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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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\begin
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\sigma_z = \tau_
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= \tau_
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= 0
\end
. The deformed structure will be in equilibrium. So the stress components should satisfy the three Below you will see that the analytical method makes assumptions that the ANSYS simulation does not.
Analytical Approach:
Assumptions made in this analysis
- long bar (length is much greater than width)
- no normal stresses in the y direction
- plane stresses
- no gravity effects
- no end effects or point load effects (i.e. uniform stresses throughout the bar)
Analysis
Assuming plane stresses:
The two dimensional equilibrium equations are:
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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
Since we are ignoring the effects of gravity; there are no body forces per unit volume.
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\begin
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F_x = F_y =0\nonumber
\end
Assuming no normal stress in the y direction:
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\begin
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\sigma_y = 0\nonumber
\end
The equilibrium equation in the y direction becomes:
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\begin
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{\partial \tau_
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\over \partial x} = 0\nonumber
\end
τ_yx must also be a constant, therefore 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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\sigma_x = constant\nonumber
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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
ANSYS simulation:
In this exercise, the geometry creation, mesh generation, boundary value setting and choosing the correct solver has been bypassed. You will be given the solution with the hope that you will explore it thoroughly to understand the numerical solution.
Please note: The ANSYS simulation solves for a 2 dimensional (x and y) boundary value problem. Compare this to what was done in the analytical section.
1. Download "Class demo1.rar" by [clicking here|^Class demo1.rar]
The zip should contain:
+ class demo1 folder
- class demo 1_files folder
- class demo 1.wbpj
Please make sure both the files and wbpj are in the folder, the program would not work otherwise. (Note: The solution was created using ANSYS workbench 12.0 release, there may be compatibility issues when opened 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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