Project
MANE-4940HEG Studies in FEM
Michael Wegrzyniak
Problem
The exact solution can be found in Timoshenko and Goodier, Theory of Elasticity , 3rd Ed, McGraw-Hill
New York, 1970. Specifically, the equation for the vertical displacement of all points in the beam is
given by:
Elements
ANSYS has a number of element types that can be used to solve this 3-D problem. Each element type
has different options that improve the accuracy and performance of the element. In addition to the
different element types, the number of elements can affect the accuracy and performance of the solution.
The more elements that a model contains the more detailed and accurate of a solution that will be
achieved. However, the additional elements cost computational time. There has to be a balance between
the accuracy of the model and the time required to solve the model.
The problem will be solved using the following elements:
Elements
Solid 5
The element has eight nodes with up to six degrees of freedom at
each node.
Solid 45
The element is defined by eight nodes having three degrees of
freedom at each node: translations in the nodal x, y, and z directions.
Solid 186
The element is defined by 20 nodes having three degrees of freedom
per node.
Solid226
The element has twenty nodes with up to five degrees of freedom
per node.
Project
MANE-4940HEG Studies in FEM
Michael Wegrzyniak
And the following and mesh densities:
1 X 10 = 10 elements
2 X 20 = 40 elements
4 X 40 = 160 elements
5 X 50 = 250 elements
10 X 100 = 1000 elements
Project
MANE-4940HEG Studies in FEM
Michael Wegrzyniak
The ANSYS Results
Since these were relatively simple models, the solution time was less than a minute for even the model
with the 1000, high order elements.
10 elements
40 elements
160 elements
250 elements
1000 elements
Solid 5
0.795E-4
0.119E-3
0.200E-3
0.240E-3
0.441E-3
Solid 45
0.795E-4
0.119E-3
0.200E-3
0.240E-3
0.441E-3
Solid 186
0.119E-3
0.200E-3
0.361E-3
0.441E-3
0.843E-3
Solid 226
0.119E-3
0.200E-3
0.361E-3
0.441E-3
0.843E-3
The MATALB Results
The exact solution is given by the formula:
𝑣(𝑥, 0) =
𝐹𝑥 3 𝐹𝐿2 𝑥 𝐹𝐿3
−
+
6𝐸𝐼
2𝐸𝐼
3𝐸𝐼
where 𝐼 =
𝑏ℎ 3
12
Gives a displacement result of 0.4E-4. When this result is compared to the ANSYS solutions, an
interesting phenomenon is observed. Instead of improving the accuracy with the increase in elements, it
appears that the solution is diverging.
Project
MANE-4940HEG Studies in FEM
This result was a surprise and needs additional investigation to validate.
Michael Wegrzyniak
Project
MANE-4940HEG Studies in FEM
Michael Wegrzyniak
Solid 5
1 X 10 = 10 elements
2 X 20 = 400 elements
4 X 40 = 160 elements
5 X 50 = 250 elements
10 X 100 = 1000 elements
Project
MANE-4940HEG Studies in FEM
Michael Wegrzyniak
Solid 45
1 X 10 = 10 elements
2 X 20 = 400 elements
4 X 40 = 160 elements
5 X 50 = 250 elements
10 X 100 = 1000 elements
Project
MANE-4940HEG Studies in FEM
Michael Wegrzyniak
Solid 186
1 X 10 = 10 elements
2 X 20 = 400 elements
4 X 40 = 160 elements
5 X 50 = 250 elements
10 X 100 = 1000 elements
Project
MANE-4940HEG Studies in FEM
Michael Wegrzyniak
Solid 226
1 X 10 = 10 elements
2 X 20 = 400 elements
4 X 40 = 160 elements
5 X 50 = 250 elements
10 X 100 = 1000 elements