Learning topic

Membrane Theory

Membrane theory assumptions, Laplace equilibrium, meridional and hoop force resultants, and stresses in thin cylindrical and spherical shells.

0 practice tasks · 0 subtopics

This topic explains membrane theory for thin shells, where bending and twisting moments and transverse shear are neglected. It covers membrane force resultants, Laplace equilibrium, cylindrical and spherical pressure shells, stress calculations, assumptions, and regions where bending effects require a more advanced shell model.

Membrane theory of shells is an approximate theory in which bending and twisting moments and transverse shear forces are neglected, while loading is carried primarily by membrane force resultants in the tangent plane of the midsurface.

Main assumptions

shell thickness $t$ is small compared with the characteristic radii of curvature;
through-thickness stresses can be represented by average membrane values;
the analyzed region is sufficiently far from local edge disturbances;
geometry and loading permit equilibrium without significant bending moments.

Membrane force resultants

In the principal midsurface directions, normal force resultants $N_1$ and $N_2$ have units of force per unit length, such as N/mm. For thickness $t$:

$$\sigma_1=\frac{N_1}{t},\qquad \sigma_2=\frac{N_2}{t}.$$

Laplace equilibrium equation

For an element of a thin shell in a membrane state under normal pressure $p$, local equilibrium in the surface-normal direction gives:

$$\frac{N_1}{R_1}+\frac{N_2}{R_2}=p.$$

$N_1$, $N_2$ — membrane normal force resultants per unit length in the two principal directions of the midsurface;
$R_1$, $R_2$ — corresponding principal radii of curvature;
$p$ — resultant normal load per unit area of the midsurface.

For a shell of constant thickness $t$, the corresponding average normal stresses are $\sigma_1=N_1/t$ and $\sigma_2=N_2/t$. Signs must follow the adopted convention for curvature, pressure, and membrane forces.

The normal equilibrium equation alone is generally insufficient to determine two unknown membrane resultants $N_1$ and $N_2$. A second relation follows from equilibrium of a cut-off portion of the shell, symmetry, or other membrane-equilibrium equations.

Cylindrical shell under internal pressure

For a cylinder, one principal radius of curvature equals $r$, while curvature along the generator is zero, so the corresponding radius is formally infinite. Normal equilibrium gives the hoop resultant $N_\theta=pr$. The longitudinal resultant $N_z$ in a closed cylinder follows from equilibrium of the end portion.

For a thin-walled closed cylindrical shell of radius $r$ and thickness $t$ under uniform internal gauge pressure $p$, sufficiently far from edges and local disturbances:

$$N_\theta=pr,\qquad \sigma_\theta=\frac{pr}{t},$$

$$N_z=\frac{pr}{2},\qquad \sigma_z=\frac{pr}{2t}.$$

The hoop stress $\sigma_\theta$ is twice the longitudinal stress $\sigma_z$.

For a thin-walled closed spherical shell:

$$N=\frac{pr}{2},\qquad \sigma=\frac{pr}{2t}.$$

$p$ — internal gauge pressure;
$r$ — shell radius;
$t$ — wall thickness;
$N_\theta$, $N_z$ — hoop and longitudinal membrane force resultants per unit length;
$\sigma_\theta$, $\sigma_z$ — hoop and longitudinal membrane stresses;
$N$, $\sigma$ — membrane resultant and stress in a spherical shell.

These are membrane approximations for $t\ll r$ and uniform pressure. Additional bending and local stresses may occur near edges, openings, supports, and abrupt geometric changes.

The calculator below uses the hoop-stress relation for a thin-walled cylinder.

Derivation of the longitudinal cylinder resultant

Pressure $p$ on the end cap produces a resultant $p\pi r^2$. It is balanced by longitudinal membrane force around the circumference:

$$2\pi rN_z=p\pi r^2,$$

hence:

$$N_z=\frac{pr}{2}.$$

Spherical shell

For a sphere, $R_1=R_2=r$ and symmetry gives $N_1=N_2=N$. From the Laplace equation:

$$\frac{N}{r}+\frac{N}{r}=p,$$

so $N=pr/2$ and $\sigma=pr/(2t)$.

Example

A thin-walled closed cylinder has internal radius $r=500\ \text{mm}$, thickness $t=10\ \text{mm}$, and internal gauge pressure $p=2\ \text{MPa}$. The membrane stresses are:

$$\sigma_\theta=\frac{2\cdot500}{10}=100\ \text{MPa},$$

$$\sigma_z=\frac{2\cdot500}{2\cdot10}=50\ \text{MPa}.$$

Thus, for this idealized model, the hoop stress $\sigma_\theta$ is more critical than the longitudinal stress $\sigma_z$.

Calculation procedure

Check that the shell is thin-walled and that bending moments may reasonably be neglected in the region being analyzed.
Define the midsurface, thickness $t$, principal directions, and radii of curvature $R_1$ and $R_2$.
Determine the external loading: normal pressure $p$, self-weight, or other distributed actions.
Write the local shell equilibrium equations and, where necessary, equilibrium of a cut-off portion of the shell.
Determine membrane force resultants $N_1$ and $N_2$ per unit length.
Calculate average membrane stresses $\sigma_1=N_1/t$ and $\sigma_2=N_2/t$.
Identify critical regions and perform the required strength check using the adopted criterion.
Assess edges, openings, supports, joints, concentrated loads, and abrupt changes in geometry separately because membrane theory may be insufficient there.

Limitations

The membrane state is commonly disturbed near rigid edges, supports, flanges, nozzles, openings, joints, concentrated loads, and abrupt changes in curvature or thickness. Bending and edge stresses then require bending shell theory or numerical analysis.