46.NA April 2019 Q.10(b)

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.

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Given 

L = 12 m

To find

1. Second moment of area

Solution

Width	SM	Product for area	Width2	SM	Product 1stmoment	Width3	SM	Product 2ndmoment
10	1	10	100	1	100	1000	1	1000
9	4	36	81	4	324	729	4	2916
7	2	14	49	2	98	343	2	686
4	4	16	16	4	64	64	4	256
1	1	1	1	1	1	1	1	1
		∑a = 77			∑m = 587			∑I = 4859

Length is given as 12 m and there are 4 equidistant spaces.
So common interval h = 12/4 = 3

Area of surface a = h/3 x Σa
                            = 3/3 x 77
                            = 77 m2

Distance of centroid from bulkhead Ӯ = Σm / (2 x Σa)
                                                            = 587 / (2 x 77)
                                                            = 3.812 m

Second moment of area about bulkhead ib = h/9 x Σi
                                                                     = 3/9 x 4859
                                                                     = 1619.7 m4

Second moment of area about centroid ig = ib - aӮ2
                                                                   = 1619.7 - (77 x 3.8122)
                                                                   = 500.78 m4