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

beam (also called a simple beam). A simply supported beam has a pin support at one end
and a roller support at the opposite end. Figure 7.3b shows a variation of the simply
supported beam in which the beam continues across the support in what is termed an
overhang. In both cases, the pin and roller supports provide three reaction forces for the
simply supported beam: a horizontal reaction force at the pin and vertical reaction forces at
both the pin and the roller. Figure 7.3c shows a cantilever beam. A cantilever beam has a
fixed support at one end only. The fixed support provides three reactions to the beam: horizontal
and vertical reaction forces and a reaction moment. These three unknown reaction
forces can be determined from the three equilibrium equations (i.e., ΣF x = 0, ΣF y = 0, and
ΣM = 0) available for a rigid body.
y
A
195
SHEAR ANd MOMENT
IN bEAMS
(a) Simply supported beam
A
(b) Simple beam with overhang
B
B
x
C
Types of Loads
Several types of loads are commonly supported by beams (Figure 7.4). Loads focused
on a small length of the beam are called concentrated loads. Loads from columns or
from other members, as well as support reaction forces, are typically represented by
concentrated loads. Concentrated loads may also represent wheel loads from vehicles
or forces applied by machinery to the structure. Loads that are spread along a portion
of the beam are termed distributed loads. Distributed loads that are constant
in magnitude are termed uniformly distributed loads. Examples of
uniformly distributed loads include the weight of a concrete floor slab or the
forces created by wind. In some instances, the load may be linearly distributed,
which means that the distributed load, as the term implies, changes
linearly in magnitude over the span of the loading. Snow, soil, and fluid
pressure, for example, can create linearly distributed loads. A beam may
also be subjected to concentrated moments, which tend to bend and rotate
the beam. Concentrated moments are most often created by other members
that connect to the beam.
A
Concentrated
load
Uniformly
distributed
load
(c) Cantilever beam
FIGURE 7.3 Types of statically
determinate beams.
Linearly
distributed
load
FIGURE 7.4 Symbols used for
various types of loads.
B
Concentrated
moment
7.2 Shear and moment in Beams
To determine the stresses created by applied loads, it is first necessary to determine the
internal shear force V and the internal bending moment M acting in the beam at any point
of interest. The general approach for finding V and M is illustrated in Figure 7.5. In this
figure, a simply supported beam with an overhang is subjected to two concentrated loads
P 1 and P 2 as well as to a uniformly distributed load w. A free-body diagram is obtained by
cutting a section at a distance x from pin support A. The cutting plane exposes an internal
shear force V and an internal bending moment M. If the beam is in equilibrium, then any
portion of the beam that we consider must also be in equilibrium. Consequently, the free
body with shear force V and bending moment M must satisfy equilibrium. Thus, equilibrium
considerations can be used to determine values for V and M acting at location x.
Because of the applied loads, beams develop internal shear forces V and bending
moments M that vary along the length of the beam. For us to properly analyze the stresses
produced in a beam, we must determine V and M at all locations along the beam span.
These results are typically plotted as a function of x in what is known as a shear-force
and bending-moment diagram. This diagram summarizes all shear forces and bending
moments along the beam, making it straightforward to identify the maximum and minimum
values for both V and M. These extreme values are required for calculating the
largest stresses.