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

17.5 Elastic Strain Energy for Flexural Deformation
Consider an arbitrary axisymmetric prismatic beam such as the one depicted in Figure 17.15a.
As the external load P acting on the beam is gradually intensified from zero to its maximum
value, the internal bending moment M acting on a differential element dx increases steadily
from zero to its final value. In response to the bending moment M, the sides of the differential
element rotate by an angle dθ with respect to each other, as shown in Figure 17.15b. If the
beam is linearly elastic, the relationship between the bending moment M and the rotation
angle dθ of the beam will also be linear, as shown in the moment–rotation angle diagram
of Figure 17.16. Therefore, the internal work and, hence, the strain energy stored in the
differential element dx is
1
dU = 2
Mdθ
The following equation relates a beam’s bending moment to its rotation angle:
d
dx
θ =
From this equation, the strain energy stored in the differential element dx can be
stated as
2
M
dU =
2EI dx
The strain energy in the entire beam is found by integrating over the length L of
the beam:
M
EI
L
2
M
U = ∫
2EI dx
0
(17.20)
Note that the bending moment M may vary as a function of x.
When the quantity M/EI is not a continuous function of x over the entire length of the
beam, the beam must be subdivided into segments in which M/EI is continuous. The integral
on the right-hand side of Equation (17.20) is then evaluated as the sum of the integrals
for all of the segments.
In the derivation of Equation (17.20), only the effects of bending moments were considered
in evaluating the strain energy in a beam. Transverse shear forces, however, are also
present in beams subjected to nonuniform bending, and these shear forces, too, will increase
the strain energy stored in the beam. Fortunately, the strain energy associated with
shear deformations is negligibly small compared with the strain energy of flexure for most
ordinary beams, and consequently, it may be disregarded.
A
v
x
dx
L
(a)
P
FIGURE 17.15 (a) Arbitrary prismatic beam. (b) Bending moments acting on a
differential beam element.
B
x
M
dx
(b)
dθ =
M
dx
EI
M
M
M
O
θ
1
U= W= 2
Mθ
FIGURE 17.16 Moment–
rotation angle diagram for a
linearly elastic material.
θ
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