Rotating Angle Shear Theories - Unified Theory of Concrete Structures - page 151

σ r cos 2 α r = σ r − σ r sin 2

Inserting α r into Equation (5.1) gives

− σ d )sin 2 − - σ + ρ f + σ r = ( σ r α r (5.5)

Similarly, inserting σ r sin 2 α r = σ r − σ r cos 2 α r into Equation (5.2) gives

σ r − σ d ) cos 2

− σ t + ρ t f t + σ r = ( α r (5.6)

Equation (5.3) remains the same

τ t = ( σ r − σ d )sin α r cos α r (5.7)

Equations (5.5)-(5.7) are the first type of expression for the equilibrium condition. These
three equations are convenient for the analysis of RC 2-D elements.
Equations (5.5)-(5.7) represent the first type of geometric relationship in the Mohr circle as
shown in Figure 5.2(a). The geometric relationship of the half Mohr circle defined by ACD is
illustrated in Figure 5.2(b). If CD in Figure 5.2(b) is taken as unity, then CE sin 2

= α r ,ED =

cos 2 α r , and AE = sin α r cos α r . These three trigonometric values are actually the ratios of the

three stresses ( − σ + ρ f + σ r ), ( − σ t + ρ t f t + σ r ) and τ t , respectively, divided by the sum

of the principal stresses ( σ r − σ d ).

Figure 5.2 Geometric relationship in Mohr stress circle for concrete in reinforced elements