OFFSHORE TECI'iOLOGY CONFERENCE 6200 North Central Expressway Dallas, Texas 75206
TEES IS A PREPRINT --- SUDJECT TO CORRECTION
\1. 4
MUSAS:
A Motion and Strength Analysis System for
Semisubmensihie Units and Floating Structures
By
G. N. C. van Oustal, D. Hans, J. W. Salomons, and J. A. van der Vlies, Koninklijke/Sheil Exploratie en Produktie Laboratorium
Copyright 1974
Offshore Technology Conference on behalf of the American Institute cf Mining, Metallurgical, and Petroleum Engineers, Inc. (Society of Mining Engineers, The ivIetal1urgical Society and Society of Petroleum Engineers), American Association of Petroleum Geologists, American Institute of Chemical Engineers, American Society of Civil Engineers, American Societyof Mechanical Engineers, Institute of Electrica.l and Electronics Engineers, Marine Technology Society, Society of Exploration
Geophysicists, and Society of Naval Architects and Marine Engineers.
This paper was prepared for presentation at the Sixth Annual Offshore Technology Conference to be held in Houston, Te:'f,, May 6-8, l974. Permission to copy is restricted to an abstract of
not more than 300 words. Illustrations may not he copied. Such use of an abstract should contain conspicuous acknosledgment of where and by whom the paper is Dresented.
PAPER
flTC'
rNUE'3ER L) I L' C.
ABSTRACT anchoring computations, the program is used to
determine the influence of vertical motions on the
A realistic approach to analysing a floating unit line tensions. is to elaborate statistically motion and
stress-transfer functions for regular waves, using measured wave-energy spectra. Accurate determination of the transfer functions is therefore of utmost importance. This paper presents a computer program in which all necessary computations have been integrated into one all-embracing system. For reasons of
efficiency, the program is divided into two parts. The motions part is comprised of the calculation of ali quantities of the motion equations and the solutior thereof. In the second part, all loading data and structural data are collected, after which a space-frame analysis program is applied to determine
stresse S.
The paper elucidates the structure of the program and its theoretical backgrounds, of which the use of potential theory for computing wave forces is
perhaps the most important.
The results of the program are illustrated by two examples. For the unit STAFLO', the computed
added-mass terms, wave forces and motions are compared with the outcome of model experiments. For a unit of the 'SEDCO-700' type, the motion-transfer as well as some stress-motion-transfer functions are given. As an example of the possih!ities of
The MOSAS' program has proved to be capable of quick and effective analysis. Contrary to previous methods, data handling is restricted to a minimum by complete integration of all computations. This approach reduces the possibility of errors and saves
a considerable amount of time.
i. INTRODUCTION
The analysis of motions and stresses for tloating units requires a large number of calculations, In view of their complexity, some of these tasks have gradually been computerised. Still, the use of individual programs involves a large amount of data
handling and processing, which in fact means oor
use of the possibilities of a computer and
unnecessary risk of making essential mistakes. n the 'MOSAS' program all computations have been
brought together into one single ailmbrang
system. The amount of input is restricted ro a mini
mum and the transfer of data between oucrcaive calculations is completely computerised. e oGiut is concise, but the storage of all results on disk
stili leaves the possibility of more deid riciysis.
Programming was performed in FOJiGG
'romotes the implementation of the p. rn r) rnc's
computers of sufficient memory cap References and illustrations at end of paper.
'MOSAS': A MOTION AND STRENGTH ANALYSIS SYSTEM FOR
722
SEMI-SUBMERSiBLE UMTS AND FLOATING STRUCTURES
'MOSASt is subdivided into two sections: analysis cf
the motions and subsequently of the stresses, both for regular long-crested waves. The first step
con-sists in the computation of the static and dynamic properties of the unit, external forces due to wind,
current and waves, and the motions for all six degrees of freedom: surge, sway, heave, roll, pitch and yaw. The effect of anchoring can also be taken into account. In the second step the tensional and bending stresses in each member are computed. Apart from some input and output subroutines, the
tress -analysis part basically comprises the existing 'ICES STRUDL' system.
The subdivision described above seemed
indicated from an efficiency point of view. The stress computations are about 40 - 50 times
more expensive than the motion computations. t. is therefore worthwhile checking the soudness
of the motion results. This approach does not
involve any serious complication for the user,
.ecause the transfer of data between both parts
remains fully computerised.
The computational methods are up -to -date as
much as possible, the features being the use
of potential theory for computing the wave
'orces and a direct proportionality between the amplitude of the waves and the amplitudes of
the resulting motions and stresses. Owing to
this linearity, it
is possible to carry out a fully
robabilistic analysis of motions and stresses in irregular seas.
'his paper gives a description of the 'MOSAS' systemB nd its possible application. Two examples will be resented. The first example gives a comparison etween the computed added mass, wave forces and
otions for the semi-submersible 'STAFLO', and he results of model experiments. The second
xample gives motion and stress-transfer functions ertaining to a unit of the SEDCO-700 series.
ppraisal of motions and stresses of these units re not dealt with in this paper. Further application f transfer functions e.g. for long- and short-term
atistics and for fatigue calculations will only be ouched upon briefly.
2. DESCRIPTION OF THE PROGRAM
simple diagram of the system is shown in Fig. 1. he main characteristics have already been indicated eparate motion and stress-analysis parts and
utomatic transfer and storage of data.
n the present version, the program can cope with inits consisting of at most 160 members and i 50 joints. The required memory capacity for the
otion part is then about 300K. Also the stress art requires a capacity of about 300 K
both parts are built up from a number of subrou-tines, each constituting a 'rounded-off' element in
3. the sequence of computations. The connection betwee
1
7I
QOO
the subroutines and their function and con
outlined below.
-,- _) z)
'
OMotion analysis . n.2
In this part of the program the six coupled
equations of motion are constructed and subsequent1 solved. In vector and matrix notation the equations read
(A1+A2)5+B+Cx+R()=F
wa wi Cu
+F.+F
(1)where x = (x1, x2, x3, x4, x5, x6) represents the six components of the motions of the centre of gravity, i. e. three translations and three rotations in the x, y and z directions. The first and second time derivatives * and 5 are the components of
velocity and acceleration.
Similarly:
= wave forces and moments wind forces and moments = current forces and moments
(restoring) anchoring forces and moments
F
wa
F.
-Wi Feu
and A1 A2 CA flow chart of the mOtions program is shown in
Fig.2a. Before entering into further detail, we
would like to make the following general remarks. Owing to the non-linearity of the anchoring forces (the forces increase progressively with motion), the equations of motion are also non-linear. This necessitates an iterative solution method by which motions and anchoring forces are computed simultaneously.
1.
2.
inherent-mass matrix (mass and mass moments of inertia)
= added-mass matrix (added mass, added-mass moments of inertia and cross-coupling terms) (linearised) damping matrix
= matrix of natural restoring spring coefficient
There is an iterative loop between 'HYSTA' and 'ANCMOT' for assessing the initial position of the unit, i. e. the static offsets in longitudinal and transverse direction, the exact draught of the unit and the static rotations. Owing to an inaccurate estimate of the draught, there will generally exist an unbalance between weight and buoyancy forces. The program will then compute the initial position, but simultaneously the pre-viously determined matrix C will have changed. A reiterative procedure is necessary to arrive
at a correct initiai position and matrix C.
Loop III allows the computatioìfs to be performed over any desired range of wave angle of attack
V.
ÒTC 2105 D. HANS, G.H.C. VAN OPSTAL, J.W. 4. So far the effect of damping has been lincarisedand the coefficients of the matrix B are not automatically generated by the program. There are two contributions to damping:
- potential damping, i.e. the energy which is taken away by radiation of waves, and - viscous damping.
Although potential damping may, in principle, be determined by analytic methods, its magnitude is usually of minor importance. On the other hand, computation of viscous damping is rather difficult. At present the damping matrix, whether based on experience or experimental science, is simply added to the input. In doing so, one should remember that damping is usually only important at resonance.
Slightly away from the natural periods, its influence is hardly perceptible.
The part of the output which is printed has been restricted to the following items:
- properties of the unit such as displacement, positions of the centres of gravity and buoyancy, longitudinal and transverse metacentric heights, the matrices A1, A2, C and the natural periods, - wave forces and moments,
- static and dynamic displacements (transfer
functions),
- a check on the equilibrium of the computed static and dynamic forces and moments,
anchoring characteristics: maximum values of line tension and scope.
ata concerning the distribution of forces over the embers and joints of the unit are stored on disk. escription of the subroutines
Input and conversion: 'INACO'
This subroutine stores all the input data necessary for the other elements of the program and, if required, converts them so that automatic transfer to these elements is possible. These data consist mainly of joint co-ordinates, member incidences, member properties, estimated draughts, anchoring
arameters, wave properties (frequency and amplitude), wind and current properties (velocity), angles of attack, water depths, added mass and damping coefficients. The input allows beam,
circular, elliptical and rectangular members to be
roces sed.
2. Mass distribution: 'MASDIS'
e mass matrix required for the equations of otion is defined in the mass-distribution subrouline.
e weight of the unit can be specified as equivalent vaIl thicknesses, i.e. the actual wall thickness plus
n addition to account for stiffeners and the like, or
SALOMONS AND J,A. VAN DER VLIES
as concentrated weights at joints. Accordingly, ti weights are converted into uniformly distributed concentrated loads. The centre of gravity is alsc computed and serves as the origin of a right-han co-ordinate system in the other subroutines.
Hydrostatics: 'HYSTA'
The subroutine 'HYSTA' computes the buoyancy forces and converts them into uniformly distributed member loads. Also computed are the centre of buoyancy, the coefficients of the matrix C and, as a measure of initial stability, the MG values.
Constant wind: 'CGWIN'
The wind forces are computed in accordance with the ABS rules for building and classing offshore mobile drilling units, but only for those members which are represented in the structural configura-tion. Forces and moments acting on other elements have to be calculated by hand and have to be
specified in the input.
Constant current: 'COCUR'
Only permanent currents are considered and the velocity is assumed to be constant with depth. The forces are computed with the conventional drag
formula.
* Added mass: 'ADMAS'
The input for the computation of the added-mass matrix A2 consists of added-mass coefficients for each member in its local principal directions. A
co-ordinate transformation, an outline of which is
given in Appendix A, is applied to convert local added masses into the global co-ordinate system.
The total added-mass matrix is approximated by
summing the contributions of the individual elements 7. Wave forces: 'WAVE'*
This subroutine computes forces and moments due to regular long-crested waves. A brief outline of
our approach is given in Appendix B. There are two contributions to the wave forces:
- a contribution from the pressure distribution over the hull of the body, and
- In both subroutines each member is treated as if all other
members are absent, neglecting possible interactions between adjacent members. It is hardly possible to take those
inter-actions into account, but to some extent this can be done by
adjusting the 'undisturbed' added-mass coefficients (see
Appendix A). 723 ie or de a *
Note on subroutines 'ADMAS' and 'WAVE'
- In Appendices A and B it is shown that for an accurate com-putation of added mass and wave forces the members of the underwater structure need to he subdivided into elements which have dimensions smaller than one-fifth of the wave length.
V
MOSAS': A MOTION AND STRENGTH ANALYSIS SYSTEM FOR72l SEMI- SUBMERSIBLE UNITS AND FLOATING STRUCTURES OTC 2105
unit ìs represented by a (supported) space frame, the damping forces will simply show up as. reaction forces at the supports. (see Section 'EXAMPLE). The hydrodynamic forces related to the added mass are computed in the same manner as described above under 'INERT'.
Strenh analysis
Owing to the linear-wave concept, and hence the proportionality of force amplitude with wave am-plitude, each individual member force, either
concentrated or uniformly distributed, can be characterised by its amplitude F and the difference in phase a with the indicent wave
F = P cos (wt + a)
The forces can also be divided into two components
F = F.cosWt + F sinwti o
the first of which is in phase with the waves and the second is 90 degrees out of phase with the
waves. Similarly, the total load system can be divided into two parts. It will then be clear that-if each of these two load system.s is separately
applied in 'STRUDL', the resulting stresses are also built up from in-phase and out-of-phase parts
a = a. cos ait ± cy0sinWt or analagously
= a cos (Uit +
being the (tensional or bending) stress amplitude and B the phase difference.
Apparently only two 'STRL'DL computations are sufficient to determine the entire stress picture during one wave period. It should be recalled that, when using a non-linear wave concept, the number of stress computations is much larger. The wave profile is stepped through the structure and forces arid stresses have to be computed for a number of positions of the wave.
The above-described procedure is repeated for all selected combinations of wave frequency and angle
of incidence. Finally, one additional 'STRUDL' run is made, applying the static load system.
The printed part of the output consists of the static and dynamic stresses for both ends of each member,
i.e. at the joints construed from the centerlines of intersecting members.
Description of the subroutines
A flow chart of the strength analysis part is shown
in Fig. 2h.
a contribution due to diffraction of the waves, which can be calculated by the acceleration of the added mass and the velocity of damping.
With the forces on each member known, the total I'orces and moments on the unit evidently follow from summation.
8 Anchoring and motions: 'ANCMOT'
The subroutine 'ANCMOT' computes the motions. There is an option for anchored and unanchored
units. If the unit is anchored, it is assumed that
the anchor lines have conventional non-linear catenary characteristics and are tangential to the sea bottom. The elastic stretch of the lines has been taken into account.
The equations of motion are non-linear. Consequent-ly, the motion response shows, apart from a
component of the same frequency as the waves, also components of lower or higher frequencies. However, it appears that the latter components are in general extremely small and can completely be neglected. In the stress analysis, the
non-linearity of the individual anchor line forces has t.o be accepted as a fact. As this analysis is
based on a presumed proportionality between force amplitude and wave amplitude, the anchor forces have been linearised. The forces are approximated by the static (mean) component and the dynamic
component of the same frequency as the waves.
9. Natural restoring forces: 'BUOYDY'
In the subroutine 'BUOYDY' the natural restoring forces [Cx in equation (1)1 are distributed over the members which pierce the still-water level. Their point of attack shifts continuously along the length of the periodically submerged and emerged parts of the members. In the program., however, it has been assumed that the forces act at the intersections with the still-water line.
i D. Inertia forces inherent mass: 'INERT' The inertia-force subroutine requires output in-formation from the motion subroutine, i.e. linear and rotational accelerations. Forces are computed for a fixed maximum increment length, which is generally less than the actual member length. The inertia forces are represented by uniformly dis-tributed member loads or, if a weight is specified that way, by concentrated point loads.
i I. Hydrodynarnie forces: 'ITYDYN'
The hydrodynamic forces result from the accelera-tion of the added mass and the velocity of damping
[A2 and Bi in equation (1)1.
As the effect of damping has been considered only for the unit as a whole, it is not possible to
distribute these forces over the members. As a consequence, in the stress-analysis part, where the
2105 D. HANS, G.H.C. VAN OPSTAL, J.W. SALOI'NS AND J.A. VAN DER VLIES 725
Data collection and conversion: 'DACAC'
All structural and loading data are brough together and arranged to an input stream for the 'STRUDL' program. Two types of loading cases are set Up: - static-load data, which class weight, buoyancy,
wind, current, natural restoring forces and anchoring forces, and,
- dynamic-load data, which class the in-phase and out-of-phase components of wave, inertia, hydro-dynamic, natural restoring and anchoring forces.
Structural analysis: 'ICES-STRUDL II'
The 'STRUDL' program carries out a conventional space-frame analysis. As the transfer of data between the motions and stress parts is computer-ised, the user has only to specify the runs to be made e.g. static analysis or a dynamic analysis using load data pertaining to specified combinations af wave frequency and angle of incidence. Individual analyses, for instance for wind, current and
restoring forces only can be carried out as well. Data processing
Two subroutines have been developed for editing the utput of the stress computations.
The subroutine 'FILE' serves to re-arrange the output of 'STRUDL' and to restrict it to the most relevant items only. For instance, the stresses are conveniently arranged in tables. The subroutine 'PROC' selects all members of which the static and dynamic stresses exceed certain specified levels. The in-phase and out-of-phase parts of the stresses are combined to stress amplitudes and phase angles, which are displayed, together with the appurtenant static
stresses, either in tables or in graphs
(stress-transfer functions). (See also Section 'EXAMPLET).
COMPARISON BETWEEN COMPUTED AND
MEASURED DATA
The approximative methods of computing added mass md wave forces (Appendices A and B) appear to be
airly accurate. To illustrate this, a comparison vill be made between computed data and the results
rom modelS experiments for the semi-submersible STAFLO'. The experiments relate to a water depth
f 40m.
The added-mass terms are pictured in Fig. 3. The suffixes are illustrated on Fig. 7. It appears that i
general the computations correspond fairly well with he measurements. Also the presumed independence rrom wave frequency seems to be justified. Only at [ow frequencies is the discrepancy sometimes rathei
arge. As in this range the wave lengths become relatively large compared with the dimensions of the
ank, the measurements could be influenced by
wall effects.
A comparison between the computed and measured wave forces and phase angles is given in
Figs.4 - 5. lt can be seen that the agreement is
good.
If the computed added-mass terms and the wave forces and moments are close to the experimental
data, it may be expected that the same holds for the motions. A comparison is given in Fig. 6.
Contrary to what has been stated above, the measurements and computations refer to a water depth of 30m, instead of 40m. It is emphasised that the figure shows the motions at the rotary table and thus represents the combined effect of linear and rotational motions. Again agreement is
good.
Finally, it should he mentioned that similar com-parisons have also been made for multi-column type units, which invariably lead to good results.
EXAMPLE: A SEDCO-700 TYPE
SEMI-SUB MERS1BLE
Notes on the input of the program
The idealisation of the structure into a space-frame is shown in Fig. 7. The extra joints and
members have been introduced to obtain a more detailed stress picture. To restore the equilibrium of forces and moments which, apart from
inevitable inaccuracies in the computation, has been wittingly disturbed by the neglection of the damping forces, the unit is fictitiously supported at three points. The effect of the reaction forces is minimised by taking appropriate locations of the supports, in the present case at the vertical columns (see Fig. 7).
For the weight distribution, use has been made of an available simplified model in which all weights were represented by point loads at the joints. Although this distribution is rather coarse, it is not expected that a more detailed weight description will affect the present results essentially.
Based on experimental data for units of comparable geometry, the following damping coefficients have been assumed
- surge, sway and heave: 0. 1 pv - roll pitch yaw 0.015 0. 020 0.015 pv L2
Current and wind act in longitudinal direction, the velocities are 2. 5 knots and 53 mph respectively. The anchoring system consists of eight 3-in chains, two at each corner making angles of 22. 5 and 45 degrees vith longitudinal direction. It is stressed
In particular its integrated structure saves a considerable amount of effort and time. NOMENCLATURE
inherent-mass matrix of total unit added-mass matrix of total unit member radius
added-mass matrix of a member acceleration of water particles damping matrix of total unit damping matrix of a member
matrix of restoring spring constants
vis cous -drag coefficient
virtualmass coefficient
force
force amplitude
F current forces and moments
Cu
F., F in-phase and out-of-phase force components
F wave force in j-th direction
pine inertia forces and moments
hyd hydrodynamic forces and moments
L'. gene mused direction cosine
F wave forces and moments - Wa
F wind forces and moments
\V1
acceleration of gravity
h wave amplitude H wave height
amplitudes of wave moments in the x, y and z directions
L length of unit
R( restoring anchor forces and moments
S surface area of a body
amplitude of oscillation in j-th direction
T wave period
's velocity of wave particles
x motion vector of centre of gravity x,y,z amplitudes of surge, sway and heave
amplitudes of longitudinal, transverse and vertical wave forces
a, ft
phase differences of forces and stresses,a. phase differences of wave forces in the
x, y and z directions
phase differences of wave moments in the x, y and z directions
A1 A2 a a.. 'j a. B b.. 'j C Cd C m F F
phase differences of surge, sway and heave phase differences of roll, pitch and yaw wave direction
p fluid density stress
stress amplitude
in-phase and out-of-phase stress
components
C4D, 6, amplitudes of rotations around the x, y and z axes
ai wave frequency y water displacement
REFERENCES
Ochi, M.K. & Bolton, Miss W. E. , Statistics for prediction of ship performance in a seaway. International Shipbuilding Progress, 20
(Feb./April/Sept., 1973), Nos 222, 224, 229.
Hans, D., \Tisser, W. & Zunderdorp, H.J.,
The stress analysis of tubular joints for offshore structure s.
Second Annual European Meeting of the Society of Petroleum Engineers of AIME, (1973), Paper SPE 4342.
Visser, W. , On the structural design of tubular
joints.
Offshore Technology Conference (1974),
Paper No OTC 2117.
Strating, J. , Fatigue and stochastic loadings.
Doctoral thesis from Delft Technical University,
f973.
Wendel, K. , Hydraudynamische Massen und
hydraudynamisehe Massentrgheits momente.
Jahrbuch Schiff sbaute chnische s Gesellschaft,
44. (1950).
G. Kochin, N.J., Kibel, l.A. & Rose, N.W. Theoretische Hydraumechanik I. Berlin, Akademie-Verlag, 1954.
Kennard, E. H. , Irrotational flow of frictionless
fluids, mostly of invariable density.
David Taylor Model Basin, Report 2299 (1967). Yamamoto, Y.. On the oscillating body below the water.
Journal of Zoseri Kiokai (1955). boIt, J. P., Hydrodynamic aspects submersible platforms.
Doctoral thesis from Delft Technical University, 1972.
of
semi-2105 .D. HANS, G.il.C. VAN OPSTAL. J.W. SALOMONS AND J.A. VAN DER VLIES
r
'MOSAS': A MOTION AND STRENGTH ANALYSIS SYSTEM FORSEMI-SUBMERSIBLE UNITS AND FLOANG STRUCTUflES OTC 2105 APPENDIX A
Added mass
For an accurate computation of the wave forces most members are longitudinally divided into a number of short sections (see Appendix B). The
subroutine 'ADMAS' makes use of the same sub-division and computes for each section the added masses in the three principal directions. By a co-ordinate transformation these are converted into the directions of the global co-ordinate system. Finally, the added-mass matrix for the total unit is
composed of all individual contributions.
The basic input consists of added-mass coefficients for all underwater members. For most member shapes, these can he found in the literature5'6'7. In general, the coefficients also depend on frequency and the distances between the member and the ,sea bottom and water surface. Several publications 8
indicate that generally, also in our case, such effectsmay, however, he neglected. Another point is the possible interference between adjacent members. Although also this effect is generally negligible, appropriate adjustment cf the 'undis-turbed' added -mass coefficients is sometimes
indicated7.
The added masses of each member or section thereof can be represented by a tensor Sjj. Any relevant textbook shows that then the transforma-tion from a (local) co-ordinate system i, j, k into
a (global) co-ordinate system i' j 'k' reads
3 3
a1,., =
E E cos (i, i') cos (j, j') a1..i=1
j=l
Ji'=l
j'=l
where the cosines refer to the angles between the old and new axes. The local added masses ajj pertain to the principal directions and therefore
= O for i j. On the other hand, as the direc-tions i'j' k' will generally not he the same as
ijk, the matrix a t t is fully populated.
Finally, the added masses ajejt äre transposed to the origin of the global co-ordinate system. As a consequence, added-mass moments of inertia and related cross-coupling ternis will show up. The total matrix results by summing all individual
contributions.
APPENDIX B
Wave forces
The use of potential theory is a linear concept: the waves and the resulting forces are sinusoidal. The amplitudes of the wave forces are directly propertional to the amplitude of the wave.
The theory starts from an ideal fluid, i. e. non-viscous and irrotational. Mainly inertia forces are
important and viscous-drag forces are completely
neglecte'. However, the ratio between wave
am-plitude (h) and member radius (a) and that between water depth (d) and wave length (X) should then fall within certain ranges. This can be illustrated with the help of Morison's well-known formula assuming a realistic ratio between the virtual-mass coefficient
Cm and the viscous-drag coefficient Cd (see Fig. 13) The physically possible maximum wave amplitude amounts to about 0. 05. Thus
H/a 0. 05 X/a = 0. 05
As an example, the above relation has been drawn in Fig. 13, assuming a water depth of 120m and member diameters of 1. 5, 3, 6 and 9m. It can be seen that only a combination of small diameter and large wave height can lead to inaccurate results.
Even then, the errors are restricted to these
par-ticular members. For instance, the underwater structure of a semi-submersible is mainly made up of large-diameter members such as columns, torpedos or footings. Hence, for the miii as a whol the inertia forces invariably dominate.
There are two contributions to the wave forces. The first is related to the potential of the
uridis-turbed waves , the so-ca]led Froude-Krilc,ff force. The second contribution results from the
diffraction of the waves by the body and is related to a potential d. From the theory it follows that for a submerged body with surface area S, the component of the force in the j-th principal
direction F. = pw ]$ ( + ) f dS j = 1, 6 (B-l) where = -1 p = fluid density w wave frequency
f. = the generalised direction cosine of an element of S
As is known, the computation of the Froude-Kriloff force involves no serious problems. On the other hand, l?,-J is known for some simple
geometries only. However, it can be shown that there exists a relation between 1d on the one hand, and added mass and damping for the body when oscillating in the absence of waves, on the other.
This leads to
3
F. = îpw j-j f. dS - - (y. f.) dS (B-2)
S S '
where
co . = the velocity potential of waves generated by
the oscillation of the body in the j-th direction s. = the amplitude of the oscillation in the j-th
y. = the vIocity of the fluid particles in the
undisturbed wave
The added mass and damping of the oscillating body are
a.. = -
im [=- j'j cp fcl'
oi i J 'Jis
andb..=2Re[
jjcp
f dSl 'J w i sThe above relations can be evaluated for bodies that are small, compared with wave length9 (see later in this appendix). In that case the velocity v may be considered to be constant over the surface S. In addition, the moments can be neglected. The
following is therefore restricted to the forces in the direction i = 1, 2, 3 only. After some calculations, one gets 1) C 21«5 = O 'J = O and thus
D. HANS, G. H. C. VAN OPSTAL, J. W. SALOMONS AND J. A. VAN DER \TLIES
It sho1he recalled that, as the velocity of the
fluid p.articles have been assumed to be constant over tile surface of the body, the relations are only valid for small' members. As to the longitudinal direction, this restriction can be easily avoided. In the program long members are divided into a number of sections and the forces are computed for each of those parts. Forces acting on connection pianes are omitted automatically and only forces on real end planes are accounted for. A similar method is, in principle, possible for the transverse directions. However, in general this will not be necessary because the cross-sectional dimensions are usually much smaller than the wave length. A criterion for this can be assessed by comparing the approximative solution of equation (B-9) with known exact solutions. We have investigated the case of a vertical circular cylinder, both analytically and experimentally. Figure 14 gives a comparison of the horizontal wave forces. It can be observed that there is a good agreement between experimental and 'exact' data. The approximative solution is fairly accurate* for ka 0. 6. The point of applica-tion of the force is pictured in Fig. 15. In general, agreement is reasonable. There is considerably more scatter, however. This is presumably due to inaccurate measurement of the overturning moment, from which the points of application have been
derived.
From the results it can be concluded that the present approach is sufficiently accurate for
ka 0. 6. With k = 2rr/X, the cross-sectional dimensions of a member (i. e. 2a) should thus be less than approximately one-fifth of the wave length. The extent of proportionality between the amplitude of the wave forces and the amplitude of the wave -or, for sinusoidal waves also the height of the wave - is illustrated in Fig. 16. It shows the hori-zontal wave force on a single pile extending to the sea bottom as a function of the wave height. The ratio of pile radius and water depth a/d = 0. 025. The results for other ratios, as high as 0. 08, show a similar picture. If the proportionality indeed exists, tac relation between force and wave height should be represented by a straight line with a slope of 45 degrees. It can be seen that this is
the case and that th.e approximative solutions are almost perfect regressions of the measurements. Quantitatively, an a/d ratio of 0. 025 is, for instance equivalent to a pile with a radius of 3m in 120m deep water. For this example Fig. 16 shows that there still exists proportionality for wave periods between 10 and 3.8 seconds in combination with wave heights between 15 and 38m.
if
j / j
± hj) (B-6)
y represents the displacement of the body. The coupling terms in and are zero for
symmetrica1 bodies. For the small bodies which are considered here, they may also be neglected in case of slight asymmetry. Then
F. = y. [ìw (pv + a..) + b..) j = 1, 3 (B-7)
I)enoting the acceleration of the water particles in the undisturbed wave by ai, so that y. = a./ÎuJ, it follows that
F. = a. (pv + a..) + y. b.. i = 1, 3 (B-8)
J J Ji J Ji
In general, the contribution due to damping is of minor importance, so ultimately
Fi = a (pv + a..) i = 1, 3 (B-9)
729
*
Difference between exact and approximate solutions 1es than five percent. s a ad
-.p-s.sj
3 E 1=1 (v.f.) dS =11 3 (Ç' 3 ..2_.-
E y. . f. dS E y. (1w a.. + s. i=1 i ,J .oJ i . 1=1 1 1) J Sîpw$
dS = îpw V V (B-5)J Stop
j
Start
Prit, t jStop
CoIL
Function Card-input dato Motion - and force - computations, storage of results Write motions ond forces Disk-input dato Stress computation storoge of results Write member stresses
Fig.
I
- MOSAS main program.
No
'I naco' 'M and in 'Cocu r' Ad mas
<'Wove 'if Yes <Buoydy' 'inert'
K'HY'
enerotion of B A1,centre of gravity Displacement, centre of buoyoncy, MG-values, C
fou A2 E Drought R (xi X X X Cx Ejne A1j fhyd Fig. 2a - Motion part. Fig. 2b
- Strs'
part. k--1.0 0.0 2 0 x 10' k 00 Axx 0.I *
*t_
',.
. 'i0'kgrn xi0 kgm 0.0 0.0 00 -1.0 I t x10 kgm 1.0 A02 M.asi.re4 Ctc1ted t t 1.0 21) W rd,Fig. 3 Comparison between caiculated and measured added mass terms for a model of drilling unit ISTAFLOr?.
lo-. 0.Ó 1.0 05 2.0'-00 -2.0 2.0-' to-Azz 0.0 -2.0 -1_o -20-0.0 5.0 0.0 -5.0 0.0 Az, I e. i 10 kgm2 x10km r-A _ _______________ xiS kgn A9 '10' kgm x1O' kgrn2
A1
xlO' kg 2 .0 1.0 OES Axi 0.0 I I L e 10' kgm 2D A29 0.0 I t t -21) OESr
1.0 Axo W rd,t 1.0 21) W rdh 0.0 i to 08 12 Wad/s K 101 Wn/m 100 50 00 o 04 08 12 Wad/s z (deç 360 270 180 90 o 350 270 180 90 04 08 12 wad/s Fig. 4
-Compartson of' calculated and measured wave forces and
Fig. 5 - Comparison of calculated
and measured wave forces and
moment transfer functions for a model of drilling unit
moment transfer functions for
a model nf drilling unit
"STAFLO.
Water depth Om
Wove direction J.&=o° Wave directicri ii 90°
2,0 1.6 12 0,8 a' 00 HEAVE o aL 0.8 1.2 w rcd/s 0 08 12 w rod/s
Fig. 6 - Measured and oalculated motion transfer functions at the rotary table for a model of drilling unit T'STAFLO1.
Pi loo 80 6O LO 2.i) on
I
-,
,ÍIr1
Tif
NormaL s Extra joints ¿ Fictitious siçport Fig. 7 - Space frame SEDCO-700 type.TOO
Strass in kg/em' p.r m wofl amplitude
200
74
0 04 08 12 rod/s
'J STATIC LOAB STRESS AND THE STRESS RESPSE OF JOINT 96, OF
MEMBER 100 tdiogcnol bracing)
j
o
Fig. 12
Stress in k/cm'r m woo. amplitud. 030
-196
Woos direction 90°
Dynamic Load cas. Woo. diritti., g0
0h 0.8 12 rod/s
1,
STATIC LOAD STRESS AF THE' STRESS RESPONSE OF JOIST 8S OF
MEMBER 115 (torpodo)
FZ TL
-Stries ii kg/cm2
200
lOO
per in wove amplitud.
Dynamic load coo. Woos d rection 90°
Stress liglem'
200-100
p.r or wove amplitude
Dynamic lood coso
Static 1,040,2.. lensil. atrios 227 kgf/cno2 Static load cas. - Tinsili stono, r -51 kgf/cm2 - - Bending atrios y 397 kgf/cm2 - - Bonding stress y r 160 kgf /om
- Bsrrding stress o 74 kgf/cm' ----8.ndimmg stress o 28 kgt tom2
io o
/
---
-ClI
0 01. 08 12r,d/s1, 0 0.4 06 t2rad/s
1,
STATIC LOAD STRESS AND THE STRESS RESPONSE OF JOINT RN, OF STATIC LOAD STRESS AND THE STRESS RESPOMS OF JÇIINTI2, OF
MEMBER 135 (Imorizontel bnaoig
MEMBER 68 column)
Stotic Load core - T.nsiLo itieis e 150 Static load co,. - Tensile stress e
29 kgf/cm2
- - Bending sjr.ss y 413 kgf /oni' -- Bending strass co 225 kg?/cm°
- bending etress z 6 kgt/cm5 - --Bending strow 53 kgfkm2
70
'w-Line tension Ctcrines) 160 120 80 ¿0 2Th. 250. 225. 200. '5. ISO. 125 '5. 0. 0.0 Fig. H H - 150m T 145s
ncIudiri heave arid pitch
- - - - Pitch and heave suppressed
TIME
Fig. IO - Effect of vertical motion on
anchor-line tensions.
\r
-B.2 - O.N 0.6 .0 1.2 .6
FREQUENCY ANO/Sl
STRESSES/WAVE AMPLITUÛE VERSUS THE FREQUENCY FOR MEPI8ER 149 (DECK MEMBER)
STEESSES ST MEMBER STRRT SIBIlO STRESSES
STRESSES MT MEMBER ENS MEMB6R SOBRI MESSEN ENO
- ROINi. STRESS - -4. -4.
.. BENOINE STRESS ERSURD T + 802. 1109.
SEMOIRS STRESS RROUNL, Z IT. 67.
Ampi, id
p.r w wo.. omptiud.
l-2 F; 0
Aplit0d. p.r ni wo,. ,n,pti lud.
ou o' 2 ,od / PITCh Ph,.. wig1. (,(dwO) 450 360 27G io 90 '-j I I L I 01 12 ,,df, Fig. 8
Motion transfer function (sEDco_700 type).
I.)
Amplitud. p.r ni Wo,.
L 12 Oi o'
Am,t,td. p., n, woo. wrpt1d.
o.'
AmpLtiudt p., nl wo'. o.np&tud.
0! 04 0, I 2rod/. 'J 4E AVE Ph... ,nqI. C,(d.g) 360 210 lia - 90 ROL L Fig. 9
to
Fig. 13 - Comparis'on of rrTagnituds inertia forc and viiscous-drag force Cm/Cd 4.0. i G 'i - £6011 450 tfl3t0(,S 00904. 01 0.2 O 04 00 06 07
Fig. 15 - Non-dimensional point of
appi cation of horizontal wave force on a vertical circular
cyl inder extending to the sea bottom.
4,3
of
60 6
Fig. 4 - Horizontal wave force on a vertical
mir-cular cylinder extending to the sea bottom.
X (N)
cl
605615 e 1.09.
00399e
i ia
Fig. 16 - Proportionality between wave force and wave height
re-sulta of tests on complete piles 1'd = 0.0251.
