Fundamentals of Shear - Unified Theory of Concrete Structures - page 128

Figure 4.2 Mohr stress circle

This set of three equations is:

σ + σ t
2

+ σ − σ t
2

σ 1 = cos φ − τ t sin φ (4.16)

σ + σ t
2

− σ − σ t
2

σ 2 = cos φ + τ t sin φ (4.17)

σ − σ t
2

τ 12 = φ + τ t cos φ

sin (4.18)

The circular property of Equations (4.16)-(4.18) can be demonstrated as follows: The first
two equations, (4.16) and (4.17), can be written as one equation, where σ 1 , 2 means either σ 1

or σ 2 :

σ 1 , 2 − σ + σ t
2

σ − σ t
2

=± cos φ − τ t sin φ (4.19)

Squaring Equation (4.19) gives
σ 1 , 2 − σ + σ t
2

2 σ − σ t

2

2

cos 2

t sin 2

2

= φ + τ φ − ( σ − σ t ) τ t sin φ cos φ (4.20)

Also squaring Equation (4.18) gives

σ − σ t
2

2

2
12

sin 2

t cos 2

2

τ = φ + τ φ + ( σ − σ t ) τ t sin φ cos φ (4.21)