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​ = −50σy​ = 80τxy​ = −30MPa

A point in a loaded part has σₓ = −50 MPa, σᵧ = 80 MPa and τₓᵧ = −30 MPa (tension positive). Find the smallest principal stress σ2.

σx = −50 MPaσy = 80 MPaτxy = −30 MPa
σ2 = MPa
2.Easy
σx​ = 10σy​ = −60MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 10 MPa and σ₂ = −60 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σ1 = 10 MPaσ2 = −60 MPa
τmax = MPa
3.Medium
σx​ = 20σy​ = −10τxy​ = 50MPa

A point has σₓ = 20 MPa, σᵧ = −10 MPa and τₓᵧ = 50 MPa (tension positive). The element is rotated 50° counter-clockwise. Find the normal stress σy′ on the faces of the rotated element that are perpendicular to the rotated y′-axis.

σx = 20 MPaσy = −10 MPaτxy = 50 MPaθ = 50°
σy′ = MPa
4.Medium
σx​ = 50σy​ = −50MPaτxy​ = ?

A point has σₓ = 50 MPa and σᵧ = −50 MPa. How large can the shear stress τxy (magnitude) become before the largest principal stress reaches 90 MPa?

σx = 50 MPaσy = −50 MPaσ1 = 90 MPa
τxy = MPa
5.Hard
σx​τxy​top surface of the shaft

A solid shaft of diameter 40 mm carries a bending moment of 500 N·m together with a torque of 400 N·m. At the top surface, where the bending stress is tensile and largest, find the maximum in-plane shear stress τmax.

d = 40 mmM = 500 N·mT = 400 N·m
τmax = MPa
6.Hard
σx​τxy​surface of the shaft

A solid shaft of diameter 50 mm carries an axial tensile force of 200 kN together with a torque of 750 N·m. At the surface of the shaft, find the largest principal stress σ1.

d = 50 mmF = 200 kNT = 750 N·m
σ1 = MPa

Answer key

  1. 1. σ2 = −56.59 MPa
  2. 2. τmax = 35 MPa
  3. 3. σy′ = −41.64 MPa
  4. 4. τxy = 74.83 MPa
  5. 5. τmax = 50.95 MPa
  6. 6. σ1 = 110.3 MPa

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