Page 91 - Aircraft Stuctures for Engineering Student

76 Energy methods of structural analysis

Also
d2v
M = -EI- (see Eqs (9.20)) (iii)
dz2
Substituting in Eq. (iii) for v from Eq. (i) and for M in Eq. (ii) from Eq. (iii)

EI ‘v;7r4
- dz
u=- lo L4sin 7rz
L
which gives
7r4 EIv;
u=-----
4L3
The total potential energy of the beam is then given by
7r4 EIv;
TPE=U+V=-- WVB
4L3
Then, from the principle of the stationary value of the total potential energy

a( u + V) - 7r4EIV~
~-
- w=o
dVB 2~3
whence
2 WL’ WL3
?/B = ~ = 0.02053 -
7r4 EI EI
The exact expression for the mid-span displacement is (Ref. 3)
WL3 WL3
VB =- = 0.02083 ~
48 EI EI
Comparing the exact (Eq. (v)) and approximate results (Eq. (iv)) we see that the
difference is less than 2 per cent. Further, the approximate displacement is less than
the exact displacement since, by assuming a displaced shape, we have, in effect,
forced the beam into taking that shape by imposing restraint; the beam is therefore
stiffer.

4.5 The principle of the stationary value of the
total complementary energy

Consider an elastic system in equilibrium supporting forces P, , P2, . . . P,, which
produce real corresponding displacements A,, A2,. . . A,,. If we impose virtual
forces SP, , SP2, . . . , CiP,, on the system acting through the real displacements then
the total virtual work done by the system is, by the argument of Section 4.4

The first term in the above expression is the negative virtual work done by the
particles in the elastic body, while the second term represents the virtual work of