Calculate the principal stresses for the state of stress:

Calculate the principal stresses and principal planes for the state of stress shown in Figure.

Figure

Solution

Given σ_x = 60 N/mm²

σ_y = 20 N/mm²

τ_xy = - 26 N/mm²

On substituting in Eq. (6), we get

∴          σ₁ = 72.8 N/mm² and σ2 = 7.2 N/mm²

Again substituting the values σ_x, σ_y and σ_xy in Eq. (2).

tan 2Φ = 2τxy/σx-σy = 2×(-26)/60-20 = -1.3

Since θ is general angle, the specific angles representing the principal planes are designated as Φ1 and Φ2.

∴          2Φ = - 52.43^º, 127.57^º

using    2Φ = - 52.43^º

σ_n = (60 + 20/2) + (60-20/2) cos (-52.43^o)- 26 sin (- 52.43^º)

=72.8 N/mm²

Hence, we recognize that Φ1 = - 52.43^º/2 defines the major principal plane and therefore, Φ2 = 127.57^o/2 should define the minor principal plane.