Centroid & second moment of area worksheet with answers

Free printable Centroid & second moment of area 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
1002016012ȳ = ?dimensions in mm

Find the height ȳ of the centroid of the T-section above its bottom edge.

bf = 100 mmtf = 20 mmtw = 12 mmhw = 160 mm
ȳ = mm
2.Easy
120 mm240 mm120 mm30 mmØ30x̄ = ?

A 120 mm × 240 mm plate has a circular hole of diameter 30 mm whose centre is 30 mm from the left edge (at mid-height). Find the distance x̄ of the centroid from the left edge.

b = 120 mmh = 240 mmd = 30 mmhole centre: x = 30 mm
x̄ = mm
3.Medium
B = 140H = 275t = 14xdimensions in mm

A hollow rectangular (box) section has outer width B = 140 mm, outer height H = 275 mm and a uniform wall thickness t = 14 mm. Find its radius of gyration r about the horizontal centroidal axis.

B = 140 mmH = 275 mmt = 14 mm
r = mm
4.Medium
150 mm200 mm140 mmØ50xIx​ = ?

A 150 mm × 200 mm plate has a circular hole of diameter 50 mm on its vertical centre line, with its centre 140 mm above the bottom. Find the second moment of area Ix about the horizontal axis through the centroid.

b = 150 mmh = 200 mmd = 50 mmhole centre: y = 140 mm
Ix = × 10⁶ mm⁴
5.Hard
120 mm240 mm156 mm30 × 20xIx​ = ?

A 120 mm × 240 mm plate has a rectangular cut-out 30 mm wide and 20 mm high, centred on the vertical centre line with its centre 156 mm above the bottom. Find the second moment of area Ix about the horizontal axis through the centroid.

b = 120 mmh = 240 mmcut-out: 30 × 20 mmhole centre: y = 156 mm
Ix = × 10⁶ mm⁴
6.Hard
1201016012xhogging Mdimensions in mm

A cast-iron beam has the T-section shown, with the flange on top. The allowable stresses are 50 MPa in tension and 100 MPa in compression. Find the largest hogging bending moment Mallow the beam can carry (give its size).

bf = 120 mmtf = 10 mmtw = 12 mmhw = 160 mmσt,allow = 50 MPaσc,allow = 100 MPa
Mallow = kN·m

Answer key

  1. 1. ȳ = 125.9 mm
  2. 2. x̄ = 60.75 mm
  3. 3. r = 97.01 mm
  4. 4. Ix = 96.33 × 10⁶ mm⁴
  5. 5. Ix = 137.4 × 10⁶ mm⁴
  6. 6. Mallow = 8.237 kN·m

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