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

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bENdINg
y
Longitudinal plane of symmetry
y
w(x)
P
z
x
z
x
Longitudinal axis
Longitudinal “fibers”
FIGURE 8.2a Longitudinal plane of symmetry.
FIGURE 8.2b The notion of longitudinal
“fibers.”
conditions, and the applied loads are symmetric with respect to this plane of symmetry.
Coordinate axes used for beams will be defined so that the longitudinal axis of the member
will be designated the x axis; the y axis will be directed vertically upward, and the z axis
will be oriented so that the x–y–z axes form a right-hand coordinate system. In Figure 8.1b,
the x–y plane is called the plane of bending, since the loads and the member deflection
occur in this plane. Bending (also termed flexure) is said to occur about the z axis.
In discussing and understanding the behavior of beams, it is convenient to imagine the
beam to be a bundle of many longitudinal fibers, which run parallel to the longitudinal axis
(or, simply, the axis) of the beam (Figure 8.2b). This terminology originated when the most
common material used to construct beams was wood, which is a fibrous material. Although
metals such as steel and aluminum do not contain fibers, the terminology is nevertheless
quite useful for describing and understanding bending behavior. As shown in Figure 8.2b,
bending causes fibers in the upper portion of the beam to be shortened or compressed while
fibers in the lower portion are elongated in tension.
pure Bending
Pure bending refers to the flexure of a beam in response to constant (i.e., equal) bending
moments. For example, the region between points B and C of the beam shown in Figure 8.3
y
P
P
h
k
x
A
B
h
k
C
D
P
a
b
∆x
a
P
P
V
V = 0
–P
M = Pa
M
FIGURE 8.3 Example of pure bending in a region of a beam.