Description
Book SynopsisBeam theories are exploited worldwide to analyze civil, mechanical, automotive, and aerospace structures. Many beam approaches have been proposed during the last centuries by eminent scientists such as Euler, Bernoulli, Navier, Timoshenko, Vlasov, etc.
Table of ContentsAbout the Authors ix Preface xi
Introduction xiii
References xvii
1 Fundamental equations of continuous deformable bodies 1
1.1 Displacement, strain, and stresses 1
1.2 Equilibrium equations in terms of stress components and boundary conditions 3
1.3 Strain displacement relations 4
1.4 Constitutive relations: Hooke’s law 4
1.5 Displacement approach via principle of virtual displacements 5
References 8
2 The Euler–Bernoulli and Timoshenko theories 9
2.1 The Euler–Bernoulli model 9
2.1.1 Displacement field 10
2.1.2 Strains 12
2.1.3 Stresses and stress resultants 12
2.1.4 Elastica 15
2.2 The Timoshenko model 16
2.2.1 Displacement field 16
2.2.2 Strains 16
2.2.3 Stresses and stress resultants 17
2.2.4 Elastica 18
2.3 Bending of a cantilever beam: EBBT and TBT solutions 18
2.3.1 EBBT solution 19
2.3.2 TBT solution 20
References 22
3 A refined beam theory with in-plane stretching: the complete linear expansion case 23
3.1 The CLEC displacement field 23
3.2 The importance of linear stretching terms 24
3.3 A finite element based on CLEC 28
Further reading 31
4 EBBT, TBT, and CLEC in unified form 33
4.1 Unified formulation of CLEC 33
4.2 EBBT and TBT as particular cases of CLEC 36
4.3 Poisson locking and its correction 38
4.3.1 Kinematic considerations of strains 38
4.3.2 Physical considerations of strains 38
4.3.3 First remedy: use of higher-order kinematics 39
4.3.4 Second remedy: modification of elastic coefficients 39
References 42
5 Carrera Unified Formulation and refined beam theories 45
5.1 Unified formulation 46
5.2 Governing equations 47
5.2.1 Strong form of the governing equations 47
5.2.2 Weak form of the governing equations 54
References 63
Further reading 63
6 The parabolic, cubic, quartic, and N-order beam theories 65
6.1 The second-order beam model, N =2 65
6.2 The third-order, N = 3, and the fourth-order, N = 4, beam models 67
6.3 N-order beam models 69
Further reading 71
7 CUF beam FE models: programming and implementation issue guidelines 73
7.1 Preprocessing and input descriptions 74
7.1.1 General FE inputs 74
7.1.2 Specific CUF inputs 79
7.2 FEM code 85
7.2.1 Stiffness and mass matrix 85
7.2.2 Stiffness and mass matrix numerical examples 91
7.2.3 Constraints and reduced models 95
7.2.4 Load vector 98
7.3 Postprocessing 100
7.3.1 Stresses and strains 101
References 103
8 Shell capabilities of refined beam theories 105
8.1 C-shaped cross-section and bending–torsional loading 105
8.2 Thin-walled hollow cylinder 107
8.2.1 Static analysis: detection of local effects due to a point load 109
8.2.2 Free-vibration analysis: detection of shell-like natural modes 112
8.3 Static and free-vibration analyses of an airfoil-shaped beam 116
8.4 Free vibrations of a bridge-like beam 119
References 121
9 Linearized elastic stability 123
9.1 Critical buckling load classic solution 123
9.2 Higher-order CUF models 126
9.2.1 Governing equations, fundamental nucleus 127
9.2.2 Closed form analytical solution 127
9.3 Examples 128
References 132
10 Beams made of functionally graded materials 133
10.1 Functionally graded materials 133
10.2 Material gradation laws 136
10.2.1 Exponential gradation law 136
10.2.2 Power gradation law 136
10.3 Beam modeling 139
10.4 Examples 141
References 148
11 Multi-model beam theories via the Arlequin method 151
11.1 Multi-model approaches 152
11.1.1 Mono-theory approaches 152
11.1.2 Multi-theory approaches 152
11.2 The Arlequin method in the context of the unified formulation 153
11.3 Examples 157
References 167
12 Guidelines and recommendations 169
12.1 Axiomatic and asymptotic methods 169
12.2 The mixed axiomatic–asymptotic method 170
12.3 Load effect 174
12.4 Cross-section effect 175
12.5 Output location effect 177
12.6 Reduced models for different error inputs 178
References 179
Index 181
