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:
and
- .
Where and are the second derivatives in - and - direction. and are the proportions of the load in - and - direction.
With the equilibrium condition one obtains the membrane equation:
- ,
where is the Laplace operator .
The boundary condition is assumed. This means that the edge is supported and does not experience any deflection.
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
- ^ 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 .
- ↑ 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 .