Flexural Analysis of Beams - Design of Reinforced Concrete - page 37

2

Flexural Analysis of Beams

2.1 INTRODUCTION

In this section it is assumed that a small transverse load is placed on a concrete beam with
tensile reinforcing and that the load is gradually increased in magnitude until the beam
fails. As this takes place we will find that the beam will go through three distinct stages
before collapse occurs. These are: (1) the uncracked concrete stage, (2) the concrete
cracked-elastic stresses stage, and (3) the ultimate-strength stage. A relatively long beam
is considered for this discussion so that shear will not have a large effect on its behavior.

Uncracked Concrete Stage
At small loads when the tensile stresses are less than the modulus of rupture (the bending
tensile stress at which the concrete begins to crack), the entire cross section of the beam
resists bending, with compression on one side and tension on the other. Figure 2.1 shows
the variation of stresses and strains for these small loads; a numerical example of this type
is presented in Section 2.2.

Concrete Cracked-Elastic Stresses Stage
As the load is increased after the modulus of rupture of the concrete is exceeded, cracks
begin to develop in the bottom of the beam. The moment at which these cracks begin to
form—that is, when the tensile stress in the bottom of the beam equals the modulus of
rupture—is referred to as the cracking moment , M cr . As the load is further increased, these
cracks quickly spread up to the vicinity of the neutral axis, and then the neutral axis be-
gins to move upward. The cracks occur at those places along the beam where the actual
moment is greater than the cracking moment, as shown in Figure 2.2(a).
Now that the bottom has cracked, another stage is present because the concrete in the
cracked zone obviously cannot resist tensile stresses—the steel must do it. This stage will
continue as long as the compression stress in the top fibers is less than about one-half of the
concrete's compression strength and as long as the steel stress is less than its yield stress.
The stresses and strains for this range are shown in Figure 2.2(b). In this stage the compres-
sive stresses vary linearly with the distance from the neutral axis or as a straight line.
The straight-line stress-strain variation normally occurs in reinforced concrete beams
under normal service-load conditions because at those loads the stresses are generally less
than To compute the concrete and steel stresses in this range, the transformed-area
method (to be presented in Section 2.3) is used. The service or working loads are the loads
that are assumed to actually occur when a structure is in use or service. Under these loads,

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