Determine the normal and shear stress elements:
The state of stress at a point is given through the stress components σ_{x} = 70 MPa, σ_{y} = 10 MPa, and τ_{xy} = - 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 25^{o}, 40^{o} and 60^{o} associatively along with the x plane.
Figure
Solution
(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
σ_{1} = 90 MPa, σ_{2} = - 10 MPa
(The radius of the Mohr's circle gives τ_{max} = 50 MPa)
(e) Measure ∠ XOA ; Here, ∠ XOA = - 52.8^{o} .
∴ Aspect angle Φ1 of major principal plane is - 26.4^{o}.
(f) Measure ∠ XOB ; Here, ∠ XOB = + 127.2^{o}
∴ Aspect angle Φ2 of major principal plane is + 63.6^{o}.
(g) Draw radial lines OR, OS and OT making angles 50^{o}, 80^{o} and 120^{o} with OX.
(h) Coordinates of R give the normal and shear stress components on the plane which makes 25^{o} with the x plane. Here,
σ_{n} = 28 MPa, τ_{nt} = - 48.5 MPa
(i) Coordinates of S given the normal and shear stress components on the plane which aspect angle 40^{o}. Here,
σ_{n} = 5.7 MPa, τ_{nt} = - 36.8 MPa
(j) Coordinates of T give the normal and shear stress components on the plane with aspect angle 60^{o}. Here,
σ^{n} = - 9.5 MPa, τ^{nt} = - 6.5 MPa