course in finite element simulations using abaqus topic 1b ...course in finite element simulations...
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Esmaeil Tohidlou
Department of Materials Engineering
University of Sistan and Baluchestan
2016
Course in finite element simulations using Abaqus
Topic 1b introduction:The Concept of Finite Element Analysis (FEA)
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Concept of Finite Element Analysis 2
Concept of FEA:
The basis of FEA relies on the decomposition of the domain into a finite number
of subdomains (Elements) for which the systematic approximate solution is
constructed by applying the variational or weighted residual methods.
The FEM requires the following major steps:
Discretization of the domain into finite number of sundomains (elements)
Selection of interpolation functions
Development of element matrix for the subdomain (element)
Asselbly of the element matrices for each subdomain to obtain global
matrix for the entire domain
Imposition of the boundary conditions
Solution of equations
Additional computations (If desired)
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Concept of Finite Element Analysis 3
The FEA representation
of practical engineering
problems
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NodeThe transformation of the practical engineering problem to a mathematical
representation is achieved by discretizing the domain of interest in to elements
(elements). These elements are connected to each other by their nodes. A node
specifies the coordinate location in space where degree of freedom and actions of the
physical problem exist. The nodal unknown(s) in the matrix system of equations
represents one (or more) of the primary field variable. Nodal variables assigned to an
element are called dgrees of freedom of the element.
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ElementDepending on the geometry and the physical nature of the problem, the domain of the
interest can be discretized by employing line, area, or volume elements. Some of the
common elements in the FEM are shown in Figure.
One-Dimensional Elements
Line
Rods, Beams, Trusses, Frames
Two-Dimensional Elements
Triangular, Quadrilateral
Plates, Shells, 2-D Continua
Three-Dimensional Elements
Tetrahedral, Rectangular Prism (Brick)
3-D Continua
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Two-Dimensional Discretization Refinement
(Discretization with 228 Elements)
(Discretization with 912 Elements)
(Triangular Element)
(Node)
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Development of Finite Element Equation
• The Finite Element Equation Must Incorporate the Appropriate Physics of the Problem
• For Problems in Structural Solid Mechanics, the Appropriate Physics Comes from
Either Strength of Materials or Theory of Elasticity
• FEM Equations are Commonly Developed Using Direct, Variational-Virtual Work or
Weighted Residual Methods
Variational-Virtual Work Method
Based on the concept of virtual displacements, leads to relations between internal and
external virtual work and to minimization of system potential energy for equilibrium
Weighted Residual Method
Starting with the governing differential equation, special mathematical operations
develop the “weak form” that can be incorporated into a FEM equation. This method is
particularly suited for problems that have no variational statement. For stress analysis
problems, a Ritz-Galerkin WRM will yield a result identical to that found by variational
methods.
Direct Method
Based on physical reasoning and limited to cases, this method is worth studying
because it enhances physical understanding of the process
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Formulation of the stiffness matrix:
Direct method
1 2
k
u1 u2
F1 F2
}{}]{[
rm Matrix Foinor
2 Nodeat mEquilibriu
1 Nodeat mEquilibriu
2
1
2
1
212
211
FuK
F
F
u
u
kk
kk
kukuF
kukuF
Stiffness Matrix Nodal Force Vector
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Bar Element
Bar element:
a) geometry,
b) nodal force applied at node 1,
c) nodal force applied at node 2,
d) nodal forces at both nodes
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Two-Dimensional
Truss Element
Degrees of freedom of a rod element in a two-dimensional space
The nodal degrees of freedom (nodal displacement) of the rod element becom four, as represented in the Figure
𝑑𝑒 = {𝐮1, 𝐯1, 𝐮2, 𝐯2}𝑇
The corresponding stifness matrix becomes:
The second and fourth columns and rows associated with transversal displacements are null since the truss member has axial deformation only.
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Another problem arises from the fact that all truss members do not have the same orientation
is that when it comes to assemble the global stiffness , we need to have the element degrees
of freedom (nodal displacements) given in terms of the common reference axes of the truss.
Truss element oriented at an arbitrary angle: a) Nodal displacement, b) Nodal forces
Figure shows two sets of nodal displacements; the first set (𝒖, 𝒗) is given in terms of the local
set of axis 𝑥, 𝑦 associated with the element, while the second set of displacements (𝑼, 𝑽) is
associated with the global set of axis 𝑋, 𝑌 .The element stifness matrix is expressed in the terms of the local displacements 𝒖 and 𝒗. In
order to assemble with the stiffness matrices of the other elements to form the global
stiffness matrix of the whole system, it should be transformed such that it is expressed in
terms of the global displacement 𝑼 and 𝐕.
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If we consider node 1, it can be seen that the displacements 𝑼1 and 𝑽1 can be written in
terms of 𝒖1 and 𝒗1 as
𝑼1 = 𝒖1𝑐𝑜𝑠𝜃 − 𝒗1𝑠𝑖𝑛𝜃𝑽1 = 𝒖1𝑠𝑖𝑛𝜃 + 𝒗1𝑐𝑜𝑠𝜃
In a similar fashion, 𝑼2 and 𝑽2 can be expressed in terms of 𝒖2 and 𝒗2 as
𝑼2 = 𝒖2𝑐𝑜𝑠𝜃 − 𝒗2𝑠𝑖𝑛𝜃𝑽2 = 𝒖2𝑠𝑖𝑛𝜃 + 𝒗2𝑐𝑜𝑠𝜃
From the above equations
Or in a more compact form as
The matrix is called transformation matrix. It is an orthogonal matrix with determinant
equal to one. Its inverse is simply equal to its transpose:
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