Rectangular membranes: notes
This form uses the theory of pure membranes, where the contribution of bending is negligible.
A polynomial expansion for w, u, and v is used as indicated in form's formulae (hit the 'Formulas' button, equations enforcing boundary conditions not shown).
The coefficients of the expansion are calculated by means of strain energy minimization. The expression used for strain energy is shown in the 'Formulas' as V.
This theory is realistic in representing plate deflections with respect to the bending theories when the center deflection is larger than plate thickness.
Please note that plate behaviour according to this theory is far from being linear: hence the principle of superposition is not applicable.
Please note also that a solution will not necessarily be found for any load value, given the geometry and material, due to the iterative method of solution required. If you don't get a meaningful result, try with a lower load. If you then get a result, this means that your load is too high for the calculation to come to the end.
Concerning the boundary conditions, the following terms are used:
-'held' means that the membrane is retained at the support in a in plane direction normal to it, and not in a direction parallel to it. This condition resembles that of a sail retained by ropes around the mast: the figure in the sheet tries to depict this condition.
-'fully held' or 'riveted' means that the membrane is retained along the support in both directions (see above). This condition resembles that of a riveted plate (but also of a welded one, of course)
It should be noted that the two conditions display quite close results, as would be expected.
Tabled figures include the location of minimum lower principal membrane stress S1m (to watch for buckling, negative=compressive), and the other (membrane) stress values.
You may also graph, by clicking the graph image, the support reactions and other values.