Page 187 - Shigley's Mechanical Engineering Design

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162 Mechanical Engineering Design
In Sec. 4–9, we will demonstrate the usefulness of singularity functions in solving
statically indeterminate problems.

4–7 Strain Energy
The external work done on an elastic member in deforming it is transformed into strain,
or potential, energy. If the member is deformed a distance y, and if the force-deflection
relationship is linear, this energy is equal to the product of the average force and the
deflection, or
F F 2
U = y = (4–15)
2 2k
This equation is general in the sense that the force F can also mean torque, or moment,
provided, of course, that consistent units are used for k. By substituting appropriate
expressions for k, strain-energy formulas for various simple loadings may be obtained.
For tension and compression, for example, we employ Eq. (4–4) and obtain
2
F l ⎫
U = (4–16)
2AE ⎪
or ⎬ tension and compression
F ⎪
2
U = dx ⎭ (4–17)
2AE
where the first equation applies when all the terms are constant throughout the length,
and the more general integral equation allows for any of the terms to vary through the
length.
Similarly, from Eq. (4–7), the strain energy for torsion is given by
2
T l ⎫
U = (4–18)
2GJ ⎪
or ⎬ torsion
T ⎪
2
U = dx (4–19)
2GJ ⎭
To obtain an expression for the strain energy due to direct shear, consider the element
with one side fixed in Fig. 4–8a. The force F places the element in pure shear, and the
work done is U = Fδ/2. Since the shear strain is γ = δ/l = τ/G = F/AG, we have
2
F l ⎫
U = (4–20)
2AG ⎪
or ⎬ direct shear
F ⎪
2
U = dx ⎭ (4–21)
2AG
Figure 4–8 O

F
d
A
F ds
l F B
dx
(a) Pure shear element (b) Beam bending element