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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 2D 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 look at 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 is an approximate solution to the 2D BVP. The approximation gets better as the elements become smaller. In contrast, the analytical solution presented above is the exact solution to the 1D BVP obtained by making approximations to the 2D BVP. In other words, in the analytic solution, we have swapped the actual 2D BVP problem for a 1D BVP problem that we can solve in closed form. Both approaches have value in engineering and complement each other. 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 "Tensile Bar Demo.zip" by [clicking here|^Tensile Bar demo.zip]
The zip should contain a Tensile Bar Demo folder with the following contents:
- Tensile Bar Demo_files folder
- Tensile Bar Demo.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 "Tensile Bar Demo.wbpj" - This should automatically open ANSYS workbench (you have to twiddle your thumbs a bit before it opens up). You will be presented with the ANSYS solution.
A tick mark against each step indicates that that step has been completed.
3. To look at the results, double click on "Results" - This should bring up a new window (again you have to twiddle your thumbs a bit before it opens up).
4. On the left-hand side there should be an "Outline" toolbar. Look for "Solution (A6)".
We'll investigate the items listed under Solution in the next step in this tutorial.
[*Go to Results*]
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