Page 179 - Shigley's Mechanical Engineering Design

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154 Mechanical Engineering Design
the results algebraically. Superposition may be applied provided: (1) each effect is
linearly related to the load that produces it, (2) a load does not create a condition that
affects the result of another load, and (3) the deformations resulting from any spe-
cific load are not large enough to appreciably alter the geometric relations of the
parts of the structural system.
The following examples are illustrations of the use of superposition.

EXAMPLE 4–2 Consider the uniformly loaded beam with a concentrated force as shown in Fig. 4–3.
Using superposition, determine the reactions and the deflection as a function of x.

Solution Considering each load state separately, we can superpose beams 6 and 7 of Table A–9.
For the reactions we find
Fb wl
Answer R 1 = +
l 2
Fa wl
Answer R 2 = +
l 2
The loading of beam 6 is discontinuous and separate deflection equations are given
for regions AB and BC. Beam 7 loading is not discontinuous so there is only one equa-
tion. Superposition yields

Fbx wx
3
2
2
2
3
2
Answer y AB = (x + b − l ) + (2lx − x − l )
6EIl 24EI
Fa(l − x) 2 2 wx 2 3 3
Answer y BC = (x + a − 2lx) + (2lx − x − l )
6EIl 24EI
Figure 4–3 y
l
F
a b
w
C
A x
B
R 1 R 2

If the maximum deflection of a beam is desired, it will occur either where the slope
is zero or at the end of the overhang if the beam has a free end. In the previous example,
there is no overhang, so setting dy/dx = 0 will yield the equation for x that locates
where the maximum deflection occurs. In the example there are two equations for y
where only one will yield a solution. If a = l/2, the maximum deflection would obvi-
ously occur at x = l/2 because of symmetry. However, if a < l/2, where would the
maximum deflection occur? It can be shown that as F moves toward the left support,
the maximum deflection moves toward the left support also, but not as much as F (see
Prob. 4–34). Thus, we would set dy BC /dx = 0 and solve for x. If a > l/2, then we
would set dy AB dx = 0. For more complicated problems, plotting the equations using
numerical data is the simplest approach to finding the maximum deflection.