Conditions for axisymmetric stress

Conditions for axisymmetric stress

Axisymmetric stress exists when all of the following hold:

1. The body is a solid of revolution about an axis (the $z$-axis).
2. The loads (surface and body forces) are symmetric about the axis, so they do not depend on the angle $\theta$.
3. The supports/boundary conditions are symmetric about the axis.
4. The material is also axisymmetric (isotropic, or the same in every radial plane).

As a result displacement, strain and stress are functions of $r$ and $z$ only, the tangential displacement is zero, and $\gamma_{r\theta}=\gamma_{z\theta}=0$, $\tau_{r\theta}=\tau_{z\theta}=0$. The non-zero stresses are $\sigma_r$, $\sigma_z$, $\sigma_\theta$ and $\tau_{rz}$. Examples: circular footing, pile, water tank, pressure vessel.

Stress-strain relations

$$\begin{Bmatrix}\sigma_r\\\sigma_z\\\sigma_\theta\\\tau_{rz}\end{Bmatrix}=\frac{E(1-\nu)}{(1+\nu)(1-2\nu)}\begin{bmatrix}1&\frac{\nu}{1-\nu}&\frac{\nu}{1-\nu}&0\\\frac{\nu}{1-\nu}&1&\frac{\nu}{1-\nu}&0\\\frac{\nu}{1-\nu}&\frac{\nu}{1-\nu}&1&0\\0&0&0&\frac{1-2\nu}{2(1-\nu)}\end{bmatrix}\begin{Bmatrix}\varepsilon_r\\\varepsilon_z\\\varepsilon_\theta\\\gamma_{rz}\end{Bmatrix}$$

with $\varepsilon_r=\partial u/\partial r$, $\varepsilon_z=\partial w/\partial z$, $\varepsilon_\theta=u/r$, $\gamma_{rz}=\partial u/\partial z+\partial w/\partial r$.