Rotating Angle Shear Theories - Unified Theory of Concrete Structures - page 164

Figure 5.7 Cracking condition at yielding of steel

5.2.4.3 Cracking Condition at Yielding of Transverse Steel

In this case of analysis, the transverse steel strain is given as ε t = ε y , and ε d is considered a

small given value. Our purpose is to express the longitudinal steel strain ε and the cracking

strain ε r as a function of ε t , ε d and the angle α r .

The longitudinal steel strain ε is related to the transverse steel strain ε t by Equation (5.49).

Rearranging Equation.(5.49) to express ε gives

ε t − ε d ) cot 2

ε = ε d + ( α r (5.63)

Substituting ε from Equation (5.63) into (5.45) provides the equation for ε r :

ε t − ε d ) cot 2

ε r = ε t + ( α r (5.64)

Dividing Equations (5.63) and (5.64) by ε y and setting ε t = ε y we obtain the nondimensional

equations for ε and ε r :

1 cot 2

ε
ε y =

ε d
ε y +

− ε d
ε y

α r

(5.65)

1 cot 2

ε r
ε y =

− ε d
ε y

1 + α r (5.66)