Membrane equation (statics)

The membrane equation statically describes a membrane using a partial differential equation .

Derivation

A load acts on a membrane that is completely flexible. The resulting curvature is absorbed by a membrane tensile force . If this membrane is divided into two vertical strips in the -direction and in the -direction , the following relationships can be established, assuming that the deflection is small: ${\displaystyle p}$ ${\displaystyle H}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle w}$

${\displaystyle H{\frac {\partial ^{2}w}{\partial x^{2}}}=-p_{x}}$

and

${\displaystyle H{\frac {\partial ^{2}w}{\partial y^{2}}}=-p_{y}}$.

Where and are the second derivatives in - and - direction. and are the proportions of the load in - and - direction. ${\displaystyle {\frac {\partial ^{2}\cdot }{\partial x^{2}}}}$ ${\displaystyle {\frac {\partial ^{2}\cdot }{\partial y^{2}}}}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle p_{x}}$ ${\displaystyle p_{y}}$ ${\displaystyle p}$ ${\displaystyle x}$ ${\displaystyle y}$

With the equilibrium condition one obtains the membrane equation: ${\displaystyle p=p_{x}+p_{y}}$

${\displaystyle \Delta w={\frac {p}{H}}}$,

where is the Laplace operator . ${\displaystyle \Delta }$

The boundary condition is assumed. This means that the edge is supported and does not experience any deflection. ${\displaystyle w=0}$

The problem is thus a Poisson equation .

application

An application with the membrane analogy of twist has Ludwig Prandtl published in 1903 and the Saint-Venant torsion linked.

Individual evidence

1. ^ ^{A b} Fritz Stüssi: Design and calculation of steel structures . Springer-Verlag, Berlin, Heidelberg 1958, ISBN 978-3-662-11682-1 , pp. 206 , doi : 10.1007 / 978-3-662-11682-1 .
2. ↑ Ludwig Prandtl Collected Treatises . Springer Verlag, Heidelberg, Berlin 1961, ISBN 978-3-662-11836-8 , on the torsion of prismatic rods, p. 79-80 , doi : 10.1007 / 978-3-662-11836-8_4 .