Introduction of Beams

Types of Beams
Among the many types of beams are joists, lintels, spandrels, stringers, and floor
beams. The discovery that steel beams as a part of a structural frame could support
masonry walls is said to have permitted the construction of today’s high-rise
buildings, Stringers are the beams in bridge floors running parallel to the roadway,
whereas floor beams are the larger beams in many bridge floors, which are
perpendicular to the roadway of the bridge and are used to transfer the floor loads
from the stringers to the supporting girders or trusses.

SECTIONS USED AS BEAMS

The W shapes will normally prove to be the most economical beam section, and they
have largely replaced channels and S sections for beam usage. The W shapes have
more steel concentrated in their flanges than do S beams and thus have larger
moments of inertia and resisting moments for the same weights. Today, they are used
primarily for special situations such as when narrow flange widths are desirable, or
where shearing. Another common type of beam section is the open-web steel joist, or
bar joist.

BENDING STRESSES

For an introduction to bending stresses, the rectangular beam and stress diagrams of Fig. 8.1
are considered. If the beam is subjected to some bending moment, the stress at any point may
be computed with the usual flexure formula" fb – Mc/I. It is to be remembered, however, that
this expression is applicable only when the maximum computed stress in the beam is below
the elastic limit. The formula is based on the usual elastic assumptions: Stress is proportional
to strain, a plane section before bending remains a plane section after bending, etc. The value
of I/c is a constant for a particular section and is known as the section modulus . The flexure
formula may then be written as follows:

Initially, when the moment is applied to the beam, the stress will vary linearly from
the neutral axis to the extreme fibers. This situation is shown in part of Fig. 8.1. The
yield moment of a cross section is defined as the moment that will just produce the
yield stress in the outermost fiber of the section.
If the moment in a ductile steel beam is increased beyond the yield moment, the
outermost fibers that had previously been stressed to their yield stress will continue to
have the same stress, but will yield, and the duty of providing the necessary
additional resisting moment will fall on the fibers nearer to the neutral axis. This
process will continue, with more and more parts of the beam cross section stressed to
the yield stress as shown by the stress diagrams of parts and , until finally a full
plastic distribution is approached, as shown in part . Any additional moment applied
at the section will cause the beam to rotate, with little increase in stress.
The plastic moment is the moment that will produce full plasticity in a member cross
section and create a plastic hinge.
PLASTIC HINGES
This section is devoted to a description of the development of a plastic hinge as in the
simple beam shown in Fig. The load shown is applied to the beam and increased in
magnitude until the yield moment is reached and the outermost fiber is stressed to the
yield stress. The magnitude of the load is further increased, with the result that the
outer fibers begin to yield. The yielding spreads out to the other fibers, away from the
section of maximum moment, as indicated in the figure.
During this same period, the interior fibers at the section of maximum moment yield
gradually, until nearly all of them have yielded and a plastic hinge is formed, as shown
in Fig. Although the effect of a plastic hinge may extend for some distance along the
beam, for analysis purposes it is assumed to be concentrated at one section. There
compact sections were defined as being those which have sufficiently stocky profiles
such that they are capable of developing fully plastic stress distributions before they
buckle locally. The student must realize that for plastic hinges to develop, the members
must not only be compact but also must be braced in such a fashion that lateral buckling
is pre- vented.

In the study of plastic behavior, strain hardening is not considered.

ELASTIC DESIGN
Until recent years, almost all steel beams were designed on the basis of the elastic theory. The
maximum load that a structure could support was assumed to equal the load that first caused a
stress somewhere in the structure to equal the yield stress of the material. The members were
designed so that computed bending stresses for service loads did not exceed the yield stress
divided by a safety factor. The design profession, however, has long been aware that ductile
members do not fail until a great deal of yielding occurs after the yield stress is first reached.

THE PLASTIC MODULUS

The yield moment My equals the yield stress times the elastic modulus. The elastic modulus
equals I/c or bd2/6 for a rectangular section, and the yield moment equals Fy bd2/6. This
same value can be obtained by considering the resisting internal couple.

The resisting moment equals T or C times the lever arm between them, as follows:
The elastic section modulus can again be seen to equal bd2/6 for a rectangular beam. The
resisting moment at full plasticity can be determined in a similar manner.The result is the so-
called plastic moment, Mp• It is also the nominal moment of the section, Mn* this plastic, or
nominal, moment equals T or C times the lever arm between them.

From the foregoing expression for a rectangular section, the plastic section modulus Z can be
seen to equal bd2/4. The shape factor, which equals Mp /My Fy Z / Fy S, or Z/S, is / = 1.50
for a rectangular section.Unless the section is symmetrical, the neutral axis for the plastic
condition will not be in the same location as for the elastic condition. The total internal
compression must equal the total internal tension. As all fibers are considered to have the
same stress in the plastic condition, the areas above and below the plastic neutral axis must be
equal.

Design of Beams for Moments

If gravity loads are applied to a fairly long, simply supported beam, the beam will bend
downward, and its upper part will be placed in compression and will act as a compression
member. The cross section of this «column» will consist of the portion of the beam cross
section above the neutral axis. For the usual beam, the «column» will have a much smaller
moment of inertia about its y or vertical axis than about its z axis. If nothing is done to brace
it perpendicular to the y axis, it will buckle laterally at a much smaller load than would
otherwise have been required to produce a vertical failure.
Lateral buckling will not occur if the compression flange of a member is braced laterally or if
twisting of the beam is prevented at frequent intervals. In this chapter, the buckling moments
of a series of compact ductile steel beams with different lateral or torsional bracing situations
are considered.

we will look at beams as follows:

1. First, the beams will be assumed to have continuous lateral bracing for their
compression flanges.
2. Next, the beams will be assumed to be braced laterally at short intervals.
3. Finally, the beams will be assumed to be braced laterally at larger and larger intervals
In Fig. 9.1, a typical curve showing the nominal resisting or buckling moments of one of
these beams with varying unbraced lengths is presented

An examination of Fig. 9.1 will show that beams have three distinct ranges, or zones, of
behavior, depending on their lateral bracing situation. If we have continuous or closely
spaced lateral bracing, the beams will experience yielding of the entire cross section and fall
into what is classified as Zone 1. Finally, with even larger unbraced lengths, the beams will
fail elastically and fall into Zone 3. A brief discussion of these three types of behavior is
presented in this section, while the remainder of the chapter is devoted to a detailed
discussion of each type, together with a series of numerical examples.

Plastic Behavior (Zone 1)

If we were to take a compact beam whose compression flange is continuously braced

laterally, we would find that we could load it until its full plastic moment «Mp» is reached at
some point or points; further loading then produces a redistribution of moments. If we now
take one of these compact beams and provide closely spaced intermittent lateral bracing for
its compression flanges, we will find that we can still load it until the plastic moment plus
moment redistribution is achieved if the spacing between the bracing does not exceed a
certain value, called Lp herein. Most beams fall in Zone 1.

Inelastic Buckling (Zone 2)

If we now further increase the spacing between points of lateral or torsional bracing, the
section may be loaded until some, but not all, of the compression fibers are stressed to «Fy» .
In other words, in this zone we can bend the member until the yield strain is reached in some,
but not all, of its compres- sion elements before lateral buckling occurs This is referred to as
inelastic buckling. The maximum unbraced length at which we can still reach F, at one point
is the end of the inelastic range. It's shown as L, its value is dependent upon the properties of
the beam cross section, as well as on the yield and residual stresses of the beam.

Elastic Buckling (Zone 3)

If the unbraced length is greater than Lr, the section will buckle elastically before the
yield stress is reached anywhere. As the unbraced length is further increased, the
buckling moment becomes smaller and smaller. As the moment is increased in such a
beam, the beam will deflect more and more transversely until a critical moment value,
Mcr, is reached.

YIELDING BEHAVIOR-FULL PLASTIC MOMENT, ZONE 1

In this section and the next two, beam formulas for yielding behavior are presented.
Formulas are presented for inelastic buckling and elastic buckling . After seeing some of
these latter expressions, the reader may become quite concerned that he or she is going
to spend an enormous amount of time in formula substitution. This is not generally true,
however, as the val- ues needed are tabulated and graphed in simple form in Part 3 of
the AISC Manual.