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

CHAPTER10
beam deflections
10.1 Introduction
Important relations between applied load and both normal and shear stresses developed
in a beam were presented in Chapters 8 and 9. However, a design is normally not complete
until the deflection of the beam has been determined for its particular load. While
they generally do not create a safety risk in themselves, excessive beam deflections may
impair the successful functioning of a structure in other ways. In building construction,
excessive deflections can cause cracks in walls and ceilings. Doors and windows may not
close properly. Floors may sag or vibrate noticeably as people walk on them. In many
machines, beams and flexural components must deflect just the right amount for gears or
other parts to make proper contact. In sum, the satisfactory design of a flexural component
usually includes a specified maximum deflection in addition to a minimum loadcarrying
capacity.
The deflection of a beam depends on the stiffness of the material and the crosssectional
dimensions of the beam, as well as on the configuration of the applied loads and
supports. Three common methods for calculating beam deflections are presented here:
(1) the integration method, (2) the use of discontinuity functions, and (3) the superposition
method.
In the discussion that follows, three coordinates will be used. As shown in Figure 10.1,
the x axis (positive to the right) extends along the initially straight longitudinal axis of the
beam. The x coordinate is used to locate a differential beam element, which has an undeformed
width of dx. The v axis extends positive upward from the x axis. The v coordinate
measures the displacement of the beam’s neutral surface. The third coordinate is y, which is
a localized coordinate with its origin at the neutral surface of the beam cross section. The y
coordinate is measured positive upward, and it is used to describe specific locations within
the beam cross section. The x and y coordinates are the same as those used in deriving the
flexure formula in Chapter 8.
v, y
w
FIGURE 10.1 Coordinate
system.
P
M
x
391