Page 188 - Shigley's Mechanical Engineering Design

bud29281_ch04_147-211.qxd 11/27/2009 7:54 pm Page 163 pinnacle s-171:Desktop Folder:Temp Work:Don't Delete (Jobs):MHDQ196/Budynas:

Deflection and Stiffness 163
The strain energy stored in a beam or lever by bending may be obtained by refer-
ring to Fig. 4–8b. Here AB is a section of the elastic curve of length ds having a radius
of curvature ρ. The strain energy stored in this element of the beam is dU = (M/2)dθ.
Since ρ dθ = ds, we have
Mds
dU = (a)
2ρ
We can eliminate ρ by using Eq. (4–8), ρ = EI/M. Thus
2
M ds
dU = (b)
2EI
.
For small deflections, ds = dx. Then, for the entire beam
M
2
U = dU = dx (c)
2EI
The integral equation is commonly needed for bending, where the moment is typically
a function of x. Summarized to include both the integral and nonintegral form, the strain
energy for bending is
2
M l ⎫
U = ⎪ (4–22)
2EI
or ⎬ bending
M ⎪
2
U = dx ⎭ (4–23)
2EI
Equations (4–22) and (4–23) are exact only when a beam is subject to pure bend-
ing. Even when transverse shear is present, these equations continue to give quite good
results, except for very short beams. The strain energy due to shear loading of a beam
is a complicated problem. An approximate solution can be obtained by using Eq. (4–20)
with a correction factor whose value depends upon the shape of the cross section. If we
use C for the correction factor and V for the shear force, then the strain energy due to
shear in bending is
2
CV l ⎫
U = ⎪ (4–24)
2AG
or ⎬ transverse shear
CV ⎪
2
U = dx ⎭ (4–25)
2AG
Values of the factor C are listed in Table 4–1.
Table 4–1
Strain-Energy Correction Beam Cross-Sectional Shape Factor C
Factors for Transverse Rectangular 1.2
Shear Circular 1.11
Source: Richard G. Budynas, Thin-walled tubular, round 2.00
Advanced Strength and Applied Box sections † 1.00
Stress Analysis, 2nd ed.,
McGraw-Hill, New York, 1999. Structural sections † 1.00
Copyright © 1999 The
McGraw-Hill Companies. † Use area of web only.