Mohr’s circle & stress transformation worksheet with answers

Free printable Mohr’s circle & stress transformation worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
σx​ = 80σy​ = 110τxy​ = 30MPa

A point in a loaded part has σₓ = 80 MPa, σᵧ = 110 MPa and τₓᵧ = 30 MPa (tension positive). Find the largest principal stress σ1.

σx = 80 MPaσy = 110 MPaτxy = 30 MPa
σ1 = MPa
2.Easy
σx​ = 20σy​ = −100MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 20 MPa and σ₂ = −100 MPa (tension positive). Find the normal stress σavg that acts on the planes of maximum in-plane shear stress.

σ1 = 20 MPaσ2 = −100 MPa
σavg = MPa
3.Medium
σx​ = 110σy​ = −60τxy​ = −60MPa

A point in a loaded part has σₓ = 110 MPa, σᵧ = −60 MPa and τₓᵧ = −60 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = 110 MPaσy = −60 MPaτxy = −60 MPa
τmax = MPa
4.Medium
σx​ = 110σy​ = −70τxy​ = 50MPa

A point has σₓ = 110 MPa, σᵧ = −70 MPa and τₓᵧ = 50 MPa (tension positive). Find the angle θs, between −90° and 90°, from the x-axis to the normal of the plane on which the shear stress has its maximum positive value +R (shear positive in the same sense as τₓᵧ).

σx = 110 MPaσy = −70 MPaτxy = 50 MPa
θs = °
5.Hard
σx​ = 130σy​ = 20τxy​ = −50MPa

A point in a loaded part has σₓ = 130 MPa, σᵧ = 20 MPa and τₓᵧ = −50 MPa (tension positive). Find the shear stress τθ on a plane whose normal is rotated 25° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).

σx = 130 MPaσy = 20 MPaτxy = −50 MPaθ = 25°
τθ = MPa
6.Hard
σx​ = −10σy​ = −70τxy​ = −20MPa

A point in a thin plate is in plane stress: σₓ = −10 MPa, σᵧ = −70 MPa and τₓᵧ = −20 MPa (σz = 0). The material has a yield strength of 355 MPa. Find the factor of safety n against yielding by the von Mises (distortion energy) criterion.

σx = −10 MPaσy = −70 MPaτxy = −20 MPaσY = 355 MPa
n =

Answer key

  1. 1. σ1 = 128.5 MPa
  2. 2. σavg = −40 MPa
  3. 3. τmax = 104 MPa
  4. 4. θs = −30.47 °
  5. 5. τθ = −74.27 MPa
  6. 6. n = 4.787

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