Solution Naval Numerical 28

Numerical 28: A box shaped vessel 65m X 12m X 8m has KG 4m and is floating in sea water upright on an even keel at 4m draft F and A. Calculate the moments of statically stability at i. 5 degrees and ii. 25 degrees heel.

Solution: The Moment of Static Stability = W x GZ

 at a Small Angle of Heel: $GZ = GM\theta $

 at a Large Angle of Heel: $GZ = (GM + \frac{1}{2} BM tan^2\theta )sin\theta$ 

W = L x B x d x 1.025    

= 65 x 12 x 4 x 1.025 = 3198 t

KB = $\frac{d}{2}$  = $\frac{4}{2}$  =  2 m

BM = $\frac{B^2}{12d}$  = $\frac{12^2}{12\times 4}$   = 3 m

KM = KB + BM = 2 + 3 = 5 m

KG = 4 m

GM = KM – KG = 5 – 4= 1 m

(i) at $5^o$ heel          

$GZ = GMsin\theta = 1\times sin 5^o$ = 0.0872 m

Moment of statical stability = W x GZ

= 3198 x 0.0872 = 278.9 t-m

(ii) at $25^o$ heel    

$GZ = (GM + \frac{1}{2} BM tan^2\theta )sin\theta$ 

$GZ = (GM + \frac{1}{2}\times 3\times tan^2 25^o )sin25^o$ = 0.56 m

Moment of statical stability = W x GZ

= 3198 x 0.56 = 1790.9 t-m