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
200 kN200 kNθ = 50°A = 800 mm²

A bar with cross-sectional area 800 mm² carries an axial tensile force of 200 kN. Find the magnitude of the shear stress |τθ| on a section whose normal makes an angle of 50° with the bar axis.

P = 200 kNA = 800 mm²θ = 50°
|τθ| = MPa
2.Easy
σx​ = −40σy​ = −60τxy​MPa

A point has σₓ = −40 MPa and σᵧ = −60 MPa, plus some shear stress. On an element rotated by 25° the normal stress in the rotated x′ direction is 2.4 MPa. Find the normal stress σy′ in the rotated y′ direction.

σx = −40 MPaσy = −60 MPaσx′ = 2.4 MPaθ = 25°
σy′ = MPa
3.Medium
σx​ = 70σy​ = 100τxy​ = −20MPa

A point has σₓ = 70 MPa, σᵧ = 100 MPa and τₓᵧ = −20 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 = 70 MPaσy = 100 MPaτxy = −20 MPa
θs = °
4.Medium
σx​ = −30σy​ = 0τxy​ = 50MPa

A point in a loaded part has σₓ = −30 MPa, σᵧ = 0 MPa and τₓᵧ = 50 MPa (tension positive). Find the angle θp from the x-axis to the direction of the largest principal stress, between −90° and 90°.

σx = −30 MPaσy = 0 MPaτxy = 50 MPa
θp = °
5.Hard
σx​ = 140σy​ = 100τxy​ = −60MPa

A point in a loaded part has σₓ = 140 MPa, σᵧ = 100 MPa and τₓᵧ = −60 MPa (tension positive). Find the shear stress τθ on a plane whose normal is rotated 45° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).

σx = 140 MPaσy = 100 MPaτxy = −60 MPaθ = 45°
τθ = MPa
6.Hard
σx​ = 50σy​ = 0τxy​ = −60MPa

A point in a loaded part has σₓ = 50 MPa, σᵧ = 0 MPa and τₓᵧ = −60 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 60° counter-clockwise from the x-axis.

σx = 50 MPaσy = 0 MPaτxy = −60 MPaθ = 60°
σθ = MPa

Answer key

  1. 1. |τθ| = 123.1 MPa
  2. 2. σy′ = −102.4 MPa
  3. 3. θs = 71.57 °
  4. 4. θp = 53.35 °
  5. 5. τθ = −20 MPa
  6. 6. σθ = −39.46 MPa

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