# Finite Elements

## Structural Analysis with Finite Elements

—–1 What are finite elements?
———-1.1 Introduction to finite elements
———-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 Greens 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
———-1.32 St. Venants 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 What are boundary elements?
———-2.1 Influence functions or Bettis theorem
———-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 Plane problems
———-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
———-4.16 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 Slabs
———-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 ReissnerMindlin plates
———-5.7 Singularities of a ReissnerMindlin plate
———-5.8 ReissnerMindlin elements
———-5.9 Supports
———-5.10 Columns
———-5.11 Shear forces
———-5.12 Variable thickness
———-5.13 Beam models
———-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 Shells
———-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 Greens identities
———-7.3 Greens functions
———-7.4 Generalized Greens 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 Greens functions
———-7.10 The dual space
———-7.11 Some concepts of error analysis.
———-7.12 Important equations and inequalities

Preface
The finite element method has become an indispensible tool in structural analysis, and tells an unparalleled success story. With success, however, came criticism, because it was noticeable that knowledge of the method among practitioners did not keep up with success. Reviewing engineers complain that the method is increasingly applied without an understanding of structural behavior.
Often a critical evaluation of computed results is missing, and frequently a basic understanding of the limitations and possibilities of the method are nonexistent.
But a working knowledge of the fundamentals of the finite element method and classical structural mechanics is a prerequisite for any sound finite element analysis. Only a well trained engineer will have the skills to critically examine the computed results.
Finite element modeling is more than preparing a mesh connecting the elements at the nodes and replacing the load by nodal forces. This is a popular model but this model downgrades the complex structural reality in such a way that instead of being helpful it misleads an engineer who is not well acquainted with finite element techniques.
The object of this book is therefore to provide a foundation for the finite element method from the standpoint of structural analysis, and to discuss questions that arise in modeling structures with finite elements.
What encouraged us in writing this book was that thanks to the intensive research that is still going on in the finite element community we can explain the principles of finite element methods in a new way and from a new perspective by making ample use of influence functions. This approach should appeal in particular to structural engineers, because influence functions are a genuine engineering concept and are thus deeply rooted in classical structural mechanics, so that the structural engineer can use his engineering knowledge and insight to assess the accuracy of finite element results or to discuss the modeling of structures with finite elements.
Just as a change in the elastic properties of a structure changes the Greens functions or influence functions of the structure so a finite element mesh effects a shift of the Greens functions.
We have tried to concentrate on ideas, because we considered these and not necessarily the technical details to be important. The emphasis should be on structural mechanics and not on programming the finite elements, and therefore we have also provided many illustrative examples.
Finite element technology was not developed by mathematicians, but by engineers (Argyris, Clough, Zienkiewicz). They relied on heuristics, their intuition and their engineering expertise, when in the tradition of medieval craftsmen they designed and tested elements without fully understanding the exact background. The results were empirically useful and engineers were grateful because they could suddenly tackle questions which were previously unanswerable. After these early achievements self-confidence grew, and a second epoch followed that could be called baroque: the elements became more and more complex (some finite element programs offered 50 or more elements) and enthusiasm prevailed. In the third phase, the epoch of enlightment mathematicians became interested in the method and tried to analyze the method with mathematical rigor. To some extent their efforts were futile or extremely difficult, because engineers employed techniques (reduced integration, nonconforming elements, discrete Kirchhoff elements) which had no analogy in the calculus of variations. But little by little knowledge increased, the gap closed, and mathematicians felt secure enough with the method that they could provide reliable estimates about the behavior of some elements.
We thus recognize that mathematics is an essential ingredient of finite element technology.
One of the aims of this book is to teach structural engineers the theoretical foundations of the finite element method, because this knowledge is invaluable in the design of safe structures.

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