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​ = 20σy​ = −40MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 20 MPa and σ₂ = −40 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σ1 = 20 MPaσ2 = −40 MPa
τmax = MPa
2.Easy
σx​ = 150σy​ = −20τxy​MPa

A point has σₓ = 150 MPa and σᵧ = −20 MPa, plus some shear stress. On an element rotated by 15° the normal stress in the rotated x′ direction is 153.6 MPa. Find the normal stress σy′ in the rotated y′ direction.

σx = 150 MPaσy = −20 MPaσx′ = 153.6 MPaθ = 15°
σy′ = MPa
3.Medium
σx​ = 80σy​ = −20τxy​ = 30MPa

A point in a loaded part has σₓ = 80 MPa, σᵧ = −20 MPa and τₓᵧ = 30 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = 80 MPaσy = −20 MPaτxy = 30 MPa
τmax = MPa
4.Medium
σx​ = 30σy​ = −50τxy​ = −50MPa

A point in a loaded part has σₓ = 30 MPa, σᵧ = −50 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 = −50 MPaτxy = −50 MPa
θp = °
5.Hard
σx​ = −60σy​ = −40τxy​ = 60MPa

A point in a loaded part has σₓ = −60 MPa, σᵧ = −40 MPa and τₓᵧ = 60 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 20° counter-clockwise from the x-axis.

σx = −60 MPaσy = −40 MPaτxy = 60 MPaθ = 20°
σθ = MPa
6.Hard
σx​ = −70σy​ = −50τxy​ = −50MPa

A point in a thin plate is in plane stress: σₓ = −70 MPa, σᵧ = −50 MPa and τₓᵧ = −50 MPa (σz = 0). Find the von Mises equivalent stress σvM.

σx = −70 MPaσy = −50 MPaτxy = −50 MPa
σvM = MPa

Answer key

  1. 1. τmax = 30 MPa
  2. 2. σy′ = −23.6 MPa
  3. 3. τmax = 58.31 MPa
  4. 4. θp = −25.67 °
  5. 5. σθ = −19.09 MPa
  6. 6. σvM = 106.8 MPa

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