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2018 | OriginalPaper | Chapter

1. Finite Element Basics with the Bar Element: Uniaxial Deformations—Interpolants, Stiffness Matrices and Nodal Loads

Author : Gautam Dasgupta

Published in: Finite Element Concepts

Publisher: Springer New York

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Abstract

In this introductory chapter, we emphasize important concepts of the finite element method using simple, intuitive examples. An undergraduate engineering mathematics background should be adequate. However, the basic operations with linear algebra and differential equations are reviewed within the context.
The finite element method, even for unbounded media, projects continuum solutions—governed by (partial) differential equations—into a finite dimensional vector space. Strikingly enough, the merit of the method permits the introduction of all basic (physical and mathematical) ideas with one-dimensional bar examples. The deformation analysis of a system of bars captures all essential aspects of thermo-mechanical behavior. Thus, this chapter provides a foundation for the topics developed in this textbook. In association with characteristic “internal forces,” which guarantee equilibrium to yield quality solutions, there are independent Rayleigh displacement modes. These finite number of basis functions (blending functions or interpolants) are the fundamental objects of the finite element method. The resulting nodal forces and displacements yield symmetric (positive semi-definite) system matrices.
The “energy minimization” concept is introduced using a single spring element (a single degree-of-freedom system). In order to reinforce the idea of degrees-of-freedom and of the energy-like scalars, the physical Rayleigh mode is introduced as the fundamental pattern of deformation.
Generalization, e.g. frame invariance concepts, in two- and three-dimensions, involves “inversion” of rectangular matrices; the associated pseudoinverse concept is introduced within that context. Discrete representation with indicial notation is described in detail, and weak solutions are introduced within a smaller dimensional vector space.
Many details (unfamiliar to advanced undergraduates), which are addressed in the successive chapters, can be skipped during introductory readings.

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next chapter Truss Analysis: A Structural Mechanics Tour of the FEM—Nodal Equilibrium and Compatibility
Footnotes
1
This is essential, otherwise the energy stored due to deformation will not be positive.
 
2
These concepts have been stated at the very beginning in Sect. 1.1.
 
3
This is not the work done by the external force; there is no \(\frac{1} {2}\) in the expression.
 
4
The Method of Finding Plane Curves that Show Some Property of Maximum or Minimum…, Laussanne and Geneva, 1744,” also “L. Euler, Opera Omnia I, Vol. XXIV, C. Carathéodory, ed. Bern, 1952.
 
5
A “function” of functions. Specifically, this is an alternate method of solving partial differential equations whose variables could be vector components.
 
6
This establishes the principle of virtual work and variational calculus to be equivalent.
 
7
John Forbes Nash extended the formalism to parabolic partial differential equations.
 
8
This has been utilized in solving problems of statics since antiquity.
 
9
We imply that the force-displacement relation is linear and the strain energy is a quadratic function of displacement.
 
10
In Fig. 1.2 the same bar model approximates the spring of Fig. 1.3 that is shown as a two-dimensional graphic of a three-dimensional model with a viewpoint of \(\left \{0,-2, 1/2\right \}\).
 
11
The mass of the system leads to the inertia related parameters; the resulting dynamic responses are outside the scope of this textbook.
 
12
The displacement profile u(x), which is a function of the coordinate x, is assumed to be small enough so that the linear theory of kinematics can be adopted.
 
13
This is also called the skin force.
 
14
The basic approximation is that u(x) is a linear combination of ϕ 1(x) and ϕ 2(x). 
 
15
We will depict the force intensity with arrows (in Fig. 1.19) to avoid clutter in Fig. 1.15.
 
16
Disregarding \(f(x)\neq 0\mbox{ or spatial variability in }\mathcal{A}(x)\ \mathcal{E}(x)\) in Eq. (1.22b).
 
17
This also points out the relation between the order of finite element strain and the number of element nodes. Even then, the finite element approximation with a two-node bar element will be a linear displacement field.
 
18
Kinematic quantities, in general.
 
19
In finite element methods, stiffness matrices \(\left [k\right ]\) occupy a very special status.
 
20
This will be elaborated in Sect. 1.5 in detail.
 
21
It is not obvious that \(\left [k\right ]\) is symmetric, and does not have any negative eigenvalue. The energy concept encompasses these crucial facts within positive semi-definiteness.
 
22
Assuming \(\left.\left (\frac{\partial A} {\partial y} \right )\right \vert _{z,x} = \left.\left (\frac{\partial B} {\partial x} \right )\right \vert _{y,z}; \left.\left (\frac{\partial B} {\partial z} \right )\right \vert _{x,y} = \left.\left (\frac{\partial C} {\partial y} \right )\right \vert _{z,x}; \left.\left (\frac{\partial C} {\partial x} \right )\right \vert _{y,z} = \left.\left (\frac{\partial A} {\partial z} \right )\right \vert _{x,y}\).
 
23
Appendix F describes the variational formulation for the steady temperature problem.
 
24
This has been repeated (several times) to anchor the FEM to the energy principle.
 
25
A multiplicative neutral does not change any object after being multiplied (in a generalized sense) by it. Hence, we can formally call \(\delta \!\left \{u\right \}\) a “unit virtual displacement.”
 
26
Related to minimization where an inner product, e.g. conjugations of forces and displacements, is crucial.
 
27
Exact interpolation of arbitrary uniform displacement fields generates equilibrium for the entre system.
 
28
This idea guarantees the correct identification of nodal forces in Chap. 7
 
29
For a positive semi-definite matrix, which is real and symmetric, the eigenvalues are either positive or zero. For positive definite matrices, all eigenvalues are positive.
 
30
In Fig. 1.22, the axial degrees-of-freedom are along the (x, y) axes.
 
31
Conventionally, we include “[]” after a Mathematica function name.
 
32
It will be shown in Chap. 8, for incompressible elements, such pseudoinverses are essential.
 
33
The stiffness matrix \(\left [k^{{\ast}}\right ],\) in Eq. (1.95), is a closed-form expression in terms of its local nodal coordinates. We can directly use this equation in 3-D computer programs.
 
34
To generalize the bar problem, we observe that all partial differential equations of mathematical physics are of even order. Hence linear interpolants are mandatory.
 
35
Plate bending problems are not considered in this textbook.
 
36
A Civil Engineer (in today’s definition) furnished the factorization in conjunction with Surveying and Cartography that inevitably used with a lot of polygons and triangulizations.
 
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Metadata
Title
Finite Element Basics with the Bar Element: Uniaxial Deformations—Interpolants, Stiffness Matrices and Nodal Loads
Author
Gautam Dasgupta
Copyright Year
2018
Publisher
Springer New York
Book
Finite Element Concepts

Print ISBN: 978-1-4939-7421-4

Electronic ISBN: 978-1-4939-7423-8

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-1-4939-7423-8_1

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