Fundamentals of Shear - Unified Theory of Concrete Structures - page 127

or

σ 1 cos σ cos

τ 12 α 1 sin α 1 τ t α 1 − sin α 1

= (4.3)

τ 21 σ 2 − sin α 1 cos α 1 τ t σ t sin α 1 cos α 1

Performing the matrix multiplications and noticing that τ t = τ t and τ 12 = τ 21 are applicable

to equilibrium condition result in the following three equations:

σ 1 = σ cos 2 α 1 + σ t sin 2 α 1 + τ t 2sin α 1 cos α 1 (4.4)

σ 2 = σ sin 2 α 1 + σ t cos 2 α 1 − τ t 2sin α 1 cos α 1 (4.5)

α 1 + τ t (cos 2

sin 2

τ 12 = − σ + σ t )sin α 1 cos α 1 − α 1 )

( (4.6)

Equations (4.4)-(4.6) can be expressed in the matrix form by one equation:
⎡ ⎤ ⎡ ⎤ ⎡ ⎤

σ 1
σ 2
τ 12

cos 2 α 1 sin 2 α 1 2sin α 1 cos α 1 σ

σ t
τ t

⎣ ⎦ = ⎣ ⎦ ⎣ ⎦

sin 2 cos 2 α 1 α 1 − 2sin α 1 cos α 1 (4.7)

(cos 2 sin 2 − sin α 1 cos α 1 sin α 1 cos α 1 α 1 − α 1 )

3 matrix in Equation (4.7) is the transformation matrix [ T ] for transforming the
stresses in the stationary

This 3 ×

t coordinate to the stresses in the rotating 1-2 coordinate. Using
the tensor notation, Equation (4.7) becomes:

−

[ σ 12 ] = [ T ][ σ t ] (4.8)

4.1.2 Mohr Stress Circle
The trigonometric functions of α 1 in Equations (4.4)-(4.6) can be written in terms of double

angle 2 α 1 by

1
2 (1

cos 2

α 1 = + cos 2 α 1 ) (4.9)

1
2 (1

sin 2 α 1 = − cos 2 α 1 ) (4.10)

1
2 sin 2

sin α 1 cos α 1 = α 1 (4.11)

cos 2 sin 2 α 1 − α 1 = cos 2 α 1 (4.12)

Substituting Equations (4.9)-(4.12) into Equations (4.4)-(4.6) gives the three transformation
equations in terms of the double angle 2 α 1 as follows:

σ + σ t
2

+ σ − σ t
2

σ 1 = α 1 + τ t sin 2 α 1

cos 2 (4.13)

σ + σ t
2

− σ − σ t
2

σ 2 = cos 2 α 1 − τ t sin 2 α 1 (4.14)

τ 12 =− σ − σ t
2

sin 2 α 1 + τ t cos 2 α 1 (4.15)

Equations (4.13)-(4.15) had been recognized by Otto Mohr to be algebraically analogous
to a set of equations describing a circle in the σ − τ coordinate, as shown in Figure 4.2(a).