Rotating Angle Shear Theories - Unified Theory of Concrete Structures - page 166

compatibility truss model , will be studied in this section. This linear model is applicable up
to the service load stage, and could even be used up to the load stage when the steel begins
to yield.
If the constitutive laws are based on the nonlinear, softened stress-strain relationships (i.e.
2-D or biaxial), then the theory will be more accurate, but more complex. This nonlinear
theory is called the rotating angle softened truss model and will be presented in Section 5.4.
This nonlinear model should be applicable to the ascending behavioral curve up to the peak
load, including both the service load stage and the ultimate load stage.

5.3.2 Summary of Equations
In the Mohr compatibility truss model, the three equilibrium equations used are Equations
(5.21)-(5.23), and the three compatibility equations are Equations (5.49), (5.45) and (5.48).
They are summarized as follows:
Equilibrium equations

ρ f = σ + τ t tan α r (5.67)

ρ t f t = σ t + τ t cot α r (5.68)

1

( − σ d ) = τ t (5.69)

α r cos α r

sin

Compatibility equations

ε t − ε d
ε − ε d

tan 2

α r = (5.70)

ε r = ε + ε t − ε d

(5.71)

γ t
2 =

( ε r − ε d )sin α r cos α r (5.72)

The constitutive laws used in the Mohr compatibility truss model are Hooke's laws. Hooke's
linear stress-strain relationships for steel and concrete are given in 1-D form as follows:

f E s

ε = (5.73)

f t
E s

ε t = (5.74)

σ d
E c

ε d = (5.75)

Similar to flexural members, the tensile stress of concrete is neglected in the design and
analysis of cracked 2-D elements, i.e.

σ r =

0 (5.76)

Equation (5.76) means that the tensile stress-strain relationship of concrete is irrelevant.
Then, we have nine equations (5.67)-(5.75) involving fourteen variables. These variables
include six stresses (

σ , σ t , τ t , σ d , f and f t ), five strains ( ε , ε t , γ t , ε r and ε d ), two cross-

sectional properties ( α r . Therefore, five variables

must be given before the remaining nine variables can be solved by the nine equations.

ρ and ρ t ) and one geometric parameter