Modeling and Dimensioning of Structures

This book provides the main topics currently used for the calculus of structures. The reference establishes a link between the traditional approach on the strength of materials and the present finite element method, details the main aspects of practical modeling, and explores numerous case studies.

Autorentext

Daniel Gay was Director of the Laboratory of Mechanical
Design at the University of Toulouse, France, for more than 15
years. He published numerous papers in international scientific
journals and several books on composites materials.

Jacques Gambelin is Professor of Mechanical Design at the
University of Toulouse, France, and taught structural design
andcalculus for aeronautical applications for more than 20
years.

Inhalt

Preface xvii

Part 1. Level 1 1

Chapter 1. The Basics of Linear Elastic Behavior 3

1.1. Cohesion forces 4

1.2. The notion of stress 6

1.2.1. Definition 6

1.2.2. Graphical representation 7

1.2.3. Normal and shear stresses 8

1.3. Hooke's law derived from a uniaxially applied force 9

1.3.1. The stretch test 9

1.3.2. Linear mechanical behavior 12

1.3.3. Elastic mechanical behavior 12

1.3.4. Interpretation of the test at a macroscopic level 13

1.3.5. Interpretation of the test at a mesoscopic level 13

1.3.6. Interpretation of the test at a microscopic level 16

1.3.7. Summary 18

1.4. Plane state of stresses 20

1.4.1. Definition 20

1.4.2. Behavior relationships for state of plane stresses 22

1.4.3. Summary 35

1.5. Particular case of straight beams 36

1.5.1. Preliminary observations 36

1.5.2. Effects linked to the resultant forces and moments 38

Chapter 2. Mechanical Behavior of Structures: An Energy Approach 51

2.1. Work and energy 51

2.1.1. Elementary work developed by a force 51

2.1.2. Elementary work developed by a moment 52

2.2. Conversion of work into energy 53

2.2.1. Potential energy of deformation 53

2.2.2. Potential energy for a spring 55

2.3. Some standard expressions for potential deformation energy 58

2.3.1. Deformation energies in a straight beam 58

2.3.2. Deformation energy under plane stresses 74

2.4. Work produced by external forces on a structure 81

2.4.1. Beam under plane bending subjected to two forces 82

2.4.2. Beam in plane bending subject to n forces 99

2.4.3. Generalization to any structure 103

2.4.4. Summary 112

2.5. Links of a structure with its surroundings 113

2.5.1. Example 113

2.5.2. Generalization 118

2.6. Stiffness of a structure 119

2.6.1. Preliminary note 119

2.6.2. Stiffness matrix 121

2.6.3. Examples 121

2.6.4. Influence of the positioning 125

2.6.5. Deformation energy and stiffness matrix 139

Chapter 3. Discretization of a Structure into Finite Elements 143

3.1. Preliminary observations 143

3.1.1. Problem faced 143

3.1.2. Practical obtaining of the deformation energy for a complex structure 144

3.1.3. Local and global coordinates 147

3.2. Stiffness matrix of some simple finite elements 153

3.2.1. Truss element loaded under traction (or compression) 153

3.2.3. Beam element under plane bending 168

3.2.4. Triangular element for the plane state of stresses 178

3.3. Getting the global stiffness matrix of a structure 191

3.3.1. Objective 191

3.3.2. Mechanism of the assembly of elementary matrices 191

3.3.1. Introduction 201

3.4. Resolution of the system {F}={K}??{d} 203

3.4.1. Linkage conditions 203

3.4.2. Generalization of the method 205

3.5. Different types of finite elements available in industrial software 207

Chapter 4. Applications: Discretization of Simple Structures 209

4.1. Stiffness matrix of a spring 209

4.1.1. Helical spring 209

4.1.2. Spiral spring. 211

4.2. Assembly of elements 213

4.2.1. Example 1 213

4.2.2. Example 2 217

4.2.3. Example 3 222

4.2.4. Assembly of a truss element and a beam element under simple plane bending 228

4.3. Behavior in the global coordinate system 232

4.3.1. Plane assembly of two truss elements 232

4.4. Bracket 246

4.4.1. Objectives 246

4.4.2. Modelizing 247

4.4.3. Calculation of the elementary stiffness matrix in the global system 252

4.4.4. Assembly of the global stiffness matrix [K]str 256 4.4.5. Establishing the linkage and load...