Assignment 4

3.23. Modified

1. Refer to the solution of Example 3.2, case (b). Determine the axial force P_x and moment M_z that the walls in Fig. 3.6b apply to the beam, in terms of integrals involving T(y).

2. Refer to case (a) of Example 3.2.  Formulate this problem as the superposition of the problem of part 1 and another problem.  Obtain the solution to that second problem, and give the final solution for the stresses and the strains.

1.  The solution for s_x for case (b) is obtained in the text in Eq. (3.27) as

s_x = -EaT

The resultant force and moment reactions at the restrained edges are

P_x = ∫s_xtdy = -Eat ∫Tdy 

M_z =  ∫-ys_xtdy = Eat∫yTdy

where the integrals are for the interval (-h, h).

2. To solve case (a) superimpose on case (b) the case obtained by reversing the reactions found above: case (c)

The solution to case (c), with relaxed B.C.'s at the left and right edges, is known:

s_x = -P_x/A + M_zy/I

where A = 2th, and I = 2th³/3

The stress in case (a)  is the sum of the stresses in (b) and (c),  or

s_x =  - EaT - P_x/A + M_zy/I  = Ea[-T + (t/A) ∫Tdy  + (ty/I)∫yTdy]

This result coincides with Eq. (3.26) in the text.

The other stresses in both cases (b) and (c) are zero.

The strains could also obtained by superposition,  but a direct use of the resulting stresses is simpler. Thus,

e_x =  aT + (s_x - ns_y)/E =  aT + (s_x - 0)/E =   a[(t/A) ∫Tdy  + (ty/I)∫yTdy]

e_y =  aT + (s_y - ns_x)/E =  aT - ns_x/E  = (1 + n)aT - na[(t/A) ∫Tdy  + (ty/I)∫yTdy]

g_xy =  0