ON THE STATISTICS OF IMPACT VELOCITIES AND HIT POSITIONS RELATED TO COLLISIONS AND MATING OPERATIONS FOR OFFSHORE STRUCTURES Ivar J. Fylling MARINTEK Division of Floating Structures P.O.Box 4 125 Valentinlyst, 7002 Trondheim, Norway ABSTRACT Impact velocities and contact point locations are important parameters in determining design criteria for accidental loads as well as operation criteria for many marine operations. The superposition of wave-induced motions on a mean approach velocity create$ complications when statistics at the impact point are wanted. An analytical expression for the impact velocity distribution is available, but only for cases where the mean velocity is larger than the wave-induced velocity. The present paper describes a time-domain approach with a scale transformation and re-sampling method that is generally applicable. Comparison with available analytical formulas is given to show the range of validity of the analytical expressions for mean impact velocity.. KEYWORDS Impact; wave induced motions; statistics; velocity. INTRODUCTION The problem of ship collisions and impact damage has been the subject of comprehensive research over a long period of time. The major part of this research has focused on collision probability, and on structural damage evaluation. Also, in traditional collision problems the ship speed is so large that the wave-induced motions play a minor role. The problem of combined mean velocity and irregular wave-induced velocity is analysed by Standing et a1 (1985) who derived an expression for the distribution-*ofcollision velocity. The expression is, however, not valid for low velocity collision where negative velocities can occur. The problem of combined mean velocity and wave induced motion is also relevant for a range of marine operations, such as offshore crane operations, mating operations involving floating vessels, 297
Transcript
ON THE STATISTICS OF IMPACT VELOCITIES AND HIT POSITIONS RELATED TO COLLISIONS AND MATING OPERATIONS FOR OFFSHORE STRUCTURES
Impact velocities and contact point locations are important parameters in determining design criteria for accidental loads as well as operation criteria for many marine operations. The superposition of wave-induced motions on a mean approach velocity create$ complications when statistics at the impact point are wanted. An analytical expression for the impact velocity distribution is available, but only for cases where the mean velocity is larger than the wave-induced velocity. The present paper describes a time-domain approach with a scale transformation and re-sampling method that is generally applicable. Comparison with available analytical formulas is given to show the range of validity of the analytical expressions for mean impact velocity..
The problem of ship collisions and impact damage has been the subject of comprehensive research over a long period of time. The major part of this research has focused on collision probability, and on structural damage evaluation. Also, in traditional collision problems the ship speed is so large that the wave-induced motions play a minor role.
The problem of combined mean velocity and irregular wave-induced velocity is analysed by Standing et a1 (1985) who derived an expression for the distribution-*of collision velocity. The expression is, however, not valid for low velocity collision where negative velocities can occur.
The problem of combined mean velocity and wave induced motion is also relevant for a range of marine operations, such as offshore crane operations, mating operations involving floating vessels,
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establishing gangway connections between flotels and platforms, etc.
It should be observed that the problems of velocity statistics of lifting or detachment operations are in many ways similar to those of the impact and mating operations.
Vinje et al (1991) have analysed the problem of lifting a load from a moving vessel. The aim is to find an optimum lift-off strategy to minirnise the risk of wire slack and impact after lift-off. The paper deals with the frequency of impact as function of distance and mean speed, and not with actual instantaneous speed at impact.
To the author's knowledge no accurate analytical expression for velocity distribution or hit point position distribution valid.for slow approach impact has been derived. The reason why it is so difficult will be illustrated in the following, and a practical approach that has been used effectively will be presented.
THE PROBLEM
Figure 1 shows a ship in waves drifting towards a platform. The x - z plot of a point on the ship is shown below for a) large drift speed, v w o%, and b) small drift speed, 7 < ob. In the first case the velocity is always positive, and impact may occur at any point on the x - z trace, In the second case negative x- velocities occur, and the x - z trace overlaps itself along the x-axis. Impacts can only occur when the velocity is positive:
Further, we are interested in the first impact. Referring to Figure Ib, the first impact will occur somewhere on the part of the trace drawn with a full line.
The wave pattern is arbitrarily located relative to the platform. Thus, we observe that the position along the x-axis is irrelevant to the problem. This is equivalent to assuming that the impact point is arbitrarily located on the x-axis.
Fig.1 Illustration of the impact problem. a) High velocity, b) low velocity case.
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The modification of the distribution of hit point position due to the mean velocity is indicated to the right in the figure.
The impact point is uniformly distributed along the x-axis. Thus, to obtain statistics on the vertical impact position, Fig. 1, the distribution fy(y) is obtained by sampling with equidistant intervals along the x-axis. The same reasoning applies also to other responses, such as velocity, accelerations, etc. The analytical expression derived by Standing et al. (1985) for impact velocity, provides an estimate for the distribution that apparently eliminates the negative velocity part (thin line in Fig. lb), but it includes the overlap part with positive velocity (dashed line). Hence, the expression is strictly valid only for the large velocity case.
THE POSITIVE VELOCITY CASE, ANALYTICAL APPROACH.
Consider a response time series yt(t) with distribution fyt(y), cfr. Fig. 2. We want to find the distribution, fyx(y), of the response at an arbitrary impact, Yx(x). This is obtained by weighting the distribution of Y, with the velocity, v = dx/dt.
Fig. 2 Transformation to derive fyx from fyt.
Assuming that the velocity is always positive this gives, after integrating over all velocities, v:
In the case that Y, and V, are statistically independent this reduces to:
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1.e. the distribution at impact is the same as the distribution of the time series.
In the case of impact velocity Y = V, and fvt (v I V, = v) is the Dirac delta function, and the eq. 1 reduces to:
This is the same distribution as derived in a slightly different way by Standing et al.(1985).
In other cases, where Y is statistically dependent on V, more complicated expressions are required to calculate the distribution of Y,.
Staying with the velocity case, eq, 3, and assuming that first order wave induced velocities are predominant, fVt(v) can be taken to be Normal distributed, N F t , bVt). The mean value and standard deviations can be calculated as:
These expressions are applicable for cases with positive velocities, vt 2 2 oV,. For other cases, involving low frequency motions, or where negative velocities occur, simple expressions can not be derived.
TIME-DOMAIN APPROACH
In order to study an arbitrary response, Y, for impact cases where the mean approach velocity is low, a time domain approach is used.
The position of a point on the vessel, moving in the x-direction towards a fixed structure, is:
. - x(t) = vtt + k(t)
where - v = mean drift velocity c(t) = wave-induced motion in x-direction.
The velocity in the x-directions is:
Impact Velocities and Hit Positions
v,(t) = i; + E(t)
Now, consider a variable, Y,, which may be a critical parameter for the impact we are studying. Time series realisations of Y,, y (t) are assumed to be available and consistent with available time series c(t). Thus, for any point in time we know both x(t) and y(t).
Fig. 3 Transformation of a steady state response Y, to represent the response Y, at an arbitrary impact point.
Referring to Fig. 3, the impact point is located at an arbitrary point on the x-axis. The response y,(t) is transformed to y,(t) so that: .
This transformation is not a one to one transformation if negative velocities occur. Fig. 3 illustrates how part of the response time series is eliminated (shadowed range), either because the velocity is negative (a), or because the vessel has been there before (b), cfr. also Fig. lb.
To obtain statistics of impact phenomena, the transformed series, y,(x) is re-sampled on constant Ax, and statistics such as mean, standard deviations and distribution functions can be obtained.
It is important to observe that the transformed series y,(x) is no longer to be regarded as a continuous process. It is the collection of all possible impact events related to the basic time series sample. Constant Ax sampling means that all points on the y,(x) series represent impact incidents with the same probability of occurrence.
Thus, the distribution of the response y,(x) is the conditional distribution, given that an impact will occur, initiated by the drift velocity, $,.
Further it should be observed that the amplitudes are not changed, and so the ranne of the distribution is not modified. The s h a ~ e of the distribution is modified, partly due to elimination of parts of the response history and partly due to increased emphasis on parts with large velocity in the collision direction.
The procedure described above can be conveniently implemented as a post-processor to any time- domain simulation program.
EXAMPLES
In order to illustrate the application of the impact analyses a few examples are shown below.
Impact velocity
Figure 4 shows mean impact velocity as function of ratio of mean drift velocity 5 to standard
deviation of wave-induced motions, C J ~ . The data are obtained from the surge motion of a floating production ship. Curve 3 shows results from the time-domain simulation. Results according to ref. Ill, shown as curve 2 are slightly above. The simple formula of eq. 4 is conservative, but fairly good for T lot t 1 .O.
For values of v 105 < 1.0. it appears that the mean impact velocity is fairly well predicted by:
P
Fig. 4 Mean impact velocity vs drift velocity to velocity standard deviation ratio. v is the mean drift velocity, o is the standard deviation of wave frequency motions, and Vcoll is the mean - - collision velocity = v,. Non-dimensional distribution of velocity, = (vx - v ) I a
Impact ~osition
Referring to Fig. 1, the vertical extension of fendering structures on the platform against accidental impact is governed by possible hit point locations. For instance the lower edge, B, should be so low that the supply boat stern, A, will not hit below B. Fig. 5 show examples of traces of the supply ship stern motions when the supply boat is drifting with the waves towards the platform. (Note that
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the Z-axis is positive downwards with origo in the still water plane in this example).
Fig. 5a shows the untransformed time series %(t) covering 2000 s. The mean position of point A is 3.6 m above the still water plane. The motion is symmetrical. Occasional extremes exceed 0.0 (goes below still water plane). Figures Sb and c shows transformed series, ~ Jx ) , for drift speeds of 0.5 and 0.1 m/s, covering 1000 m and 200 m, respectively, in the same time period. The skewing of the response is clearly seen. For v = 0.5 rnls, the stern rarely hits below - lm and for v = 0.1 m/s it rarely hits below -2.5 m.
Supply boat o tern case 6 T cme troce of Z mot ron
Z - x troce ve loc i t y .5 m / s , cov*r;nj 2008 d pm'c.( P d u r Xirl COIC;LT ~ 2 % *&;o nb
t B e
7
t 17 lpIlY[ '1'1 Ill FIT1
l " " I " " 1 " " l " " ~ " 0 200 400 6M) Ba) loo0 12m t 400
Z - x troce V = 0. .( m / s , C O u w ; y 2000 9 / r ~ , ' ~ 4 - , s&clbA c0lCl'Sioh PUS c ' C I ' d e
L
0 400 600 8M) I2m 1400 Is00 2000
Fig. 5 Examples of vertical hit positions of supply ship stem. Note: Z axis is positive downwards.
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Gannwav disconnect
A telescopic gangway connection from a flotel to a fixed platform is constructed for automatic lift- off if the flotel comes too close to the platform, either due to anchorline breakage or due to DP system mal-function. Critical responses when an automatic disconnect operation is carried out are the motions of the gangway tip as a structure rigidly connected to the flotel. Transient drift velocities due to anchor line breakage may be of the order 0.2 d s .
Figure 6 shows the cumulative distributions of vertical gangway tip motions. Probabilities of exceeding a limit of 0.5 d s are indicated. The probability decreases with increasing drift speed for the present case. A similar check on downwards (negative) velocity would show increased probability with increasing drift speed. This effect depends on the correlation of the horizontal, the vertical and the angular motions of the flotel.
I
36 37 3 8 3 9 4 0 41 42 z - position (m)
I
0.0 1 .O 2.0 3.0 z - velocity (rnls)
Fig. 6 Example of vertical velocity and position at gangway disconnect due to stroke limit exceedance.
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DESIGN CRITERIA
The transformed series y,(x) represents samples of impact incidents with equal probability of occurrence. Hence, in a statistical interpretation, the fractiles of the distribution should be used directly to calculate impact states. This is in contrast to normal wave-induced motions and loads, where the distribution of maxima is used.
Thus, the velocity, voSl that is not exceeded in 9 / 10 of impacts is determined by:
Consider a collision incident with a frequency, pi , of The velocity, v,, that is exceeded with a frequency, p , of e.g. lo4 year is found from the equation:
Thus, v, is found as the .99 fractile in the distribution.
As a second example, consider the landing of a structure on the seabed from a crane with wave- induced vertical motions. Assuming a "blind" landing where the operator can not adjust the crane velocity within a few wave cycles, the above analysis will apply. In this case the impact is a certain event, and the velocity, v,, that is exceeded with a probability of is determined by:
As a reference to a Gaussian distribution, wich would apply for a large velocity impact, this would give a velocity:
CONCLUSIONS
Simple expressions for velocity and hit point distributions of impact between floating vessels and fixed structures can only be derived when the mean velocity is larger than wave-induced velocities.
A straightforward time-domain procedure to derive impact statistics has been described and demonstrated.
The expressions for impact velocity distribution derived for positive velocities have been compared with results from time-domain simulations for a range of mean velocities. For the mean value of impact velocities the correspondence is fairly good for 5 1 qt 2 1.0
For smaller mean velocities it is recommended to use the time-domain transformation and re- sampling procedure to determine the statistics. This procedure can be used for any response and also accounts for the low frequency motions.
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TERMINOLOGY
distribution density function for the variable Y. Ply < Y < y + Sy] = fy(y) Sy. - a stochastic variable, Y sampled on constant time increment (Y(t)). - a stochastic variable, Y, sampled on constant position increments, (Y(x)) - distribution function. P[Y < y] = Fy(y)
probability of the event E. - expected value of Y.
OY - - standard deviation of Y. Y - mean value of Y. X - coordinate in direction of impact to be studied. x - velocity in direction x.
5 - wave-induced motions in x-direction.
6 - wave induced velocity in x-direction.
REFERENCES
1 Standing, R.G., Brendling, W. (1985): "Collision of Attendant Vessels with Offshore Installations, Part 2, Detailed calculations" OTH 84209, HMSO, London.
2 Vinje, T., Kaalstad, J.P., Daniel, D.W. (1991): "A statistical method for evaluation of heavy lift operations offshore", ISOPE, Edinburgh.
