Calculate the principal stresses for the state of stress:
Calculate the principal stresses and principal planes for the state of stress shown in Figure.
Given σx = 60 N/mm2
σy = 20 N/mm2
τxy = - 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 σx, σy and σxy 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 = - 52.43º, 127.57º
using 2Φ = - 52.43º
σn = (60 + 20/2) + (60-20/2) cos (-52.43o)- 26 sin (- 52.43º)
Hence, we recognize that Φ1 = - 52.43º/2 defines the major principal plane and therefore, Φ2 = 127.57o/2 should define the minor principal plane.