Classical and Advanced Theories
This text proposes a new original unified approach to beam theory that includes almost all classical and advanced models for beams.
1 Fundamental Equations of Continuous Deformable Bodies.
1.1 Displacement, strain and stresses.
1.2 Equilibrium Equations in term of Stress Components and Boundary Conditions.
1.3 Strain Displacement Relations.
1.4 Constitutive Relations: Hooke`s Law.
1.5 Displacement Approach via Principle of Virtual Displacements.
2 The Euler-Bernoulli and Timoshenko Theories.
2.1 The Euler-BernoulliModel.
2.1.1 Displacement field.
2.1.3 Stresses and stress resultants.
2.2 The Timoshenko Model.
2.2.1 Displacement Field.
2.2.3 Stresses and stress resultants.
2.3 Bending of a cantilever beam: EBBT and TBT solutions.
2.3.1 EBBT solution.
2.3.2 TBT solution.
3 A refined beam theory with in-plane stretching: the complete linear expansion case, CLEC.
3.1 The CLEC displacement field.
3.2 The importance of linear stretching terms.
3.3 A Finite Element based on CLEC.
4 EBBT, TBT, and CLEC in Unified Form.
4.1 Unified Formulation of CLEC.
4.2 EBBT and TBT as particular cases of CLEC.
4.3 Poisson Locking and its Correction.
4.3.1 Kinematic considerations on strains.
4.3.2 Physical considerations on strains.
4.3.3 First remedy: use of higher-order kinematics.
4.3.4 Second remedy: modification of elastic coefficients.
5 Carrera Unified Formulation and Refined Beam Theories.
5.1 Unified Formulation.
5.2 Governing Equations.
5.2.1 Strong Form of the Governing Equations.
5.2.2 Weak Form of the Governing Equations.
6 The parabolic, cubic, quartic and N-order beam theories.
6.1 The second-order beam model, N = 2.
6.2 The third-order, N = 3, and the fourth-order,N = 4, beam models
6.3 N-order beam models.
7 CUF Beam Finite Element Models: Programming and Implementation Issue Guidelines.
7.1 Preprocessing and Input Descriptions.
7.1.1 General FE Inputs.
7.1.2 Specific CUF Inputs.
7.2 FEM Code.
7.2.1 Stiffness and Mass Matrix.
7.2.2 Stiffness and Mass Matrix Numerical Examples.
7.2.3 Constraints and Reduced Models.
7.2.4 Load vector.
7.3.1 Stresses and Strains.
8 Shell Capabilities of Refined Beam Theories.
8.1 C-Shaped Cross-Section and Bending-Torsional Loading.
8.2 Thin-Walled Hollow Cylinder.
8.2.1 Static Analysis: Detection of Local Effects due to a Point Load.
8.2.2 Free Vibration Analysis: Detection of Shell-Like Natural Modes.
8.3 Static and Free Vibration Analyses of an Airfoil-Shaped Beam.
8.4 Free Vibrations of a Bridge-Like Beam.
9 Linearized Elastic Stability.
9.1 Critical Buckling Load Classic Solution.
9.2 Higher-Order CUF Models.
9.2.1 Governing equations fundamental nucleus.
9.2.2 Closed form analytical solution.
10 Beams Made of Functionally Graded Materials.
10.1 Functionally Graded Materials.
10.2 Material Gradation Laws.
10.2.1 Exponential gradation law.
10.2.2 Power gradation law.
10.3 Beam Modeling.
11 Multi-Model Beam Theories via the Arlequin Method.
11.1 Multi-Model Approaches.
11.1.1 Mono-theory approaches.
11.1.2 Multi-theory approaches.
11.2 The Arlequin Method in the context of the Unified Formulation.
12 Guidelines and Recommendations.
12.1 Axiomatic and Asymptotic Methods
12.2 The Mixed Axiomatic/Asymptotic Method.
12.3 Load effect.
12.4 Cross-section effect.
12.5 Output location effect.
12.6 Reduced models for different error inputs.