Navier Solution for a Singularly Loaded Kirchhoff Rectangular Plate

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.

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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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