Determine the normal and shear stress elements:
The state of stress at a point is given through the stress components σx = 70 MPa, σy = 10 MPa, and τxy = - 40 MPa. Using Mohr's circle find (i) Principal stresses, and (ii) Principal planes. Also, determine the normal and shear stress elements on planes making 25o, 40o and 60o associatively along with the x plane.
(a) Choose (σ, - τ) coordinate system to a suitable scale.
(b) Mark the points X (70, - 40) and Y (10, + 40).
(c) Draw a circle with XY as diameter. This circle cuts σ axis at A and B.
(d) Measure the coordinates of A and B to obtain principal stresses
σ1 = 90 MPa, σ2 = - 10 MPa
(The radius of the Mohr's circle gives τmax = 50 MPa)
(e) Measure ∠ XOA ; Here, ∠ XOA = - 52.8o .
∴ Aspect angle Φ1 of major principal plane is - 26.4o.
(f) Measure ∠ XOB ; Here, ∠ XOB = + 127.2o
∴ Aspect angle Φ2 of major principal plane is + 63.6o.
(g) Draw radial lines OR, OS and OT making angles 50o, 80o and 120o with OX.
(h) Coordinates of R give the normal and shear stress components on the plane which makes 25o with the x plane. Here,
σn = 28 MPa, τnt = - 48.5 MPa
(i) Coordinates of S given the normal and shear stress components on the plane which aspect angle 40o. Here,
σn = 5.7 MPa, τnt = - 36.8 MPa
(j) Coordinates of T give the normal and shear stress components on the plane with aspect angle 60o. Here,
σn = - 9.5 MPa, τnt = - 6.5 MPa