Catamaran Design Formulas

Part 2: With permission from Terho Halme – Naval Architect

While Part 1 showcased design comments from Richard Woods, this second webpage on catamaran design is from a paper on “How to dimension a sailing catamaran”, written by the Finnish boat designer, Terho Halme. I found his paper easy to follow and all the Catamaran hull design equations were in one place.  Terho was kind enough to grant permission to reproduce his work here.

Below are basic equations and parameters of catamaran design, courtesy of Terho Halme. There are also a few references from ISO boat standards. The first step of catamaran design is to decide the length of the boat and her purpose. Then we’ll try to optimize other dimensions, to give her decent performance. All dimensions on this page are metric, linear dimensions are in meters (m), areas are in square meters (m2), displacement volumes in cubic meters (m3), masses (displacement, weight) are in kilograms (kg), forces in Newton’s (N), powers in kilowatts (kW) and speeds in knots. 

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Length, Draft and Beam

There are two major dimensions of a boat hull: The length of the hull L_H and length of waterline L_WL . The following consist of arbitrary values to illustrate a calculated example. 

L_H = 12.20      L_WL = 12.00

After deciding how big a boat we want we next enter the length/beam ratio of each hull, L_BR. Heavy boats have low value and light racers high value. L_BR below “8” leads to increased wave making and this should be avoided. Lower values increase loading capacity. Normal L_BR for a cruiser is somewhere between 9 and 12. L_BR has a definitive effect on boat displacement estimate.  

B WL= LWL/ LBR	In this example LBR = 11.0 and beam waterline B WL will be:
Figure 2
BWL= 1.09	A narrow beam, of under 1 meter, will be impractical in designing accommodations in a hull.
B TR = B WL / T c	A value near 2 minimizes friction resistance and slightly lower values minimize wave making. Reasonable values are from 1.5 to 2.8. Higher values increase load capacity. The deep-V bottomed boats have typically B TR between 1.1 and 1.4. B TR has also effect on boat displacement estimation.
T c = B WL / B TR   T c = 0.57	Here we put B TR = 1.9 to minimize boat resistance (for her size) and get the draft calculation for a canoe body T c (Figure 1).
	Midship coefficient – C m
C m = A m / T c (x) B WL	We need to estimate a few coefficients of the canoe body. where A m is the maximum cross section area of the hull (Figure 3). C mdepends on the shape of the midship section: a deep-V-section has C m = 0.5 while an ellipse section has C m = 0.785. Midship coefficient has a linear relation to displacement. In this example we use ellipse hull shape to minimize wetted surface, so C m = 0.785
Figure 3

	Prismatic coefficient – Cp
Cp=D / A m× LWL	where D is the displacement volume (m 3) of the boat. Prismatic coefficient has an influence on boat resistance. Cp is typically between 0.55 and 0.64. Lower values (< 0.57) are optimized to displacement speeds, and higher values (>0.60) to speeds over the hull speed (hull speed   ). In this example we are seeking for an all round performance cat and set C p := 0.59
	Water plane coefficient – Cw
Cw = Aw / BWL× LWL	where Aw is water plane (horizontal) area. Typical value for water plane coefficient is Cw = 0.69 – 0.72. In our example Cw = 0.71
	Fully loaded displacement – mLDC
mLDC = 2 × BWL x LWL× T c   × C p × C m× 1025  mLDC = 7136	At last we can do our displacement estimation. In the next formula, 2 is for two hulls and 1025 is the density of sea water (kg/m3). Loaded displacement mass in kg’s
	Length/displacement -ratio – LDR
LDR = 6.3	LDR near five, the catamaran is a heavy one and made from solid laminate. Near six, the catamaran has a modern sandwich construction. In a performance cruiser LDR is usually between 6.0 and 7.0. Higher values than seven are reserved for big racers and super high tech beasts. Use 6.0 to 6.5 as a target for LDR in a glass-sandwich built cruising catamaran. To adjust LDR and fully loaded displacement mLDC , change the length/beam ratio of hull, LBR .
	Empty boat displacement – m LCC
mLCC= 0.7 × mLDC  mLCC= 4995	We can now estimate our empty boat displacement (kg): This value must be checked after weight calculation or prototype building of the boat.
	Light loaded displacement – m moc
mmoc= 0.8 × m LDC  mmoc = 5709	The light loaded displacement mass (kg); this is the mass we will use in stability and performance prediction:
	Beam of sailing catamaran
	The beam of a sailing catamaran is a fundamental thing. Make it too narrow, and she can’t carry sails enough to be a decent sailboat. Make it too wide and you end up pitch-poling with too much sails on. The commonly accepted way is to design longitudinal and transversal metacenter heights equal. Here we use the height from buoyancy to metacenter (commonly named B M ). The beam between hull centers is named B CB (Figure 4) and remember that the overall length of the hull is L H .
Figure 4

	Length/beam ratio of the catamaran – L BRC
LBRC = LH / BCB	If we set L BRC = 2.2 , the longitudinal and transversal stability will come very near to the same value. You can design a sailing catamaran wider or narrower, if you like. Wider construction makes her heavier, narrower means that she carries less sail.
B CB = LH / LBRC B CB = 5.55	Beam between hull centers (m) – B CB
BM T = 2[(BWL3 × LWL x Cw2 / 12) +( LWL × BWL × Cw x (0.5BCB )2 )] × (1025 / mLDC ) BMT = 20.7	Transversal height from the center of buoyancy to metacenter, BMT can be estimated
BML = (2 × 0.92 x L WL3 × B WL x C w2 ) / 12 x (1025 / m LDC ) BML = 20.9	Longitudinal height from the center of buoyancy to metacenter, BML can be estimated. Too low value of BML (well under 10) will make her sensitive to hobby-horsing
BH1 = 1.4 × BWL	We still need to determine the beam of one hull BH1 (Figure 4). If the hulls are asymmetric above waterline this is a sum of outer hull halves. BH1 must be bigger than BWL of the hull. We’ll put here in our example:
B H = BH1 + BCBB H = 7.07	Now we can calculate the beam of our catamaran B H (Figure 4):
Z WD = 0.06 × L WL  Z WD = 0.72	Minimum wet deck clearance at fully loaded condition is defined here to be 6 % of L WL :
	EU Size factor
SF=1.75 x m mocSF = 82 x 103	While the length/beam ratio of catamaran, L BRC is between 2.2 and 3.2, a catamaran can be certified to A category if SF > 40 000 and to B category if SF > 15 000.
	Engine Power Requirements
P m = 4 x (mLDC /1025)P m = 28	The engine power needed for the catamaran is typically 4 kW/tonne and the motoring speed is near the hull speed. Installed power total in Kw
V m = 2.44V m = 8.5	Motoring speed (knots)
Vol = 1.2(Rm / Vm )(con x Pm ) Vol = 356	motoring range in nautical miles R m = 600, A diesel engine consume on half throttle approximately: con := 0.15 kg/kWh. The fuel tank of diesel with 20% of reserve is then