Page 124 - Aircraft Stuctures for Engineering Student

108 Energy methods of structural analysis

It 6z (1 + at)

R

(a) (b)
Fig. 4.29 (a) Linear temperature gradient applied to beam element; (b) bending of beam element due to
temperature gradient.

the element will increase in length to 6z( 1 + at), where a is the coefficient of linear
expansion of the material of the beam. Thus from Fig. 4.29(b)
R R+h
-=
Sz Sz(1 +at)
giving

R = h/at (4.32)
Also

so that, from Eq. (4.32)
Szat
60 = - (4.33)
h
We may now apply the principle of the stationary value of the total complementary
energy in conjunction with the unit load method to determine the deflection A,, due
to the temperature of any point of the beam shown in Fig. 4.28. We have seen that the
above principle is equivalent to the application of the principle of virtual work where
virtual forces act through real displacements. Therefore, we may specify that the
displacements are those produced by the temperature gradient while the virtual
force system is the unit load. Thus, the deflection ATe,B of the tip of the beam is
found by writing down the increment in total complementary energy caused by the
application of a virtual unit load at B and equating the resulting expression to zero
(see Eqs (4.13) and (4.18)). Thus