Beam Structures - Classical and Advanced Theories

About 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