Geoscience Reference
In-Depth Information
P
q
(A)
(B)
M
(C)
FIGURE 11.16
Classification of beams: (A) simple beam, concentrated load; (B) cantilever beam, distrib-
uted load; and (C) continuous beam, concentrated moment.
When determining the load that can be carried, two properties of the beam are important: the
moment of inertia (
I
) and the section modulus (
Z
). The moment of inertia (
I
) is the sum of differen-
tial areas multiplied by the square of the distance from a reference plane (usually the neutral axis)
to each differential area. Note that the strength of a beam increases rapidly as its cross section is
moved farther from the neutral axis because the distance is squared. This is why a rectangular beam
is much stronger when it is loaded along its thin dimension than along its flat dimension. The section
modulus (
Z
) is the moment of inertia divided by the distance from the neutral axis to the outside of
the beam cross-section.
Of special interest to environmental engineers is the type of beams and allowable loads on
beams. Allowable loads differ from maximum loads that produce failure by some appropriate fac-
tor of safety (Brauer, 1994). The following types of beams are approximations of actual beams used
in practice (Tapley 1990):
•
Simple beams
are supported beams that have a roller support at one end and pin support
at the other. The ends of a simple beam cannot support a bending moment but can support
upward and downward vertical loads. Stated differently, the ends are free to rotate but can-
not translate in the vertical direction. The end with the roller support is free to translate in
the axial direction (see Figure 11.16A for concentrated load).
•
Cantilever beams
are beams rigidly supported on only one end. The beam carries the
load to the support, where it is resisted by moment and shear stress (see Figure 11.16B for
distributed load).
•
Continuously supported beams
rest on more than two supports (see Figure 11.16C for
concentrated moment).
•
Fixed beams
are rigidly fixed at both ends.
•
Restrained beams
are rigidly fixed at one end and simply supported at the other.
•
Overhanging beams
project beyond one or both ends of their supports.
11.6.7 b
ending
m
oment
A bending moment (internal torque) exists in a structural element when a moment is applied to the
element so that the element bends. In engineering design (and safety engineering), it is important
to determine the points along a beam where the bending moment (and shear force) is maximum
since it is at these points that the bending stresses reach their maximum values. When a beam is in
equilibrium, the sum of all moments about a particular point is zero. To determine the maximum
bending moment (uniform loading), we use the following equation:
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