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

v
w
P
v
w
v
P
A
L
B
v B
x
=
A
L
B
( v B ) w
x
+
A
L
B
( v B ) P
x
(a) Total load (b) Distributed load only (c) Concentrated load only
FIGURE 10.8 Superposition principle applied to beam deflections.
mecmovies
Consider a cantilever beam subjected to a uniformly distributed load and a concentrated load
at its free end. To compute the deflection at B (Figure 10.8a), two separate deflection calculations
can be performed. First, the cantilever beam deflection at B is calculated considering
only the uniformly distributed load w (Figure 10.8b). Next, the deflection caused by the
concentrated load P alone is computed (Figure 10.8c). The results of these two calculations
are then added algebraically to give the deflection at B for the total load.
Beam deflection and slope equations for common support and load configurations
are frequently tabulated in engineering handbooks and other reference materials. A table
of equations for frequently used simply supported and cantilever beams is presented in
Appendix C. (This table of common beam formulas is often referred to as a beam table.)
Appropriate application of these equations enables the analyst to determine beam deflections
for a wide variety of support and load configurations.
Several conditions must be satisfied if the principle of superposition is to be valid for
beam deflections:
1. The deflection must be linearly related to the loading. Inspection of the equations found
in Appendix C shows that all load variables (i.e., w, P, and M) are first-order variables.
2. Hooke’s law must apply to the material, meaning that the relationship between stress
and strain remains linear.
3. The loading must not significantly change the original geometry of the beam. This
condition is satisfied if beam deflections are small.
4. Boundary conditions resulting from the sum of individual cases must be the same as
the boundary conditions in the original beam configuration. In this context, boundary
conditions are normally deflection or slope values at beam supports.
ExAmpLE
m10.7 Introduction to the superposition method with two
elementary examples, one of a cantilever beam and the other
of a simply supported beam.
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Applying the Superposition method
The superposition method can be a quick and powerful method for calculating beam deflections;
however, its application may initially seem more like an art than an engineering