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Pre-Analysis and Start-Up
UNDER CONSTRUCTION
In the Pre-Analysis step, we'll review the following:
- Mathematical Model: We'll look at the governing equations + boundary conditions and the assumptions contained within the mathematical model.
- Hand-calculations of expected results: We'll use an analytical solution of the mathematical model to predict the expected stress field from ANSYS. We'll pay close attention to additional assumptions that have to be made in order to obtain an analytical solution.
Mathematical Model
Assumptions
We'll assume that:
- Plane stress conditions apply since the bar is thin
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[
\sigma_
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= \tau_
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= \tau_
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= 0
]
- Gravity effects can be neglected i.e. no body forces.
Governing equations
Since we are assuming plane stress conditions, we can use the 2D version of the equilibrium equations. When the bar is in equilibrium, the stresses will satisfy the 2D equilibrium equations with no body forces.
((2D Equilibrium equations with body forces zeroed out))
Boundary Conditions
The domain over which we'll solve the governing equations is a rectangle.
((Equation and Figure))
The boundary conditions are:
((equations))
((Old Stuff Below: To be deleted eventually))
Since we don't expect significant variation of stresses in the z direction, it is reasonable to assume plane stress:
[
\sigma_z = \tau_
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= \tau_
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= 0
]
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The deformed structure will be in equilibrium. Thus, the 2D stress components should satisfy the 2D equilibrium equations:
\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
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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.
[
F_x = F_y =0
]
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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
\begin
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\sigma_y = \tau_
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= 0\nonumber
\end
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Then the equilibrium equation in the x-direction becomes:
\begin
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= 0\nonumber
\end
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Therefore,
\begin
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\sigma_x = constant\nonumber
\end
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Apply Boundary Conditions: If we make a vertical cut in the geometry, then the stress must be P/A. Therefore,
\begin
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\sigma_x = P/A
\end
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This is of course a well-known result. For this problem, we have
\begin
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\sigma_x = 2000/(10*1) = 200 \ N/mm^2 = 200 \ MPa
\nonumber
\end
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Numerical Solution using FEA
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.
Load FEA Solution obtained using ANSYS
As mentioned before, we are providing the FEA solution obtained using ANSYS so that you can focus on comparing the analytical and numerical solutions (which is the goal of this exercise). 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, otherwise the solution will not load into ANSYS properly. (Note: The solution provided was created using ANSYS workbench 13.0 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 then be presented with the ANSYS solution in the project page.
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 of this tutorial.
[*Go to Results*]
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