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84 Energy methods of structural analysis
Ip +Lp +*
4 0,t 2 C,t
-
- 2 L
+ L
:z
Fig. 4.1 1 Deflection of a simply supported beam by the method of complementary energy.
Example 4.3
Calculate the vertical displacements of the quarter and mid-span points B and C of
the simply supported beam of length L and flexural rigidity EI loaded, as shown in
Fig. 4.1 1.
The total complementary energy C of the system including the fictitious loads PB.f
and Pc,f is
L
c = IL jbMdsdM - PB,& - PqAC - lo Awdz (0
Hence
dC
dM
-=/Lde~-A~=O
(ii)
apB;f
and
dM
--=hdsap,,-AC=O
dC
(iii)
aPc,f
Assuming a linearly elastic beam, Eqs (ii) and (iii) become
From A to B
thus
