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concrete compressive stress distribution becomes non-linear when the extreme fibre stress
exceeds about .
A flexural member which is designed to exhibit ductility at failure usually has failure of the
critical section preceded by yielding of the bonded tensile steel, i.e. by effectively exhausting
the capacity of the tensile steel to carry any additional force. Such a member is said to be
under-reinforced. Because the stress-strain curve for the prestressing steel has no distinct yield
point and the stress increases monotonically as the strain increases (see Figure 2.13), the
capacity of the prestressing steel to carry additional force is never entirely used up until the
steel actually fractures. However, when the steel strain ε p exceeds about 0.01 (for wire or
strand), the stress–strain curve becomes relatively flat and the rate of increase of stress with
strain is small. After yielding, the resultant internal tensile force, T(=Ts+Tp) in Figure 4.1,
remains approximately constant (as does the resultant internal compressive force C, which is
equal and opposite to T). The moment capacity can be further increased by an increase in the
lever arm between C and T. Under increasing deformation, the neutral axis rises, the
compressive zone becomes smaller and smaller and the compressive concrete stress increases.
Eventually, perhaps after considerable deformation, a compressive failure of the concrete
above the neutral axis occurs and the section reaches its ultimate capacity. It is, however, the
strength of the prestressing tendons and the non-prestressed reinforcement in the tensile zone
that control the strength of a ductile section. In fact, the difference between the moment at
first yielding of the tensile steel and the ultimate moment is usually small.
A flexural member which is over-reinforced, on the other hand, does not have significant
ductility at failure and fails without the prestressed or non-prestressed tensile reinforcement
reaching any form of yield. At the ultimate load condition, both the tensile strain at the steel
level and the section curvature are relatively small and, consequently, there is little
deformation or warning of failure.
Because it is the deformation at failure that defines ductility, it is both usual and reasonable
in design to define a minimum ultimate curvature to ensure the ductility of a cross-section.
This is often achieved by placing a maximum limit on the depth to the neutral axis at the
ultimate load condition. Ductility can be increased by the inclusion of non-prestressed
reinforcing steel in the compression zone of the beam. With compressive steel included, the
internal compressive force C is shared between the concrete and the steel. The volume of the
concrete stress block above the neutral axisis therefore reduced and, consequently, the depth to the neutral axis is decreased. Some
compressive reinforcement is normally included in beams to provide anchorage for transverse
shear reinforcement.
Ductility is desirable in prestressed (and reinforced) concrete flexural members. In
continuous or statically indeterminate members, ductility is particularly necessary. Large
curvatures are required at the peak moment regions in order to permit the inelastic moment
redistribution that must occur if the moment diagram assumed in design is to be realized in
practice.
Consider the stress distribution caused by the ultimate moment on the section in Figure 4.1.
The resultant compressive force of magnitude C equals the resultant tensile force T and the
ultimate moment capacity is calculated from the internal couple,
(4.1)
The lever arm l between the internal compressive and tensile resultants (C and T) is usually
about 0.9d, where d is the effective depth of the section and may be defined as the distance