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TABLE 8.2 Deflection and Bending Moments of Clamped Plate Under Uniform Load q
[Evans 1939]
b/a W(x = 0, y = 0) M x (x = a/2, y = 0) M y (x = 0, y = b/2) M x (x = 0, y = 0) M y (x = 0, y = 0)
4
1 0.00126qa /D −0.0513qa 2 −0.0513qa 2 0.0231qa 2 0.0231qa 2
4
1.5 0.00220qa /D −0.0757qa 2 −0.0570qa 2 0.0368qa 2 0.0203qa 2
4
2 0.00254qa /D −0.0829qa 2 −0.0571qa 2 0.0412qa 2 0.0158qa 2
4
∞ 0.00260qa /D −0.0833qa 2 −0.0571qa 2 0.0417qa 2 0.0125qa 2

FIGURE 8.4 Thin plate subjected to positive pressure q.
read out using a conventional resistive bridge circuit. The initial pressure sensors were fabricated via
anisotropic etching of silicon, which results in a rectangular diaphragm. Figure 8.4 shows a thin-plate,
subjected to normal pressure q, resulting in out-of-plane displacement w(x, y). The equilibrium condition
for w(x, y) is given by the thin plate theory [Timoshenko 1959]:

4 4 4
q
∂ w
∂ w 2---------------- + ∂ w ----, (8.28)
--------- =
--------- +
2
∂x 4 ∂x ∂y 2 ∂y 4 D
3
2
where D = Eh /12(1 − ν ) is the flexural rigidity, E is the Young’s modulus, ν is the Poisson ratio, and h
is the thickness of the plate. The edge-moments (moments per unit length of the edge) and the small
strains are
2
2
2


∂ w
∂ w
∂ w
(
M x x, y( ) = – D --------- – n--------- , e xx x, y, z) = – z---------

2 
∂x 2 ∂y ∂x 2
2
2
2


∂ w
∂ w
∂ w
(
M y x, y) = – D --------- – n--------- , e yy x, y, z) = – z--------- (8.29)
(

2 
∂y 2 ∂x ∂y 2
2
2
∂ w
∂ w
(
M xy x, y) = D(1 n)------------, e xy x, y, z) = – z------------
(
–
∂x∂y ∂x∂y
Using (8.29), one can calculate the maximum strains occurring at the top and bottom faces of the plate
in terms of the edge-moments:
12
e xx x, y, z( ) = 12z nM y ) = -------- M x –( nM y )
-------- M x –(
max
Eh 3 z=h Eh 2
(8.30)
12
max
(
-------- M y –(
e yy x, y, z) = 12z nM x ) z=h = -------- M y –( nM x )
Eh 3 Eh 2
In the case of a pressure sensor with a diaphragm subjected to a uniform pressure, the boundary conditions
are built-in edges: w = 0, ∂w/∂x = 0 at x = ±a/2 and w = 0, ∂w/∂y = 0 at y = ±b/2, where the diaphragm
has lateral dimensions a × b. The solution of this problem has been obtained by [Evans 1939], showing
that the maximum strains are at the center of the edges. The values of the edge-moments and the
displacement of the center of plate are listed in Table 8.2.
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