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

8.4 Analysis of Bending Stresses in Beams
In this section, the flexure formula will be applied in the analysis of bending stresses for
statically determinate beams subjected to various applied loads. The analysis process
begins with the construction of shear-force and bending-moment diagrams for the specific
span and loading. The cross-sectional properties of the beam will be determined next.
Essential properties include
(a) the centroid of the cross section,
(b) the moment of inertia of the cross-sectional area about the centroidal axis of bending, and
(c) the distances from the centroidal axis to both the top and bottom surfaces of the beam.
After these prerequisite calculations have been completed, bending stresses can be
calculated from the flexure formula at any location on the beam.
Beams can be supported and loaded in a variety of ways; consequently, the distribution
and intensity of positive and negative bending moments are unique for each beam. Understanding
the significance of the bending-moment diagram as it relates to flexural stresses is
essential for the analysis of beams. For instance, consider a reinforced-concrete beam with an
overhang, as shown in Figure 8.8. Concrete is a material with substantial strength in compression,
but very low strength in tension. When concrete is used to construct a beam, steel bars
must be placed in those regions where tensile stresses occur, in order to reinforce the concrete.
In some portions of the overhang beam, tensile stresses will develop below the neutral
axis, while tensile stresses will occur above the neutral axis in other portions. The engineer
must define these regions of tensile stress so that the reinforcing steel is placed where it is
needed. In sum, the engineer must be attentive not only to the magnitude of bending stresses,
but also to the sense (either tension or compression) of stresses that occur above and below
the neutral axis and that vary with positive and negative bending moments along the span.
cross-Sectional Shapes for Beams
Beams can be constructed from many different cross-sectional shapes, such as squares,
rectangles, solid circular shapes, and round pipe or tube shapes. A number of additional
shapes are available for use in structures made of steel, aluminum, and fiber-reinforced
M
Steel reinforcing bars
Tension on top
Tension on bottom
FIGURE 8.8 Reinforced-concrete beam.
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