Related Variational and Energy Principles - Mechanics of Structures: Variational and Computational Methods - page 154

Thus, by setting δ P k = 1 , Eq. (3.15) takes the form

L

M
EI δ

w k = Mdx (3.17)

0

Similar relations apply for the effects of axial extension and shear deformation. In using Eq.
(3.17), remember that the moment M is due to the actual loads on the structure, whereas
the moment

δ δ

M is due to the virtual (unit) load. If the slope at location k is desired, M in

Eq. (3.17) should be due to a virtual (unit) moment at k .

EXAMPLE 3.12 Statically Determinate Beam
Use the unit load method to find the vertical deflection at the free end of a cantilevered beam
carrying a distributed load p 0 (Fig. 3.10a). Show how this solution should be extended to
encompass the beam of Fig. 3.10c with a sudden jump in cross-section. Also, find the slope
at the free end.
Apply a virtual (unit) force at the point where the deflection is sought (Fig. 3.10b). From
the equilibrium condition, the moment Similarly,

the summation of moments for a segment with the actual loading p 0 (Fig. 3.10a) gives
M

δ M generated by this force is δ M =− x .

p 0 x 2

=− / 2 . We find

L L

L
0 =

p 0 x 2 p 0 x 4

8 EI

p 0 L 4
8 EI

δ MM

EI

x ( / 2 )

w = dx = dx = (1)

0

EI

0 0

as the deflection at the free end.
In utilizing the unit load method, both the real and the virtual internal forces were chosen
to satisfy equilibrium.
Extend this solution to a beam of piecewise constant varying cross-section by computing
the internal complementary virtual work for each segment of the beam separately and

FIGURE 3.10
Cantilevered beam for unit load method examples.