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Problems
for Chapter 8
Theory and
Implementation of the Finite Element Method
8.1. Generalized
FEM for Static Linear Elasticity
8.1.1.
Consider a
one-dimensional isoparametric quadratic element, illustrated in the figure,
and described in more detail in Section 8.1.5.Â
8.1.1.1.
Suppose that
the nodes have coordinates  ,
,
. Using a parametric plot, construct graphs
showing the spatial variation of displacement in the element, assuming that
the nodal displacements are given by (a) ,
(b) ,
(c) ,
(d)
8.1.1.2.
Suppose that
the nodes have coordinates ,
,
. Plot graphs showing the spatial variation
of displacement in the element for each of the four sets of nodal
displacements given in 1.1.
8.1.2.
Consider the
finite element scheme to calculate displacements in an axially loaded 1D bar,
described in Section 8.15. Calculate
an exact analytical expression for the 3x3 stiffness matrix
for a quadratic 1-D element illustrated in the
figure. Write a simple code to integrate the stiffness matrix numerically,
using the procedure described in 8.1.5 and compare the result with the exact
solution (for this test, choose material and geometric parameters that give  ) Try
integration schemes with 1, 2 and 3 integration points.
8.1.3.
 Consider the mapped, 8 noded isoparametric
element illustrated in the figure.
Write a simple program to plot a grid showing lines of  and  in the mapped element, as shown. Plot the
grid with the following sets of nodal coordinates
·
,
,
,
,
,
,
,
·
,
,
,
,
,
,
Note that for the latter case, there is a region in
the element where .
This is unphysical. Consequently, if
elements with curved sides are used in a mesh, they must be designed
carefully to avoid this behavior.  In
addition, quadratic elements can perform poorly in large displacement
analyses.
Â
8.1.4.
Set up an input
file for the general 2D/3D linear elastic finite element code provided to
test the 6 noded triangular elements. Â
Run the test shown in the figure (dimensions and loading are in
arbitrary units) and use a Young’s modulus and Poissons ratio .
Compare the FEA solution to the exact solution.
8.1.5.
Set up an input
file for the general 2D/3D linear elastic finite element code provided to
test the 4 noded tetrahedral elements. Â
Run the test shown in the figure. Take the sides of the cube to have
length 2 (arbitrary units), and take .
Run the following boundary conditions:
8.1.5.1.
 at node 1,  at node 2, and  at nodes 3 and 4.  The faces at  subjected to uniform traction  (arbitrary units)
8.1.5.2.
 at node 1,  at node 5, and  at nodes 3 and 6. The face at  subjected to uniform traction  (arbitrary units)
In each case compare the finite element solution
with the exact solution (they should be equal)
8.1.6.
Add lines to
the 2D/3D linear elastic finite element code to compute the determinant and
the eigenvalues of the global stiffness matrix. Calculate the determinant and eigenvalues
for a mesh containing a single 4 noded quadrilateral element, with each of
the boundary conditions shown in the figure.Â
Briefly discuss the implications of the results on the nature of
solutions to the finite element equations.
8.1.7.
Set up an input
file for the mesh shown in the figure.Â
Use material properties  and assume plane strain deformation. Run the following tests
8.1.7.1.
Calculate the
determinant of the global stiffness matrix
8.1.7.2.
Change the code
so that the element stiffness matrix is computed using only a single
integration point. Calculate the
determinant and eigenvalues of the global stiffness matrix.
8.1.8.
Extend the
2D/3D linear elastic finite element code provided to solve problems involving
anistropic elastic solids with cubic symmetry (see Section 3.2.16 for the
constitutive law). This will require
the following steps:
8.1.8.1.
The elastic
constants  for the cubic crystal must be read from the
input file
8.1.8.2.
The orientation
of the crystal must be read from the input file. The orientation of the crystal can be
specified by the components of vectors parallel to the [100] and [010]
crystallographic directions.
8.1.8.3.
The parts of
the code that compute the element stiffness matrix and stress (for
post-processing) will need to be modified to use elastic constants  for a cubic crystal. The calculation is complicated by the fact
that the components of  must be expressed in the global coordinate system, instead of a
coordinate system aligned with the crystallographic directions. You will need to use the unit vectors in
6.2 to calculate the transformation matrix  for the basis change, and use the basis
change formulas in Section 3.2.11 to calculate .
8.1.8.4.
Test your code
by using it to compute the stresses and strains in a uniaxial tensile
specimen made from a cubic crystal, in which the unit vectors parallel to
[100] and [010] directions have components .
Mesh the specimen with a single cubic 8 noded brick element, with side length
0.01m. Apply displacement boundary
conditions to one face, and traction boundary conditions on another
corresponding to uniaxial tension of 100MPa parallel to the  axis. Run the following tests: (i) Verify
that if  are given values that represent an istropic
material, the stresses and strains the element are independent of .
(ii) Use values for  representing copper (see Section
3.1.17). Define an apparent axial
Young’s modulus for the specimen as . Plot a graph showing  as a function of .
8.1.9.
Extend the
2D/3D linear elastic finite element code to solve problems involving body
forces. This will require the
following steps
8.1.9.1.
You will need
to read a list of elements subjected to body forces, and the body force
vector for each element. The list can
be added to the end of the input file.
8.1.9.2.
You will need
to write a procedure to calculate the contribution from an individual element
to the global system of finite element equations. This means evaluating the integral
over the volume of the element. The integral should be evaluated using
numerical quadrature  the procedure (other than the integrand) is
essentially identical to computing the stiffness matrix.
8.1.9.3.
You will need
to add the contribution from each element to the global force vector. It is simplest to do this by modifying the
procedure called globaltraction().
8.1.9.4.
Test your code
by using it to calculate the stress distribution in a 1D bar which is
constrained as shown in the figure, and subjected to a uniform body
force. Use 4 noded plane strain
quadrilateral elements, and set  (arbitrary units) and take the body force to
have magnitude 5. Compare the
displacements and stresses predicted by the finite element computation with
the exact solution.
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8.1.10. Implement the linear 3D wedge-shaped element shown
in the figure the 2D/3D linear elastic finite element code. To construct the shape functions for the
element, use the shape functions for a linear triangle to write down the
variation with ,
and use a linear variation with . Assume that . You will need to add the shape functions
and their derivatives to the appropriate procedures in the code. In addition, you will need to modify the
procedures that compute the traction vectors associated with pressures acting
on the element faces.  Test your code
by meshing a cube with two wedge-shaped elements as shown in the figure.  Take the sides of the cube to have length
2 (arbitrary units), and take .
Run the following boundary conditions:
8.1.10.1.
 at node 1,  at node 2, and  at nodes 3 and 4.  The faces at  subjected to uniform traction  (arbitrary units)
8.1.10.2.
 at node 1,  at node 5, and  at nodes 3 and 6. The face at  subjected to uniform traction  (arbitrary units)
In each case compare the finite element solution
with the exact solution (they should be equal).
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