Rectangular thin plates in bending with large deflections: notes
This form uses the theory of plates with large deflections, where the forces acting in the plane of the plate are not 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 .
This theory is more realistic in representing plate deflections with respect to the pure bending theory when the center deflection is larger than a fraction of plate thickness.
Tabled figures include the lower principal membrane stress S1m (to watch for buckling, negative=compressive), and the stress intensity (Tresca stress) for membrane stress SIm and membrane+bending stress SIm+b (the latter is the higher of top and bottom values).
You may also graph, by clicking the graph image, other stress values and the transverse (r) and in plane (s) support reactions.
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.
Due to the highly unstable behavior in calculating plates with large deformations when the sides are free in the plane, in these forms the condition of free sides is enforced only globally, not locally, for each side: in other words, is set to zero only the integral of the membrane stress along each side and perpendicular to the side itself.