RECOMMENDED
PRACTICE
NUMBER xx
Date Issued: May xx, 2009
Serving the Aerospace - Shipbuilding - Land Vehicle
and Allied Industries
Executive Director
P.O. Box 60024, Terminal Annex
Los Angeles, CA 90060
Weight Distribution and Moments of Inertia for
Marine Vehicles
Revision Issue No. draft
Prepared by
Marine Systems
Government – Industry Workshop
of the
Society of Allied Weight Engineers
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To view the latest revisions of this recommended practice go to www.sawe.org. Send questions,
suggestions or comments to the Government/Industry Committee Chairman.
SAWE Recommended Practice xx
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Change Record
Issue No
Date
------
May xx, 2009
Title/Brief Description
Initial Issue
Entered By
David Hansch
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Contents
Section
Page Number
1.0 Introduction / Scope
2.0 Units of Measure
3.0 Terms
4.0 Longitudinal Weight Distribution Prediction and Calculation Methodologies
4.1 Parametric Estimates without complete Centers of Gravity for Line Items
4.2 Parametric Estimates/ Rough Estimates with Centers of Gravity for Line Items
4.3 Direct Estimates and Calculations with Centers of Gravity for Line Items
4.4 Representative Shapes
4.5 Distributing Margins
4.6 Reporting of Weight Distribution
5.0 Gyradii Prediction and Calculation Methodologies
5.1 Parametric Methods
5.2 Calculation Methods
5.3 Gyradii Calculation for Hull Modifications during Overhauls
5.4 Reporting of Weight Moments of Inertia and Gyradii
6.0 References
APPENDIX A: Longitudinal Weight Distribution Parametrics
APPENDIX B: Gyradii Parametrics
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4
4
5
6
6
7
7
7
8
8
8
8
12
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13
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22
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1.0 Introduction / Scope
This Recommended Practice documents the preferred methods of estimating and
calculating the distributive mass properties of marine vehicles at various stages of weight
database maturity. Distributive mass properties include mass (weight) distributions and
mass (weight) moments of inertia which are dependant on the mass distributions. These
mass moments of inertia are typically reported as Gyradii.
At the earliest stages of design parametric methods are required to estimate distributive
mass properties. These parametric methods are briefly discussed and supplied in the
appendices; however, the original sources are referenced for complete application and
applicability.
2.0 Units of Measure
Distributive marine mass properties are generally discussed using either Imperial Units
(weight, feet and Long Tons) or soft Metric (weight often used to mean mass, meters and
Tonnes). This document is primarily written using Imperial Units, however; the gyradii
calculation example is given in Metric Units.
3.0 Terms
Term
Gyradius
Line Item
Mass Moment of
Inertia
Radius of
Gyration
3-digit weight
estimate
Transference
Inertia, IT
Own Moment of
Inertia
Definition
Radius of Gyration
Most detailed subdivision of data in a given
weight database
The inertia of a rigid rotating body with
respect to its rotation. The integral of
differential mass multiplied by the distance to
the object's center of gravity squared
The square root of the Mass Moment of
Inertia of an object divided by the Mass of the
object
Weight estimate at the 3 digit level of detail in
the ESWBS system.
The inertia caused by the distance of an
object from the location about which the
inertia is calculated. IT = mass or weight
times distance squared
The inertia of an object with respect to the
axis in question about its own center of
gravity.
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4.0 Longitudinal Weight Distribution Prediction and Calculation
Methodologies
Longitudinal Weight distributions are primarily used in the determination of loads for the
longitudinal strength calculation. The longitudinal strength calculation considers the ship
as a beam of varying sectional inertia and depth. The loads on this beam are determined
from the longitudinal weight distribution and the buoyancy curve.
Generally the longitudinal weight distribution is simplified to 22 segments (one forward
of the FP, one aft of the AP and then in between each of the 20 ship stations). The
weights of these segments are represented by trapezoids with weight per length on the
vertical scale versus length on the horizontal scale. Generally the longitudinal weight
distribution is calculated for the full load and lightship condition. For ships with
intermediate operating conditions that differ largely in longitudinal distribution or ships
which are highly stressed, additional conditions should also be considered. Typical
additional conditions include a “hogging” load, where all tanks and storerooms forward
or abaft the ship’s quarterpoints are loaded and the midship tanks and storerooms are
empty, and a “sagging” load where the midship tanks and storerooms are full and those in
the ends of the ship are empty. Tankers frequently have loading cases where every other
tank is loaded.
Because the longitudinal strength calculation is a prime determinant of the scantlings of
the hull girder and thus a large portion of the ship’s weight, estimates of the longitudinal
weight distribution of a vessel must be performed very early in the ship design process
and continually updated. The preferred method for estimating or calculating the
longitudinal weight distribution varies with the fidelity of the weight estimate/calculation.
While these levels of fidelity could be tied to the differing design phases, exact
definitions of the phases can vary, therefore; it is more appropriate to refer to them based
on the levels of detail that are included in the weight database. These levels are:
1. Parametric Estimates without complete Centers of Gravity for Line Items
2. Parametric Estimates / Rough Calculations with Centers of Gravity for Line Items
3. Direct Estimates and Calculations with Centers of Gravity for Line Items
Each level of fidelity of the weight estimate elicits a different preferred method of
distribution.
Transverse and vertical weight distribution are primarily used to support the integral
method of calculating moments of inertia and can be determined using the same
methodologies outlined in sections 4.2 and 4.3.
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4.1 Parametric Estimates without complete Centers of Gravity for Line
Items
In this case, the parametric methods of estimating the weight distribution are the only
option. There are two methods of parametric estimation, general and parent ship.
General parametrics are determined based on vessel type and only distribute hull weight.
The weight of heavy machinery and deadweight must be added to such distributions to
build up the total weight distribution for the vessel in question.
Parent ship parametrics use the complete weight distribution of a similar vessel as a
starting point for the estimation of the new weight distribution. The weights of the parent
distribution should be scaled to the new ship and then any significant differences in
weight or arrangement should be accounted for by removing the parent items and
replacing them with the new ship items using simple representations such as rectangles or
trapezoids. For example, a larger deckhouse could be accounted for by removing a
rectangle with the weight and extents of the original deckhouse from the scaled parent
distribution and then adding a rectangle of heavier weight and larger extents.
Such estimates are useful for quickly determining if the new vessel will differ
significantly in its loading, which is useful for initial estimation of the scantlings and in
feasibility analyses. However, when more detailed information is available this estimate
must be updated.
Reference (1) contains an appendix which presents a large number of both types of
parametrics. These parametrics have been included in Appendix A for convenience.
4.2 Parametric Estimates/ Rough Estimates with Centers of Gravity for
Line Items (i.e. 3-digit weight estimate)
A reasonably accurate weight distribution can be calculated based on a 3-digit weight
estimate and a general arrangement drawing. Based on the arrangements drawing,
extents for the 3-digit weights can be estimated. Using the weight, center of gravity and
extents, each 3-digit weight estimate should be distributed, using representative shapes.
Depending on the perceived weight distribution, triangles, trapezoids, rectangles,
parabolas or coffin shapes can be used to distribute the weight of each group. Once
distributed, the weight of each group at each longitudinal location of the ship is summed
to determine the total longitudinal weight distribution. This distribution is then
subdivided and summed to determine a Twenty Station Weight Distribution.
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4.3 Direct Estimates and Calculations with Centers of Gravity for Line
Items
With this greatest level of fidelity of the weight database, a very detailed distribution can
be created. This distribution is the result of the summation of the individual distributions
of every weight element in the weight database. Representative shapes calculated from
the weight, center of gravity and extents of each item provide the individual distributions.
If extents for each individual line item in the weight database are not available, partial
summations of systems can be made in order to provide weight entities of known extents.
For example, all of the 2nd deck plating between frames 17 and 44 can be combined into
one entry for distribution purposes to provide known extents. Certainly calculations
based on the true extents of each line item are more accurate and preferred, but the
creation of partial summations yields an acceptable result.
The individual distributions of either the line items or the partial summations are then
summed at each location along the length of the ship to create the ship’s weight per foot
curve. This weight-per-foot curve is then numerically integrated to yield the weights
between stations for the Twenty Station Weight Distribution. A summation interval on
the order of 0.05% of the length of the ship gives good results. The exact interval should
be chosen to land evenly on the stations. The station weights are then corrected to give
the correct total ship weight by smearing the numerical integration uncertainty by
percentage across each station.
4.4 Representative Shapes
Trapezoids are the basic representative shape of weight distributions. However,
trapezoids can only represent items whose center is in the middle one third of the item’s
length. It may be necessary to utilize compounded shapes such as combinations of
triangles and trapezoids to distribute some items. An example of such a shape is shown
in Figure 1. Other shapes and compounds beyond trapezoids and triangles are acceptable
and may prove advantageous in specific applications.
Figure 1: An Example of a Compound Weight Distribution
A Compound Weight Distribution
4.5 Distributing Margins
There are three acceptable methods of applying the weight margin to the weight
distribution. The first, most preferable and most accurate method is to distribute the
margins the same way as the weights to which they are applied. This method essentially
requires distributing the weight of the ship twice, the first time using the weight of each
item as the weight and the second time using the margin applied to each item in place of
the weight of the item. The second and simplest method is to distribute the margin by
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percentage along the length of the ship. In this case if the margin is two percent of the
weight of the ship, the weight of each station will be increased by two percent of its
weight. The third method is to distribute the margin as a trapezoid to correct any
calculation discrepancies in between longitudinal center of gravity of the weight
distribution and that of the weight report. This corrective method is the least preferred
because while it gives a result that appears perfect, it introduces the greatest amount of
uncertainty into the calculation.
4.6 Reporting of Weight Distribution
The weight distribution report should be completed as required in Reference (4).
5.0 Gyradii Prediction and Calculation Methodologies
The gyradius (K) is mathematically defined as the square root of the quantity of the mass
moment of inertia about an axis (I) divided by the displacement (Δ), K = √I/Δ. The
optimum method of predicting and calculating the gyradii of ships depends on the
amount of detail that is available about the ship. Initial estimates are parametric
estimates of gyradii based on similar ships; later when a detailed weight database is
available there are calculation methods available which allow for greater accuracy.
Weight moments of inertia are calculated via these methods and then converted to
gyradii.
5.1 Parametric Methods
The gyradii of a ship can be estimated parametrically by a percentage of the beam for the
roll gyradius and a percentage of length for the pitch and yaw gyradii. These percentages
should be chosen based on the known gyradii of similar ships. In this case a similar ship
is one which has a similar weight distribution. For roll gyradius estimation the vertical
and transverse weight distributions are most important. For pitch and yaw gyradii
estimation the longitudinal weight distribution is the most important. References 45) and
(6) provide parametric gyradii for merchant and military ships respectively. For
convenience these parametrics are reprinted in Appendix B. It should be noted when
using other parametrics that it is very important to be sure that the parametrics are for the
gyradii of the ship in air, as damping and added mass effects in water can dramatically
change gyradii value predictions.
5.2 Calculation Methods
There are two basic methodologies for directly calculating gyradii. The first method
involves the calculation of the transference inertia of every weight item in the weight
database as well as the own moments of inertia of an appreciable amount of those weight
items. This method is useful for databases which contain sufficient amounts of own
moment of inertia data. The second method is an integral method which does not require
own moment of inertia data.
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A. Transference and Own Moments of Inertia Method:
This method involves summing the transference inertia of each item in the weight
database as well as the own moment of inertia of each item. This method is dependant
upon sufficient own moment of inertia data being available. In Reference (6) Cimino and
Redmond suggest the use of shape factors to allow for the determination of the own
moments of inertia of weight entries. Reference (6) suggests that all weight items that are
≥ 0.1% of the ships total weight should have own moment of inertia data associated with
them in order to provide an overall moment of inertia accurate to within +/- 0.5%. The
ship’s moments of inertia in this methodology are calculated in accordance with the
following equations:
roll inertia: Ixx
or Ixx  Itx
 [wn(yn2  zn2 )]   ioxn
 Iox
pitch inertia: Iyy
n
n
 [wn(xn2  z n2 )]   ioyn
n
or Iyy
 Ity  Ioy
yaw inertia: Izz 
[wn(xn2  yn2 )]  ioz
n
n
or
n
n
Izz  Itz  Ioz
where;
wn = weight of the nth element
xn = longitudinal distance of the nth element from the ship's or submarine's overall CG to
the item's CG along the x-axis
yn = transverse distance of the nth element from the ship's or submarine's
overall CG to the item's CG along the y-axis
zn = vertical distance of the n element from the ship's or submarine's overall CG to the
item's centroid CG along the z-axis
ioxn = weight moment of inertia of the nth element about an axis parallel to the
x-axis and passing through the CG of the nth element
ioyn = weight moment of inertia of the nth element about an axis parallel to the
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y-axis and passing through the CG of the nth element
iozn = weight moment of inertia of the nth element about an axis parallel to the
z-axis and passing through the CG of the nth element
Iox = sum of the item weight moments of inertia for all elements about an axis
parallel to the x-axis
Ioy = sum of the item weight moments of inertia for all elements about an axis
parallel to the y-axis
Ioz = sum of the item weight moments of inertia for all elements about an axis
parallel to the z-axis
Itx = sum of the transference weight moments of inertia for all elements about
an axis parallel to the x-axis
Ity = sum of the transference weight moments of inertia for all elements about
an axis parallel to the y-axis
Itz = sum of the transference weight moments of inertia for all elements about
an axis parallel to the z-axis
B. Integral Method
The second method of calculating gyradii eliminates the need for own moments of
inertia. This is accomplished by treating the whole ship as a single entity of varying
density and determining the gyradii of this entity by integration. This integration of the
transference inertia of every differential unit of volume inherently captures the own
moment of inertia of each item on the ship. This is effective assuming that the
differential volume is small enough. The mass moment of inertia is calculated based on
the equation:
Itotal = ∫ w(r) * r dr
Where:
Itotal = the total moment of inertia about a given axis
r = the radial distance from the axis
w(r) = the weight at location r
Through the use of the Pythagorean Theorem, this equation can be broken into two
equations such that:
Itotal = ∫ w(a) * a da + ∫ w(b) * b db
Where:
Itotal = the total moment of inertia about a given axis
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a = the distance along the a axis from the axis about which the inertia is taken
w(a) = the weight at distance a
b = the distance along the b axis from the axis about which the inertia is taken
w(b) = the weight at distance b
A and b are orthogonal axes to the axis the inertia is taken about. For example, if I is
taken about the longitudinal axis (for roll inertia), a and b are the vertical and transverse
axes respectively. The creation of highly detailed weight (or mass) per foot curves for
each axis via representation of details or partial summations as described in the
longitudinal weight distribution process above is necessary to use this approach. These
curves are then numerically integrated and combined in pairs to yield each gyradius. The
interval of integration for the vertical and transverse axes may need to be smaller than the
interval of integration for the longitudinal axis. This is because the distributions of these
axes often have higher uncertainties associated with the numerical integration than the
longitudinal axes. The direct integration of the weight distribution along each axis should
result in a weight within 1% of the true weight. This is important because unlike in the
creation of the Twenty Station Weight Distribution, the integration uncertainty should not
be smeared back into the results of these calculations. In order to eliminate the effect on
the gyradii of an error in the mass moment of inertia due to integration uncertainty with
the weight of the ship, the weight calculated by integrating the weight distribution along
each axis should be averaged in order to determine the weight to divide the mass moment
of inertia by. Also, it is often easier to take the moments of inertia about the ship’s origin
and covert to the ship’s center of gravity through the use of the parallel axis theorem.
Table 1 shows how the results of the numerical integrations of the mass moment of
inertia equation along each axis can be combined to determine the mass moments of
inertia and gyradii for a vessel. This included correcting the moments to the ship’s center
of gravity.
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Table 1: Example of the Combination of Moments due to Distributions along Axes to Determine
Mass Moments of Inertia and Gyradii
Corrected to Ship
Uncorrected
Centers
Moments
Mass Moments
Mass Moment of
due to
about FP, BL, CL
Inertia
distribution
(Kilogram(KilogramMass
Center of
along axis
Meter^2)
Meter^2)
(Kilogram)
Gravity
Moment
Vertical
1,890,000,000
655,800,000
10,200,000
11.00
112,200,000
Longitudinal
77,460,000,000
11,540,000,000
10,300,000
80.00
824,000,000
Transverse
420,000,000
420,000,000
10,400,000
Mass Moment
Displacement
Gyradius
-
0
Gyradius
Coefficient
Pitch
12,195,800,000
10,250,000
34.49
0.221
* Length
Yaw
11,960,000,000
10,350,000
33.99
0.218
* Length
Roll
1,075,800,000
10,300,000
10.22
0.393
* Beam
Length
Beam
Depth
Ship Characteristics
156
meters
26
meters
18.4
meters
5.3 Gyradii Calculation for Hull Modifications during Overhauls
Typically gyradii are reported as being a function of the ship’s beam for roll gyradii and
ship’s length for yaw and pitch gyradii. This is an effective method of expressing gyradii
and is very useful for parametric estimates and many ship motions analysis programs
accept gyradii inputs in this form. However, when dealing with ships that are to be
plugged or blistered, great care must be taken when reporting gyradii in this manner. In
such cases it is best to convert the gyradii that are dependant on the changing parameter
(roll for blistering, pitch and yaw for plugging) to moments of inertia and then add to it
the calculated additional moment of inertia of the new section. The resultant moment of
inertia can then be converted to a gyradius and normalized against length or beam if the
data is needed in this format. This process is necessary because the plug or blister will
probably not contribute the same amount to the moments of inertia as the original
portions of the ship. Thus the normalized gyradii should change as the ship parameters
they are normalized against change.
5.4 Reporting of Weight Moments of Inertia and Gyradii
The weight moments of inertia and gyradii should be reported as described in Reference
4).
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6.0 References
1. Hansch, David L. “Methods of Determining the Longitudinal Weight Distribution
of a Ship”. Presented at the 67th Annual Conference of the Society of Allied
Weight Engineers. May 2008.
2. Marine Vehicle Weight Engineering. Edited by Cimino, Dominick and David
Tellet. Society of Allied Weight Engineers, Inc. USA. 2007.
3. Hansch, David L. “Weight Distribution Method of Determining Gyradii of
Ships”. Presented at the Chesapeake Bay Regional Conference of the Society of
Allied Weight Engineers. November 2006.
4. Society of Allied Weight Engineers “Recommended Practice 12: Weight Control
Technical Requirements for Surface Ships”
5. Peach R. W. and A. K. Brook. “The Radii of Gyration of Merchant Ships”.
Transactions of the North East Coast Institution of Engineers and Shipbuilders,
Vol. 103, No. 3, pg. 115-117.
6. Cimino, Dominick and Mark Redmond. “Naval Ships’ Weight Moment of Inertia
– A Comparative Analysis”. Society of Allied Weight Engineers, Paper 2013 May
1991.
7. Comstock, John P. Introduction to Naval Architecture. Simmons-Boardman
Publishing Corporation. New York, New York. 1944.
8. Munro-Smith, R. Applied Naval Architecture. American Elsevier Publishing
Company, Inc. New York 1967.
9. Principles of Naval Architecture Volume I. Edited by Rossell, Henry E. and
Lawrence B. Chapman. The Society of Naval Architects and Marine Engineers.
New York. New York. 1941.
10. Hughes, Owen. Ship Structural Design: A Rationally-Based, Computer-Aided
Optimization Approach. Society of Naval Architects and Marine Engineers.
Jersey City, New Jersey. 1988.
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APPENDIX A: Longitudinal Weight Distribution Parametrics
All of the following figures show distributions of Hull Weight. Comstock (6) defines
Hull Weight as “the steel, woodwork, fittings and outfit except anchors and cables, hull
engineering except windlass and steering gear, any spread-out items of deadweight, such
as passengers and crew, and designer’s margin.” Comstock goes on to note that, “The
diagram must be proportioned that not only will the area be correct but also the LCG.”
The cargo should be, “distributed over the length of the cargo holds as trapezoids, and so
on until the diagram includes all the weights in the loaded ship.”
Figure A-1
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Figure A-2
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Figure A-3
Type of Ship
Tankers
Full Cargo Ships Without Erections
Fine Cargo Ships Without Erections
Full Cargo Ships With Erections
Fine Cargo Ships With Erections
Small Passenger Ships
Large Passenger Ships
X / (W/L)
0.75
0.65
0.60
0.55
0.45
0.40
0.30
H / (W/L)
1.125
1.175
1.20
1.225
1.275
1.30
1.35
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Figure A-4
This distribution is intended for vessels which don’t have parallel middle body. The
centroid of the distribution is shifted a desired distance, D, from midships by “swinging
the parabola”. As PNA explains, “Through the centroid of the parabola draw a line
parallel to the base and in length equal to twice the shift desired (forward or aft).
Through the point thus determined draw a line to the base of the parabola at its midlength. The intersection of this line with the horizontal drawn from the intersection of the
midship ordinate with the original parabolic contour determines the location of on point
on the corrected curve. Parallel lines drawn at other ordinates … determine the new
curve.”
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Figure A-5
K
X
H
Y
D
Fine Ships
Merchant Type
0.333 W/L
0.567 W/L
1.195 W/L
0.653 W/L
0.0052 L
Full Ships
Merchant Type
0.333 W/L
0.596 W/L
1.174 W/L
0.706 W/L
0.0017 L
Great Lakes
Bulk Freighters
0.250 W/L
0.572 W/L
1.125 W/L
0.676 W/L
0.0054 L
This approximation is useful for ships with parallel middle body.
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Figure A-6
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20 Station Distributions by ship type from Reference (2)
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APPENDIX B: Gyradii Parametrics
Naval Ship Parametrics from Reference (2)
DDG 51
ARS 52
FFG 60
CG 62
MCM 1
LHD 2
CVN 73
LPD 17
CONVENTIONAL SURFACE SHIPS
ROLL
PITCH
(% B)
(% L)
38.90%
25.20%
36.50%
25.10%
36.20%
24.40%
40.40%
25.30%
38.10%
24.30%
42.00%
25.60%
40.90%
23.20%
40.50%
23.80%
MEAN, CONVENTIONAL HULL
FORMS
TOLERANCES (+/-)
TAGOS 19
TAGOS 23
MEAN SWATHS
LSV NSURFACE
SSN 756 NSURFACE
SSBN 737 NSURFACE
SSN 756 NSUBMERGES
SSBN 737 NSUBMERGES
MEAN, SUBMARINES
TOLERANCES (+/-)
YAW
(% L)
25.10%
24.90%
24.30%
25.20%
24.40%
25.60%
23.40%
23.80%
38.90%
24.70%
24.70%
2.10%
0.80%
0.70%
SWATHS
ROLL
(% B)
43.30%
43.60%
PITCH
(% L)
30.50%
27.70%
YAW
(% L)
32.80%
29.40%
43.45%
29.10%
31.10%
SUBMARINES
ROLL
(% B)
37.40%
36.40%
36.70%
34.70%
34.90%
PITCH
(% L)
22.90%
24.00%
24.30%
25.70%
26.30%
YAW
(% L)
22.90%
23.90%
24.20%
25.70%
26.30%
36.00%
24.60%
24.60%
1.20%
1.40%
1.40%
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Merchant ship Parametrics per Reference (5)
Kroll = 0.30 x (B2 + D2)1/2
As given in RW Peach’s MarAd report (Peach and Brook)
Kroll/B = 0.289 √1+4(KG/B)2
Proposed by Bureau Veritas as reported by Brook (Peach and Brook)
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