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Plate and Shell Analyses 191

Et 3
D = 2 (the bending rigidity of the plate ) (6.12)
(
12 1 − ν )
q = lateral distributed load (force per unit area).
Note that Equation 6.11 is of analogous form with the following 1-D equation for straight
beam:

4
dw
EI = qx() (6.13)
dx 4
The fourth-order partial differential equation, given in Equation 6.11 and in terms of
the deflection w(x,y), needs to be solved under certain given boundary conditions. Typical
boundary conditions for plate bending include:

∂ w
Clamped: w = 0, = 0 (6.14)
∂ n

Simply supported: w = 0, M n = 0 (6.15)

Free: Q n = 0, M n = 0 (6.16)

where n is the normal direction of the boundary (Figure 6.5). Note that the given values in
the boundary conditions shown above can be nonzero values as well.

6.2.2.1 Example: A Thin Plate
A square plate (Figure 6.6) with four edges clamped or hinged, and under a uniform load
q or a concentrated force P at the center C.
For this simple geometry, Equation 6.11 with the boundary condition of Equation 6.14 or
6.15 can be solved analytically. The maximum deflections are given in Table 6.1 for the four
different cases. These values can be used to verify the FEA solutions.

s n

Boundary

FIGURE 6.5
The boundary of a plate.