Structural Analysis with Finite Elements

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• Structural Analysis with Finite Elements
• Preface
• Contents
• 1. What are finite elements?
• 1.1 Introduction
• 1.2 Key points of the FE method
• 1.3 Potential energy
• 1.4 Projection
• 1.5 The error of an FE solution
• 1.6 A beautiful idea that does not work
• 1.7 Set theory
• 1.8 Principle of virtual displacements
• 1.9 Taut rope
• 1.10 Least squares
• 1.11 Distance inside = distance outside
• 1.12 Scalar product and weak solution
• 1.13 Equivalent nodal forces
• 1.14 Concentrated forces
• 1.15 Green’s functions
• 1.16 Practical consequences
• 1.17 Why finite element results are wrong
• 1.18 Proof
• 1.19 Influence functions
• 1.20 Accuracy
• 1.21 Why resultant stresses are more accurate
• 1.22 Why stresses at midpoints are more accurate
• 1.23 Why stresses jump
• 1.24 Why finite element support reactions are relatively accurate
• 1.25 Gauss points
• 1.26 The Dirac energy
• 1.27 How to predict changes
• 1.28 The influence of a single element
• 1.29 Retrofitting structures
• 1.30 Local errors and pollution
• 1.31 Adaptive methods
• 1.32 St. Venant’s principle
• 1.33 Singularities
• 1.34 Actio = reactio?
• 1.35 The output
• 1.36 Support conditions
• 1.37 Equilibrium
• 1.38 Temperature changes and displacement of supports
• 1.39 Stability problems
• 1.40 Interpolation
• 1.41 Polynomials
• 1.42 Infinite energy
• 1.43 Conforming and nonconforming shape functions
• 1.44 Partition of unity
• 1.45 Generalized finite element methods
• 1.46 Elements
• 1.47 Stiffness matrices
• 1.48 Coupling degrees of freedom
• 1.49 Numerical details
• 1.50 Warning
• 2.1 Influence functions or Betti’s theorem Beams
• 2.2 Structural analysis with boundary elements
• 2.3 Comparison finite elements—boundary elements
• 3. Frames
• 3.1 Introduction
• 3.2 The FE approach
• 3.3 Finite elements and the slope deflection method
• 3.4 Stiffness matrices
• 3.5 Approximations for stiffness matrices
• 3.6 Cables
• 3.7 Hierarchical elements
• 3.8 Sensitivity analysis
• 4.1 Simple example
• 4.2 Strains and stresses
• 4.3 Shape functions
• 4.4 Plane elements
• 4.5 The patch test
• 4.6 Volume forces
• 4.7 Supports
• 4.8 Nodal stresses and element stresses
• 4.9 Truss and frame models
• 4.10 Two-bay wall
• 4.11 Multistory shear wall
• 4.12 Shear wall with suspended load
• 4.13 Shear wall and horizontal load
• 4.14 Equilibrium of resultant forces
• 4.15 Adaptive mesh refinement
• 4.16 Plane problems in soil mechanics
• 4.17 Incompressible material
• 4.18 Mixed methods
• 4.19 Influence functions for mixed formulations
• 4.20 Error analysis
• 4.21 Nonlinear problems
• 5.1 Kirchhoff plates
• 5.2 The displacement model
• 5.3 Elements
• 5.4 Hybrid elements
• 5.5 Singularities of a Kirchhoff plate
• 5.6 Reissner–Mindlin plates
• 5.7 Singularities of a Reissner–Mindlin plate
• 5.8 Reissner–Mindlin elements
• 5.9 Supports
• 5.10 Columns
• 5.11 Shear forces
• 5.12 Variable thickness
• 5.13 Beam models
• 5.14 Wheel loads
• 5.15 Circular slabs
• 5.16 T beams
• 5.17 Foundation slabs
• 5.18 Direct design method
• 5.19 Point supports
• 5.20 Study
• 5.21 Sensitivity analysis
• 6.1 Shell equations
• 6.2 Shells of revolution
• 6.3 Volume elements and degenerate shell elements
• 6.4 Circular arches
• 6.5 Flat elements
• 6.6 Membranes
• 7. Theoretical details
• 7.1 Scalar product
• 7.2 Green’s identities
• 7.3 Green’s functions
• 7.4 Generalized Green’s functions
• 7.5 Nonlinear problems
• 7.6 The derivation of influence functions
• 7.7 Weak form of influence functions
• 7.8 Influence functions for other quantities
• 7.9 Shifted Green’s functions
• 7.10 The dual space
• 7.11 Some concepts of error analysis
• 7.12 Important equations and inequalities
• References
• Index

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