# Structural mechanics simulation using FLUENT compared to analytical solution

This is a continuation of my previous post, 1D analytical solution vs FEM solution for a bar under compression. For some reason, I cannot comment in it.

The analytical solution to the 1-D static compression problem is:

\begin{align} u = \frac{du}{dx}x \end{align}

where $x=0$ is at the fixed end, and $x=L$ is at the loaded end.

My question this time deals with meshing. In FLUENT, you cannot simulate a structural problem with only 1 column of elements. I believe you need at least 2 elements in each direction.

Say you simulate a problem with 9 columns for a square rod, where the square cross-section is divided into 9 smaller square elements. Would you expect that the displacement solution in the compression direction to be the same in all 9 elements or would you expect it to vary?

Let's consider an orthohedric bar defined by

$$(x,y,z)\in [-w/2,w/2]\times[-d/2,d/2]\times[0, h]\, .$$

A vertical load $P$ is applied on the top ($z=h$) surface and the following boundary conditions on the bottom surface:

\begin{align} &u_x(0, y, z) = 0\\ &u_y(x, 0, z) = 0\\ &u_z(x, y, 0) = 0\, , \end{align}

that would imply that your bottom surface can slide freely (except for the midpoint), the solutions for the differential equations are:

\begin{align} &u =\frac{P}{E}(\nu x, \nu y, -z)\, ,\\ &\epsilon = \frac{P}{E}\begin{bmatrix}\nu &0 &0\\ 0 &\nu &0\\ 0 &0 &-1\end{bmatrix}\, ,\\ &\sigma = P\begin{bmatrix}0 &0 &0\\ 0 &0 &0\\ 0 &0 &-1\end{bmatrix}\, , \end{align}

where $P$ is the load, $E$ the Young modulus and $\nu$ the Poisson's ratio. Based on this solution we can see that the displacement is larger for points further away from the center due to the Poisson's effect (except for $\nu=0$).

• Let's continue this conversation in chat. – nicoguaro Apr 24 '18 at 14:26