A forward deep tank 12 m long extends from a longitudinal bulkhead to the ship's side. The widths of the tank surface measured from the longitudinal bulkhead at regular intervals are 10, 9, 7, 4 and 1 m. Calculate the second moment of area of the tank surface about a longitudinal axis passing through its centroid.

 A forward deep tank 12 m long extends from a longitudinal bulkhead to the ship's side. The widths of the tank surface measured from the longitudinal bulkhead at regular intervals are 10, 9, 7, 4 and 1 m. Calculate the second moment of area of the tank surface about a longitudinal axis passing through its centroid.

Reference: REED'S NAVAL ARCHITECTURE FOR MARINE ENGINEERS (4^th edition), Exercise – 3.9

 

Solution:

width	SM	f(A)	Width2	SM	FM	Width3	SM
10	1	10	100	1	100	1000	1000
9	4	36	81	4	324	729	2916
7	2	14	49	2	98	343	689
4	4	16	16	4	64	64	256
1	1	1	1	1	1	1	1
		∑f(A) = 77			∑f(FM) = 587		∑f(SM) = 4862

 

Common interval, h = 12/4 = 3

SM about a longitudinal axis from waterline = h/3* 1/3 * ∑f(SM)

                                                                        = 3/3 * 1/3 * 4862

                                                                        = 1620.67 m⁴

Area of surface = h/3 * ∑f(A)

                          = 3/3 * 77

                          = 77 m²

Centroid from centerline = FM/area

                                         = (h/3 * ½ * ∑f(FM)) / (h/3 * ∑f(A))

                                         = ½ * 587 / 77

                                         = 3.812 m

SM about centroid = I_centerline – AX²

                               = 1620.67 – 77*3.812²

                               = 501 m⁴