Rotating Angle Shear Theories - Unified Theory of Concrete Structures - page 159

Figure 5.5 Geometric relationship in Mohr stress circle

It should be noted that the magnitude of ε d is an order of magnitude smaller than the other

three strains, ε r , ε , and ε t , because of cracking. If we neglect ε d , Equation (5.45) states that the

principal tensile strain (or cracking strain) ε r is the sum of the steel strains in the longitudinal

and transverse directions ( ε + ε t ).

The state of strain in a RC 2-D element can be illustrated by Mohr strain circles in Figure
5.5(a). In the

− t coordinate, point A represents the reference -direction with strains of ε

and γ t / 2, while point B represents the t -direction with strains of ε t and − γ t / 2. The principal

tensile strain, ε r , is denoted as point C which is located at an angle of 2 α r away from point A.

ε d , is denoted as point D located at an angle of 180 ◦ away from The principal compressive strain,

point C.

5.2.2 First Type of Compatibility Equations
In view of the fact that

ε d is much smaller than the other four strains, our interest will naturally
be focused on the relationship among the four strains, γ t , while considering

ε d as a small secondary strain. With this aim in mind, the three transformation equations,
(5.41)-(5.43), can be changed into three explicit relationships among the four strains

ε , ε t , ε r and

ε , ε t ,

γ t and ε r .

Inserting ε d sin 2 α r = ε d − ε d cos 2 α r into Equation (5.41) gives

ε r − ε d ) cos 2

( ε − ε d ) = ( α r (5.46)