Shells - Solution methods

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A shell element

The shells differential equation can, in principle, be solved in several ways. Three different methods will here be outlined:

1. The inverse method

By solution after the inverse method, one choose an equation that satisfy the shells differential equation, that is:


After this, one have to inspect what types of boundary conditions this equation satisfy. This means that you have a solution, and try to find out to which problems it is suitable.

2. The semi-inverse method

By solution after the semi-inverse method, one have to assert a certain pattern for the deformation or stress distribution in the shell. From this, a stress function is established. This stress function is then controlled for consistence with the boundary conditions.

3. Serial solutions

The final stress function is made up of a set of weighted functions:


where each equation satisfies the differential equation


for each n=1,2,....,N

The coefficients Cn are then determined so that the boundary conditions for the problems are fulfilled to the best.

Summing up

When the stress function are established, the stresses, strains and displacements can be determined from the following equations:

Stresses:


Strains:


Displacements:


The integration constants F1(y) and F2(x) are determined by differentiating them, and putting them in the same equation by expressing the strains by the stress function. Then you have to integrate again, and the new integration constants are determined from the boundary conditions.