MOI of composite area about Y axis

Reading of the figure: origin O at the bottom-left; the x axis is 100 + 100 = 200 mm long. The area is bounded by: the sloping line from O to (200, 125), the right edge x = 200 from y = 125 up to y = 275 (150 mm long), the top edge from x = 200 to x = 100 at y = 275, a step down of 75 mm to y = 200 at x = 100, then the top edge back to the y axis at y = 200, and down the y axis to O.

Division into simple shapes

• Rectangle A: 100 × 200 (x from 0 to 100)
• Rectangle B: 100 × 275 (x from 100 to 200)
• Triangle C (to be removed): vertices (0, 0), (200, 0), (200, 125), the part below the sloping line

Moment of inertia about the Y axis

For a rectangle with one side on the Y axis, I_y = h b³/3 (b the width along x). For the rectangle B, I_y = h (x₂³ − x₁³)/3.

Rectangle A:  I_y = 200 × 100³ / 3             =  66.667 × 10⁶ mm⁴
Rectangle B:  I_y = 275 × (200³ − 100³) / 3
                  = 275 × 7 × 10⁶ / 3           = 641.667 × 10⁶ mm⁴
Triangle C (below the slope line y = 0.625 x):
              I_y = ∫₀²⁰⁰ x² (0.625 x) dx
                  = 0.625 × 200⁴ / 4            = 250.000 × 10⁶ mm⁴   (subtract)

I_y = 66.667 + 641.667 − 250.000 = 458.33 × 10⁶ mm⁴ (4.58 × 10⁸ mm⁴)

(Check: the area is 20 000 + 27 500 − 12 500 = 35 000 mm², centroid x̄ = 98.8 mm, and I_y = 116.6 × 10⁶ + 35 000 × 98.8² = 458.3 × 10⁶ mm⁴.)