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Pre-Analysis and Start-Up
Since we don't expect significant variation of stresses in the z direction, it is reasonable to assume plane stress:
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[
\sigma_z = \tau_
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= \tau_
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= 0
]
The deformed structure will be in equilibrium. Thus, the 2D stress components should satisfy the two dimensional equilibrium equations:
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\begin
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+ {\partial \tau_
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\over \partial y} + F_x = 0 \nonumber
{\partial \tau_
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\over \partial x} +
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+ F_y = 0 \nonumber
\end
We need to solve these equations in our rectangular domain and impose the appropriate boundary conditions: imposed displacement constraints at the left end and applied force at the right end. In effect, we have to solve a boundary value problem (BVP). Recall that the elements of a BVP are:
- Governing differential equations
- Domain
- Boundary conditions
You probably have solved simple BVPs before in your math classes. We will first review the analytical approach to solving this BVP. We'll then lookat the FEA approach.
Analytical Solution:
Since we are ignoring the effects of gravity; there are no body forces per unit volume.
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[
F_x = F_y =0
]
Since the length is much larger than the width, we ignore end effects and neglect variations in the y direction. Plugging and chugging into the equilibrium equations yields
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\begin
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\sigma_y = \tau_
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= 0\nonumber
\end
Then the equilibrium equation in the x-direction becomes:
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\begin
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= 0\nonumber
\end
Therefore,
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\begin
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\sigma_x = constant\nonumber
\end
Apply Boundary Conditions: If we make a cut at "A", as indicated in the problem specification
, then the stress in A must be P/A. Therefore,
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\begin
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\sigma_x = P/A
\end
This is of course a well-known result.
Numerical Approach:
In the numerical solution using FEA, we solve the 2D BVP directly by dividing the structure into small elements and approximating the solution for these small elements. Unlike the analytical approach, we do not assume that there is no variation in the y direction. Also, end effects are not neglected. The FEA solution tends towards the exact solution as the elements become very small. We have checked that the FEA solution presented to you is reasonably accurate.
The following figure summarizes the contrasts between the analytical and numerical approaches.
ANSYS Solution:
Without further ado, let's download the ANSYS solution and load it into ANSYS.
1. Download "Class demo1.zip" by [clicking here|^Class demo1.zip]
The zip should contain a class demo1 folder with the following contents:
- class demo 1_files folder
- class demo 1.wbpj
Please make sure both these are in the folder, the program would not work otherwise. (Note: The solution was created using ANSYS workbench 12.1 release, there may be compatibility issues when attempting to open with other versions). Be sure to extract before use.
2. Double click "Class Demo1.wbpj" - This should automatically open ANSYS workbench.
3. Double click on "Results" - This should bring up a new window.
4. On the left-hand side there should be an "Outline" toolbar
5. Look for "Solution (A6)" - Inside the tree structure should be
- Solution information
- Displacement
- sigma_x
- sigma_y
- tau_xy
The last four are where you can find the results you need to investigate for the next step in this tutorial.
[*Go to Results*]
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