A thick-walled cylinder cannot be represented by the membrane-stress approximation used for thin shells; its stress and displacement vary through the wall. This case simultaneously exercises curved pressure loading, cylindrical stress distributions, symmetry conditions, and the plane-strain constitutive relation.

What this case verifies

• Sign and magnitude of pressure traction normal to the inner wall
• Radial and hoop stress variation through the thickness
• The plane-strain condition for a long cylinder
• Roller constraints on a quarter-symmetry model
• Mesh-convergence order of a smooth displacement quantity

Problem definition

The cylinder has inner radius $a=0.05$ m and outer radius $b=0.1$ m. Internal pressure $p_i$ is applied, the external pressure is zero, and roller conditions are imposed on the two cut faces of the quarter model. The model uses plane strain to represent a sufficiently long, axially restrained cylinder.

Reference solution

The axisymmetric Lamé stress field has the form

$$\sigma_r(r) = A - \frac{B}{r^2}, \qquad \sigma_\theta(r) = A + \frac{B}{r^2}$$

For internal pressure only,

$$A = \frac{p_i a^2}{b^2-a^2}, \qquad B = \frac{p_i a^2b^2}{b^2-a^2}$$

Using tension-positive signs, $\sigma_r(a)=-p_i$ and $\sigma_r(b)=0$ , exactly satisfying the two radial traction conditions. Under plane strain, the radial displacement is

$$u_r(r) = \frac{1+\nu}{E} \left[(1-2\nu)Ar + \frac{B}{r}\right]$$

The pressure/constraint junction can generate corner singularities. The smooth inner-wall displacement $u_r(a)$ is therefore used as the primary acceptance quantity instead of a single local peak stress.

Results

The inner-wall displacement agrees with theory to within 0.25%. Under mesh refinement, the observed order $p_{obs}$ is approximately 1.8–2, consistent with the expected second-order convergence of the smooth displacement field. Convergence at the expected rate is stronger implementation evidence than a small error on only one mesh.

Engineering significance

This case is a baseline for pressure vessels, hydraulic cylinders, bearing or bushing fits, and rotor sleeves, where through-thickness stress gradients matter and the inner and outer surfaces must be evaluated separately.

Reference

• MIT OpenCourseWare, Introduction to Structural Mechanics — “Thick-Walled Cylinder under Radial Pressure.”

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