- 2069 Chaitra · 8 marks
Radius of gyration of angle section
Short answer
Divide into 3×9 cm and 3×3 cm rectangles. Centroid at (2.25, 3.75) cm. I_xx = 249.75 cm⁴, A = 36 cm²; radius of gyration r_x = 2.63 cm.
Determine radius of gyration (rx) of the angle section shown in figure below about centroidal x-axis. [Figure: L-shaped angle section, overall height 9 cm and overall width 6 cm, both legs 3 cm thick.]
Reading of the figure: an L (angle) section, overall height 9 cm and overall width 6 cm, both legs 3 cm thick. Origin at the outer bottom-left corner.
Divide into: vertical leg 3 × 9 cm and horizontal leg 3 × 3 cm (the part of the bottom leg beyond the vertical leg, 6 − 3 = 3 cm long).
Centroid
| Part | A (cm²) | x (cm) | y (cm) | A·x | A·y |
|---|---|---|---|---|---|
| Vertical leg 3 × 9 | 27.00 | 1.50 | 4.50 | 40.5 | 121.5 |
| Horizontal leg 3 × 3 | 9.00 | 4.50 | 1.50 | 40.5 | 13.5 |
ΣA = 36.00 cm² x̄ = 2.250 cm, ȳ = 3.750 cm
Moment of inertia about the centroidal x axis
I_xx = Σ [ b h³/12 + A (y − ȳ)² ]
Vertical leg : 3×9³/12 + 27 × (4.5 − 3.75)² = 182.25 + 15.19 = 197.44 cm4
Horizontal leg: 3×3³/12 + 9 × (1.5 − 3.75)² = 6.75 + 45.56 = 52.31 cm4
I_xx = 249.75 cm4
Radius of gyration
r_x = √(I_xx / A) = √(249.75 / 36) = √6.9375 = 2.63 cm
✓ Answer checked by Claude Opus 5.5 (high effort)
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