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Steel Designers' Manual - 6th Edition (2003)
Fig. 9.27 Unit load method
jPi r2 jFs
B C D
iagram
IA
11
B
qram
Analysis of skeletal structures 315
The total strain energy of an elastic system is given by the sum of the strain
energies stored in each member due to bending, shear, torsion and axial
loading.
The use of this will be illustrated by considering a simply-supported beam of
length l subject to an external loading (see Fig. 9.27). Strain energy stored in the
beam is predominantly flexural and is given by
l
2 M dx
Ú 2EI
0
x1= deflection under Pl=
P
∂ Ê
Á
∂ Ë
∂M
is the bending moment due to a unit load and is denoted by m.
∂P1
Hence the procedure of the unit load method can be outlined (see Fig. 9.27):
El
(9.21)
M
(1) diagram due to the external loading is obtained.
EI
(2) The external loads are now removed and the moment diagram (m) due to a
unit load applied at the point of required deflection is drawn.
(3) These two diagrams should now be integrated; in other words, the ordinates of
the two diagrams are multiplied to obtain the deflection, given by
l
x
M
= Ú mdx EI
0
1
l l
2
M dxˆ
˜ =
2EI
¯
Ú Ú
M M
EI P x
∂
d
∂
0 0 1