Problem 3.10

3.10.  Add part (d): Check overall equilibrium.

a) Determine the stresses for the stress function F = px³y.

s_x = F,_yy = 0

s_y = F,_xx = 6pxy

t_xy = - F,_xy = -3px²

b) Show the boundary stresses (tractions) on the plate.

c)  Resultants of boundary tractions

For each diagram the resultant force is equal to the area under the diagram.  For a triangular, normal- traction diagram, the equivalent resultant is at the two-third point from the zero value.

Top edge:

R_x is shown in the figure.

R_y = (6pab)(a)/2 = 3pa²b at 2a/3 from left edge.

Right edge:

x-force = 0

y-force = p_yb = -3pa²b

Bottom edge:

x-force =  pa³

y-force = 0

Left edge: Edge is free

d) Check overall equilibrium

SF_x = pa³ - pa³ = 0

SF_y = 3pa²b -3pa²b = 0

SM = moment of two couples = - (a/3)(3pa²b) + b(pa³) = 0