Hencky Bar-chain/Net for Structural Analysis
- Length: 308 pages
- Edition: 1
- Language: English
- Publisher: Wspc (Europe)
- Publication Date: 2020-05-14
- ISBN-10: 1786347989
- ISBN-13: 9781786347985
- Sales Rank: #12583849 (See Top 100 Books)
As an emerging discrete structural model, the Hencky bar-chain/net model (HBM) has shown its advantages over other numerical methods in some problems. Owing to the discrete properties of HBM, it is also a suitable model for nano-scale structures which are currently a very hot research topic in mechanics.
This book introduces the concepts and previous research of the Hencky bar-chain/net model, before demonstrating how beams, columns, arches, rectangular plates and circular plates could be successfully modelled by HBM. HBM comprises rigid bars connected by frictionless hinges with elastic rotational springs (and a system of torsional springs in the cells for plates). In the treatment of the above-mentioned structures, HBM is found to be mathematically equivalent to the first order central finite difference method (FDM). So HBM may be regarded as the physical structural model behind the FDM.
This book is a compilation of the authors’ research on the development of the Hencky bar-chain/net model, and is organized according to the development and application of HBM for beams, columns, frames, arches and rings, and plates. Exercises are provided at the end of each chapter to aid comprehension and guide learning. It is a useful reference for students, researchers, academics and practitioners in the field of structural analysis.
Readership: Senior undergraduate students, graduate students, researchers and practitioners in the field of structural analysis, academics who are teaching courses on structural stability and vibration.
CONTENTS Preface About the Authors 1. Hencky Bar-Chain Model 1.1. Introduction 1.2. Literature Survey on HBM 1.2.1. Beams and columns 1.2.2. Frames 1.2.3. Arches and rings 1.2.4. Plates 1.3. Layout of the Book References 2. Uniform Beams 2.1. Introduction 2.2. Uniform Beams with Internal and End Elastic Restraints 2.2.1. Problem definition 2.2.2. FDM for uniform beams with elastic restraints 2.2.3. HBM for uniform beams with elastic restraints 2.2.4. Equivalent spring stiffnesses of FDM and HBM 2.2.5. Illustrative example problems for HBM with elastic springs 2.2.5.1. Pinned–pinned beam with internal rigid support at midspan 2.2.5.2. Clamped–clamped HBM with an internal hinge at midspan 2.2.5.3. Pinned–pinned HBM with internal elastic springs at midspan 2.2.6. Exact buckling and vibration solutions for HBM with elastic restraints 2.2.6.1. Exact buckling loads of HBM in closed-form 2.2.6.2. Exact vibration frequencies of HBM in closed-form 2.3. Uniform Beams on Winkler Foundation 2.3.1. Problem definition 2.3.2. FDM for beam on partial variable foundation 2.3.3. HBM for beam on partial variable foundation 2.3.4. Example problems for beams on partial variable elastic foundation 2.3.4.1. Solutions for vibration problem 2.3.4.2. Solutions for buckling problem 2.3.4.3. Special case of beam on full uniform elastic foundation 2.3.5. Exact buckling and vibration solutions for HBM on partial elastic foundation of constant stiffness 2.3.5.1. Exact buckling loads of HBM in closed-form 2.3.5.2. Exact vibration frequencies of HBM in closed-form 2.4. Columns Under Self-Weight 2.4.1. Problem definition 2.4.2. FDM for columns under self-weight and end axial load 2.4.3. HBM for columns under self-weight and end axial load 2.4.4. Exact buckling solution for HBM under self-weight 2.5. Concluding Remarks 2.6. Exercises Appendix A. Stiffness Matrices K1,K2,K3,K4,K5,K6 for Exact Buckling Solutions for HBM with Elastic Restraints Appendix B. Stiffness Matrices K1,K2,K3,K4,K5,K6 for Exact Vibration Solutions for HBM with Elastic Restraints Appendix C. Stiffness Matrices K1,K2,K3,K4,K5,K6 for Exact Buckling Solutions for HBM on Elastic Foundation Appendix D. Stiffness Matrices K1,K2,K3,K4,K5,K6 for Exact Vibration Solutions for HBM on Elastic Foundation References 3. Non-uniform Beams 3.1. Introduction 3.2. Development of HBM for Non-uniform Beams 3.2.1. Problem definition 3.2.2. FDM for non-uniform beams 3.2.3. HBM for non-uniform beams 3.2.4. Equivalence between FDM and HBM 3.2.4.1. Internal spring stiffnesses of HBM based on FDM 3.2.4.2. End spring stiffnesses of HBM based on FDM 3.2.5. Example problems involving non-uniform beams 3.2.5.1. Solutions for vibration problem 3.2.5.2. Solutions for buckling problem 3.2.5.3. Special case of two stepwise beam 3.3. Exact Buckling Solution for Non-uniform HBM with a Specific Class of Spring Stiffnesses 3.4. HBM for Shape Optimization of Columns Against Buckling 3.4.1. Semi-analytical formulation for optimization of column under axial load 3.4.1.1. Clamped-free column 3.4.1.2. Pinned–pinned column 3.4.1.3. Clamped–spring-supported column 3.4.2. Semi-analytical formulation for optimization of column under uniform distributed axial load 3.5. Concluding Remarks 3.6. Exercises References 4. Frames 4.1. Introduction 4.2. HBM for Buckling and Vibration of Typical Member 4.2.1. Problem definition 4.2.2. Development of HBM 4.2.3. HBM matrix formulations 4.2.3.1. Elastic stiffness matrix for elastic strain energy 4.2.3.2. Geometric stiffness matrix for potential energy due to axial force 4.2.3.3. Mass matrix for kinetic energy 4.2.3.4. Total energy and matrix transformations 4.2.3.5. Governing equation from energy principle 4.3. HBM for Buckling of General Frame with Multi-member Connection 4.3.1. Problem definition 4.3.2. Development of HBM 4.3.3. Energy formulations for multi-member connections 4.3.3.1. Four-member connection 4.3.3.2. Two- and three-member connections 4.3.3.3. Simplified expression for two-member connection 4.3.4. HBM energy formulations in matrix 4.3.4.1. Elastic strain energy 4.3.4.2. Potential energy due to axial force 4.3.4.3. Total energy and matrix transformations 4.3.4.4. Governing equation by applying energy principle 4.4. Illustrative Examples 4.4.1. Fixed-base column with/without top restraint 4.4.1.1. Formulation using HBM 4.4.1.2. Results 4.4.1.3. Discussions 4.4.2. Buckling and vibration of non-symmetric portal frame 4.4.2.1. Formulation using HBM 4.4.2.2. Buckling results 4.4.2.3. Vibration results 4.4.2.4. Buckling mode shapes 4.4.3. Buckling of trapezoidal portal frame 4.4.3.1. Formulation using HBM 4.4.3.2. Results 4.4.4. Buckling of two-storey frame with semi-rigid beam–column connections 4.5. Comparing HBM, Traditional Matrix Method ,and FEM 4.5.1. Traditional matrix stiffness method (MSM) 4.5.1.1. Matrix stiffness method 4.5.1.2. MSM for buckling of non-symmetric portal frame 4.5.2. Portal frame with local structural changes 4.5.2.1. Portal frame with local damage or stiffening in column 4.5.2.2. Portal frame with internal hinge in column 4.5.3. Portal frame with non-uniform columns 4.5.3.1. Formulation using HBM 4.5.3.2. Buckling results 4.5.4. Advantage of HBM matrix method over MSM and FEM 4.6. Concluding Remarks 4.7. Exercises Appendix. MATLAB Code for Stiffness Matrices in HBM Matrix Method References 5. Arches and Rings 5.1. Introduction 5.2. Assumptions on Load Behavior in Arch Buckling 5.3. Funicular Arch Buckling Problem 5.3.1. Problem definition 5.3.2. Condition for buckling and axial force expressions 5.4. HBM for Buckling Analysis of Arches 5.4.1. Description of HBM 5.4.2. HBM stiffness matrix formulation 5.4.3. Governing equation considering compatibility conditions 5.4.4. Preprocessing of input parameters 5.5. Illustrative Problems 5.5.1. Pinned-ended circular arch under uniform radial pressure 5.5.2. Fixed-ended circular arch under uniform radial pressure 5.5.3. Rotationally restrained circular arch under uniform radial pressure 5.5.4. Symmetric catenary arch under uniform vertical load along arc length 5.5.5. Non-symmetric catenary arch under uniform vertical load along arc length 5.5.6. Catenary arch with an internal rotational spring 5.6. Buckling of Triangular Arches 5.7. Buckling of Trapezoidal Arches 5.8. Buckling of Rings 5.9. Concluding Remarks 5.10. Exercises Appendix. MATLAB Code for Buckling Analysis of Arches Using HBM References 6. Plates 6.1. Introduction 6.2. Rectangular Plates with Elastic Restraints 6.2.1. Governing equation for HBM 6.2.2. Boundary spring stiffness of HBM 6.2.3. Mixed boundary condition and point support for HBM 6.2.3.1. Mixed elastic restraints 6.2.3.2. Point supports 6.2.4. Illustrative examples of HBM for buckling and vibration of rectangular plates 6.2.4.1. Simply supported plate with two short edges elastically restrained 6.2.4.2. Square plate with different kinds of point supports 6.2.4.3. Simply supported plate with one edge partially clamped 6.2.4.4. Simply supported plate with two opposite edges partially clamped 6.2.4.5. Simply supported plate with two short edges mixed elastically restrained 6.2.4.6. Plate with mixed boundary condition and interior point support 6.2.5. Exact buckling and vibration solutions for HBM 6.2.5.1. SSSS plate 6.2.5.2. SCSCplate 6.3. Circular and Annular Plates with Elastic Restraints 6.3.1. Governing equation for HBM 6.3.2. Edge spring stiffnesses of HBM 6.3.2.1. Outer edge 6.3.2.2. Inner edge 6.3.3. Solutions for HBM 6.3.3.1. Annular HBM 6.3.3.2. CircularHBM 6.4. Exercises References Index
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