Question

Explain the behavior of rc beam (load-deflection or Moment-curvature) under the flexural loading highlighting different stages of loading till failure; i.e. un-cracked section, cracked section, service condition, yielding and ultimate loading condition.

Answer 1

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 as shown in Figure below. According to ACI 318-11 code the modulus of rupture is:
??=0.62?√??′
Where ? =
1 for normal concrete.
0.85 For sand light weight concrete.
0.75 For all light weight concrete.

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, Mcr. As the load is further increased, these
cracks quickly spread up to the vicinity of the neutral axis, and then the neutral axis begins 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 (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 (b). In this stage, the compressive stresses vary linearly with the distance from the neutral axis or 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 than0.5??'. To compute the concrete and steel stresses in this range, the transformed-area method 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, moments develop that are considerably larger than the cracking moments. Obviously, the tensile side of the beam will be cracked.

Beam Failure—Ultimate-Strength Stage
As the load is increased further so that the compressive stresses are greater than0.5??', the tensile cracks move farther upward, as does the neutral axis, and the concrete compression stresses begin to change appreciably from a straight line. For this initial discussion, it is assumed that the reinforcing bars have yielded. The stress variation is much like that shown in Figure below.

To further illustrate the three stages of beam behavior that have just been described, a moment–curvature diagram is shown in Figure below. For this diagram, θ is defined as the
angle change of the beam section over a certain length and is computed by the following expression in which ε is the strain in a beam fiber at some distance, y, from the neutral axis
of the beam:

The first stage of the diagram is for small moments less than the cracking moment, Mcr, where the entire beam cross section is available to resist bending. In this range, the strains
are small, and the diagram is nearly vertical and very close to a straight line. When the moment is increased beyond the cracking moment, the slope of the curve will decrease a
little because the beam is not quite as stiff as it was in the initial stage before the concrete cracked. The diagram will follow almost a straight line from Mcr to the point where the
reinforcing is stressed to its yield point. Until the steel yields, a fairly large additional load is required to appreciably increase the beam’s deflection. After the steel yields, the beam has
very little additional moment capacity, and only a small additional load is required to substantially increase rotations as well as deflections. The slope of the diagram is now very
flat.