Module 8
Beams
A. Bending Stress and the Plastic Moment
Beams are structural members that support transverse loads and are therefore
subjected primarily to flexure, or bending. If a substantial amount of axial load is also
present, the member is referred to as a beam–column (beam–columns are considered in
Chapter 6). Although some degree of axial load will be present in any structural member,
in many practical situations this effect is negligible and the member can be treated as a
beam. Beams are usually thought of as being oriented horizontally and subjected to
vertical loads, but that is not necessarily the case. A structural member is considered to be
a beam if it is loaded so as to cause bending. Commonly used cross-sectional shapes
include the W, S, and M shapes. Channel shapes are sometimes used, as are beams built
up from plates, in the form of I or box shapes. For reasons to be discussed later, doubly
symmetric shapes such as the standard rolled W, M, and S shapes are the most efficient.
To be able to determine the nominal moment strength Mn, we must first examine
the behavior of beams throughout the full range of loading, from very small loads to the
point of collapse. Consider the beam shown in Figure 5.2a, which is oriented so that
bending is about the major principal axis (for an I shape, it will be the x–x axis). For a
linear elastic material and small deformations, the distribution of bending stress will be as
shown in Figure 5.2b, with the stress assumed to be uniform across the width of the
beam.
For a homogeneous material, the neutral axis coincides with the centroidal axis.
Equation.5.3 is based on the assumption of a linear distribution of strains from top to
bottom, which in turn is based on the assumption that cross sections that are plane before
bending remain plane after bending. In addition, the beam cross section must have a
vertical axis of symmetry, and the loads must be in the longitudinal plane containing this
axis. Beams that do not satisfy these criteria are considered in Section 5.15. The
maximum stress will occur at the extreme fiber, where y is maximum. Thus there are two
maxima: maximum compressive stress in the top fiber and maximum tensile stress in the
bottom fiber. If the neutral axis is an axis of symmetry, these two stresses will be equal in
magnitude.
