# Navier Solution for a Singularly Loaded Kirchhoff Rectangular Plate

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This Demonstration illustrates the approximate Navier solution for the bending and stress analysis of a simply supported Kirchhoff rectangular plate subject to a concentrated singular load applied at given coordinates. The deformation of the middle surface is represented on a 3D plot, colored according to its von Mises equivalent stress.

Contributed by: Jorge García Tíscar (September 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Start from the classical equation for pure bending of a simply supported Kirchhoff plate with dimensions , Young modulus , and Poisson modulus subject to a load :

(flexural rigidity).

Assume a general sinusoidal load of the form

.

Assume a solution for the deformation of the form

.

Consider the boundary conditions of the simply supported plate:

at ,

at .

Then you get the following general solution:

.

In this case, the load is a singular one of value applied at coordinates :

.

Follow Navier's procedure by expanding the load as a Fourier series:

.

Solving for (using orthogonality) you get

.

Introducing the coefficients in the prior general solution and applying the principle of superposition, you can derive a general expression for the deformation under unitary load () applied at coordinates :

.

You can then calculate the deformation , multiplying by the load value :

.

Now you can compute the stresses at the top of the plate ), following the classic theory of Kirchhoff plates:

,

,

.

You can then find the two principal stresses (Kirchhoff theory assumes plane stress):

Finally, you can obtain the von Mises equivalent stress:

Reference

[1] S. P. Timoshenko, *Theory of Plates and Shells*, New York: McGraw–Hill, 1959.

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