Mechanics of deformable solids Definition

Membrane theory of shells

Builds on Shell theory Second fundamental form First fundamental form

Definition

Let $S$ be the middle surface of a thin shell, with metric $a_{\alpha\beta}$ , the first fundamental form, and covariant derivative $\nabla_\beta$ . Let $b_{\alpha\beta}$ be its second fundamental form with respect to a unit normal $a_3$ . Let the load per unit area of $S$ have tangential components $p^\alpha$ and normal component $p^3$ along $a_3$ . The membrane theory of shells is the shell theory in which the bending moments and transverse shear forces are zero. The symmetric membrane forces $N^{\alpha\beta}$ satisfy $\nabla_\beta N^{\alpha\beta} + p^\alpha = 0, \qquad b_{\alpha\beta} N^{\alpha\beta} + p^3 = 0.$ Indices run over $1, 2$ and repeated indices are summed.

Remarks

Example: a sphere of radius $R$ under an internal pressure $p$ . With $a_3$ the inward normal, $b_{\alpha\beta} = a_{\alpha\beta}/R$ and the load is $p^3 = -p$ . The uniform tension $N^{\alpha\beta} = N a^{\alpha\beta}$ has $\nabla_\beta N^{\alpha\beta} = 0$ and $b_{\alpha\beta}N^{\alpha\beta} = 2N/R$ , so $N = pR/2$ . For a closed circular cylinder of radius $R$ the normal equation gives the hoop force $pR$ , and the equilibrium of an end cap gives the axial force $pR/2$ .

Non-example: a flat plate under a transverse load has $b_{\alpha\beta} = 0$ , so the normal equation forces $p^3 = 0$ ; a membrane state cannot carry the load, which the plate must carry by bending.

The equations are three equations for three unknowns $N^{11}, N^{12}, N^{22}$ , so the membrane forces are statically determinate given suitable edge conditions. The sign of the normal equation depends on the choice of $a_3$ .

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Tag	`Q4YP8M4HY0`
URI	`m2://Q4YP8M4HY0`
Kind	definition
Writing	W0 stub
MSC	74K25, 74K15
Defines	Membrane theory of shells
Uses	Shell theory, Second fundamental form, First fundamental form
Changed	3 November 2026

Membrane theory of shells

Definition

Remarks

Reading path

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