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
σy​ = −90MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 0 MPa and σ₂ = −90 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σ1 = 0 MPaσ2 = −90 MPa
τmax = MPa
2.Easy
τxy​ = −90MPapure shear

A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = −90 MPa (σₓ = σᵧ = 0). Find the normal stress σθ on a plane whose normal is rotated 60° counter-clockwise from the x-axis.

τxy = −90 MPaθ = 60°
σθ = MPa
3.Medium
σx​ = 90σy​ = 20MPaτxy​ = ?

A point has σₓ = 90 MPa and σᵧ = 20 MPa. How large can the shear stress τxy (magnitude) become before the largest principal stress reaches 145 MPa?

σx = 90 MPaσy = 20 MPaσ1 = 145 MPa
τxy = MPa
4.Medium
σx​ = −40σy​ = −20τxy​ = −10MPa

A point in a thin plate is in plane stress: σₓ = −40 MPa, σᵧ = −20 MPa and τₓᵧ = −10 MPa, with σz = 0. Find the absolute maximum shear stress τabs at the point (considering all planes in three dimensions).

σx = −40 MPaσy = −20 MPaτxy = −10 MPaσz = 0
τabs = MPa
5.Hard
σx​ = −60σy​ = 100τxy​ = −20MPa

A point in a thin plate is in plane stress: σₓ = −60 MPa, σᵧ = 100 MPa and τₓᵧ = −20 MPa (σz = 0). The material has a yield strength of 250 MPa. Find the factor of safety n against yielding by the Tresca (maximum shear stress) criterion.

σx = −60 MPaσy = 100 MPaτxy = −20 MPaσY = 250 MPa
n =
6.Hard
σx​ = 150τxy​ = 60MPa

A point in a thin plate is in plane stress: σₓ = 150 MPa, σᵧ = 0 MPa and τₓᵧ = 60 MPa (σz = 0). The material has a yield strength of 350 MPa. Find the factor of safety n against yielding by the Tresca (maximum shear stress) criterion.

σx = 150 MPaσy = 0 MPaτxy = 60 MPaσY = 350 MPa
n =

Answer key

  1. 1. τmax = 45 MPa
  2. 2. σθ = −77.94 MPa
  3. 3. τxy = 82.92 MPa
  4. 4. τabs = 22.07 MPa
  5. 5. n = 1.516
  6. 6. n = 1.822

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