Problem 3.6

3.6.  Let the boundary conditions at the top and bottom walls be conditions of smooth contact ( not fixed, as the figure may indicate).

Problem is one of plane stress

1) Strain-displacement relations of plane stress

e_x = u,_x =  -(1 - n²)s_o/E

e_y = v,_y =  0

g_xy = u,_y + v,_x = 0

Additional relation

e_z = w,_z =  n(1 + n)s_o/E   (1)

2) Stress-strain relations of plane stress

s_x = (E/1-n²)(e_x + ne_y) = -s_o

s_y = (E/1-n²)(e_y + ne_x) = -ns_o

t_xy = Gg_xy  = 0

Auxiliary equation obtained from condition s_z = 0:

e_z =  -n(s_x + s_y)/E  =  n(1 + n)s_o/E  agrees with   (1)

3) Equilibrium equations

Stresses are constant, and thus satisfy the differential equations of equilibrium

4) Boundary conditions

At x = ± a:

s_x = -s_o, satisfied.

t_xy = 0, satisfied.

At top and bottom edges, assuming smooth contact:

v = 0, satisfied.

t_yx = 0, satisfied.

Conclusion: All governing equations and boundary conditions are satisfied by the given displacements.