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​ = 140σy​ = −30MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 140 MPa and σ₂ = −30 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σ1 = 140 MPaσ2 = −30 MPa
τmax = MPa
2.Easy
σx​ = 160σy​ = −20τxy​ = 40MPa

A point in a loaded part has σₓ = 160 MPa, σᵧ = −20 MPa and τₓᵧ = 40 MPa (tension positive). Find the smallest principal stress σ2.

σx = 160 MPaσy = −20 MPaτxy = 40 MPa
σ2 = MPa
3.Medium
σx​ = −50σy​ = 100τxy​ = −40MPa

A point in a loaded part has σₓ = −50 MPa, σᵧ = 100 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 = −50 MPaσy = 100 MPaτxy = −40 MPa
θp = °
4.Medium
σx​ = −10σy​ = −60MPaτxy​ = ?

A point has σₓ = −10 MPa and σᵧ = −60 MPa. How large can the shear stress τxy (magnitude) become before the largest principal stress reaches 10 MPa?

σx = −10 MPaσy = −60 MPaσ1 = 10 MPa
τxy = MPa
5.Hard
σx​ = −70σy​ = 0τxy​ = 10MPa

A point in a loaded part has σₓ = −70 MPa, σᵧ = 0 MPa and τₓᵧ = 10 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 30° counter-clockwise from the x-axis.

σx = −70 MPaσy = 0 MPaτxy = 10 MPaθ = 30°
σθ = MPa
6.Hard
σx​ = 90σy​ = 50τxy​ = −30MPa

A point in a loaded part has σₓ = 90 MPa, σᵧ = 50 MPa and τₓᵧ = −30 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 = 90 MPaσy = 50 MPaτxy = −30 MPaθ = 60°
τθ = MPa

Answer key

  1. 1. τmax = 85 MPa
  2. 2. σ2 = −28.49 MPa
  3. 3. θp = −75.96 °
  4. 4. τxy = 37.42 MPa
  5. 5. σθ = −43.84 MPa
  6. 6. τθ = −2.321 MPa

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