Axisymmetric stress analysis formulation

Axisymmetric problems

A body of revolution (geometry and material symmetric about an axis) with loads and supports also symmetric about the same axis can be analysed on a single plane section, as a 2D problem in $r$-$z$. All quantities are independent of the circumferential angle $\theta$. Examples: pressure vessels, pipes, flywheels, rotors, pistons and shrink fits.

        z (axis)
        |    +------+
        |    | mesh |  <- triangular ring
        |    +------+     elements
        +------------- r

Each element is really a ring of material of volume $2\pi\bar r A$.

Unknowns and strains

Displacements: radial $u$ and axial $w$. Because $u$ displaces a ring, the circumference changes, producing a hoop strain.

$$\{\varepsilon\} = \begin{Bmatrix} \varepsilon_r \\ \varepsilon_\theta \\ \varepsilon_z \\ \gamma_{rz} \end{Bmatrix} = \begin{Bmatrix} \partial u/\partial r \\ u/r \\ \partial w/\partial z \\ \partial u/\partial z + \partial w/\partial r \end{Bmatrix}$$

For the triangular element with $u = \sum N_iu_i$:

$$[B] = \frac{1}{2A}\begin{bmatrix} \beta_1 & 0 & \beta_2 & 0 & \beta_3 & 0 \\ \frac{\alpha_1}{r} + \beta_1 + \frac{\gamma_1z}{r} & 0 & \frac{\alpha_2}{r} + \beta_2 + \frac{\gamma_2z}{r} & 0 & \frac{\alpha_3}{r} + \beta_3 + \frac{\gamma_3z}{r} & 0 \\ 0 & \gamma_1 & 0 & \gamma_2 & 0 & \gamma_3 \\ \gamma_1 & \beta_1 & \gamma_2 & \beta_2 & \gamma_3 & \beta_3 \end{bmatrix}$$

with $\alpha_i$, $\beta_i = z_j - z_k$ and $\gamma_i = r_k - r_j$ defined as for the CST using $(r, z)$.

Stress-strain matrix

$$[D] = \frac{E}{(1 + \nu)(1 - 2\nu)}\begin{bmatrix} 1 - \nu & \nu & \nu & 0 \\ \nu & 1 - \nu & \nu & 0 \\ \nu & \nu & 1 - \nu & 0 \\ 0 & 0 & 0 & \frac{1 - 2\nu}{2} \end{bmatrix}$$

Stiffness matrix

$$[k] = 2\pi\int_A [B]^T[D][B]\,r\,dr\,dz$$

The hoop-strain row of $[B]$ contains $1/r$ and $z/r$ terms, so $[B]$ varies with position, unlike the CST in the plane problem. The integral also contains the factor $r$. Therefore $[k]$ is not constant. In practice it is approximated by evaluating $[B]$ at the centroid $(\bar r, \bar z)$:

$$[k] \approx 2\pi\,\bar r\,A\,[\bar B]^T[D][\bar B]$$

This is accurate when elements are small compared with their distance from the axis. Loads are applied as total forces on the full ring, $2\pi r$ times the load per unit circumference.