Page 178 - Shigley's Mechanical Engineering Design

bud29281_ch04_147-211.qxd 11/27/09 2:55PM Page 153 ntt 203:MHDQ196:bud29281:0073529281:bud29281_pagefiles:

Deflection and Stiffness 153
(uniform loading), variable intensity q(x), to Dirac delta functions (concentrated
loads).
The intensity of loading usually consists of piecewise contiguous zones, the
expressions for which are integrated through Eqs. (4–10) to (4–14) with varying
degrees of difficulty. Another approach is to represent the deflection y(x) as a Fourier
series, which is capable of representing single-valued functions with a finite number of
finite discontinuities, then differentiating through Eqs. (4–14) to (4–10), and stopping
at some level where the Fourier coefficients can be evaluated. A complication is the
piecewise continuous nature of some beams (shafts) that are stepped-diameter bodies.
All of the above constitute, in one form or another, formal integration methods,
which, with properly selected problems, result in solutions for q, V, M, θ, and y. These
solutions may be
1 Closed-form, or
2 Represented by infinite series, which amount to closed form if the series are rapidly
convergent, or
3 Approximations obtained by evaluating the first or the first and second terms.
The series solutions can be made equivalent to the closed-form solution by the use of a
1
computer. Roark’s formulas are committed to commercial software and can be used on
a personal computer.
There are many techniques employed to solve the integration problem for beam
deflection. Some of the popular methods include:
• Superposition (see Sec. 4–5)
• The moment-area method 2
• Singularity functions (see Sec. 4–6)
• Numerical integration 3
The two methods described in this chapter are easy to implement and can handle a large
array of problems.
There are methods that do not deal with Eqs. (4–10) to (4–14) directly. An energy
method, based on Castigliano’s theorem, is quite powerful for problems not suitable for
the methods mentioned earlier and is discussed in Secs. 4–7 to 4–10. Finite element
programs are also quite useful for determining beam deflections.
4–5 Beam Deflections by Superposition

The results of many simple load cases and boundary conditions have been solved
4
and are available. Table A–9 provides a limited number of cases. Roark’s provides
a much more comprehensive listing. Superposition resolves the effect of combined
loading on a structure by determining the effects of each load separately and adding

1 Warren C. Young and Richard G. Budynas, Roark’s Formulas for Stress and Strain, 7th ed., McGraw-Hill,
New York, 2002.
2 See Chap. 9, F. P. Beer, E. R. Johnston Jr., and J. T. DeWolf, Mechanics of Materials, 5th ed., McGraw-Hill,
New York, 2009.
3 See Sec. 4–4, J. E. Shigley and C. R. Mischke, Mechanical Engineering Design, 6th ed., McGraw-Hill,
New York, 2001.
4 Warren C. Young and Richard G. Budynas, Roark’s Formulas for Stress and Strain, 7th ed., McGraw-Hill,
New York, 2002.