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More About This Title Beam Structures - Classical and Advanced Theories
English
Beam Structures: Classical and Advanced Theories proposes a new original unified approach to beam theory that includes practically all classical and advanced models for beams and which has become established and recognised globally as the most important contribution to the field in the last quarter of a century.
The Carrera Unified Formulation (CUF) has hierarchical properties, that is, the error can be reduced by increasing the number of the unknown variables. This formulation is extremely suitable for computer implementations and can deal with most typical engineering challenges. It overcomes the problem of classical formulae that require different formulas for tension, bending, shear and torsion; it can be applied to any beam geometries and loading conditions, reaching a high level of accuracy with low computational cost, and can tackle problems that in most cases are solved by employing plate/shell and 3D formulations.
Key features:
- compares classical and modern approaches to beam theory, including classical well-known results related to Euler-Bernoulli and Timoshenko beam theories
- pays particular attention to typical applications related to bridge structures, aircraft wings, helicopters and propeller blades
- provides a number of numerical examples including typical Aerospace and Civil Engineering problems
- proposes many benchmark assessments to help the reader implement the CUF if they wish to do so
- accompanied by a companion website hosting dedicated software MUL2 that is used to obtain the numerical solutions in the book, allowing the reader to reproduce the examples given in the book as well as to solve other problems of their own www.mul2.com
Researchers of continuum mechanics of solids and structures and structural analysts in industry will find this book extremely insightful. It will also be of great interest to graduate and postgraduate students of mechanical, civil and aerospace engineering.
English
Erasmo Carrera, Politecnico di Torino, Italy
Erasmo Carrera is Professor of Aerospace Structures and Computational Aeroelasticity and Deputy Director of Department of Aerospace Engineering at the Politecnico di Torino. He has authored circa 200 journal and conference papers. His research has concentrated on composite materials, buckling and postbuckling of multilayered structures, non-linear analysis and stability, FEM; nonlinear analysis by FEM; development of efficient and reliable FE formulations for layered structures, contact mechanics, smart structures, nonlinear dynamics and flutter, and classical and mixed methods for multilayered plates and shells.
Gaetano Giunta, Centre de Recherche Public Henri Tudor, Luxembourg
Gaetano Giunta is a research scientist in the Department of Advanced Materials and Structures in the Centre de Recherche Public Henri Tudor.
Marco Petrolo, Politecnico di Torino, Italy
Marco Petrolo is a research scientist in the Department of Aeronautics and Space Engineering at the Politecnico di Torino.
English
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, andN-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
