- 2073 Shrawan · 8+4 marks
MOI of composite area with circular hole
Short answer
Dividing into rectangle 400×650 mm, triangle 300×500 mm, minus circular hole r=25 mm, centroid is x̄=267.56 mm, ȳ=290.08 mm. About centroidal axes: I_xx=11.639×10⁹ mm⁴, I_yy=9.071×10⁹ mm⁴; radii of gyration k_x=186.9 mm, k_y=165.0 mm.
Calculate the moment of inertia of the composite area as shown in the figure about its centroidal axes. Define centroid, center of gravity, axis of symmetry and radius of gyration. [Figure: plane area with top width 400 mm, right vertical edges 150 mm and 500 mm, bottom width 700 mm, left height 200 mm, left offset 200 mm, and a circular hole of radius r = 25 mm; axes X, Y at lower-left.]
Reading of the figure: origin at the bottom-left corner. Overall height = 150 + 500 = 650 mm. Top width 400 mm, bottom width 700 mm. The right edge runs down 150 mm from the top right corner, then slopes (taken as straight) to the bottom right corner at x = 700. The circular hole (r = 25 mm) has its centre 200 mm from the left and 200 mm from the bottom.
Divide into: rectangle 400 × 650, right triangle with base 300 and height 500 (vertical side at x = 400), minus the circular hole.
Centroid
| Part | A (mm²) | x (mm) | y (mm) | A·x | A·y |
|---|---|---|---|---|---|
| Rectangle 400 × 650 | 260,000.00 | 200.00 | 325.00 | 52,000,000.0 | 84,500,000.0 |
| Triangle 300 × 500 | 75,000.00 | 500.00 | 166.67 | 37,500,000.0 | 12,500,000.0 |
| Circular hole r = 25 (hole, −) | -1,963.50 | 200.00 | 200.00 | -392,699.1 | -392,699.1 |
ΣA = 333,036.50 mm² x̄ = 267.56 mm, ȳ = 290.08 mm
Moments of inertia about centroidal axes
Own values: rectangle bh³/12, hb³/12; triangle bh³/36, hb³/36; circle πr⁴/4.
| Part | I_x own (mm⁴) | A·dy² | I_y own | A·dx² |
|---|---|---|---|---|
| Rectangle 400 × 650 | 9,154,166,666.7 | 317,041,756.5 | 3,466,666,666.7 | 1,186,737,605.9 |
| Triangle 300 × 500 | 1,041,666,666.7 | 1,142,317,796.9 | 375,000,000.0 | 4,052,120,892.9 |
| Circular hole r = 25 | -306,796.2 | -15,932,675.8 | -306,796.2 | -8,962,130.2 |
I_xx = 11.639 × 10⁹ mm⁴ I_yy = 9.071 × 10⁹ mm⁴ (k_x = 186.9 mm, k_y = 165.0 mm)
Definitions
- Centroid: the point at which the whole area of a plane figure can be assumed to be concentrated. It is the geometric centre of the area and depends only on shape: x̄ = ∫x dA / A, ȳ = ∫y dA / A.
- Centre of gravity: the point through which the whole weight of a body acts, whatever its orientation. For a body of uniform density and in a uniform gravity field it coincides with the centroid of its volume (or area for a thin plate).
- Axis of symmetry: a line that divides a figure into two parts that are mirror images. The centroid always lies on an axis of symmetry, and the product of inertia about it is zero.
- Radius of gyration: the distance k from an axis at which the whole area could be concentrated to give the same moment of inertia, I = A k², so k = √(I/A).
✓ Answer checked by Claude Opus 5.5 (high effort)
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