Compute the stresses σxx, σyy and σxy for the plate of Example 13.1.Example 13.1: Square P...

Compute the stresses σxx, σyy and σxy for the plate of Example 13.1.

Example 13.1: Square Plate Subject to Sinusoidally Distributed Pressure

A square plate is simply supported on all edges (Figure 13.7) and is loaded by gravel such that

(a) Determine the maximum deflection and its location.

(b) Determine the maximum values of the moments Mxx, Myy.

(c) Determine the maximum values of the Kirchhoff shear forces Vx, Vy.

Solution:

The boundary conditions for simply supported edges are

Since w = 0 around the plate boundary,  for edges parallel to the x axis and likewise for edges parallel to they axis. Hence, noting the expressions for Mxx Myy in Eq. 13.54, we may rewrite the boundary conditions, Eqs. (b), in the form (note that b =a)

(a) Equations (c) may be satisfied by taking win the form

where w0 is a constant that must be chosen to satisfy the plate equation (Eq. 13.56), namely, with Eq. (a),

Substitution of Eq. (d) into Eq. (e) yields

By Eq. (d), we see that the maximum deflection of the plate occurs at x = y = a/2. Thus, the maximum deflection of the plate is

(b) To determine the maximum values, o f moment s Mxx, Myy,we find from Eqs. 13.54 with Eqs. (d) and (f)

The maximum values of Mxx and Myy occur at x = y = a/2. Thus,

(c) To calculate the Kirchhoff shear forces, we have by Eqs. 13.54 with Eqs. (d) and (f)

We see that the maximum values, of Vx and Vy occur along the edges of the plate. Thus, by Eqs. (j),

FIGURE 13.7 Simply supported rectangular plate.

 (13.54)

 (13.56)