Ocean Engng, Vol. 17, No 1/2, pp. 155-169, Ï990. 0029-8018/90 $3.00 + .00
Printed in Great Britain. S Pergamon Press pic
HYDROSTATIC STABILITY CALCULATIONS BY
PRESSURE INTEGRATION
S. SCHALCK and J. BAATRUP
The Department of Ocean Engineering. The Technical University of Denmark, DK-2800 Lyngby, Denmark
AbstractIt is the purpose of this paper to present a new method for the calculation of
hydrostatic properties of intact and damaged ship hulls and other floating structures The geometry of the floating structure is modelled as a set of compartments bounded by flat
panels. Hydrostatic properties are derived by pressure integration on these panel elements. By application of Greens integration theorem the area integration is transformed into line integrals around the contour of each element The line integrals can be expressed analytically such that the final result is that all the hydrostatic properties can be determined as a summation of easily evaluated expressions. This calculation procedure has the advantage of being well suited for computer calculation and is easily applied to truly arbitrarily shaped floating structures Another advantage is that it is based on a geometrical model which is equally suited for finite element strength analysis and hydrodynamic calculations based on a panel method. The application of the procedure is demonstrated by an example.
NOMENCLATURE
A area
äA boundary curve of area A
B letter to symbolise any hydrostatic characteristic F = (Fi, F)., F) force in XYZ system of coordinates
F buoyancy force
g acceleration of gravity
i,j,k index
rn number of compartments
n number of panels in a compartment n = (n, n),, n) normal vector to a panel
p number of edges in a panel
P pressure with respect to the free surface
S Wetted surface
T draught of struçture
(x,y,z) set of coordinates in global system of coordinates
center of buoyancy in global coordinates
(X,Y,Z)g center of gravity in global coordinates
(X,Y,Z)inp set of coordinates in input system of coordinates (x,y,z)o origin of panel system, expressed in global coordinates (X,y,Z)p set of coordinates in panel system of coordinates
XYZ global system of coordinates
input system of coordinates
X.YZP panel system of coordinates
a,13 constants in the equation of a line
O angle of trim p density of water angle of heel V submerged volume. 155 TECHNISCHE IJNIVERSITEIT Laboratorium voor Scheepshydromechanlca Archief Mekelweg 2,2828 CD Deift TeL 015-786873- Fax 015-781838
Hydrostatic characteristics:
A water plane area
-
-moments of inertia of the water plane area
mixed moment of the water plane area
moments of the submerged volume moments of the: water plane area.
.
1. INTRODUCTION
IN THE classical procedure forcalculation of the hydrostatic
properties of a ship the
first step is to calculate the Bonjean-curves for a number of transverse sections. As Bonjean-curves express the submerged sectional areas as a function of draught it s
possible to determine the hydrostatic properties at a given condition by integration over
the length of the ship. Primarily Simpsons rule or a planimeter was used but it has now been replaced by computer-aided numerical integration. However, the theory behind most of the current stability programs is still based on Bonjean-curves.
However, this method has some disadvantages. Usually the righting moment is
determined only for angles of heel between O and 600 as the superstructure does not interfere with the water plane in that interval. The reason for this limited interval is that analysing structures with a complicated geometry will entail certain difficulties.
Typically complicated structures are mostly considered as being box-type vessels such
as semisubmersibles, but ordinary ships also have a complicated geometry if deck and
superstructure are to be modelled. For special vessels, such as rescue vessels, it is
insufficient to analyse only between O and 600 büt necessary to carry out stability calculations between O and 1800or even - 180 and 1800 in the case of a non-symmetrical structure. Therefore, there is a need for a calculation procedure which is able to handle complicated structures. Recent years have seen a number of publications in which a direct pressure evaluation method has been presented, see van Santen (1986), McIver
et ql. (1983), Radwan (1983) and Witz and Patel (1985). They present different ways to carry out the pressure integration, but all are based on a discretisatjon of the outer
surface into fiat panel elements. Witz and Patel (1985) suggest a summation of a series
of line integrals, where the line integration is carried out over a panel strip. The line
integrals are evaluated, analytically and are exact, but
a numerical inaccuracy is intro-duced during the summation as the line integral is said to have a finite width, equal
to
the width of the panel
strip. Mclver et al. (1983)use triangular elements where the rectangular coordinates are converted into "area coordinates" as described in Zienkiewicz (1985). The method described by van Santen (1986)
makes it possible to use panels of arbitrary shape as the method relies on integration around the contour of each panel.
All the objects analysed. in Mclver et al. (1983), Radwan
(1983), van Santen (1986) and Witz and Patel (1985) are assembled by simple geometrical bodies,such as cylinders
and boxes. In this paper it will be shown that the theory is equally suitable for more
traditional vessels such as ships. In order to model the outer surface of a ship
as fiat
panels a procedure for automatical discretisation of
a surface into fiat panels is
presented. The panels are generated such that they have straight edges. Thereby it is
possible to evaluate all the conventional hydrostatic properties at a given condition
directly from analytical integration of line integrals. The discretised surface, used for hydrostatic analysis, can also be used as a basis for
FIG. 1. Definition of global system of coordinates.
Hydrostatic stability calculatiòns by pressure integration 157 panel method (see Newman and Sclavounos, 1988) and strength analysis based
on a
finite element method.To obtain an exact method with as few geometrical limitations as possible, the method
suggested by van Santen (1986) has been adopted in the present study.
2. HYDROSTATIC CHARACTERISTICS
In order to solve the equations of equilibrium, derived in Appendix A, it is necessary
to determine hydrostatic characteristics, such as volume, volume moments etc.. It is assumed that the geometry of the vessel is modelled bycompartments5 each consisting
of flat panels The characteristics can then be determined by a summation Since
the basic summation procedure is equal for all characteristics, let B symbolise any characteristic. The summatiön is as follows
m n
B=
6B1J. (1)i=J j=I
The number of compartments is m, the number of panels in a compartment is n and B1 is the contribution from the jth panel in the ith compartment. All characteristics are çxpressed as surface integrals
B=
f
B(x,y)dA= f
B(x,y)thdy (2)JA JA
where the position of the global XYZ-system is shown in Fig. 1,.
The basic principle in the present method is analytical integration around the
perimeter of the individual panel Therefore, surface integrals are transformed using 'Greens integratiön theorem
JA \ ay âxJ .L)A
(Pdx±Qdy).
(3)The surface over which to integrate is denoted A and A is the boundary curve of surface A. If all boundaries are straight lines the use of Equation (3) is simple and
analytical expressions for the hydrostatic properties can be obtained.
F1G 2. A panel projected on the water plane.
Two different kinds of hydrostatic properties are needed, namely characteristics
depending on the water plane and characteristics depending on the submerged volume he characteristics depending on the watet plane can be determined as shown by Fig. 2.
The contribution from a single panel
to thç total water plane area is found by
projecting the submerged part of the panel on the water surface (see Fig. 2). The contour around the area A,,, shown in Fig 2, consists of a piecewise O-curve, suchthat Greens theorem [Equation (3)] can be applied As an example let us consider the
moment of inertia, I,, of the water plane area
Ï
J y2dA,
1=1 A1
ydx,
(4)i=i /=1 aA,.1
The contour of A,1 is assumed to consist ofp straight line segments, which means that
y along the segment k can be expressed as
y=ax+3
XE[Xk;xk+l];y[yk;yk1I
(5)where
Yk+lYk
Xk+i Xk
Substituting Equation (5) into (4) yields
'x=
f
(ctx±f3)3thi=1 j=1 JaA11 'xx
Hydrostatic stability calculations by pressure integration 159 n p x =
(J .(ax±
dx) ¡ I j i k lxk ¡j =±
(+ a23x3 + af32x2 + I33x)I1).
¡=1 j=i k=i ¡j
All characteristics involving properties of the water plane area can be found by using this method, and the results are given in Appendix B.
When determining hydrostatic characteristics depending on the submerged volume, it is ùseful to introduce a new system of coordinates X,,Y,,Z,,.
The origin in this system is placed in one of the vertices of the panel, with the
Y,,Z,,-plane in the panel Y,,Z,,-plane and the Xe-axis perpendicular to the panel (see Fig. 3). The global coordinates (x,y,z) can be expressed by the panel coordinates (x,y,z) as
The normal vector to the panel is denoted ñ = (ni,. ns,, n) and the transformation
matrix in Equation (7) is orthogonal.
The pressure P with respect to the free surface at any point (x,y,z) below the free
surface is
P=pgz.
(8)There are two forces acting on the vessel, one the weight of the vessel acting vertically downwards, the other due to the hydrostatic pressure acting on the vessels submerged surface. The incremental färce dF acting on the vessel due to the pressure is
dtpgzñdS
(9) _nynxnz.
-/(n+n) j(1n)
nx n).n.j(n+n) iJ(1n)
OFIG. 3. Panel system vs global system of coordinates.
(6) nx X X0
Y =
Yo + fl Z Z0 nz Xp Yp (7) Zp(10)
(14) where p is the density of the water and g is the acceleration of gravity. By integration
over the submerged surface the total fOrce F = (Fi, F, F) is
F=
1pgzñdS.J5
As the surface consists of flat panels, it is possible to integrate separately on each surface element. This yields
1=1 j=1 A,1
Using the transformation in Equation (7) it is possible to express z as
Z = Z)+1lzXp+\/(1fl)Z. (12)
In the panel system x1, is always situated in the YZ1,-plane, thus
Z = Z+v(1fl)Zp.
(13)As z is a function ofza,, the surface integral in Equation (11) must be converted from
the global system XYZ into the pançl system X)YOZp which can be done using the Jacobian J
,
û(x,y,z)
c3(x,y,z)=1
Equation (11) now becomes
pgzdA11 ¡=1 j=1 A1 ,fl n = I J Pg(z0z +
j(1 n)z) dy
iI j=I
Ajj >ñ,1pg[z(aiy, +
i=l J=I k=I
+
1_n)(a2y + a3zy
± 13 Y)]1Jk where Z,,kf 1ZpAy,,;a -
; 13 = Z/( - YPk. YPk+I YPkThe moment of the submerged volume with respect to z,L, is determined as = n2. J fZz dz dA, ¡=1 1=1 A1 O
nJ
zdy
i=I j=I
aAHydrostatic stability calculations by pressure integration 161
where n.. is the z-component of the normäl vector to the panel.. By use of Equation (13) it is possible to express Equation (16) in a förm similar to the previous equations.
The moment of the submerged volume with respect to x and y are determined in a slightly different way, as L and L can be expressed as
= XObV
L = YCOhV (17)
where (x,y)CØb are the global coordinates of the center of buoyancy. The results are given in Appendix B.
An advantage of the present method is the simple way in which the calculation of damage stability is included. It is simply done by identifying the compartments being damaged. By subdividing the structure into a number of compartments it is possible to model all kinds of damage situations. The contribution from a damaged compartment
is
found by
multiplying the contribution from an intact compartment with(1 permeability).
3. DEFINITION OF GEOMETRY
So far the objects analysed by pressure integration have mostly been box-type
structures, such as semisubmersibles. The reason is that they can usually be subdivided
into a number of simple geometrical elements such as boxes and cylinders This
facilitates the definition of geometry of the structure as it is possible to define these geometrical elements, rotate and move them to different places and make copies and thereby build up vessels piece by piece. This procedure of definition is unsuitable for ships, as the hull cannot be subdivided in the same way The definition of geometry described in the following was originally developed for use in the structural analysß of
ships, built of sandwich elements, but it is equally suited to approximate ship hull structures for hydrostatical calculations by pressure integration. The basis for the
method is the classical definition of the ship form by a number of sections perpendicular
to the longitudinal axis as shown by Fig. 4
These sections can be defined by most of the existing ship form definitiön programs.
In the example to follow they are defined by the programs described in Jensen and Andersen (1986). The ship hull surfáce is approximated by a number of triangles, all with the vertices on the ship hull surface. The hull is in the present case defined as a
single compartment, but when doing damage stability calculations it can be subdivided
X
FIG. 4. Traditional definition of ship hull by sections perpendicular to the longitudinal axis, in this case a
Fio. 5. The rescue vessel modelled by flat triangular panels.
into a number of compartments by watertight bulkheads. The approximation of the
ship surface is done separately for the hull and the superstructure in the following way On a user-selected section the number of tnangles along the section contour is specified,
and this is used to calculate a target value for the side length of all triàngles. With the target value of .the side length in mind corner points for the triangles are generated
automatically on all sections. In the longitudinal direction. the surface is approximated by straight lines between the sections, and the number of triangles between tWO sections
is either calculated from the target side length or user-specified With this information
triangles are generated for both the hull and the superstructure, and afterwards the
stem and stern surfaces are approximated in the same way About 90% of a ship hull surface can be automatically subdivided by this method The remaining part must be manually approximated by flat panels This is done using a general pre-processor
module where the triangles are moved, added or deleted, one by one. Figure 5 shows a rescue vessel approximated by about 1100 panels.
After approximation and storing of the ship surface data the hydrostatic calculation
module has direct access to the data stored in a data base. The advantage of this
integration of structural and hydrostatical analysis is that nearly the same element mesh
can be shared. However, doors, windows etc. described as cut-outs in the structural model must be added in the hydrostatic model Furthermore, it is possible to transfer the hydrostatic pressure on each panel to the structural mOdel.
4. EXAMPLE
The method has been tested on the simplest possible geometrical form, a floating box.
The floating box case offers a direct comparison between the analytical conventional hydrostatic results and those given by the pressure integration technique The box was
rotated through 180° and GZ was calculated by both methods; full agreement was
found.
A more complicated geometry analysed is the vessel RF.2. It is a rescue vessel,
constructed in 1980 for operating in the North Sea Not long after it was launched the
vessel capsized in heavy sea. The crew were lost and the accident initiated a
governmental investigation. In this investigation a GZ-curve of the vessel was calculated
Hydrostatic stability calculations by pressure integration 163
in that interval. The calculation was originally carried out in tWo parts, one for 0-90°
and one for 90-180°. It required two different definitions of the vessel, öne upright and
one upside down. Various problems occured at 90° as it was necessary to do some curve-fitting in order to obtain a continuous curve.
During the
last seven yearsthere have been two additional governmental
investigations of RF 2, so that altogether three different GZ-curves have been produced
Therefore it was natural to choose RF2 for testing the present theory on a real
construction. The vessel geometry used for this analysis is basèd on the model used in the latest investigation, but several changes have been made. Whçn defining the outer geometry by many fiat panels it is possible to obtain a better description of the hull and superstructure, as it is possible to include more details than by merely using theclassical section-by-section definition.
When looking at Fig 5 one sees a chute at the stern, that the stern is not vertical,
and that the front of the wheel house is circular These properties were not previously taken intO accOunt, and of course they affect the final results. On Fig. 6 and in Table
i are shown the GZ-curves calculated by the classical and the present method. One observes that from O to 80° the two curves are almost identical, but from 80 to 180° differences appear The largest difference appears when the vessel is almost turned
upside down. As can be seen by the present method, GZ becomes less than zero
between 170 and 180°. This is in conflict with the results published in the Governmental
Report (1988).
5 CONCLUSION
A theory has been presented for the calculation of hydrostatic characteristics by
pressure integration over the submerged surface. The theory uses a description of the
surface in terms of flat panels with straight edges. Since all expressions for the
hydrostatic characteristics are derived analytically, the solution is exact even for large
angles of rotation and for vessels with complicated geometries There are neither
limitations on the size nor on the number of vertices in a single panel, i.e. there is a
.30 Metres .20 .10 .00 .10 .20 -.30
Ftc. 6. GZcurve calculated using the generated model, shown in Fig. 5. The punctuated curve is taken from the Governmental Report (1988).
.
.00- 45.00 180.00
large degree of freedom when subdividing the geometry of the vessel into plane
elements. Hence, very fine details of the hull or the superstructure can be modelled. The test case considered illustrates the usefulness of the method when calculating
hydrostatic stability, especially because the method offers several advantages compared
to the classical way of calculation when considering geometrical complicated floating
structures.
Acknowledgement-The authors would like to thank J. A. van Santen for the help given during the theoretical developments of the technique described. This paper is a contribution towards a Ph.D degree, financed by
the research program Marine Structures and sponsored by The Danish Technical Research
Council.
REFERENCES
GOVERNMENTAL REPORT 1988. Governmental investigation on the disaster of RF.2. Report No. 1131. The Department of Industry, Copenhagen (in Danish).
JENSEN J J andANDERSEN P 1986 ISH SHIP DESIGN PACKAGE an Interactive preliminary ship design system. Proceedings of CA DM0 86, Washington, DC, September ¡986, pp. 33-43. Springer-Verlag. MCIVER D B TAGGERT S STARSMORE N andLILLYWHITE PA 1983 The development of stability analysis
of semisubmersibles Second SSPA Symposium on Ocean Engineering and Ship Handling Sweden 1983 pp. 363-374. SSPA, Gothenburg.
NEWMAN J N and SCLAVOUNOS P D 1988 The computation of wave loads on large offshore structures
Proceedings of BOSS 88 Trondheim June ¡988 pp 605-622 Tapir Publishers Trondheim
RADWAN A M 1983 A different method to evaluate the intact stability of floating structures Mar Technol 20, 21-25.
SANTEN J A VAN 1986 Stability calculations for jack ups and semi submersibles Proceedings of CA DM0
1986 Washington DC September ¡986 pp 519-550 Springer Verlag
WITZ, J.A. and PATEL, M.H. 1985. A pressure integration technique for hydrostatic analysis. RINA Sup-plementary Papers, 1985, Vol. 127, pp. 285-294. RINA, London.
ZÌENKIEWICZ, O.C. 1985. The Finite Element Method inEngineering Science. McGraw-Hill, New York.
Angles of heel
(deg.) GZ (present method)(m) GZ [GovernmentalReport (1988)]
(m) 0.0 0.000 0.000 10.0 0:110 0.106 20.0 0.160 0.153 30.0 0.183 0.173 40.0 0.173 0.167 50.0 0.114 0.111 60.0 0.042 0.042 70.0 -0.041 -0.043 80.0 -0.124 -0.118 90.0 -0.143 -0.123 100.0 -0.159 -0.163 110.0 -0.157 -0.165 120.0 -0.128 -0.139 130.0 - 0.060 -0.074 140.0 0.018 0.005 150.0 0.Ó40 0.043 160.0 0.030 0.043 170.0 0.00 1 0.014 180.0 0.000 0.000
Hydrostatic stability calculations by pressure integration 165
APPENDIX A
Determination of, the position of equilibriüm for a rigid body, floating in a calm sea, requires information about its weight, center of gravity and its geometry.
The system of coordinates is chosen such that the X1-axis is parallel to the longitudinal axis, if any. But, the orientation of the input system of coordinates, XYZI, is of no importance for the calculations of equilibrium itself. In Fig. 7 the iñput system XYZ is compared with the global system of coordinates XYZ. The global system, XYZ, has the XY-plane situated in the
still water level and the Z-axis pointing out of the water. The global coordinates can be expressed
with respect to the input system as
x = x1cosO +y,sinOsini - z1,sinOcos,
y=
z = x1sinO y,cosOsin + zcosOcos4i T. (18)
The angles of heel and trim are denoted O and IJJ, respectively. The draught, T, is the translation
of the system in the direction of the Z-axis, but does not necessarily express the actual draught. The draught T depends on the chosen reference point, and therefore it can become negative.
The position of equilibrium is obtained from the three conditions for equilibrium
F1MpV =0
F2 xgMpx = 0 (19) F3 ygMpLy = O where L f L). f ydV l- = f zdV. JV iv ivThe resulting force in the direction of the Z-axis is denoted F1 and the resulting moments
around the X- and Y-axis are denoted F2 and F3 respectively. The coordinates of the center of gravity, (x y)g are with respect to the global system The submerged volume is denoted V and
the volume moments are L, L) and L Since it is assumed that the weight and the center of gravity are known, the conditions of Equation (19) are functions of (T,O i1) alone and assuming F1, F2 and F differential, a Taylor senes around an arbitrary position (T,O,IJ)() yields
FIG. 7. Definition of draught T, trim O and heel ,. Global system XYZ, input system XYZ,,J,
F1(T,0,,)= Fj±07
+
(Ji -)
(20)i 1,2,3
truncated after the first-order terms. The notation means calculation at (T,04)1). In the
following this notation is ignored to simplify the expressions The first estimate (T O 1i), will normally not satisfy the conditions in Equation (19) but using Equation (20) a better estimate, (T,0,41)1, can be found by solving the equation
0V
= sxÇosO
whereas G(x,y1z-) x gives: aL 0T aL1 00 OLX !1cos0+Lsin0 (TT1'1) + (0-00)
The position of equilibrium is found iteratively using Equation (21).
To determine the denvatives of Equation (19) the integral J over the submerged volume V of an arbitrary function G(x,y,z) is anaIysed
j
=fJf
G(x,y,z) dV=Jf J G(x,y,z)dx dy dz. (22)
The integration over J goes from the oûter surface, z = z(x,y) of the vessel, to the still water
level, z = O
if
[jrny
dz]dx dy. (23)
The water plane, A, is defined as the area of the part of the plane Z = O which is inside the body.
If Equation (22) is differentiated with respect to a (T,0,i) the result is
OaIIA{oa')]Y
JJJ0Q'Y'dV.
By setting G(x,y,z) 1, the Equation (24) gives the following results
8V A 0V aO_ sy
JLTV
(26) 8F1 8F1 8F -(T1T0)(0 0)
IJo) = - F1F2
F (21) aT 00 8F2 0F2 8F2 aT 80 8F 8F1 0F1 .OT 80- pScos0
psy
Hydrostatic stability caiculätions by pressure integration 167
'n
=;
((2+13x)Ixk+I)
jI k=I
'JThe water plane área moments and S
s=Ï
JYdAIJ ¡=1 ¡=I Ajj ( !(a2x3+a13x2+2x)rk+1) ¡=1 j=1 k=I i; AH,= : dA1 j=11A11 Pl? I?and Gx,y,z) = y gives:
aT s (27)
I
00
= 1,.cos8 - Lsin0+L cos0+TVcosO.
Combining Equations (25), (26) and (27) with the derivatives of Equation (19) the coefficients in Equation (21) can be expressed as
0F1
pA.
0F1--=
psy 0F1 a") 0F2 aT 0F2 00 (Zg+T)M+p(1+L+ TV) f_2 = ygMsinop(Ix,.cos0+L3sin0) (28) ¿3F psX 0F3 = PJ!XY = M(xgsinø+zgcos0+Tcos0)- p(Icos9 - Lsin0+L2cosO+TVcosO).
APPENDIX B
In this appendix all the hydrostatic characteristics are given. The waterplane area A. is calculated as
Ixy = in = ±1 y2dA j=1 A.1
(:
a3x4+a23x3+j=I k!
1=
x2d.A i=1 j=i A1 ( .a3y4+a23y3+ a132x2+133x)Il). i= ¡=1 kI 'IThe mixed moment of the water plane area
ÎfxYdAii
i=I J=iAq =± (
(ia2x4+af3x3+2x2)Irk+I)¡=1 jI k1
'i In all cases: Yk+1Yky=c.x+;a=
;13=yk-cY-rk. Xk+1 XkThe moments of the submerged volume, L and L, are calculated as L = XC(,hV
L3. = YC0hV
where the center of buoyancy, (X.Y,Z)Ch is determined as follows.
First the point of action of the hydrostatic force is determined för each panel and is then transformed to the global system of coordinates. In order to find the point of action, it is
necessary to evaluate the moment around the Ya,,- and Zr-axis (see Fig. 3). The moment with respect toY,, is
p
M=
JzjdF
= (31) (32)+ hj(1n)
(36)The moment with respect to the X,, axis is
M =
2
dF xdA,1 1j1
A.1 = ( a2y3+a43y2+132y)Ik+1)iI j=I
k=I 'jThe moments of inertia I,. and 1. of thewater plane area
Ypa = Zpa = Mi,, F M F
Hydrostatic stability calculations by pressure integration 169
= pg
{zo)(ay
+ yp) + (.aL» ++
( i n) [y)(a2iy +ay +
I32y) (37)±
}k
Zpk IZPk
=
zay; 4y =
YPk+IYPkYPk+I YPk
The point of action for the hydrostatic pressure force F in the panel system, denoted (x,y,z pa' can then be determined as
xpu = O
As the coordinates (X,Y,Z)PU are transformed into global coordinates, denoted (x,y,z)ga, it is possible to determine the global moments
-"! (FzXga + Fxzga)q ¿=1 j=1 (Fzyga - Fvzga)ii i=I j1 flZq
f
fzz dzdAq ¿=1 ¡=1 A- (1 m n p = > nzii i=I j1 k=I+ À(1 n) (ai
+a2y3 +a32y + 1331Yp)}1Jk.The center of buoyancy can now be determined as M Xc(,l, =
