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

The maximum force P that can be applied without causing any permanent deformation
will be controlled by the smaller of these two areas. Therefore, P is calculated from
the yield strength σ Y and A 2 :
2 2
P = σ A = (124 N/mm )(176.715 mm) = 21,912.61 N
Y 1
Now use Equation (17.14) to calculate the strain energy of each segment, as well as
the total strain energy in the brass rod:
U
n 2
Fi
Li
F1 2 L1
F2 2 L2
= ∑ = +
2AE
2AE
2A E
i=
1
i
i
1 1
2 2
2
2
(21,912.61 N) (900 mm) (21,912.61 N) (1,100 mm)
= +
2 2
2 2
2(490.874 mm )(115,000 N/mm ) 2(176.715 mm )(115,000 N/mm )
= 3,827.7 Nmm ⋅ + 12,995.1 Nmm ⋅
= 16.82 Nmm ⋅ = 16.82 J
Ans.
17.4 Elastic Strain Energy for Torsional
Deformation
A
FIGURE 17.13 Prismatic shaft
in pure torsion.
L
ϕ
T
B
x
Consider a circular prismatic shaft of length L, and suppose the shaft is subjected to a
torque T as shown in Figure 17.13. If the torque is applied gradually, the free end B of the
shaft rotates through an angle φ. If the bar is linearly elastic, the relationship between
the torque T and the rotation angle of the shaft will also be linear, as shown in the torque–
rotation diagram of Figure 17.14 and as given by the torque–twist relationship φ = TL/JG,
where J is the polar moment of inertia of the cross-sectional area. The external work W
done by the torque as it rotates through the angle φ is equal to the area of the shaded triangle.
From the principle of conservation of energy, and with no dissipation of energy in
the form of heat, the strain energy of the circular shaft is thus
U = W =
1 2
Tφ
From the torque–twist relationship φ = TL/JG, the strain energy in the shaft can be
expressed in terms of the torque T as
T
U
T 2
= L
(17.16)
2JG
or in terms of the rotation angle φ as
T
O
ϕ
1
U= W= 2
Tϕ
FIGURE 17.14 Torque–
rotation diagram for a linearly
elastic material.
ϕ
U
JG φ
= 2
(17.17)
2L
Notice the parallels in form between Equations (17.12) and (17.13), on the one hand,
which express the strain energy in a prismatic bar with a constant axial load, and Equations
(17.16) and (17.17), on the other, which give the strain energy for a prismatic shaft with a
constant torque.
The total strain energy of a shaft that consists of several segments (each having a
constant torque, polar moment of inertia, and shear modulus) is equal to the sum of the
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