## CAE and river dam

 Three Gorges Dam China Three gorges river dam 2, China

Dams are classified as timber dams, arch-gravity dams, embankment dams or masonry dams, with several subtypes. We have computing some simple dam during our termomechanics course. I’ll try to program a short program that visualizes the air and water tension that a dam has to cope with during my holidays. The forces are as following: we have air pressure that comes from up and water pressure that acts on a dam. The picture below describes a case that I have computed. We set coordinate system

$x_{1},x_{2},x_{3}$ as shown in the picture below.

Water Dam Schema. Pressure and forces are acting on the back, striped wall and the upper one. There is an air pressure acting on water dam from above and water from behind. A construction must sustain both. In other words in order a river dam to sustain it has to sustain stresses and distortions to some computed level. Read below to know how to compute stresses and distortions.

Plane stress

## Deriving mathematical model:

$\sigma_{ij}=2\mu\epsilon_{ij}+\lambda\sigma_{ij}\epsilon_{kk}$

$\sigma_{11}=\sigma_{22}=\sigma_{33}=-P$

$P=-(\lambda+\frac{2}{3}\mu)e = -Ke$

$\epsilon_{ij}=\frac{1}{2\mu}(\sigma_{ij}-\frac{\lambda}{3\lambda+2\mu}\sigma_{kk}\sigma_{ij})$

E- Young.

## Using constitutive equations you get:

$\begin{vmatrix} \sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{12} \\ \sigma_{23} \\ \sigma_{31} \end{vmatrix}=\begin{vmatrix} \lambda+2\mu & \lambda & \lambda & & & \\\lambda & \lambda+2\mu & \lambda & & & \\\lambda & \lambda & \lambda+2\mu & & & \\ & & &\lambda & \\ & & & &\lambda & \\ & & & & &\lambda \end{vmatrix}$

$e=\epsilon_{11} + \epsilon_{22} + \epsilon_{33} = \frac{v-v_{0}}{v_{0}}$

$\sigma_{kk}=(3\lambda+2\mu)\epsilon_{kk}=(3\lambda+2\mu)e$

## Plane stress state:

$\begin{vmatrix} \sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{vmatrix} = \frac{E}{(1+\gamma)(1-2\gamma)}\begin{vmatrix} 1-\gamma & \gamma & 0 \\ \gamma & 1-\gamma & 0 \\ 0 & 0 & \frac{1-2\gamma}{2} \end{vmatrix}$

$\sigma_{33}=\frac{E\gamma}{(1+\gamma)(1-2\gamma)}(\epsilon_{11}+\epsilon_{22})$

$\begin{vmatrix} \epsilon_{11} \\ \epsilon_{12}\\ 2\epsilon_{12}\end{vmatrix}=\frac{1+\gamma}{E}\begin{vmatrix} 1-\gamma & -\gamma & 0 \\-\gamma & 1-\gamma & 0 \\ 0 & 0 & 2 \end{vmatrix} \begin{vmatrix} \sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{vmatrix}$

$\sigma_{33}=\gamma(\sigma_{11}+\sigma_{22})$

## Example

$E=80 000 MPa \\ \mu=\frac{1}{3} \\ \epsilon_{11}=0.002 \\ \epsilon_{22}=-0.001\\ \epsilon_{12}=0.001$

$\begin{vmatrix} \sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{vmatrix} = \frac{80000}{1-(\frac{1}{3})^2}\begin{vmatrix} 1 & \frac{1}{3} & \\\frac{1}{3} & 1 & \\ & & \frac{1}{3} \end{vmatrix} \begin{vmatrix} 0.002 \\ -0.001 \\ 0.002 \end{vmatrix}=\begin{vmatrix}90 & 30 & \\30 & 90 & \\ & & 30 \end{vmatrix}=\begin{vmatrix}150 \\ -30 \\ 60 \end{vmatrix}MPa \\ \epsilon_{33}=-\frac{1}{2}(0.001)=-0.0005$

$180000\begin{vmatrix}\frac{2}{3} & \frac{1}{3} & \\ \frac{1}{3} & \frac{2}{3} & \\ & & \frac{1}{6} \end{vmatrix}\begin{vmatrix} 0.002 \\ -0.001 \\ 0.002 \end{vmatrix}=\begin{vmatrix}120 & 60 & \\ 60 & 120 & \\ & & 30\end{vmatrix}$

$\sigma_{33}=\frac{1}{3}(180)=60$

Combining those computations to geometrical model of dam we can easily visualize stresses and distortion the pressure creates. As you can see, the computational model I have published here is not sophisticated. Nonetheless is true and provide a lot of fun and models quite good real dam (concerning it as a student exercise).