Civil Engineering Reference
In-Depth Information
concept of how a reinforced concrete beam resists shear after cracking.
Second, the effect of prestress can be included in a logical way.
Consequently, the whole range of prestressing from nonprestressed
structures to fully prestressed structures can be unified. Third, the
interaction of bending and axial load with shear can be easily managed. The
combination is quite consistent and comprehensible. Fourth, it can serve as a
basis for the formulation of general design codes.
The original truss model concept was first proposed to treat shear
problems by Ritter (1899) and Morsch (1909) at the turn of the twentieth
century. It was extended to treat torsion problems by Rausch (1929) in 1929.
In these theories, a concrete element reinforced with orthogonal steel bars
and subjected to shear stresses will develop diagonal cracks at an angle
inclined to the steel bars. These cracks will separate the concrete into a
series of diagonal concrete struts, which is assumed to resist axial
compression. Together with the steel bars, which are assume to take only
axial tension, they form a truss action to resist the applied shear stresses. For
simplicity, the concrete struts are assumed to be inclined at 45° to the steel
bars. Consequently, these theories are known as the 45° truss model.
The rudimentary truss model of Ritter, Morsch and Rausch is very
elegant and the equations derived from the equilibrium conditions are
simple. Unfortunately, the predictions from these equations did not agree
with the test results. For the case of pure torsion, the theory may
overestimate the test values by 30%. For the case of low-rise shear walls,
the overestimation may exceed 50%.
In order to improve the predictions of the truss model, the theory had
undergone three major developments. The first important development was
the generalisation of the angle of inclination of the concrete struts by Lampert
and Thurlimann (1968). They assumed that the angle of inclination may
deviate from 45°. On this basis, three basic equilibrium equations had been
derived, which could explain why longitudinal and transverse steel with
different percentages can both yield at failure. Their theory was known as the
variable-angle truss model. The second development was the derivation of the
compatibility equation by Collins (1973) to determine the angle of inclination
of the concrete struts. Since this angle is assumed to coincide with the angle of
inclination of the principal compression stress and strain, this theory is also
known as the compression field theory. In this theory, the average strain
condition should satisfy Mohr's strain circle and the stress in the concrete
struts should satisfy Mohr's stress circle. The third development was the
discovery of the softening of concrete struts by Robinson and Demorieux
(1968) and the quantification of this phenomenon by Vecchio and Collins
(1981). Vecchio and Collins proposed a softened stress-strain curve, in which
the softening effect depends on the ratio of the two principal strains.
Combining the equilibrium, compatibility and softened stress-strain
relationships, a theory was developed which can predict with good accuracy the
test results of various types of reinforced concrete structures subjected to shear
