Beam Structures
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Beam 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.  Most of these models are problem dependent: they provide reliable results for a given problem, for instance a given section and cannot be applied to a different one.

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

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.


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.


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, 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