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​ = 40σy​ = 100τxy​ = 30MPa

A point in a loaded part has σₓ = 40 MPa, σᵧ = 100 MPa and τₓᵧ = 30 MPa (tension positive). Find the largest principal stress σ1.

σx = 40 MPaσy = 100 MPaτxy = 30 MPa
σ1 = MPa
2.Easy
σx​ = 90σy​ = 120τxy​ = −40MPa

A point in a loaded part has σₓ = 90 MPa, σᵧ = 120 MPa and τₓᵧ = −40 MPa (tension positive). Find the smallest principal stress σ2.

σx = 90 MPaσy = 120 MPaτxy = −40 MPa
σ2 = MPa
3.Medium
σx​ = 20σy​ = 10τxy​ = −60MPa

A point in a loaded part has σₓ = 20 MPa, σᵧ = 10 MPa and τₓᵧ = −60 MPa (tension positive). Find the angle θp from the x-axis to the direction of the largest principal stress, between −90° and 90°.

σx = 20 MPaσy = 10 MPaτxy = −60 MPa
θp = °
4.Medium
σx​ = 100σy​ = −50MPaτxy​ = ?

A point has σₓ = 100 MPa and σᵧ = −50 MPa. How large can the shear stress τxy (magnitude) become before the largest principal stress reaches 145 MPa?

σx = 100 MPaσy = −50 MPaσ1 = 145 MPa
τxy = MPa
5.Hard
40°D = 1000 mm, t = 6 mm, p = 3 MPap

A thin-walled cylindrical tank of inner diameter 1000 mm and wall thickness 6 mm holds gas at a gauge pressure of 3 MPa. It is made from plate welded along a helical seam at 40° to the axis of the cylinder. Find the normal stress σn acting across the weld seam.

D = 1000 mmt = 6 mmp = 3 MPaα = 40°
σn = MPa
6.Hard
σx​ = 20σy​ = 40τxy​ = −20MPa

A point in a loaded part has σₓ = 20 MPa, σᵧ = 40 MPa and τₓᵧ = −20 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 45° counter-clockwise from the x-axis.

σx = 20 MPaσy = 40 MPaτxy = −20 MPaθ = 45°
σθ = MPa

Answer key

  1. 1. σ1 = 112.4 MPa
  2. 2. σ2 = 62.28 MPa
  3. 3. θp = −42.62 °
  4. 4. τxy = 93.67 MPa
  5. 5. σn = 198.4 MPa
  6. 6. σθ = 10 MPa

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