Draft:Elastic membrane

An elastic membrane is a structural element of small thickness and negligible bending stiffness that can only resist tensile stress.

Geometric description

Geometrically, a membrane is characterized by having a curved middle surface and a small thickness on either side of this middle surface. Typically, the points of a shell are represented by three parameters (u, v, z), where two of them (u, v) represent the middle surface and the third, in the perpendicular direction, represents the thickness. Thus, the points of a membrane can be represented by the position vector:

$\mathbf {r} (u,v,z)=\mathbf {R} (u,v)+z\mathbf {n}$

Where:

$\mathbf {R} (u,v)$ is the position vector of a point on the middle surface.

$\mathbf {n}$ is the unit normal vector at each point of the middle surface.

$-h/2\leq z\leq h/2$ is the coordinate along the thickness.

$h\,$ is the total thickness of the membrane.

When forces are applied from the convex side and in the direction of the normal vector to the middle surface, the membrane deforms under the effect of these normal stresses. At each point on this surface, the stresses in the membrane are related to the acting forces and the radii of curvature along two perpendicular directions.

Equations of equilibrium

In terms of the normal pressure acting on the middle surface of a membrane, the stresses along two perpendicular directions are related by:

${\frac {\sigma _{u}}{\rho _{u}}}+{\frac {\sigma _{v}}{\rho _{v}}}={\frac {p}{h}}\qquad {\text{(1)}}$

Where:

$u,v\,$ are two orthogonal coordinates on the middle surface of the membrane.

$\rho _{u},\rho _{v}\,$ are the radii of curvature along the coordinate lines u and v.

$\sigma _{u},\sigma _{v}\,$ are the stresses along the coordinate lines u and v.

$p\,$ is the pressure perpendicular to the membrane.

$h\,$ is the thickness of the membrane.

Prandtl's membrane

A particular case of a membrane is a nearly flat membrane (i.e., one whose deviation from a flat shape is small) with a constant tension per unit length at its boundary. Under this assumption, equation **(1)** can be written by calculating the inverse of the radius of curvature from the second derivative of the deflection with respect to the flat state. This yields the equation:

${\frac {\partial ^{2}w}{\partial x^{2}}}+{\frac {\partial ^{2}w}{\partial y^{2}}}={\frac {p}{t}}\qquad {\text{(2)}}$

Where:

$w\,$ is the deflection of the membrane from the original plane.

$p\,$ is the pressure perpendicular to the membrane.

$t\,$ is the uniform tension per unit length at the boundary of the membrane.

Prandtl's analogy

The above membrane equation was used by Ludwig Prandtl (1903) to study torsion stresses (of the pure Saint-Venant type) in a structural element with a non-circular cross-section. Specifically, Prandtl proved that if a membrane has the same shape as the cross-section whose torsion is to be studied and is subjected to a pressure difference between its two sides, the shape assumed by the membrane shows the stress distribution for the case of torsion. This is known as Prandtl's membrane analogy.

To see Prandtl's membrane analogy in detail, consider the vector field of shear stresses associated with torsion, which satisfy the equation:

${\boldsymbol {\tau }}=(\tau _{zx},\tau _{zy}),\qquad {\frac {\partial \tau _{zx}}{\partial x}}+{\frac {\partial \tau _{zy}}{\partial y}}=0$

For a simply connected cross-section with a smooth boundary curve, a scalar stress function can be defined such that:

$\tau _{zx}={\frac {\partial \phi }{\partial y}},\quad \tau _{zy}=-{\frac {\partial \phi }{\partial x}},\qquad {\frac {\partial ^{2}\phi }{\partial x^{2}}}+{\frac {\partial ^{2}\phi }{\partial y^{2}}}=-2G\theta \qquad {\text{(3)}}$

Where $\theta \,$ is the angle of twist per unit length under the effect of torsion, and $G$ is the shear modulus. This equation is analogous to equation **(2)** if the following identifications are made:

Fluid storage tanks

Fluid storage tanks are characterized by a state of biaxial tension. In tanks containing gases, the pressure can be considered uniform over all the walls, whereas in tanks containing liquids, the pressure exerted on the walls varies with height.

Cylindrical tank under uniform pressure

In a cylindrical tank of radius R and height H, subjected to a uniform pressure p, the maximum stress occurs in the circumferential (hoop) direction, and the minimum stress in the longitudinal direction; specifically, the former is twice the latter. Using cylindrical coordinates and exploiting the fact that the radius of curvature in the longitudinal direction is infinite, we have:

$\lim _{R_{z}\to \infty }\left({\frac {\sigma _{z}}{R_{z}}}+{\frac {\sigma _{\theta }}{R_{\theta }}}\right)={\frac {\sigma _{\theta }}{R}}={\frac {p}{e}}\qquad \Rightarrow \qquad \sigma _{\theta }={\frac {pR}{e}}$

Where:

$e\,$ is the thickness of the tank wall.

$\sigma _{\theta },\sigma _{z}\,$ are the circumferential (hoop) and longitudinal stresses, respectively.

The longitudinal stress can be obtained by calculating the tensile stress of one half of the tank on the other. The equilibrium equation yields:

$(2\pi Re)\sigma _{z}=(\pi R^{2})p\qquad \Rightarrow \qquad \sigma _{z}={\frac {pR}{2e}}$

Spherical tank under uniform pressure

In a spherical tank of radius R, subjected to a uniform pressure p, the maximum stress is identical in all directions and is given by:

${\frac {\sigma }{R}}+{\frac {\sigma }{R}}={\frac {p}{e}}\qquad \Rightarrow \qquad \sigma ={\frac {pR}{2e}}$

Where:

$e\,$ is the thickness of the tank wall.

Cylindrical tank for liquids

In a cylindrical tank of radius R, filled with a liquid of density $\rho$ up to a height H, the pressure increases linearly below the free surface of the fluid. The stress in the circumferential direction at a height y is calculated straightforwardly as:

$\lim _{R_{z}\to \infty }\left({\frac {\sigma _{z}}{R_{z}}}+{\frac {\sigma _{\theta }}{R_{\theta }}}\right)={\frac {\sigma _{\theta }}{R}}={\frac {\rho g(H-y)}{e}}\qquad \Rightarrow \qquad \sigma _{\theta }=\rho g(H-y){\frac {R}{e}}$

Where:

$e\,$ is the thickness of the tank wall.

$\sigma _{\theta },\sigma _{z}\,$ are the circumferential (hoop) and longitudinal stresses, respectively.

The longitudinal stress in this case depends heavily on how the tank is supported.

References

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Categories:

• Structural elements
• Strength of materials