Analysis of Thin-Walled Beams

This book presents a comprehensive introduction to an advanced beam theory applicable to thin-walled beams of rectangular and arbitrarily-shaped cross-sections. Furthermore, it describes a unique beam-based approach to handling joint structures consisting of thin-walled beams, compiled here for the first time. This higher-order beam theory (HoBT), developed by the authors over the past two decades, uses more than six degrees of freedom (DOFs) in contrast to the classical theories, which use only six DOFs. The additional degrees of freedom describe sectional deformations such as warping and distortion. This book presents a novel systematic procedure to derive the sectional deformations analytically for rectangular cross-sections and numerically for arbitrarily-shaped cross-sections. This book is a must for structural/mechanical engineers who wish to understand and design structures involving thin-walled beams. Preface Contents 1 Introduction 1.1 Beam Analysis for a Conceptual Design 1.2 Vlasov Beam Theory 1.2.1 Vlasov Beam Theory for Thin-Walled Open-Section Beams 1.2.2 Vlasov Beam Theory for Thin-Walled Closed-Section Beams 1.3 Higher-Order Beam Theory (HoBT) 1.4 Other Higher-Order Beam Theories References 2 Torsional Warping and Torsional Distortion as Fundamental Deformable Section Modes 2.1 Vlasov Beam Theory for Thin-Walled Open Cross-Sectioned Beams 2.1.1 Kinematic Assumptions, Displacements, and Strains 2.1.2 Stresses 2.2 Vlasov Beam Theory for Thin-Walled Closed Cross-Sectional Beams 2.3 Torsional Warping and Torsional Distortion of the HoBT 2.3.1 Simple Torsion Problem of a Thin-Walled Rectangular Cross-Sectional Beam 2.3.2 Derivation of the Shape Functions of Torsional Warping and Distortion References 3 One-Dimensional Governing Equations and Finite Element Formulation 3.1 Analysis of the Torsion of Thin-Walled Box Beams Using Three Modes θ, W and χ 3.1.1 Three-Dimensional Displacements, Strains, and Stresses at a General Point 3.1.2 Governing Equations 3.1.3 Finite Element Formulation 3.1.4 Examples of Finite Element Solutions 3.1.5 Analytic Solutions* 3.2 Finite Element Implementation Using an Arbitrary Number of Section Mode Shapes for General Loading 3.2.1 Three-Dimensional Displacements, Strains, and Stresses at a General Point for General Cases 3.2.2 Finite Element Formulation 3.2.3 Governing Equations* References 4 Sectional Shape Functions for a Box Beam Under Torsion: Membrane Field 4.1 General Field Relationship for the Higher-Order Deformable Section Modes of a Wall-Membrane Field 4.2 Generalized Force-Stress Relationship for Zeroth-Order Modes 4.3 Derivation of Higher-Order Deformable Section Modes by Means of a Recursive Analysis 4.3.1 Derivation of ψsχ1 (Shape Function of the First-Order Distortion Mode) 4.3.2 Derivation of ψzW1 (Shape Function of the First-Order Warping Mode) 4.3.3 Derivation of ψsχ2 4.3.4 Derivation of ψzW2 References 5 Sectional Shape Functions for a Box Beam Under Torsion: Wall-Bending Field 5.1 General Field Relationship for Higher-Order Deformable Section Modes of a Wall-Bending Field 5.2 Generalized Force-Stress Relationship 5.3 Derivation of Sectional Shape Functions ψnxk of Unconstrained Distortion Mode χk 5.3.1 Derivation of ψnχk 5.3.2 Relationship Between the Generalized Force (Sk) and Stress (overlineσzz) 5.4 Derivation of the Sectional Shape Functions { ψnoverlinek ,ψnk } of Constrained Distortion Modes { overlinek ,k } 5.4.1 Derivation of ψnoverline1 5.4.2 Derivation of ψnoverline2 5.4.3 Derivation of ψnoverlineN 5.4.4 Correction for Corner Conditions 5.4.5 Finite Element Formulation 5.5 Case Studies 5.5.1 Case Study 1: Static Wall-Membrane Response by Torsional Moment Mz 5.5.2 Case Study 2: Coupled Response of a Wall Membrane and Wall Bending by Surface Traction { tzz ,tzn ,tzs } 5.5.3 Case Study 3: Free Vibration Response References 6 Sectional Shape Functions for a Box Beam Under Extension 6.1 General Field Relationships for Higher-Order Deformable Section Modes of a Wall-Membrane Field 6.1.1 Displacements, Stress, Strain Fields, and Generalized Forces 6.1.2 Generalized Force-Stress Relationship for the Zeroth-Order Mode 6.2 Derivation of ψsχk and ψzWk by Means of a Recursive Analysis 6.2.1 Derivation of ψsχ1 6.2.2 Derivation of ψzW1 6.2.3 Derivation of ψsχ2 6.2.4 Derivation of ψzW2 6.2.5 Derivation of ψsχk and ψzWk for k ge3 6.3 n-Directional Displacements and Resulting Stress and Strain Fields 6.4 Generalized Force-Stress Relationships for Constrained Distortion Modes { overlinek ,k } 6.5 Derivation of ψnχk of Distortion Mode χk and Related Analysis 6.5.1 Derivation of ψnχk 6.5.2 Relationship Between the Generalized Force (Sk) and Stress (overlineσzz) 6.6 Derivation of the Sectional Shape Functions { ψnoverlinek ,ψnk } of Constrained Distortion Modes { overlinek ,k } 6.7 Finite Element Formulation 6.8 Case Studies 6.8.1 Case Study 1: Static Wall-Membrane Response by Axial Force Fz 6.8.2 Case Study 2: Mixed Response of Wall-Membrane and Wall-Bending Deformations by Surface Traction { tzz ,tzn } 6.8.3 CaseStudy3: Free Vibration Response References 7 Sectional Shape Functions for a Box Beam Under Flexure 7.1 General Field Relationships for Higher-Order Deformable Section Modes of a Wall-Membrane Field 7.1.1 Displacements, Stress, Strain Fields, and Generalized Forces 7.1.2 Generalized Force-Stress Relationship of the Zeroth-Order Modes 7.2 Derivation of ψsχk ( s ) and ψzWk ( s ) by Means of a Recursive Analysis 7.2.1 Derivation of ψsχ1 7.2.2 Derivation of ψzW1 7.2.3 Derivation of ψsχ2 7.2.4 Derivation of ψzW2 7.3 n-Directional Displacements and Resulting Stress and Strain Fields 7.4 Derivation of ψnχk and the Generalized Force-Stress Relationship for Mode χk 7.4.1 Derivation of ψnχk 7.4.2 Relationship Between the Generalized Force (Sk) and Stress (overlineσzz) 7.5 Derivation of { ψnoverlinek ,ψnk } of Mode { overlinek ,k } 7.6 Case Studies 7.6.1 Case Study 1: Static Wall-Membrane Response by Vertical Force Fy 7.6.2 Case Study 2: Mixed Response of Wall-Membrane and Wall-Bending by Surface Tractions { tzz ,tzn } 7.6.3 Case Study 3: Free Vibration Response References 8 Bridging Between Rectangular Cross-Sections and Generally Shaped Cross-Sections 8.1 Higher-Order Modes for Cross-Sections with General Thin-Walled Shapes 8.2 Recursive Equations to Derive Sectional Shape Functions 8.2.1 Recursive Relationships to Derive Distortion Modes χk 8.2.2 Recursive Relationships to Derive Warping Modes Wk 8.2.3 Recursive Relationships to Derive Wall-Bending Modes k 9 Sectional Shape Functions of Thin-Walled Beams with General Cross-Section Shapes 9.1 Displacement, Strain, and Stress Fields at a Generic Point 9.2 Generalized Force-Stress Relationships 9.2.1 Shear Stress 9.2.2 Axial Stress 9.2.3 Wall-Bending Stress 9.3 Non-Deformable Section Modes 9.4 Fundamental Deformable Section Modes: Linear Warping and Inextensional Distortion 9.4.1 Deriving ψzW0 and ψsχ0 as Solutions to an Eigenvalue Problem 9.4.2 n-directional Shape Function ψnχ0 9.5 Higher-Order Deformable Section Modes 9.5.1 Higher-Order Unconstrained Distortion Modes ( χ) Involving Wall Extension 9.5.2 Non-Linear Higher-Order Warping Modes (W) 9.5.3 Higher-Order Constrained Distortion Modes (Η) Not Involving Wall Extension 9.6 Case Studies 9.6.1 Static Analysis: A Cantilever Beam with an Open Cross-Section 9.6.2 Static Analysis: A Simply Supported Beam with an Open Cross-Section 9.6.3 Static Analysis: A Cantilever Beam with a Closed Cross-Section 9.6.4 Modal Analysis: A Beam with a Flanged Cross-Section with a Free-free Support Condition References 10 Joint Structures of Box Beams 10.1 Higher-Order Beam Theory for Out-of-Plane Bending Problems of a Box Beam 10.2 Sectional Resultants and Edge Resultants 10.3 Generalized Force Equilibrium Conditions 10.4 Field Variable Matching Conditions 10.5 Verification with Numerical Examples 10.5.1 Finite Element Equations 10.5.2 Case Study 1: Two-Box Beam Joint System 10.5.3 Case Study 2: T-Joint System 10.5.4 Case Study 3: N Box Beam-Joint System References 11 Joint Structures of Thin-Walled Beams with General Section Shapes 11.1 Joint Section and Connection Points 11.2 Kinematics of a Joint Section 11.2.1 Rotations Calculated on a Joint Section 11.2.2 Displacements Calculated on a Joint Section 11.3 Implementation 11.4 Numerical Examples 11.4.1 Two-Beam-Joint Structure with a Uniform Rectangular Cross-Section 11.4.2 Two-Beam-Joint Structure Having an I-shaped Cross-Section 11.4.3 T-joint Structure with Mixed Cross-Section Shapes 11.4.4 Simplified Vehicle Frame Appendix References Index