Plates - Levy's solution

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As mentioned, this method can be used on a plate with two pinned edges. The other two edges can have arbitrary boundary conditions:

Levy chose to express the displacement in the plate as a single sine Fourier series:

and this function will satisfy the boundary conditions for the two sides.

The displacement function must also satisfy the differential equation for the plate, and the boundary conditions at y = 0 and y = b. To establish a solution, it is practical to divide the displacement function into one homogeneous solution and one particular solution:

w(x,y) = wH(x,y)+wP(x,y)

After some calculations, you will end up with an expression for the displacement:

The expression contains four unkown constants for each m, Am, Bm, Cm and Dm. The boundary conditions on the two sides with arbitrary support is used to determine these constants.

The general steps in Levy's solution is therefore:

  • Calculate the coefficients in the load Fourier series.
  • Establish the particular solution for all m.
  • Superposition of the homogeneous and particular solution.
  • Us the boundary conditions on the sides with arbitrary support to determine the constants Am, Bm, Cm and Dm.
  • The final solution is then expressed as above