## 25 Plates

2.5.1 Bending of Thin Plates

A plate whose thickness is small compared to the other dimensions is called a thin plate. The plane parallel to the faces of the plate and bisecting the thickness of the plate, in the undeformed state, is called the middle plane of the plate. When the deflection of the middle plane is small compared with the thickness h, it can be assumed that

1. There is no deformation in the middle plane.

2. The normals of the middle plane before bending are deformed into the normals of the middle plane after bending.

3. The normal stresses in the direction transverse to the plate can be neglected.

Based on these assumptions, all stress components can be expressed by deflection w of the plate. w is a function of the two coordinates (x, y) in the plane of the plate. This function has to satisfy a linear partial differential equation, which, together with the boundary conditions, completely defines w.

Figure 2.31a shows a plate element cut from a plate whose middle plane coincides with the xy plane. The middle plane of the plate subjected to a lateral load of intensity q is shown in Figure 2.31b. It can be shown, by considering the equilibrium of the plate element, that the stress resultants are

(b)
FIGURE 2.31 (a) Plate element and (b) stress resultants.

02w 0x 0y

0x3 0x 0y

92w s2 0y 0x2 0y

where

Mx, My = bending moments per unit length in the x and y directions, respectively Mxy, Myx = twisting moments per unit length

Qx, Qy = shearing forces per unit length in the x and y directions, respectively

Vx, Vy = supplementary shear forces in the x and y directions, respectively

R = corner force

D = Eh3/(12(1 — v )), flexural rigidity of the plate per unit length v = Poisson's ratio

The governing equation for the plate is obtained as

04w 04w 04 w q

Any plate problem should satisfy the governing Equation 2.41 and boundary conditions of the plate.

2.5.2 Boundary Conditions

There are three basic boundary conditions for plates. These are the clamped edge, simply supported edge, and free edge.

### 2.5.2.1 Clamped Edge

For this boundary condition, the edge is restrained such that the deflection and slope are zero along the edge. If we consider the edge x = a to be clamped, we have

2.5.2.2 Simply Supported Edge

If the edge x = a of the plate is simply supported, the deflection w along this edge must be zero. At the same time this edge can rotate freely with respect to the edge line. This means that

2.5.2.3 Free Edge

If the edge x = a of the plate is entirely free, there are no bending and twisting moments and also vertical shearing forces. This can be written in terms of w, the deflection, as q2w n 02w\ _o 0x2 n 0y2/ "

2.5.3 Bending of Rectangular Plates

A plate bending problem may be solved by referring to the differential Equation 2.41. The solution, however, depends on the loading and boundary conditions. Consider a simply supported plate subjected to a sinusoidal loading as shown in Figure 2.32. The differential Equation 2.41 in this case becomes

The boundary conditions for the simply supported edges are

The deflection function becomes px py w = w0 sin — sin— (2.47)

ab which satisfies the boundary conditions in Equations 2.46. w0 must be chosen to satisfy Equation 2.45. Substitution of Equation 2.47 into Equation 2.45 gives

The deflection surface for the plate can, therefore, be found to be q0 px py w = ——77—^tt-;—"ttttv sin — sin — 2.48

Using Equations 2.48 and 2.36 we find expression for moments as q0 /1 v\ . px . py Mx =-7 + sin—sin-f-

x p2((1/a2) + (1/b2 ))2V a2 bV a b q0 / n 1 \ . px . py

Maximum deflection and maximum bending moments that occur at the center of the plate can be written by substituting x — a/2 and y — b/2 in Equations 2.49 as q0

If the plate is square, then a = b and Equations 2.50 become

0 0