Form Coefficients

WPA = Waterplane area L = Length overall B = maximum breadth d = maximum depth

$$C_w$$

= coefficient of waterplane area

$$C_m$$

= coefficient of midship area

$$WPA = L\times B \times C_w$$$$Area Amidships = B \times d\times C_m$$

when displacement is constant ( box shaped vessels)

$$\frac {New Displacement}{Old Displacement} = \frac{New Density}{Old Density}$$$$TPC_{SW}= \frac {WPA} {97.56}$$$$FWA = \frac {W} { 4 times TPC_{SW}}$$

change in draft

$$= \frac {FWA \times (1025-\rho_{DW}) }{25}$$

Trim

$$MCTC = \frac {W \times GM_L}{100 \times L}$$$$GM_L = KM_L-KG$$

$$KM_L = KB + BM_L$$

$$BM_L = \frac{I_{LL}}{\nabla}$$

Change in Trim $\Delta_T$

$$\Delta_T = \frac{Trimming moment}{MCTC}= \frac{ W \times (LCB_{foap}-LCG_{foap})}{MCTC}$$

Change of draft Aft $\delta_{aft} = \frac{1}{L} \times \Delta_T$

distance ’d’ to keep aft draft constant $\frac{MCTC \times L}{TPC \times l}$

Simpson’s Rules

Common Interval = h

• $$Area = \frac{1}{3} \times h \times \Sigma_1$$ Multipliers (1,4,1)
• $$Area = \frac {3h}{8} \times \Sigma_1$$ Multipliers (1,3,3,1)
• $$Area = \frac{h}{12}(5a+8b-c)$$

Second Moment of Inertia (I)

Essentially moment of an Area about an axis . Here the formulas are given for its own centroidal axis.

rectangle$B \times L$

For rectangular waterplane Area

$$I_{CG} = \frac { L \times B^3}{12}$$$$BM = \frac{I}{V}$$

triangle base =‘B’, height =‘H’

$$I_{CG}= \frac{B\times H^3}{36}$$

Circle of Radius ‘R’

$$I_{CG} = \frac{\pi \times R^4}{4}$$

Polar moment is (for rudder stock problem) $J = \frac{\pi \times R^4}{2}$

Prism base ‘B’,‘C’ and height ‘H’

$$I_{CG}= \frac {H^3}{36} \times \frac{B^2+4BC+C^2}{B+C}$$

for the Sectional Modulus Problem:

where b is the horizontal length and h can be height or can be thickness

$$I_{CG} = \frac {bh^3}{12}$$

General Formula (Inclined Surface)

$$h_{CP} = \bar{h} + \frac{I_G \sin^2\theta}{A\bar{h}}$$

For vertical surfaces $\theta = 90°$: $\quad h_{CP} = \bar{h} + \dfrac{I_G}{A\bar{h}}$

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Shape Properties & Center of Pressure

Shape	Area $A$	Centroid Depth $\bar{h}$*	$I_G$ (about centroid)	$h_{CP}$ (Vertical Surface)
Rectangle width $b$, height $h$	$bh$	$\frac{h}{2}$ (top at surface)	$\frac{bh^3}{12}$	$\frac{2h}{3}$
Triangle (base $b$, height $h$, apex up)	$\frac{1}{2}bh$	$\frac{2h}{3}$ (top/apex at surface)	$\frac{bh^3}{36}$	$\frac{3h}{4}$
Circle (radius $R$)	$\pi R^2$	$R$ (top at surface)	$\frac{\pi R^4}{4}$	$\frac{5R}{4}$
Trapezoid (bases $B$ bottom, $C$ top, height $h$)	$\frac{B+C}{2}h$	$\frac{h}{3}\cdot\frac{B+2C}{B+C}$ (top at surface)	$\frac{h^3}{36}\cdot\frac{B^2+4BC+C^2}{B+C}$	$\bar{h} + \frac{h^2}{18\bar{h}}\cdot\frac{B^2+4BC+C^2}{(B+C)^2}$
Semicircle (radius $R$, flat side up)	$\frac{\pi R^2}{2}$	$\frac{4R}{3\pi}$ (diameter at surface)	$0.1098R^4$	$\bar{h} + \frac{0.1098R^4}{\frac{\pi R^2}{2}\cdot\bar{h}}$

*Measured from the free surface to the shape’s centroid. For submerged cases, add the depth of the top edge.

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Quick Exam Shortcuts

• Rectangle (top at surface): $h_{CP} = \frac{2}{3}h$ (measured from top)
• Triangle (apex at surface): $h_{CP} = \frac{3}{4}h$ (measured from apex)
• Circle (top at surface): $h_{CP} = \frac{5}{4}R$ (measured from top)
• Universal Check: $h_{CP}$ is always below the centroid ( $\bar{h}$) for vertical/inclined surfaces.

Bilging Connection (from your previous point)

• Sinkage: $\dfrac{\text{Lost Volume}}{\text{Intact Waterplane Area}}$
• New $GM_T$: Use $I_{T,\text{intact}}$ (about centerline) $\rightarrow BM_T = \frac{I_T}{\nabla}$
• New $GM_L$: Use $I_{L,\text{intact}}$ (about midships) $\rightarrow BM_L = \frac{I_L}{\nabla}$

Pro Tip: If the surface is inclined, remember the $\sin^2\theta$ factor. If the tank wall is curved, find the horizontal/vertical force components first; the center of pressure formula above applies only to plane surfaces.

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Simpson’s Rules for Moment of Inertia

Area = A, y is athwartship, x is longitudinal, da is an elemental strip of the area.

1st moment of Area about X axis:

2nd moment of inertia about X axis: For a small area dA the moment of inertia about its centroid is

Therefore, when taking moments about X Axis:

Therefore, for Simpson’s Rule 1: Where ‘h’ is the common interval along X axis and ‘y0, y1, y2, y3, y4’ are y values along the X axis .

1st Moment of Area about Y Axis:

2nd Moment of Area about Y Axis: (ignoring first term as it is too small)

For  (stations ) and common interval h,

Now, if Y is Longitudinal and X is athwartship then switch x for y. Therefore, for TRANSVERSE items like Centre of Floatation use For Longitudinal items use