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{include: Tensile Bar - Panel}
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h1. Pre-Analysis and Start-Up

In the _Pre-Analysis_ step, we'll review the following:

* *Mathematical Modelmodel:* 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.
* *Numerical solution procedure in ANSYS.*

h2. Mathematical Model

We'll first list the assumptions in the mathematical model. Then, we'll review the governing equations and boundary conditions that form the mathematical model. Note that this type of a mathematical model where you have a set of differential equations together with a set of additional restraints at the boundaries is called a [_Boundary Value Problem (BVP)_|http://en.wikipedia.org/wiki/Boundary_value_problem]. A lot of practical problems that are solved using ANSYS and other FEA software are BVP's. You should have encountered simple BVP's in your math courses, problems of the kind that involve solving a differential equation with a set of boundary conditions (I was never good at these math problems and it showed in my math grades to the displeasure of my parents .... fortunately that is now a distant memory\!). You can think of the BVP considered in this tutorial as a souped-up version of simpler BVP's you have encountered in math courses (and either liked or hated\!).

h4. 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}

h4. 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} = 0 \nonumber
\\
{\partial \tau_{xy} \over \partial x} + {\partial \sigma_y \over \partial y} = 0 \nonumber
\end{eqnarray}
{latex}

h4. Boundary Conditions

We solve these equations in a rectangular domain and impose the appropriate boundary conditions. At every point on the boundary, either the displacement or the traction must be prescribed.

!Traction-displacement.png|width=500!

The bottom and top edges are free. If a boundary location is not constrained and can move freely, it can expand and contract without incurring stress. Thus, traction on the free edges is zero and we get
\\
{latex}
\[
\sigma_y = \tau_{xy} = 0 \:\: at \: y = 0 \: and \: y = H
\]
{latex}


The left end is fixed. So both components of displacement are zero at this end:
\\
{latex}
\[
u = v = 0 \:\: at \: x = 0
\]
{latex}

At the right end, the traction is specified at the mid-point where the point load is applied. The applied traction at all other points on the right boundary is zero. For brevity, we won't write out the corresponding equations at the right boundary. We'll simplify this boundary condition in our hand calculations below (to make the problem tractable) but the ANSYS solution provided uses the full set of boundary conditions. Another complication is that since we have a point load, the specified traction at the mid-point of the right end is infinite. We'll later discuss the effect of this in the ANSYS solution. Do keep in mind that there are no point loads in practice, it's just an idealization that can lead to weird behavior that we need to be aware of.

h2. Hand Calculations

Now that we have reviewed the mathematical model for our problem, let's hold off diving into ANSYS just yet and first make some hand calculations of expected results. We'll use these hand calculations to check ANSYS results (like an expert engineer would\!). In order to make the problem solvable by hand, we need to make additional assumptions. The ANSYS solution does _not_ make these additional assumptions.

h4. Additional Assumptions in Hand Calculations

# We'll simplify the right boundary condition. Instead of a point load, we'll assume that the load is distributed over the entire right boundary. So the traction condition at the right boundary becomes
\\
{latex}
\[
\sigma_x = P/(H \, t), \: \: \tau_{xy} = 0 \: \: \: at \: x = L
\]
{latex}
Here, _t_ is the thickness.
\\
\\
# Away from the left and right ends, we expect a uni-axial state of stress with zero shear (OK, this is a bit of a leap of the imagination but it's plausible). So we'll assume that everywhere
\\
{latex}
\[
\tau_{xy} = 0
\]
{latex}

h4. Analytical Solution

With these additional assumptions in hand, we can easily solve the BVP and we get the following analytical solution:
\\
{latex}
\[
\sigma_x = P/(H \, t), \: \: \: \sigma_y = 0
\]
{latex}
This is the well known (P/A) result but we have arrived at it somewhat carefully, accounting for the additional assumptions we made in the process. We'll need to keep these additional assumptions in mind when comparing the hand calculations with the ANSYS solution. For the values given in the problem statement, we have
\\
{latex}
\begin{eqnarray}
\sigma_x = 2000/(10*1) = 200 \ N/mm^2 = 200 \ MPa
\nonumber
\end{eqnarray}
{latex}

h2.The Numericalcorresponding Solution Procedurestrain in ANSYS

In the numerical solution using ANSYS, we divide or "the x-direction can be calculated from Hooke' law:
\\
{latex}
\[
\epsilon_x = \frac{sigma_x}{E} - \nu \, \frac{sigma_y}{E} = 1 \times 10^{-6}
\]
{latex}
The strain is tiny since the material is very stiff with an Young's modulus of 200 GPa.

The displacement at the right end can be estimated by integrating the constant x-strain:
{latex}
\[
\epsilon_x = \frac{\partial u}{\partial x} \\
u(x=L) = \int_0^L{\frac{\partial u}{\partial x}} dx = 0.05 mm
\]
{latex}

The above hand calculations give us expected values of stress, strain and displacement which we'll compare with the ANSYS results.

h2. Numerical Solution Procedure in ANSYS

In the numerical solution using ANSYS, we divide or "discretize" the domain into small rectangles or "elements" (hence the name _finite element analysis_). ANSYS obtains the numerical solution to the BVP in the discrete domain. ANSYS directly solves for the _u_ and _v_ displacements at selected points called "nodes". Everything else such as the stress variation is derived from these nodal displacements through interpolation. The nodes in our case are the corners of the elements as shown below. As you can imagine, the numerical solution should get better as you increase the number of elements.

!Elements and nodes.png|width=350!

The following figure summarizes the contrasts between the hand calculations and ANSYS's approach. One important point to keep in mind that both start with the same mathematical model but use different assumptions and approximations to solve it.
\\
\\  [!tensile_bar_hc_vs_ANSYS.png|width=350!|^tensile_bar_hc_vs_ANSYS.png]\\
\\

h4.This brings us to the end of the _Pre-Analysis_ section.

h2. Start-Up: Load Solution into ANSYS

As mentioned before, we are providing the ANSYS solution so that you can focus on comparing the hand calculations with the ANSYS results (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]
Unzip the file at a convenient location. You will see a folder called _Tensile Bar Demo_ with the following contents:
* *Tensile Bar Demo_files* (this is a folder)
* *Tensile Bar Demo.wbpj*

Please make sure both these objects are in the unzipped 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).

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 (A6)_ in the next step of this tutorial.
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*[Go to Step 2: Numerical Results|Tensile Bar - Numerical Results]*

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