Rotating Angle Shear Theories - Unified Theory of Concrete Structures - page 160

ε d cos 2 α r = ε d − ε d sin 2

Similarly, inserting α r into Equation (5.42) gives

ε r − ε d )sin 2

( ε t − ε d ) = ( α r (5.47)

Equation (5.43) remains the same
γ t
2 =

( ε r − ε d )sin α r cos α r (5.48)

Equations (5.46)-(5.48) are the first type of expression for the compatibility condition. In
this type of expression, the three strains in the − t coordinate, ε , ε t and γ t , are each related

individually to the principal tensile strain ε r , or more accurately the absolute-value sum of the

principal strains ( ε r − ε d ). These three equations are convenient for the analysis of RC 2-D

elements.
Equations (5.46)-(5.48) represent the first type of geometric relationship in the Mohr circle,
as shown in Figure 5.5(a). The geometric relationship of the half Mohr circle defined by ACD
is illustrated in Figure 5.5(b). If CD in Figure 5.5(b) is taken as unity, then CE

=

sin 2

α r ,

=

cos 2

α r , and AE = α r cos α r . These three trigonometric values are actually the ratios

ED sin

of the three strains ( ε − ε d ), ( ε t − ε d ) and γ t /2, respectively, divided by the sum of the

principal strains ( ε r − ε d ).

Dividing Equation (5.47) by Equation (5.46) we have

ε t − ε d
ε − ε d

tan 2

α r = (5.49)

α r . Its derivation signaled
the arrival of the Mohr compatibility truss model and the rotating angle softened truss model.
These two models will be studied in Sections 5.3 and 5.4, respectively.
Multiplying Equations (5.46) and (5.47) gives

Equation (5.49) is the compatiblity equation to determine the angle

ε r − ε d ) 2 sin 2 α r cos 2

( ε − ε d )( ε t − ε d ) = ( α r (5.50)

Squaring Equation (5.48) gives
γ t
2

2

ε r − ε d ) 2 sin 2 α r cos 2

= ( α r (5.51)

Equating Equations (5.50) and (5.51) and taking square root result in
γ t
2

(

=± ε − ε d )( ε t − ε d ) (5.52)

Equation (5.52) shows that the shear strain γ t / 2 can be calculated without knowing the

angle α r . γ t / 2 is simply the square root of the product of ( ε − ε d ) and ( ε t − ε d ).

5.2.3 Second Type of Compatibility Equations
The three compatibility equations, (5.46)-(5.48), can be expressed in another form. The three
strains ε , ε t and ε r can each be related individually to the shear strain γ t . Substituting