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​ = 60σy​ = −70τxy​ = −40MPa

A point in a loaded part has σₓ = 60 MPa, σᵧ = −70 MPa and τₓᵧ = −40 MPa (tension positive). Find the largest principal stress σ1.

σx = 60 MPaσy = −70 MPaτxy = −40 MPa
σ1 = MPa
2.Easy
50 kN50 kNθ = 50°A = 1000 mm²

A bar with cross-sectional area 1000 mm² carries an axial tensile force of 50 kN. Find the magnitude of the shear stress |τθ| on a section whose normal makes an angle of 50° with the bar axis.

P = 50 kNA = 1000 mm²θ = 50°
|τθ| = MPa
3.Medium
σx​ = −60σy​ = 90τxy​ = 40MPa

A point in a loaded part has σₓ = −60 MPa, σᵧ = 90 MPa and τₓᵧ = 40 MPa (tension positive). Find the angle θp from the x-axis to the direction of the largest principal stress, between −90° and 90°.

σx = −60 MPaσy = 90 MPaτxy = 40 MPa
θp = °
4.Medium
σx​ = −30σy​ = 80τxy​ = −20MPa

A point in a thin plate is in plane stress: σₓ = −30 MPa, σᵧ = 80 MPa and τₓᵧ = −20 MPa, with σz = 0. Find the absolute maximum shear stress τabs at the point (considering all planes in three dimensions).

σx = −30 MPaσy = 80 MPaτxy = −20 MPaσz = 0
τabs = MPa
5.Hard
σx​ = −80σy​ = 80τxy​ = 40MPa

A point in a loaded part has σₓ = −80 MPa, σᵧ = 80 MPa and τₓᵧ = 40 MPa (tension positive). Find the shear stress τθ on a plane whose normal is rotated 60° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).

σx = −80 MPaσy = 80 MPaτxy = 40 MPaθ = 60°
τθ = MPa
6.Hard
σx​τxy​top surface of the shaft

A solid shaft of diameter 50 mm carries a bending moment of 1500 N·m together with a torque of 250 N·m. At the top surface, where the bending stress is tensile and largest, find the largest principal stress σ1.

d = 50 mmM = 1500 N·mT = 250 N·m
σ1 = MPa

Answer key

  1. 1. σ1 = 71.32 MPa
  2. 2. |τθ| = 24.62 MPa
  3. 3. θp = 75.96 °
  4. 4. τabs = 58.52 MPa
  5. 5. τθ = 49.28 MPa
  6. 6. σ1 = 123.1 MPa

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