Structural Concrete - Hassoun

88 Chapter 3 Flexural Analysis of Reinforced Concrete Beams
Figure 3.2 Shape of beam 1 at shear failure (top) and beam 2 at bending moment
failure (bottom).
part in resisting the principal stresses. Cracks did not extend along the horizontal main steel bars,
as in beam 1. On increasing the load, diagonal cracks on the other end of the beam developed at a
load of 13,250 lb (59.6 kN). Failure did not occur at this stage because of the presence of stirrups.
Stage 5. When the load on beam 2 was further increased, strains increased rapidly until the maximum
carrying capacity of the beam was reached at ultimate load, P u = 16,200 lb (72.9 kN).
In beam 2, the amount of steel reinforcement used was relatively small. When reached, the
yield strain can be considered equal to yield stress divided by the modulus of elasticity of steel,
ε y = f y /E s ; the strain in the concrete, ε c , was less than the strain at maximum compressive stress,
f c. ′ The steel bars yielded, and the strain in steel increased to about 12 times that of the yield strain
without increase in load. Cracks widened sharply, deflection of the beam increased greatly, and the
compressive strain on the concrete increased. After another very small increase of load, steel strain
hardening occurred, and concrete reached its maximum strain, ε ′ c, and it started to crush under load;
then the beam collapsed. Figure 3.2 shows the failure shapes of the two beams.
3.4 TYPES OF FLEXURAL FAILURE AND STRAIN LIMITS
3.4.1 Flexural Failure
Three types of flexural failure of a structural member can be expected depending on the percentage
of steel used in the section.
1. Steel may reach its yield strength before the concrete reaches its maximum strength, Fig. 3.3a.
In this case, the failure is due to the yielding of steel reaching a high strain equal to or
greater than 0.005. The section contains a relatively small amount of steel and is called a
tension-controlled section.
2. Steel may reach its yield strength at the same time as concrete reaches its ultimate strength,
Fig. 3.3b. The section is called a balanced section.
3. Concrete may fail before the yield of steel, Fig. 3.3c, due to the presence of a high percentage
of steel in the section. In this case, the concrete strength and its maximum strain of 0.003 are
reached, but the steel stress is less than the yield strength, that is, f s is less than f y . The strain in
the steel is equal to or less than 0.002. This section is called a compression-controlled section.