Page History

{include: Tensile Bar - Panel}
{include: ANSYS Google Analytics}

h1. Pre-Analysis and Start-Up

{note}
UNDER CONSTRUCTION
{note}

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.

h2. Mathematical Model

TheWe'll deformedfirst structurelist willthe beassumptions in the mathematical equilibriummodel. ThusThen, we'll review the governing equations 2Dand stressboundary componentsconditions shouldthat satisfyform the 2Dmathematical equilibriummodel. equations:
\\
{latex}
\begin{eqnarray}
{\partial \sigma_x \over \partial x} + {\partial \tau\_{yx} \over \partial y} + F_x = 0 \nonumber
\\
{\partial \tau\_{xy} \over \partial x} + {\partial \sigma_y \over \partial y} + F_y = 0 \nonumber
h3. Assumptions

We'll assume that:

# Plane stress conditions apply since the bar is thin, thus we don't expect significant variation of stresses in the z direction:\\
{latex}\[
\sigma_{zz} = \tau_{xz} = \tau_{yz} = 0
\]
{latex}

# Gravity effects can be neglected i.e. no body forces.
{latex}
\[
F_x = F_y =0
\]
{latex}

h3. Governing Equations

Since we are assuming plane stress conditions, we can use the 2D version of the equilibrium equations. When the deformed structure reaches equilibrium, the 2D stress components should satisfy the 2D equilibrium equations with zero body forces:
\\
{latex}
\begin{eqnarray}
{\partial \sigma_x \over \partial x} + {\partial \tau\_{yx} \over \partial y} \nonumber
\\
{\partial \tau\_{xy} \over \partial x} + {\partial \sigma_y \over \partial y} \nonumber
\end{eqnarray}
{latex}

h3. Boundary Conditions

We solve these equations in a rectangular domain and impose the appropriate boundary conditions: displacement=0 at the left end (which is fixed) and applied point force at the right end. 

((Equation and Figure))

The bottom and top edges are free. Since traction on free edges are zero, we get
\\
{latex}
\begin{eqnarray}
\sigma_y = \tau_{xy} = 0 \:\: at \: y = 0 and y = w \:\: (top, bottom, right \: sides)
\end{eqnarray}
{latex}

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. The left end is fixed. So both components of displacement are zero at this end:
\\
{latex}
\begin{eqnarray}
u = v = 0 \:\: at \: x = 0 \:\: (left side)
\end{eqnarray}
{latex}


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

h4. Assumptions

We'll assume that:

1. Plane stress conditions apply since the bar is thin, thus we don't expect significant variation of stresses in the z direction:
\\

{latex}\[
\sigma_{zz} = \tau_{xz} = \tau_{yz} = 0
\]
{latex}
2. Gravity effects can be neglected i.e. no body forces.
{latex}
\[
F_x = F_y =0
\]
{latex}

3. x and y displacement along the fixed end of the bar is assumed to equal zero.
{latex}
U = V = 0 \: at \: x = 0
{latex}

h4. Additional Assumptions

Since the length is much larger than the width, we ignore end effects and neglect stress variations in the y direction.

Because they are free surfaces, stress in the y direction equals zero at both the top (y = H) and bottom (y = 0) surfaces of the bar.
If a surface is not constricted and can move freely, it can expand and contract without incurring stress.

All regions other than the left and right ends, experience pure uniaxial stress, thus we assume stress in the y direction and shear in the xy direction equals zero.
{latex}
\begin{eqnarray}
\sigma_y = \tau_{xy} = 0 \: \: at \: x \not = 0\nonumber
\end{eqnarray}
{latex}



h4. 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.

{latex}
\begin{eqnarray}
{\partial \sigma_x \over \partial x} + {\partial \tau\_{yx} \over \partial y} + 0 = 0 \nonumber
\\
{\partial \tau\_{xy} \over \partial x} + {\partial \sigma_y \over \partial y} + 0 = 0 \nonumber
\end{eqnarray}
{latex}

h4. Boundary Conditions

The domain over which we'll solve the governing equations is a rectangle.

((Equation and Figure))

The boundary conditions are:

{latex}
\begin{eqnarray}
\sigma_y = \tau_{xy} = 0 \:\: at \: x \not = 0 \:\: (top, bottom, right \: sides)
\\
U = V = 0 \:\: at \: x = 0 \:\: (left side)
\end{eqnarray}
{latex}

h2. Solving the BVP

1.  After applying the boundary condition (1) at the top and bottom edges:
{latex}
\begin{eqnarray}
{\partial \sigma_x \over \partial x} = 0\nonumber
\\ \nonumber
\\ \nonumber
\sigma_y = 0\nonumber
\end{eqnarray}
{latex}
\\
Therefore,
{latex}
\begin{eqnarray}
\sigma_x = constant\nonumber
\end{eqnarray}
{latex} 
\\
\\
Note that because of the additional assumptions, we are unable to satisfy the boundary condition (2). The additional assumptions were used to simplify the BVP and hand calcuation. Later on, boundary condition (2) will be used to explain edge effects in the Verification and Validation section of the tutorial.

h2. Hand Calculations

Stress is found using the equation:
{latex}
\begin{eqnarray}
\sigma_x = Force/Area \nonumber
\end{eqnarray}
{latex}

Because we chose to neglect end effects, we make a vertical cut in the bar to determine the stress. Therefore,
\\
{latex}
\begin{eqnarray}
\sigma_x = P/A = P/(H*Thickness) \nonumber
\end{eqnarray}
{latex}
 This is of course a well-known result. For this problem, we have
\\
{latex}
\begin{eqnarray}
\sigma_x = 2000/(10*1) = 200 \ N/mm^2 = 200 \ MPa
\nonumber
\end{eqnarray}
{latex}

h2. 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.


!Tensile Bar breakdown small.png!\\

h4. 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 - Pre-analysis and Start-up^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.
!tensile_bar_wb.jpg!
A tick mark against each step indicates that that step has been completed.

3. To look at the results, double click on {color:purple}{*}{_}Results{_}{*}{color} \- 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 {color:purple}{*}{_}Solution (A6)_{*}{color}.

!tensile_bar_outline.jpg!
We'll investigate the items listed under Solution in the next step of this tutorial.
\\

*[Go to Step 2: Numerical Results|Tensile Bar - Numerical Results]*

[Go to all ANSYS Learning Modules|ANSYS Learning Modules]