Introduction - Unified Theory of Concrete Structures - page 11

equations for a 2-D element. The steel and concrete stresses in these three equations should
satisfy the Mohr stress circle.
By assuming that all the steel bars in the 2-D element will yield before the crushing of
concrete, it is possible to use the three equilibrium equations to calculate the stresses in the
steel bars and in concrete struts at the ultimate load stage. This method of analysis and design
is called the equilibrium (plasticity) truss model .
Since the strain compatibility condition is irrelevant under the plasticity condition, the
equilibrium truss model becomes very powerful in two ways: First, it can be easily applied to
all four types of actions (bending, axial loads, shear and torsion) and their interactions. The
interactive relationship of bending, shear and torsion were elegantly elucidated by Elfgren
(1972). Second, this model can easily be incorporated into the strength design codes, such as
the ACI Code and the European Code.
Looking at the weakness side of not utilizing the compatibility condition and the constitutive
laws of materials, the equilibrium (plasticity) truss model could not be used to derive the load-
deformation relationship of RC beams subjected to shear and torsion. More sophisticated
theories will have to be developed for shear and torsion that takes care of all three principles
of the mechanics of materials.
In this topic, the equilibrium (plasticity) truss model will be presented in detail in
Chapter 2.

1.3.2.5 Shear Theory

The derivation of three equilibrium equations for 2-D elements was soon followed by the
derivation of the three strain compatibility equations by Bauman (1972) and Collins (1973).
The steel and concrete strains in these three compatibility equations should satisfy Mohr's
strain circle.
Combining the 2-D equilibrium equations, Mohr's compatibility equations, and Hooke's
law, a linear shear theory can be developed for a 2-D element. This linear model has been called
the Mohr compatibility truss model . It could be applied in the elastic range of a 2-D element
up to the service load stage. Nonlinear shear theory is required to describe the behavior of 2-D
shear elements up to the ultimate load stage.
When an RC membrane element is subjected to shear, it is essentially a 2-D problem because
the shear stress can be resolved into a principal tensile stress and a principal compressive stress
in the 45 ◦ direction. The biaxial constitutive relationship of a 2-D element was a difficult task,
because the stresses and strains in two directions affect each other.
The most important phenomenon in a 2-D element subjected to shear was discovered
by Robinson and Demorieux (1972). They found that the principal compressive stress was
reduced, or 'softened', by the principal tensile stress in the perpendicular direction. However,
without the proper equipment to perform biaxial testing of 2-D elements, they were unable to
formulate the softened stress-strain relationship of concrete in compression.
Using a biaxial test facility called a 'shear rig', Vecchio and Collins (1981) showed that the
softening coefficient of the compressive stress-strain curve of concrete was a function of the
principal tensile strain ε 1 , rather than the principal tensile stress. Incorporating the equilibrium

equations, the compatibility equations, and using the 'softened stress-strain curve' of concrete,
Collins and Mitchell (1980) developed a 'compression field theory' (CFT), which could predict
the nonlinear shear behavior of an element in the post-cracking region up to the peak point.