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/mm^{2}
σ_{y} = 20 N/mm^{2}
τ_{xy} = - 26 N/mm^{2}
On substituting in Eq. (6), we get
∴ σ_{1} = 72.8 N/mm^{2} and σ2 = 7.2 N/mm^{2}
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/mm^{2}
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.