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
100 mm240 mm84 mmØ40ȳ = ?

A 100 mm × 240 mm plate has a circular hole of diameter 40 mm on its vertical centre line, with its centre 84 mm above the bottom edge. Find the height ȳ of the centroid above the bottom edge.

b = 100 mmh = 240 mmd = 40 mmhole centre: y = 84 mm
ȳ = mm
2.Easy
200 mm200 mm50 mmØ30ȳ = ?

A 200 mm × 200 mm plate has a circular hole of diameter 30 mm on its vertical centre line, with its centre 50 mm above the bottom edge. Find the height ȳ of the centroid above the bottom edge.

b = 200 mmh = 200 mmd = 30 mmhole centre: y = 50 mm
ȳ = mm
3.Medium
xD = 240d = 208dimensions in mm

A circular tube has outside diameter D = 240 mm and inside diameter d = 208 mm. Find its radius of gyration r about the horizontal centroidal axis.

D = 240 mmd = 208 mm
r = mm
4.Medium
120 mm300 mm180 mmØ40xIx​ = ?

A 120 mm × 300 mm plate has a circular hole of diameter 40 mm on its vertical centre line, with its centre 180 mm above the bottom. Find the second moment of area Ix about the horizontal axis through the centroid.

b = 120 mmh = 300 mmd = 40 mmhole centre: y = 180 mm
Ix = × 10⁶ mm⁴
5.Hard
100 mm200 mm70 mm30 × 20xIx​ = ?

A 100 mm × 200 mm plate has a rectangular cut-out 30 mm wide and 20 mm high, centred on the vertical centre line with its centre 70 mm above the bottom. Find the second moment of area Ix about the horizontal axis through the centroid.

b = 100 mmh = 200 mmcut-out: 30 × 20 mmhole centre: y = 70 mm
Ix = × 10⁶ mm⁴
6.Hard
2601010012xM = 25 kN·mdimensions in mm

The T-section is used as a beam with the flange on top and carries a sagging bending moment of 25 kN·m. Find the size of the bending stress σtop at the top edge.

bf = 260 mmtf = 10 mmtw = 12 mmhw = 100 mmM = 25 kN·m
σtop = MPa

Answer key

  1. 1. ȳ = 122 mm
  2. 2. ȳ = 100.9 mm
  3. 3. r = 79.4 mm
  4. 4. Ix = 268.7 × 10⁶ mm⁴
  5. 5. Ix = 66.09 × 10⁶ mm⁴
  6. 6. σtop = 159.5 MPa

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