Modeling a Simply Supported Rectangular Plate

Let’s say that there is a rectangular plate that is simply supported, and it has dimensions of a = 48 in, and b = 24 in, along the x and y axes respectively. Let’s say it has a geometric loading of

From this geometric loading, I can determine:

A. Computer-generated plot of q(x,y)

B. The final (reduced) equation that describes the load coefficient q_mn and the location and magnitude of the maximum load.

C. The final (reduced) equation that describes the deflection w(x,y).

D. Computer-generated plot of the deflection w(x,y) assuming the plate has the properties of E = 8 X 106 psi, ν = 0.3, and t = 0.125 in.

E. The location and magnitude of the maximum deflection.

Part A

Part B

The load coefficient q_mn is equal to q0 which is 30 psi whenever the double sine values are their maxima or minima.

The ranges for -> m <- are from 1 to 5 in finite integer increments.

The ranges for -> n <- are from 2 to 2 in finite integer increments.

The location of the maximum load, q_max, occurs at 24 inches in the positive x-direction and 18 inches in the positive y-direction with a maximum load of 30 psi.

Part C

Part D

Part E

The location of the maximum deflection, w_max, occurs at 24 inches in the positive x-direction and 18 inches in the positive y-direction with a maximum deflection of 0.679707 inches.

Here is a list of the assumptions used for arriving at a solution.

Assumption # 1: We are dealing with small deflections

• All deflection components (u, v, w) are small compared to the plate thickness (t), although deflections can be several times thickness as long as the plate remains elastic with small curvatures.

Assumption # 2: Material is linearly-elastic

· such that the normal and shear stress components do not exceed the elastic limits of the material.

Assumption # 3: Isotropic Conditions

Assumption # 4: Laterally distributed load

Assumption # 5: Simply Supported Boundary Conditions

• The supported edges are free to move in the plane of the plate. Hence, there is no membrane action, and the plate's middle surface has no deformation (i.e., it is the neutral surface). With this assumption, all the strain components in the midplane of the plate go to zero.

Assumption # 6: Transverse-shear strains γ xz = γ yz = 0

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