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