Fig50 5 See Fig 50 and attached 45 points a Determine the mo

Fig.5.0 5) See Fig. 5.0 and attached. (45 points] (a) Determine the moment of inertia of the cross-sectional area of the s0 channel with respect to the x and y axes. the oment of inertia of the caross-sectional area of (b) Hence, determine the moment of inertia of the cross-sectional area of 50mm the channel with respect to the x, and yb axes. 300mm so mm 200mm C: Centroid of the Channel Cross-sectional Area

Solution

The given cross-section can be splitted as given as Top, Segment and Bottom. Each shape is rectangle so it has its centre of gravity at the point of intersection of its diagnoals.

a) Moment of inertia of top portion about its centroid along horizontal axis = 50³x200/12 = 2.083 x 10⁶ mm⁴

Moment of inertia of top portion about x-axis = 2.083 x 10⁶ + 50 x 200 x 325² = 1.0583 x 10⁹ mm⁴

Moment of inertia of top portion about its centroid along vertical axis = 200³x50/12 = 33.333 x 10⁶ mm⁴

Moment of inertia of top portion about y-axis = 33.333 x 10⁶ mm⁴

Moment of inertia of segment about its centroid along horizontal axis = 200³x50/12 = 33.333 x 10⁶ mm⁴

Moment of inertia of segment about x-axis = 33.333 x 10⁶ + 50 x 200 x 150² = 2.583 x 10⁸ mm⁴

Moment of inertia of segment about its centroid along vertical axis = 50³x200/12 = 2.083 x 10⁶ mm⁴

Moment of inertia of segment about y-axis = 2.083 x 10⁶ mm⁴

Moment of inertia of bottom portion about its centroid along horizontal axis = 50³x200/12 = 2.083 x 10⁶ mm⁴

Moment of inertia of bottom portion about x-axis = 2.083 x 10⁶ + 50 x 200 x 25² = 8.333 x 10⁹ mm⁴

Moment of inertia of bottom portion about its centroid along vertical axis = 200³x50/12 = 33.333 x 10⁶ mm⁴

Moment of inertia of bottom portion about y-axis = 33.333 x 10⁶ mm⁴

Moment of inertia of whole cross-section about x-axis = 1.0583 x 10⁹ + 2.583 x 10⁸ + 8.333 x 10⁹

= 1.324 x 10⁹ mm⁴

Moment of inertia of whole cross-section about y-axis = 33.333 x 10⁶ + 2.083 x 10⁶ + 33.333 x 10⁶

= 68.75 x 10⁶ mm⁴

b) Moment of inertia of whole cross-section about x_b axis = 1.324 x 10⁹ + 50 x 200 x (375² - 325²) + 50 x 200 x (200² - 150²)

= 1.85 x 10⁹ mm⁴

Moment of inertia of whole cross-section about y_b axis = 68.75 x 10⁶ + 3 x 50 x 200 x 100²

= 3.68 x 10⁸ mm⁴