Problem 2.40

2.40.  In answering the question concerning the boundary conditions, please do the following:

a) Check that the boundary conditions on the bottom and side edges are satisfied.

b) Show that the conditions of a fixed top edge, as implied by the figure, are not satisfied.

c) Check that the relaxed boundary conditions at the top edge are satisfied.

Field Equations

The 15 equations of elasticity consist of:

a) 6 strain - displacements relations

let c = g/E

e_x = u,_x = -ncz

e_y = v,_y = -ncz

e_z = w,_z = cz

g_xy = u,_y + v,_x = 0

g_yz = v,_z + w,_y =  -ncy + ncy = 0

g_zx = w,_x + u,_z = ncx - ncx = 0

b) 6 stress - strain relations

s_ij = 2Ge_ij + l(e_kk)d_ij

e_kk = e₁₁ + e₂₂ + e₃₃  = (1 - 2n)cz

d_ij = 1 if i = j, 0 otherwise

l = nE/(1+n)(1-2n)

G = E/2(1+n)

le_kk =  nEcz/(1+n) =  ngz/(1+n)

s_x = 2Ge_x + l(e_kk) =  -nczE/(1+n) + ngz/(1+n)  =  - ngz/(1+n) + ngz/(1+n) = 0

s_y = 2Ge_y + l(e_kk) = 0 since e_y = e_x

s_z = 2Ge_z + l(e_kk) = Ecz/(1+n)  + ngz/(1+n) = gz/(1+n)  + ngz/(1+n) = gz

All shear stresses are zero since all shear strains are zero

c) 3 equilibrium equations

s_ij,_i + F_j = 0

We have F₁ = F₂ = 0, and

F₃ = -g

For j = 1,2, obtain 0 = 0, and for j = 3, obtain

s₃₃,₃ + F₃ =  g -  g = 0

All 15 equations are satisfied.

Boundary Conditions

a) Check that the boundary conditions on the bottom and side edges are satisfied.

The bottom edge is free. The conditions are [(s_z , t_zx , t_zy ) at z = 0]  = 0.  Satisfied.

The side edges are free. The conditions are [(s_y , t_yx , t_yz ) at y = h and y = -h] = 0.  Satisfied.

b) Show that the conditions of a fixed top edge, as implied by the figure, are not satisfied.

At the top edge, z = a.  Thus 

u = -ncax

v = -ncay

w = cn(x² + y²)/2

The top edge is not fixed

c) Check that the relaxed boundary conditions at the top edge are satisfied.

The relaxed boundary conditions at the top edge are
x-resultant force = 0. Satisfied since t_zx = 0.

y-resultant force = 0. Satisfied since t_zy = 0.

z-rezultant force upward = weight

weight = g(volume) = g(2h)(b)(a)

z-rezultant force upward = s_z(area) = g(a)(2h)(b) = weight