p, p, A SCIENCE PUBLISHERS BOOK A SCIENCE PUBLISHERS BOOK First edition published 2022 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2022 Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. 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For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data (applied for) ISBN: 978-0-367-55912-0 (hbk) ISBN: 978-0-367-55915-1 (pbk) ISBN: 978-1-003-09569-9 (ebk) DOI: 10.1201/9781003095699 Typeset in Times New Roman by Radiant Productions In Memory of Professor Lingyi Lu Preface On 13th of July, 2019, I was informed about sad news of the passing away of Professor Lingyi Lu, after battling pancreatic cancer for more than half a year. Just two days before this, I visited him at a hospital in Shanghai, and he informed me about the progress of this book and asked me to finalize it. Together with efforts of Dr. Zhuo Tang, a former PhD student of Prof. Lu, the book has finally been finished with a maximum preservation of Professor Lu’s academic thoughts. This book covers essential topics on structural mechanics and dynamics. It is mainly based on the contents of an undergraduate course in structural mechanics taught by Professor Lu at Southeast University, from the 1990s in the last century to 2019, and is an important contribution of Professor Lu’s lifetime academic and pedagogic activities that he has been deeply engaged in, reflecting the forefront education status of structural mechanics in China during that period. Modern structural mechanics originated from Galileo Galilei, Isaac Newton, and Robert Hooke. It is the computation of deformations, deflections, and internal forces or stresses within structures, either for design or for performance evaluation of existing structures. With the objective of providing principles of structural mechanics, topics covered in this book include both basic and advanced ones. Basic topics include geometric stability, internal forces and deflections of statically determinate structures, force and displacement method, and influence lines. Advanced topics include matrix displacement method for structural analysis, dynamics of structures, and limit load analysis. The book serves as a classroom textbook in structural mechanics. It is written in such a way that it can be followed by anyone with a basic knowledge of classical and material mechanics. While the book does not seek to promote any specific “school of thought,” it inevitably reflects the authors’ best practices and working habits. This is particularly apparent in the topics selected and the level of detail devoted to each of them, the choice of mathematical treatments and symbolic notations. It should not deter readers from seeking to find their own best practices and working habits. Many individuals have provided valuable supports to finalize the book. Those contributions come from the family members of Professor Lu: his wife, Professor Hong Zhuang, and his daughter, Hongling Lu, and from his former students: Hongsheng Wang, Jie Zhang, Yuanzhi Liu, Min Chen, and Wensong Hu. Preface v Professor Gang Wu from Southeast University has inquired about the book status several times and expressed an inspiring encouragement on behalf of Southeast University. When I visited Professor Lu in the hospital in his last stage, he had requested me to write a preface for this book, as both his friend and former undergraduate student, I came up with this preface with emotion, and with a deep memory of my undergraduate study at Southeast University, that has had a strong influence on my life and career after I left the University. Hope this book will be useful for both students and academic professionals. Junbo Jia Contents Preface iv 1. Geometric Stability and Types of Structures 1 1.1 Classifications of Structural Members and Connections 1 1.2 Introduction to Geometrically Stable and Unstable Systems 5 1.3 Rigid Degrees of Freedom and Constraints 6 1.4 Rules for Constructing Geometrically Stable Systems 9 1.4.1 Two-Body Rule 9 1.4.2 Three-Body Rule 11 1.4.3 The Dual-Link Rule 12 1.4.4 The Simply-Supported Rule 14 1.5 Examples of Geometric-Stability Analysis 16 2. Internal Forces in Statically Determinate Structures 22 2.1 Analysis of Beams and Frames 22 2.1.1 Properties of Moment Diagrams 22 2.1.2 Constructions of Moment Diagrams by the Superposition 24 Method in Segments 2.1.3 Calculations of Internal Forces through Moment Diagrams 29 2.1.4 Analysis of Frames 30 2.2 Analysis of Trusses and Composite Structures 36 2.2.1 Introduction 36 2.2.2 Analysis of Statically Determinate Trusses 36 2.2.3 Analysis of Statically Composite Structures 39 2.3 Analysis of Statically Determinate Arches 40 2.3.1 Comparison of Internal Forces in Arches and the 40 Corresponding Beams 2.3.2 The Ideal Axis of Parabolic Arch 42 3. Deflections of Statically Determinate Structures 45 3.1 Virtual-Work Principle for Rigid Bodies and Its Applications 45 3.1.1 Work and Virtual Work 45 3.1.2 Principle of Virtual Displacements for Rigid Bodies 46 3.1.3 Principle of Virtual Forces for Rigid Bodies 49 3.2 Principle of Virtual Forces for Elastic Structures 51 Contents vii 3.3 Deflections Caused by External Loads 53 3.3.1 Calculation of Structural Displacements to Loads by 53 the Unit-Load Method 3.3.2 Graphic Multiplication and Its Applications 56 4. Force Method 62 4.1 Statically Indeterminate Structures 62 4.2 General Procedure of the Force Method 63 4.3 Analysis of Statically Indeterminate Structures under Loads 66 4.4 Symmetric Structures and Their Half-Structures 68 4.4.1 Symmetry of Structures, Loads and Responses 68 4.4.2 Half-Structures of Symmetric Structures 71 4.5 Analysis of Statically Indeterminate Structures Having Thermal 78 Changes, Fabrication Errors and Support Settlements 4.6 Deflections of Statically Indeterminate Structures 80 5. Displacement Method 85 5.1 Beams with Support Displacements and Slope-Deflection Equations 85 5.2 Displacement Method for Analyzing Frames under Nodal Loads 87 5.2.1 The Procedure Using Slope-Deflection Equations 87 5.2.2 The Procedure Directly Using Primary Systems 88 5.2.3 Unknown Degrees of Freedom of the Displacement Method 89 5.3 The Analysis of Frames Under In-Span Loading 93 5.3.1 Fixed-End Forces 93 5.3.2 Processing of In-Span Loads and Nodal Equivalent Loads 95 5.4 Examples of Frames with In-Span Loads 98 5.5 Moment Distribution Approach 103 5.5.1 Moment Distribution Approach for SDOF Structures 103 5.5.2 Moment Distribution Approach for MDOF Beams 105 6. Influence Lines for Statically Determinate Structures 112 6.1 Introduction 112 6.2 Influence Lines for Beams 112 6.2.1 Constructing Influence Lines by the Principle of 113 Virtual Displacements 6.3 Influence Lines for Trusses 115 6.4 Maximum Response at a Specific Point 116 6.4.1 Maximum Response at a Point under Live Loads 116 6.4.2 Maximum Response at a Point under a Set of Concentrated 117 Moving Loads 6.5 Moment Envelopes and Absolute Maximum Moments of Members 119 6.5.1 Definition of Moment Envelope 119 6.5.2 Moment Envelopes of Beams under Moving Loads 120 viii Structural Mechanics: Analytical and Numerical Approaches for Structural Analysis 7. Matrix Displacement Analysis 124 7.1 An Introductory Example 124 7.1.1 A Beam Element Type with Two Degree-of-Freedoms 124 (DOFs) – Beam1 7.1.2 Pre-processing – Discretizing and Digitizing of the 125 Continuous Beam 7.1.3 Calculate the Element Stiffness Matrices of the 126 Continuous Beam 7.1.4 Assembling Stiffness Equation of the Structure by the 126 Direct Stiffness Method 7.1.5 Solving and Post-processing 128 7.1.6 MATLAB Codes – Beam1 Package 129 7.2 Boundary Conditions and the Beam Element with Six DOFs 129 7.2.1 Analyzing an Unrestrained Continuous Beam by 129 Beam2 Element 7.2.2 Post-Imposing of Boundary Conditions and 134 Support Settlements 7.3 Frames Subjected to Nodal Loading – Change of Coordinates 142 7.3.1 Introduction 142 7.3.2 Change of Coordinates 142 7.3.3 Element Stiffness Matrices in Global Coordinates and the 144 Assembling Rules 7.3.4 Analysis of the Frames Subjected to Nodal Loads 145 7.4 Frames Subjected to In-span Loads and Equivalent Nodal Loads 152 7.4.1 The Equivalent Nodal Loads 152 7.4.2 The Stiffness Equation of the Structure 156 7.4.3 The End Displacement and Forces of the Structure 157 8. Dynamics of Structures 159 8.1 Introduction to Structural Dynamics 159 8.1.1 What is Structural Dynamics? 159 8.1.2 Models for Dynamic Analysis 160 8.1.3 Equations of Motion and Initial Conditions 162 8.1.4 Free Vibrations and Dynamic Properties 164 8.1.5 Dynamic Responses to External Excitations 165 8.1.6 Summary 167 8.2 Equations of Motion 167 8.2.1 Stiffness Method: The Dynamic-Equilibrium Procedure 167 8.2.2 Stiffness Method: The Virtual Constraint Approach 170 8.2.3 Flexibility Method to Formulate Equations of Motion 176 8.2.4 Stiffness Method: The Matrix Displacement Approach 179 and Static Condensation 8.2.5 Damping in Structures 188 Contents ix 8.3 Dynamic Properties of Structures 191 8.3.1 Vibrations of SDOF Systems: Natural Frequency and 191 Damping Ratio 8.3.2 Undamped Free Vibrations of MDOF Systems: Normal Modes 198 8.3.3 Properties of Modes 202 8.3.4 Rayleigh Damping Matrix 205 8.4 Analysis of Dynamic Responses Using the Mode 207 Superposition Method 8.4.1 Transient Responses of Uncoupled MDOF Systems to 207 Combined Excitations 8.4.2 Steady-State Responses of Uncoupled MDOF Systems to 212 Combined Excitations 8.4.3 Responses of Coupled MDOF Systems: Mode 215 Superposition Method 8.5 Analysis of the Dynamic Response Using the MATLAB ODE 221 Solver: “ode45” 8.6 Steady-State Responses to Separable Excitations 222 8.6.1 Response Spectra 224 8.6.2 Peak Responses to Separable Excitations and Modal 225 Combination Rules 8.7 Appendix for Structural Dynamics 227 8.7.1 Steady-State Responses to Space-time Coupled Excitations 227 9. Limit Loads of Structures 231 9.1 Introduction 231 9.2 Theorems of Plasticity 235 9.3 Applications of the Upper Bound Theorem 236 9.4 Limit Analysis by Linear Programing 240 Appendix 245 Index 247