78 
Mil. L, N. G. FILON OX AX APPHOXIMATE SOLUTIOX FOR BEXDIXG A 
§ 9. Part of the Solutions Corresponding to the Terinr a,,, Co, ^o- 
Ill the first place it is obvious, liaving regard to the conditions of rigid equilibrium, 
that if the ends a; = d: « a-i’e free from stress, then ag must = /Bq. If ^ /3g Ave 
must have a shear over the two ends in order to balance the excess of the pressure 
on the one side over the pressure on the other side, and this will require special 
investigation. The solution arising from such conditions is discussed in §§ 39—40. 
For the present let us confine ourselves to olq = /3g. This corresponds to a uniform 
traction along the axis y and introduces the following additional terms :— 
E ’ 
P = 0 , Q = ao> 
V — “"ll- 
^ r 
S = 0 j 
(53). 
Now turning to the terms in and 6q, it is easy to verify that the additional 
terms 
TT — (X + 2/x) \ 3X + 4;U, , y \ 
16 yu, (pJ + fjb) 
'7/ 
h 
and therefore 
Q = 0, 
P = - 
r,, - 
2h 
s = 
Cl) fo 
2h 
y 
(54), 
satisfy the conditions that 8 shall have constant values over the tAvo boundaries 
y = dszh, these values being equal in magnitude and ojiposite in sign. The effect of 
these shears is balanced by the pressure and tension (Co — ^o) over the tAvo ends, 
and the conditions of rigid equilibrium are satisfied. 
Finally, if Ave have equal shears over the boundaries, the sign being the same (so 
that the external impressed forces act in opposite directions), the solution 
^ . . . . (oo) 
P = 0, Q = (1, S = i (i„ + 0„) 
Avill satisfy all conditions over the boundaries ^ = dz ‘^nd Avill introduce over the 
boundaries .t = d= « a system of shear necessary to maintain rigid equilibrium. 
Adding together the solutions (54) and (55), Ave find that the conditions Q = 0 
over y = p- h, S = Co ^^er y = -^ h, S = 9 q over y = — b are all satisfied. 
This completes the solution of the problem proposed, with the exception of the case 
ag ;8g, Avhich can be reduced to the problem of a beam uniformly loaded along the 
top and free along the bottom, the load being taken by shears oA’er the ends. 
