This module calculates the wall stresses and deflections in homogeneous, linear elastic, isotropic, cylinders and spheres subjected to internal and/or external pressure. Stress and deflection results found by this module are valid only for wall sections remote from any stress concentration effects. Stresses and deflections in the vicinity of end caps or vessel openings must be calculated by other methods (see Reference 3).

Figure 1. Module input form

• Hoop (circumferential) Stress
• Radial Stress
• Axial (longitudinal) Stress (Thick-walled cylinder only)
• Radial Deflection

Plotted Output

Radial and hoop (circumferential) stresses can be plotted versus radial station using the standard ETB plot package shown in Figure 2.

Figure 2. Module plot results.

Text Output

Input parameters and all calculated results are tabulated in the standard ETB text output window shown in Figure 3.

Figure 3. Module tabulated results.

Figure 4. Thick-walled cylinder

Normal Stresses

The geometry and stress definitions for the thick-walled cylinder problem are shown in Figure 4.  The stresses in the cylinder wall are functions of the inner radius, r_i, the outer radius, r_o, the inner pressure, P_i, the outer pressure, P_o, and a radial coordinate, r, where r_i <= r <= r_o.

If the cylinder ends are closed and the resulting axial loads are carried by the cylinder (not by the cylinder support structure) the normal stress in the longitudinal direction, or axial stress, is given by:

If the cylinder ends are free, the axial stress is zero.

The normal stress in the circumferential direction, or hoop stress, at a radial distance r, where r_i <= r <= r_o, is given by:

The normal stress in the radial direction, or radial stress, at a radial distance r, where r_i <= r <= r_o, is given by:

Radial Deflections

Radial deflections are derived from the calculated stresses using basic geometry and Hooke's law.  Given a linear isotropic material with an Elastic Modulus, E, and a Poisson's Ratio, n, the radial deflection can be calculated from:

Figure 5. Thick-walled sphere

Normal Stresses

The geometry and stress definitions for the thick-walled sphere problem are shown in Figure 5.  The stresses in the sphere wall are functions of the inner radius, r_i, the outer radius, r_o, the inner pressure, P_i, the outer pressure, P_o, and a radial coordinate, r, where r_i <= r <= r_o.

The normal stresses in the circumferential directions are given by:

The normal stresses in the radial direction is given by:

Radial Deflection:

Radial deflections are derived from the calculated stresses using basic geometry and Hooke's law.  Given a linear isotropic material with an Elastic Modulus, E, and a Poisson's Ratio, n, the radial deflection can be calculated from: