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Chapter 7
Introduction to Finite Element Analysis in Solid Mechanics
7.2 A simple Finite Element program
The goal of this section is to provide some insight into the theory and algorithms that are coded in a finite element program. Here, we will implement only the simplest possible finite element code: specifically, we will develop a finite element method to solve a 2D (plane stress or plane strain) static boundary value problem in linear elasticity, as shown in the picture.Â
We assume that we are given: 1. The shape of the solid in its unloaded condition 2. Boundary conditions, specifying displacements To simplify the problem, we will make the following assumptions
We
then wish to find a displacement field
These steps are discussed in more detail in the sections below.
7.2.1 The finite element mesh and element connectivity
For simplicity, we will assume that the elements are 3 noded triangles, as shown in the picture. The nodes are numbered 1,2,3…N, while the elements are numbered 1,2…L. Element numbers are shown in parentheses.
The position of the ath node is specified by its
coordinates
The element connectivity specifies the node numbers attached to each element. For example, in the picture the connectivity for element 1 is (10,9,2); for element 2 it is (10,2,1), etc.
7.2.2 The global displacement vector
We
will approximate the displacement field by interpolating between values at
the nodes, as follows. Let
The unknown displacement components will be determined by minimizing the potential energy of the solid.
7.2.3 Element interpolation functions
To
calculate the potential energy, we need to be able to compute the
displacements within each element. This
is done by interpolation. Consider a
triangular element, with nodes a, b, c
at its corners. Let
These shape functions are constructed so that
We then write
One
can readily verify that this expression gives the correct values for
Of course, the shape
functions given are valid only for 3 noded triangular elements
7.2.4 Element strains, stresses and strain energy density
We
can now compute the strain distribution within the element and hence
determine the strain energy density. Since we are solving a plane strain
problem, the only nonzero strains are
The
factor of 2 multiplying the shear strains in the strain vector has been
introduced for convenience. Note that,
for linear triangular elements, the matrix of shape function derivatives
Now, we can compute the strain energy density within the element. Begin by computing the stresses within the element. For plane strain deformation, have that
For plane stress, the result is
Recall
(see Sect 3.1.7) that the strain energy density is related to the stresses
and strains by
where
Now, express these results in terms of the nodal displacements for the element
We
can now compute the total strain energy stored within the element. Because
Hence, the total strain energy of the element is
7.2.5 The element stiffness matrix
The strain energy can be simplified by defining the element stiffness matrix
so that
Observe
that, because the material property matrix
7.2.6 The Global stiffness matrix
The total strain energy of the solid may be computed by adding together the strain energy of each element:
It
is more convenient to express W in
terms of the vector
If we wanted to, we could add the missing terms to each element displacement vector:
We can now collect together corresponding terms in the two element stiffness matrices to express this as
 We can therefore write
where
Because the element stiffness matrix is symmetric, the global stiffness matrix must also be symmetric.
To assemble the global stiffness matrix for a plane strain or plane stress mesh with N nodes, we use the following procedure.
1. Note that for 2. Next, begin a loop over the elements. 3. For the current element, assemble the element stiffness matrix
4. Add the element stiffness matrix to the global
stiffness matrix, using the following procedure. Let a,
b, c denote the numbers of the nodes on the 3 corners of the
element. Let
Here, the symbol += means that the term on the left is incremented by the term on the right, following standard C syntax.
5. Proceed to the next element.
We have now found a way to compute the strain energy for a finite element mesh. Next, we need to compute the boundary term in the potential energy.
Boundary loading will be specified as follows: 1. The element on which the loading acts 2. The face of the element which is loaded 3. The traction vector t (force per unit area) that acts on the face of the element. The traction is assumed to be constant on the face of any one element.
Now, we compute the contribution to the potential energy due to the traction acting on the face of one element.
For the element shown, the contribution to the potential energy would be
Recall that the displacements vary linearly within a 3 noded triangle. Therefore, we can write
So, since the tractions are uniform
Abbreviate this as
7.2.8 Global residual force vector
The total contribution to the potential energy due to boundary loading on all element faces is
It is more convenient to
express this in terms of the global displacement vector
where
The global residual force vector for a mesh with N nodes is assembled as follows. 1. The residual force vector has length 2N (2 entries per node). Reserve storage for a vector of length 2N and initialize to zero 2. Loop over elements 3. Determine which face of the element is loaded. Let a,
b denote the node numbers attached to this face. Determine the residual force vector for the
element face. Let
7.2.9 Minimizing the Potential Energy
We have set up the following expression for the potential energy of a finite element mesh
Now, minimize V
where we have noted that
Simplify this by noting
that
This is a system of 2N simultaneous linear equations for the 2N unknown nodal displacements. Standard computational techniques such as Gaussian elimination, Cholesky factorization or conjugate gradient methods may be used to solve the system of equations.
7.2.10 Eliminating prescribed displacements
So far, we have seen how to calculate displacements in a finite element mesh which is subjected to prescribed loading. What if displacements are prescribed instead?
If this is the case, the stiffness matrix and residual are first assembled exactly as described in the preceding section. They are then modified to enforce the constraint. The procedure is best illustrated using an example. Suppose that the finite element equations after assembly have the form
To
prescribe displacements for any node, we simply replace the equation for the
appropriate degrees of freedom with the constraint. For example, to force
Thus, the equation for
This procedure works, but has the disadvantage that the modified stiffness matrix is no longer symmetric. It is preferable to modify the stiffness and residual further, to retain symmetry. To do so, we eliminate the constrained degrees of freedom from all rows of the stiffness matrix. This is best illustrated by example. Suppose our modified stiffness matrix has the form
Now, we wish to set each entry in the second column (apart from the diagonal) to zero, so that the stiffness is symmetric. Recall that we can add and subtract equations in the system from one another without affecting the solution. Therefore, to symmetrize the stiffness matrix in our example, we can subtract appropriate multiples of the second row so as to set each entry in the second column to zero.
The result of Sections 7.2.1-7.2.10 is a set of simultaneous linear equations of the form
These
can be solved for the unknown displacements
Once the displacements have been computed, the strain in each element can be computed, and so the stress distribution can be deduced. The procedure is as follows 1. For the element of interest, extract the displacement of each node from the global displacement vector 2. Calculate the strains using the procedure in 7.2.4
3. The stresses can then be determined from the stress-strain equations
where
A simple implementation of the procedure is provided in the file FEM_conststrain_mws.
The code reads an input file. A very simple input file (with just two elements) can be found in FEM_conststrain_input.txt.
The data file solves the problem illustrated in the picture. A rectangular block with Young’s modulus 100 and Poisson’s ratio 0.3 is meshed with two elements. Node 1 is pinned; node 4 is constrained against horizontal motion, and the right hand face of element 2 is subjected to a constant horizontal traction with magnitude 10.Â
Note that
1. Nodes are numbered sequentially
2. The element connectivity specifies the node numbers attached to each element, using a counterclockwise numbering convention. It doesn’t matter which node you use for the first one, as long as all the others are ordered in a counterclockwise sense around the element. For example, you could use (2,4,1) instead of (1,2,4) for the connectivity of the first element.
3. To fix motion of a node, you need to enter the node number, the degree of freedom that is being constrained, (1 for horizontal, 2 for vertical), and a value for the prescribed displacement.
4. To specify tractions acting on an element face, you need to enter (a) the element number; (b) the face number of the element, and (c,d) the horizontal and vertical components of traction acting on the face. The face numbering scheme is determined by the element connectivity, as follows. Face (1) has the first and second nodes as end points; face (2) has the second and third nodes; and face (3) has the third and first nodes as end points. Since connectivity for element (2) was entered as (2,3,4), face 1 of this element has nodes numbered 2 and 3; face 2 connects nodes numbered 3 and 4, while face 3 connects nodes numbered 4 and 1.
To run the code, you need to open the .mws file with Maple, then follow these steps: 1. Edit the code to insert the full path for the input file in the line near the top of the code that reads > # Change the name of the file below to point to your input file > infile :=fopen(`insert full path of input file `,READ):
2. Scroll down near the bottom to the line that reads > # Print nodal displacements, element strains and stresses to a file > Â Â Â outfile := fopen(`insert full path of output file``,WRITE): and enter a name for the output file.
3. Return to the top of the file, and press <enter> to execute each MAPLE block. If all goes well, you should see that, after reading the input data, MAPLE plots the mesh (just as a check).Â
4. If you continue to the end, you should see a plot of the displaced mesh (red) superimposed on the original mesh (green), as shown below.
5. Finally, open the output file. It should contain the results shown below
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(c) A.F. Bower, 2008 |