Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

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dEFLECTIONS OF bEAMS by
THE VIRTuAL-wORk METHOd
A
x
dx
B
Δ
L
C
θ
Deflection due to
real external loads
x
D
A
x
dx
B
L
x
D
(a) Prismatic beam subjected to
an arbitrary real loading
m
M
dθ= dx
EI
dx
(c) Internal work of virtual moment m
FIGURE 17.23 Virtual-work method for beams.
m
A
v
(b) Virtual external load required
in order to determine D at B
x
dx
L
(d) Virtual external moment required
in order to determine θ at C
C
1
x
D
deformations created by the real external loads (in Figure 17.23a), the virtual external work
performed by the virtual external load as the beam moves downward through the real
deflection D will be
W 1 ve = ⋅D (a)
To obtain the virtual internal work, recall from Section 17.5 that the internal work of
a beam is related to the moment and the rotation angle θ of the beam. Now consider a
differential beam element dx located at a distance x from the left support, as shown in
Figures 17.23a and 17.23b. When the real external loads are applied to the beam, bending
moments M rotate the plane sections of the beam segment dx through an angle
M
d θ = EI dx
(b)
When the beam with the virtual unit load (Figure 17.23b) is subjected to the real rotations
caused by the external loading (Figure 17.23a), the virtual internal bending moment m acting
on the element dx performs virtual work as the element undergoes the real rotation dθ, as
shown in Figure 17.23c. For beam element dx, the virtual internal work dW vi performed by the
virtual internal moment m as the element rotates through the real internal rotation angle dθ is
dW
vi
= mdθ
(c)
Note that the virtual moment m remains constant during the real rotation dθ; therefore,
Equation (c) does not contain the factor ½. (Compare Equation (c) with the expression for
work in Figure 17.16.)
Now substitute the expression for dθ in Equation (b) into Equation (c) to obtain
⎛ ⎞
dW = m⎜
M vi ⎟ dx
(d)
⎝ EI ⎠