Building Design and Construction Handbook - Merritt - Ventech!

STRUCTURAL THEORY 5.99
Similarly, a beam made of a ductile material continues to carry more load after
the stresses in the outer surfaces reach the yield point. However, the stresses will
no longer vary with distance from the neutral axis, so the flexure formula [Eq.
(5.54)] no longer holds. However, if simplifying assumptions are made, approximating
the stress-strain relationship beyond the elastic limit, the load-carrying ca-
pacity of the beam can be computed with satisfactory accuracy.
FIGURE 5.79 Stress-strain relationship for a
ductile material generally is similar to the curve
shown in (a). To simplify plastic analysis, the
portion of (a) enclosed by the dash lines is approximated
by the curve in (b), which extends
to the range where strain hardening begins.
Modulus of rupture is defined as
the stress computed from the flexure
formula for the maximum bending moment
a beam sustains at failure. This is
not a true stress but it is sometimes used
to compare the strength of beams.
For a ductile material, the idealized
stress-strain relationship in Fig. 5.79b
may be assumed. Stress is proportional
to strain until the yield-point stress ƒ y is
reached, after which strain increases at
a constant stress.
For a beam of this material, the following
assumptions will also be made:
1. Plane sections remain plane, strains
thus being proportional to distance
from the neutral axis.
2. Properties of the material in tension
are the same as those in compression.
3. Its fibers behave the same in flexure
as in tension.
4. Deformations remain small.
Strain distribution across the cross
section of a rectangular beam, based on
these assumptions, is shown in Fig. 5.80a. At the yield point, the unit strain is � y
and the curvature � y, as indicated in (1). In (2), the strain has increased several
times, but the section still remains plane. Finally, at failure, (3), the strains are very
large and nearly constant across upper and lower halves of the section.
Corresponding stress distributions are shown in Fig. 5.80b. At the yield point,
(1), stresses vary linearly and the maximum if ƒ y. With increase in load, more and
more fibers reach the yield point, and the stress distribution becomes nearly constant,
as indicated in (2). Finally, at failure, (3), the stresses are constant across the
top and bottom parts of the section and equal to the yield-point stress.
The resisting moment at failure for a rectangular beam can be computed from
the stress diagram for stage 3. If b is the width of the member and d its depth, then
the ultimate moment for a rectangular beam is
2 bd
Mp � ƒ y
(5.141)
4
Since the resisting moment at stage 1 is M y � ƒ ybd 2 /6, the beam carries 50% more
moment before failure than when the yield-point stress is first reached at the outer
surfaces.