Prediction of slamming characteristics and hull responses for ship design

Prediction of Slamming Characteristics and Hull

Responses for Ship Design

Michel K. Ochi,^ Member, and Lewis E. IWotter,^ Visitor

The paper presents a method to predict the necessary information on slamming characteris-tics and hull responses of a ship at an early design stage using only the ship lines. The pre-diction includes (a) frequency of occurrence of slam impacts; (b) time interval between suc-cessive impacts; (c) elapsed time before the next severe impact will occur; (d) magnitude of impact pressure; (e) the largest (extreme value) impact pressure most lil<ely to occur; (f) the extreme pressure for design consideration; (g) ship speed at which bottom plate damage is most likely to occur; (h) ship speed free from slam damage; (i) limiting speed for which slam impact is tolerable for the crew; (f) spatial distribution of impact pressure and resulting force; and (k) main hull girder responses, such as whipping stress and acceleration, to slam impact. Principles and detailed procedure of the prediction for each subject are discussed, and application of the method to practical ship design is given using numerical examples made on the Mariner, assuming that she is operated as a cargo liner following two different routes in the North Atlantic Ocean and a winter route in the North Pacific Ocean.

Introduction

I N THE PAST FORTY YEARS since the slamming phenomenon first came to the attention of naval architects with the advent of the diesel engine in the marine engineering field, more than 300 papers on ship slamming have been published including both theoretical and experimental approaches [1].^ I n par-ticular, the latter approach covers various aspects of the sub-ject such as observation aboard a ship at sea, model experi-ment* in waves generated in the towing tank, and water-entry drop tests on two-dimensional models. Deterministic as well as statistical approaches have been made in an attempt to clarify tlie basic natxire of the impact and to find means of overcoming the distress resulting from one of the most com-plicated ijhenomena experienced by a ship operated in rough seas. Through these studies significant progress has certainly been achieved; nevertheless, a complete method to estimate slamming characteristics and hull responses of a ship at the design stage, including the estimation of ship speed for which damage ou bottom plating could possibly occur, is not avail-able at present.

I t may be appropriate here to give a brief summary of cin-rently available, though isolated, methods to obtain hmited information on slamming for ship design. Perhaps one of the most important pieces of information needed is the magnitude of the impact pressure associated with slam impact. Although several formulas are available to evaluate the magnitude of the impact pressure, most of them are based on either two-dimensional impact theories on wedges or two-two-dimensional models dropped onto the smooth water surface. Recent

' Ship Performance Department, Naval Ship Research and Development Center, Bethesda, Maryland.

2 Numbers in brackets designate References at end of paper. Presented at the Annual Meeting, New York, N . Y., No-vember 15-17, 1973, of T H E SOCIETY OF NAV.VL AUCHITECTS AND M.\BINE E.NQINBERS.

studies, however, have revealed that pressure magnitudes from the two-dimensional approach are several times higher than those obtained from the three-dimensional approach— seakeeping tests for example—for the same impact velocity

[2, 3, 4]. Therefore, results obtained from either full-scale trials or seakeeping model experiments appear to provide the most appropriate information on slamming pressure for design use. Watanabe developed a method to estimate the magni-tude of impact pressure applicable for shipUke sections [5, 6]. Apparently his method is developed using empirical data on structural damage observed at sea; however, the method carries several assumptions and no justification has been given regarding the vaHdity of these assumptions. Abra-hamsen proposes a method to evaluate the pressure including correction factors for speed and draft [7]. The proposed method is for long-term prediction and his design value is for a probabiUty level of 10~« considering 20 years of ship service. However, this does not necessarily mean the value represents the largest pressure (extreme value) experienced by a ship in her lifetime. For the estimation of impact force dehvered to the ship's bottom, the spatial distribution of pressure must be obtained as a function of time. Unfortunately, no appro-priate estimation method for impact force is currently avail-able. On the other hand, many useful studies on bottom plate response to slam impact and associated problems have been carried out, and an outhne of these studies may be found in references [8, 9]. All these, however, pertain to response to a given impact pressure, and no attempt has been made to correlate the plate response (including damage) to ship speed, which appears to be extremely important for design consider-ation, Hull response to slam impact, such as a sudden change of bending moment and acceleration, has been studied by using mathematical models representing the hull structural characterLstics. Results of these studies, nonetheless, are pertinent to extremely Umited cases such as a single impact force applied to an arbitrary location along the ship length.

Ia summary, as this brief review suggests, many problems on various aspects are yet to be solved for ship design.

Information needed on slamming required at the initial stage of ship design is in two areas—loadings and responses. The former is a function of ship motions, sustained speed in waves, and hull form, particularly the shape of the bottom portion of the forward sections. On the other hand, the latter entirely depends on the structural characteristics of the ship such as the thickness of the bottom plate forward and bending rigidity of the entire hull. These two areas, however, cannot be treated separately in the design consideration. For ex-ample, the damage of bottom plating due to slam impact is directly associated with sustained ship speed in a seaway; and hence, in determining the thickness of the bottom plate, con-sideration must be given to the speed expected at sea, and this in turn depends on the seakeeping characteristics of the ship.

The purpose of this paper is to present a method to predict the necessary information on the slamming characteristics and hull responses of a ship at an early design stage using only the ship lines. The basic underlying concept is that the ship has to withstand the severest impact expected in rough seas in the service area. The severest impact is a function of sea severity, ship speed, and operation time in that sea.

A statistical approach to the development of a prediction method is taken in this paper. Order statistics is applied for estimating the ex-treme values of impact pressure, and the magnitude of the extreme pressure most hkely to occur is derived along with that to be considered in design practice. Furthermore, the ship speed for which bottom plate damage due to slam impact is likely to occur and the speed free from slam damage are estimated; the latter is compared with the limiting speed for which slam impact is tolerable for the crew. Slamming impact force is evaluated by integrating the spatial distribution of the impact pressure on the ship bottom, taking into account its duration and traveling time in the longitudinal direction as well as outboard as the bow submerges. Using the above impact force as an input, hull responses are ob-tained by a mathematical model respresenting the hull structural characteristics.

The paper consists of two parts: Part 1 describes the principles and detailed procedure of the prediction method for each subject of interest. Part 2 discusses the apphcation of the prediction method to practical ship design using numerical examples for the Mariner, assuming that she is operated as a cargo Uner in the North Atlantic as well as in the North Pacific Oceans. Estimations of severe sea states in the North Atlantic and North Pacific Oceans are discussed in Appendices

1 and 2, re,speotively.

Part i : Principles and Procedure

of Prediction

Outline of the prediction method

The method of predicting slamming characteristics and hull responses discussed in this paper covers a variety of subjects; hence i t may be well to outhne briefly the overall prediction method prior to discussing the individual subjects in detail.

Figure 1 illustrates the prerequisites required for prediction and the subjects of interest for prediction therefrom at the design stage.

Two prerequisites, both of which can be obtained from ship Unes, have to be prepared in advance. One is the infor-mation on ship motions in the desired sea state. I n particular, information on acceleration at the bow and motions and

velocities relative to waves at several forward locations are required. This is obtainable from the offsets of sections for a given draft by using ship motion computer programs, many of which are available. The information should be obtained for various ship speeds so that the sea-speed relationship as-sociated with slamming can be predicted. I n selecting sea state for evaltiating ship motions, the service area of the ship has to be considered, and an example showing the selection of sea state in the North Atlantic as well as in the North Pacific Oceans is shown later in the second part of this paper together with the information given in Appendices 1 and 2.

The other prerequisite is the evaluation of the form coef-ficient associated with slam impacts. This can be obtained by a regression equation (shown later) that can be evaluated from the offsets of the bottom portion, i.e., one tenth of the design draft, of the several forward sections. A computer program to evaluate the coefficient for a given section shape is available in reference [10]. For a quick evaluation of the coefficient, however, an estimation chart is included in this paper.

With these two prerequisites in hand, the following pre-dictions can be made:

1 Freqxwncy of occurrence of slam impact. This can be easily obtained from the information on ship motion and velocity relative to waves.

2 Limiting speed for which slamming impact is tolerable. From results obtained in Item 1 as a function of ship speed, the limiting speed for which slam impact is tolerable—hereafter referred to as tolerable speed for slam impact—is estimated as the speed for which either the jirobability of slam impact at Station 3 reaches a level of 0.03 or the significant amplitude of the vertical bow acceleration reaches a level of 0.4 g. This subject is discussed later in the section on ship speed and slamming damage.

3 Time interval between successive impacts. From results obtained in Item 1, the time interval between successive im-pacts can be estimated.

4 Slamming pressure. From the two prerequisites, the probabihty function necessary for predicting impact pre.ssure is established, from which the average and significant values of impact pressure can be predicted.

5 Elapsed time before the next severe impact. The time be-tween one severe impact and the next at a specific location along the ship length can be estimated from the results ob-tained in Items 1 and 4.

6 Most probable extreme pressure. By applying order statistics, the magnitude of the largest impact pressure most hkely to occur in a specified ship operation time in a given sea can be predicted at any location along the ship length from the results obtained in Item 4.

7 Ship speed ai which bottom plate damage is most likely to occur. By equating the pressure magnitude obtained in Item 6 to that which will cause permanent set of a rectangular plate, the speed is estimated at which bottom plate damage is most likely to occur.

8 Extreme pressure for design consideration. The magni-tude of extreme pressure for design consideration can be esti-mated by applying order statistics to the probability function obtained in Item 4. The extreme value is controlled by a pre-assigned small probability of being exceeded that is specified by the designer.

9 Ship speed free from slam damage. By equating the pressure magnitude obtained in Item 8 to that which will cause permanent set of a rectangular plate, the attainable (maximum) speed below which no plate damage would occur is estimated. The probability of occurrence of damage is the small number assigned in Item 8.

10 Slam impact force. The magnitude of impact force Prediction of Slamming Cfiaractenstics and Hull R e s p o n s e s for Ship Design ₁₄₅

MAIN HULL GIRDER RESPONSES TO IMPACT

Fig. 1 Flow chart for prediction of slamming characteristics and hull responses

can be estimated from Items 6 or 8, taking spatial distribution and traveling time of the prassm-e into consideration. The force evaluated using the extreme jM'essure in Item 8 is used for design consideration.

11 Main hull girder responses io impaci. Using the infor-mation obtained in Item 10 as an input, the main hull girder responses such as wliipping stress and deceleration due to impact are estimated by solving a mathematical model respresenting the hull structural characteristics. Available computer programs for dynamic response of the hull may be used with necessary alterations.

Prediction of these items enables the ship designer to de-termine the bottom panel dimensions required to withstand severe slam impact, and to examine the sufficiency of the hull girder dimensions against \«hipping stress and deceleration as-sociated with impact. I t may also serve to indicate how much reduction in the magnitude of impact pressure is expected and thereby to what extent the speed free from slanv damage may be increased by a small alteration of the hues near the ship forward bottom without causing any significant effects on the performance characteristics.

Frequency of occurrence of slamming

The statistical prediction of slamming phenomenon can be

achieved by assuming that the phenomenon is a sequence of events occurring in time according to the Poisson process. The validity of this assumption was made earlier through model experiments [11], and further confirmation was ob-tained i n full-scale trials as wiU be shown in the next section. Occurrence of impact is considered to be an e.xample of the Bernoulli trials, since the outcome of the event (wave en-counter) is either slam or uo-slam for each encounter.

The frequency of occurrence of impact at a station is a function of the vertical motion and velocity at that station relative to waves. Hereafter they will be called the relative motion and velocity, respectively. Results of e.xperimental studies have revealed that the necessary and sufficient condi-tions leading to slam impact are (a) bottom emergence, and

(6) certain magnitude of relative velocity, called the threshold

velocity. That is, for slam impact at a specific location along the ship length, the ship bottom at least at the location has emerged from the water surface. However, this is not a sufficient condition for a ship moving in waves, since the im-pact at reentry may still be insignificant if the bottom just breaks the wave surface.

Taking these conditions into consideration, the following formula for the frequency of slam impact can be derived theoretically by applying either the concept of the threshold crossing problem [12] or the method of the phase-plane dia-146 Prediction of Slamming Characteristics and Hull R e s p o n s e s for Ship Design

gram [11] in stochastic processes: Pr{ slam impact} = e where

\Rr' ^ Rf'/

(1)

H = ship draft at location considered f* =

Rr' =

threshold relative velocity, 12 fps for a 520-ft ves-sel. Froude law is used for a ship of different length

twice variance of the relative motion twice variance of the relative velocity

The variances of relative motion and velocity in the above equation are equal to the areas under the spectral density functions of relative motion and velocity, respectively, at the desired location, and they can be evaluated at the design stage from the ship lines by using available ship motion com-puter programs. Many programs compute the response amplitude operator of the relative motion at the desired loca-tions [13-16]. There are many more computer programs available which provide response amplitude operators of pitching and heaving motions only. For these programs, the response amplitude operator of the relative motion at an ar-bitrary location along the ship length can be evaluated by applying the following formula;

\Hr{o^)\^ = 1 + \HM\' - cos - 6 ^ ) (2) where

\H,{<J3)\^ = response amphtude operator of vertical motion of a ship at point x

= \H,(o})\^ + x'\Ho{o>)\' + 2xHMH0{u) cos (fe — 62)

€j = phase between vertical motion at point x and wave at center of gravity (CG)

_ Hzjo}) sin -I- xHejo}) sin eg Hi{ui) cos 62 + xHeiu) cos ea X = distance between point x and CG of ship CO = frequency, cps

g = gravity constant

7/2 (co) ^ = response amplitude operator of heaving motion

Hel<J) ' = response amplitude operator of pitching motion «2 = phase between heaving and wave crest at CG,

positive if heave leads wave

eg = phase between pitching and wave crest at CG, positive if pitch leads wave

The spectral density function of relative motion is simply obtained by midtiplying equation (2) by the desired sea spectrum, and the spectrum of the relative velocity is ob-tained by taldng the second moment at the motion spectrum.

The magnitude of threshold relative velocity, r*, may be taken as 12 fps for a 520-ft vessel, which is obtained empiri-cally through model experiments, and the Froude law is ap-phed for ships of different length. Although the magnitude of threshold velocity, 12 fps for a 620-ft vessel, has not yet been fully confirmed through full-scale trials, i t may be of interest to note here that Aertssen found a threshold velocity of 18 fps in trials carried out on the Jordaem (length = 480 ft) [17]. I t is of interest to point out, however, that Aertssen defines slamming not as every impact but as a severe impact which may cause the captain to substantiaUy reduce speed in a storm. This implies that Aertssen's value may be signifi-cantly reduced if every impact is considered as is the case in the present study.

I t may be more meaningful and convenient for ship design

to express the probabihty given in equation (1) in terms of the number of occurrences of slam impacts as a function of time. For this, the foUowing formulas may be used:

Number of impacts per unit time 1

Ir

''

^ 2Tr\R'' ^'^^^^^^ impact} Number of impacts i n T-hours

(3a)

N{T) = (3.6 X W) £ yj^^, Pr{slam impact} (3&) where T is the ship oj^eration time.

Time interval between slam impacts

I t may often be necessary to know the time interval be-tween successive slam impacts or the time elapse before the next severe impact occurs. This information can be obtained by using the properties of the Poisson process. In general, for the Poisson process, the probabihty that the event wiU occur exactly n-times in a specified time is evaluated by the fol-lowing probabiUty function:

Pr{X = n\ = e- V (4)

where X =

n = V =

number of occurrences of the event, random variable integer, 0 , 1 , 2, etc.

average number of occurrences in a specified time U I n applying the above equation to slamming, let X be the number of impacts in time ^o, and let v be equal to N^k, where Ns is evaluated from equation (3o). I t should be noted, how-ever, that io should be chosen such that P is not large, since the Poisson distribution is approximated by the normal distribu-tion if V is large.

Figure 2 shows the comparison between number of impacts evaluated by equation (4) and that obtained in trials on the Wolveririe State (Length = 496 ft.) in Beaufort 9 head seas [18]. Observations were made over a period of 20 min in which the number of slam impacts in successive 20-sec inter-vals was coimted. The y-value used i n the computation

(0.77) was obtained from a total of 47 slams in sixty-one 20-sec intervals. Good agreement can be seen in the figure. Since good agreement between predicted and observed values was also obtained earlier in model experiments [11], i t may safely be concluded that slamming is a sequence of events occurring in time obeying the Poisson process.

z 0.4 o S8 < s 0.2 0.1 V OBSERVATIC N THEORY \ N V 1 2 3

NO. OF SLAMS IN 20-SEC. OBSERVATION

Fig. 2 Comparison between theoretical and observed number of slam impacts [18 ]

Following the properties of the Poisson process, the time interval between successive impacts can be estimated from the followmg probabihty density function [11]:

(5) where = natural pitching period.

The probabihty that a specified time to or longer will elapse before the next impact can be estimated by

Pr 'Time interval between

^successive impact > fej = ƒ f{t)dt = e^'^f*- •(») (6) Comparison of the results from equation (5) and the time interval between successive slams observed on the Wolverine State (from a set of same data previously used) is shown in Fig. 3 [18]. Although some discrepancy can be seen in the figure, the agreement between the predicted and observed values appears to be acceptable.

The method to predict the waiting time before the next severe impact is discussed in detail in reference [11]. Here, severe impacts are defined as impacts for which the pressure exceeds a certain value.' For example, if one is concerned with an impact exceeding 50 psi being apphed to the ship bottom at a particular location along her length, i t is possible to predict the time elapsed before the next impact—with a pressure of 50 psi or greater—comes at the location. For this prediction, i t is necessary iu advance to estimate the im-pact pressure characteristics at the desired station, which is given in the next section. For convenience, however, the formula to predict the probability that a time 4, or more, wiU elapse before the next severe slam occurs is given as follows:

Time elapse before the next Pr severe impact with pressure > to

greater than pi (7) where 77i = X = k = h = Pl = Po = to = g X ( p . —7>i>) l/{kRf') = 2/(phR/)

form coefficient [see equation (12) ]

nondimensional form coefficient [see equation (13)] specified pressure magnitude

threshold pressure [see equation (9) ] specified time

Slamming pressure

For estimating the magnitude of impact pressure, i t is necessary to obtain the functional relationship between the pressure and velocity. Here, the velocity is the vertical com-ponent of the velocity at a specific location of a ship relative to waves. For the pressure-velocity relationship, results ob-tained from seakeeping model e.xperiments will be used for design consideration due to the reason mentioned in the I n -troduction.

I t may be well to state here several properties of slamming pressure which will be taken into consideration in the subse-quent derivation of the estimation method. These are:

1 Slamming pressure is approximately proportional to the square of the relative velocity at the instant of impact [11]. I n other words, the pressure is expressed by,

20 30 40 50 6 0 7Ö TIME BETWEEN SLAMS IN SEC.

Fig. 3 Comparison between theoretical and observed time interval between successive slams [18 ]

p = kf-' = ykif^ (8)

where k = h = r =

dimensional constant depending upon section shape nondimensional A-value

relative velocity between wave and ship bow at in-stant of impact

p = density of water

2 The ^;-value and hence the ^.-rvalue are not functions of ship speed up to SF = 0.2 and are independent of sea severity and wave irregularity [19]. Hence, they may be obtained from model tests in either regular or irregular waves.

3 The k and fci-values are a function only of the hull sec-tion shape, particularly the shape of the bottom porsec-tion below about one tenth of the design draft [2, 19]. I f the bottom of the section is above or below the baseline, then the distance between LWL and the bottom is substituted for the design draft.

I t is noted that the relative velocity, r, in equation (8) is that at the moment of impact, and the statistical estimation of f may be achieved by applying the threshold crossing problem in stochastic processes. Since this evaluation is rather com-plicated, the impact relative velocity is approximated by the amphtude of relative velocity, which is also a random variable but follows a very simple statistical law, the Rayleigh proba-bihty distribution. Although the amplitude of relative velocity is somewhat larger than the impact velocity, the difference does not appear to cause any serious effect in the statistical prediction of impact pressure, as can be seen i n several e.xamples in reference [11] where measured and pre-dicted impact pressures are compared.

Since the amphtude of relative velocity follows the Ray-leigh probability law, i t can be shown with the aid of equation

(8) that the impact pressure has the fohowing truncated

ex-ponential probabihty distribution [11]:

fip) = \e-^<f-P''> Po è p < (9) where

p = slamming pressure (random variable) Po = threshold pressure = kf;^'

X = l / m / ) = ViphRt')

In the above equation, the truncation of the pressure is due to the threshold velocity below which no appreciable impact pressure is imparted to the ship bottom.

Comparison between the theoretical probabihty density function and the histogram of impact pressure obtained from 148 Prediction of Slamming Characteristics and Hull R e s p o n s e s for Ship Design

model experiments has shown some discrepancy for lower pressures; however, agreement between them appears to be reasonable, in general [11]. The histogram obtained from fuU-scale trials has also shown a trend similar to that given in equation (9) [20].

From the probability density function given in equation (9), the average pressure, p, the average of the one third highest (significant) pressui-e, p</„ can be obtained as follows:

p = Hu' + Rf')

=

hUU'

+

Rf') ^^^^

PV3 = W*"" + 2.10«r') = hHr*'' + 2.10B/) Thus, the statistical values of impact pressure can be esti-mated for a desired ship speed and sea state if the ^;-value (or All-value) for a given ship section is known. The following discussion, therefore, pertains to the estimation of the form coefficient, A>value, at the design stage where no experimental results are available for a given huh form.

Using the third property listed above, the section shape below one tenth of the design draft is mapped onto a circle by conformal transformation with three parameters as fol-lows [21]:

. =

<'(f

+ f + f. +

r.)

a.)

%vhere

Z = X + iy (coordinate for section) i = ^ + iri (coordinate for circle) IJ = scale ratio

01, 03, Qs = real numbers

Since the bottom portion of a ship section is generally fair with no point of inflection, equation (11) is considered to be an adequate mapping for the present study.

Multiple regression analysis was performed in reference [10] to establish the best regression equation for the fc-value on various variables oi, 03, and as,. I n the analysis, not only hnetir models but also intrinsically nonhnear regression models (equations) were considered. Since the number of available experimentally obtained fc-values was limited, several precautions were taken in selecting the regression equation, and the equation was examined in detail from the statistical viewpoint (see reference [10]). Of the twenty-some candidate regression models (equations) for the fc-values, the foUowing regression equation was selected after a com-prehensive search for the best fit to the available data:

k = exp {-3.599 -^ 2.419ai - 0.873a3 - f 9.62405} (12) For the nondimensional expression, only the first term should be changed. That is,

fc, = exp {1.377 + 2.419ai - 0.873o3 + 9.624a5} (13) Table 1 shows a comparison between sample values and values estimated by equation (13). As can be seen in the table, although there is one sample (Sam])le No. 15) for which the difference between actual (observed) and estimated values is more than 50 percent, the proposed regression equa-tion fits the experimental fci-values with an average difference of 16.2 percent. Although this difference a])pears to be some-what large, some errors in the experimental determination of fci-value are inevitable since the fci-value is associated with a transient unsteady phenomenon.

At an early stage of ship design, i t may often be extremely convenient to evaluate expeditiously the A'l-value using simple parameters such as sectional area and size of flat bottom, even if .some inaccuracy may be involved in the evaluation.

Lewison proposed a method to exia-ess the authors'

ki-Table 1 Comparison between ki values obtained by experiments and those by regression equation

fci-Value • . ,

Sample Experiment licgression ^5 jn % number ka k, kr 1 3.04 3.47 12.4 2 5.06 5.06 0 3 5..50 5.06 8.7 4 7.67 6.66 15.2 5 10.13 13.46 24.7 6 11.14 11.87 6.2 7 11.14 13.02 14.4 8 11.14 13.02 14.4 !) 12.44 11.43 8.8 10 14.47 17.36 16.7 11 18.09 18.09 0 12 19.82 14.76 34.3 13 20.98 19.97 5.1 14 15 22.43 26.77 16.2 14 15 28.36 17.07 66.1 Average 16.2

Fig. 4 Definition of curvature, R, and deadrise angle, |3 [22 ]

value with curvature, R, and effective deadrise of the section, (3 (see Fig. 4), and presented a chart from which the fci-value may be estimated by two parameters, R and /3 [22]. Although Lewison's method is simple, there is some reservation as to the accuracy of fitting an appropriate circular curve to the bottom .shape.

In the present study, the authors have categorized hnes representing the bottom portion (one tenth of the design draft) of ordinary-type vessels into three series, and the ki-values are computed by equation (13) for a total of 105 diiïerent hnes with various combinations of the vsddth-draft ratio, h/d, and size of flat bottom. I t should he noted that the width/draft ratio pertains to the section shape below one tenth of the design draft.

Figure 5 illustrates an example of three series of lines, A, B, and C, for b/d = 2. Results of the computed fci-values are l^reseuted in Fig. 6 as a function of width/draft ratio, b/d, non-dimensional flat bottom-width, 6*/&, and sectional area coef-ficient, Ca = A^:/2bd, where is the half width of the flat bottom and yl* is the sectional area below the one tenth of the design draft. I t can be seen in Fig. 6 that the width/draft ratio is the parameter which significantly contributes to the yfci-value, and that an increase in the size of flat bottom for a given width/draft ratio results in a substantial increase in the fci-value.

As an example of a practical application of the chart shown in Fig. 6, the effect of modification of the section shape shown in Fig. 7 on the A'l-value will be examined. The section shape (T) in Fig. 7 represents Station 3 of the Mariner having a flat 149

bottom of 5.2 f t , whicli is equal to 30 percent of the width at one tenth of the design draft. The hnes © and ® are both similar to that of the original hne but the widths of the flat bottom are reduced by 25 and 50 percents, respectively, of the original width. The section shape of the line ®, on the other hand, has an extreme V-shape and has much less flat bottom (1.92 f t as compared with the original 5.2 ft), but the area below the LWL is held constant.

The A;i-values for these lines are evaluated by using the chart given in Fig. 6 and the results are tabulated in Table 2 together with parameters necessary for the evaluation. The sectional coefficient, Ca, was obtained by using a planimeter m this case. Included also in Table 2 are the fci-values com-puted from equation (13). As can be seen in the table, the values estimated by using the chart are somewhat greater than the computed values, but the results indicate that a quick estimation of the Ai-value may be made with reasonable accuracy by using the chart shown in Fig, 6, The table sug-gests that the reduction of the fci-value and thereby the mag-nitude of impact pressure by 10 to 15 percent may easily be achieved through a small modification of the hnes without causing any change in the ship performance characteristics. The effect of modification of overaU section shape on slamming characteristics may be found in references [23-26].

Extreme values of impact pressure

The magnitude of impact pressure varies randomly from one impact to the next following the probabihty law given iu equation (9), and the statistical average pressure, the one third highest pressure, etc., can be evaluated by the formulas given in equation (10). These statistical values, however, may not be sufficient to permit us to predict the largest (e.\-treme) pressure that wiU occur once in a certain ship opera-tion time. For design consideraopera-tion, predicopera-tion of the e.xtreme value is of utmost importance, since a ship must withstand such high impacts as might otherwise cause structural

d SERIES A

^—

^—

0

0.6 d SERIES B

/

J

W

r SERIES C 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 b

Fig. 5 Tiiree series of lines used for computation of k^-value

Fig. 7 Lines of Mariner Station 3 and three modifications

Table 2 Estimation of hi values Lines b , f t d = H/XO, f t 6., f t h/d b,/b C„ fci From eq. (13)

®

(D

®

®

8.81 8.50 8.17 5.63 2.97.5 2.975 2.975 2.976 2.60 1,95 1.30 0.96 2.96 2.86 2.75 1.89 0.30 0.23 0.16 0.17 0.73 0.70 0.68 0.63 18.1 16.4 15.0 10.4 16.7 15.6 13.8 9.7

damage of bottoiii plating or severe vibration of the entire huU.

The statistical prediction of the extreme values of impact pressure can be achieved by further apphcation of order statistics to the foregoing probabihty distribution [27]. That is, by letting p„ be the extreme value of pressure in n finite number of impacts, the probabihty density function of p„ is given by fiPn) = [nmlFip)]''-^]^,^ Po è Pn< ^ (14) where F{p)

- r

f(j))dp

I t is possible from equation (14) to derive the magnitude of extreme pressure most likely to occur in n-observations (the most probable value), p„, and the magiutude of extreme pres-sure for which the probability of being exceeded is a, desig-nated by p„(a) [27]. For a better miderstanding of the dif-ference between two extreme values, p„ and pnia), Fig. 8 was prepared showing a pictorial sketch of the probabihty density function of the extreme pressure. As can be seen, the extreme value has a rather sharply tuned probabihty

distri-P n ( « )

Fig. 8 Explanatory sketch of probability density function of extreme pressure

bution, and the value p„ for which the probability density function has a peak is called the most probable extreme pres-sure in M-observatious. I t can be obtained by letting the derivative of equation (14) with respect to p„ be zero. That is

dfiPn)

dpn 0

From equations (14) and (15), p„ is obtained as Vn = Po - f In M = ^ pfci(r*^ + Ri' In ?0

(15)

(16) Since the most probable extreme pressure, p„, is the modal value of the density function, i t may be reasonable to com-pare p„ with the largest value observed in experiments. How-ever, the probabihty that a pressure higher than p„ will occur is rather high; theoretically i t is 0.632. Hence, i t is not ap-propriate to use the extreme pressure p„. for design considera-tion. Then the question arises as to the amount of margin above this value which should be considered for safe design. To answer this question, the ex-treme pressure for which the probabihty of being exceeded is a preassigned small value, a, will be introduced. Tliis e.xtreme pressure is denoted by p„(a), and may be obtained by finding the solution of the following equatibn:

Jvn

f{Pn)dpn = a (17)

From equations (14) and (17), p„{a) becomes

PnM = Po - ^ h i . { l - (1 - a ) ! ' " }

= ^ ph[r*'- Rf'In {I - (I - ay'"}] (18) where a is a preassigued constant.

The extreme pressure p„ia.) thus defined is controUed by a preassigned small probability, a, which is specified by the designer. Suppose a is chosen to be 0.01, then i t is possible to estimate the extreme value with 99 percent assurance. I n other words, only one .ship in 100 sister ships operating under the same envhonment may suffer from a value greater than Pn(a). I f greater assurance for safety is required, one may choose a smaller a depending on the particular situation. Although the decision concermug the selection of the a-value depends on the designer's discretion, a — 0.01 ajjpears to be reasonably safe [28].

The foregoing discussion on the e.xtreme pressure pertams to that which will occm- in a given number of observations, Prediction of S l a m m i n g Characteristics and Hull R e s p o n s e s for Ship Design 151

since order statistics deals with the extreme value in a certain number, n. Hence, n is the number of slams and is not the number of wave encountere; therefore, n is significantly in-fluenced by the severity of a sea in which the shij) is operated. For practical purposes, it may be more meaningful to express the e.xtreme values in terms of time by replacing n in equa-tions (16) and (18) by the number of slams in T-hours ship operation time, N{T), given in equation (36).

Several examples of the computation of the extreme values of impact pressure by equations (16) and (18) are shown for

the Mariner in Figs. 9 through 12. AU examples (except Fig. 12) pertain to a Sea State 7 represented by Bretschneider's spectral formulation having a significant wave height of 25 f t and a modal period of 12.6 sec. Details of the sea states used in the present study are given in Appendix 1.

Figure 9 shows the distribution of p„ and p„(a = 0.01) along the ship length for a 10-knot speed as a function of ship operation time. Included also in the figure is the frequency of impact at various locations. As can be seen, the frequency of occurrence of impact pressure reduces significantly with

-Nomenclature.

6* =

C Ca

A* = sectional area below Ho

of design draft

Up = width of ship bottom panel

(short side)

Cl, oj, 05 = real numbers in regression

equation

bp = length of ship bottom

panel (long side) & = half width of station at

Ho of design draft waterfine

half width of flat bottom structural damping

coeiB-cient

constant for evaluating pa sectional area coefficient,

A*/2bd

d = Ho of design draft E = modulus of elasticity, 30

X 10« psi for steel

EI = bending rigidity

£F = Froude number

F{x) = cimimulative probability

distribution function of random variable x

Fi(x, t) = traveling impact force f(x) — probability density

func-tion of random variable

x

g = gravity constant H = ship draft at location

con-sidered

| H„ ( « ) ) | 2 = response amplitude opera-tor of vertical motion

\Hr(w)\' = response amphtude

opera-tor of vertical motion relative to waves |/fj(«))|* = response amplitude

opera-tor of heaving motion |//fl(i/;)|^ = response amplitude

opera-tor of pitching motion

hp = thickness of ship bottom

plate

hi = hmiting depth for impact

force consideration

k = dimensional constant

as-sociated with impact pressure

h = nondimensional fc-value KAG = shear rigidity

L = ship length

M = bending moment

m = expf/iCp - po)}

nio = area under the spectral density function wij = area under the 2nd

mo-ment of spectral density function

m„ = virtual mass at a section

associated with huh vi-bration

N, = number of slam impacts

per unit time

N(T) = number of slam impacts in

r-hours

n = integer, 0, 1, 2, etc.

Prix <a\ = probability that the

ran-dom variable x is less than or equal to a slamming pressure average pressure

average of H highest (significant) pressures Clarkson's allowable

pres-sure

Wood's collapse pressure for a clamped perfectly plastic rectangular plate extreme value of slam

pressure in n finite num-ber of slams

threshold pressure = kr*' specified pressure

magni-tude

magnitude of extreme pressure most likely to occur in n-observations magnitude of extreme

pressure for which the probabihty of being ex-ceeded is a

shearing force at a section twice the variance of the

relative motion

Rf' = twice the variance of the

relative velocity

f = relative. velocity between

wave and ship bow ?••« = threshold relative velocity,

12 fps for a 520-ft vessel

T = ship operation time Tt, = expected period of

wave-induced stress of a ship (o = specified time for com-puting time interval be-tween slams p = P = PVa = Pa = Vc = Vn = Po pl Vn Ma) Q Rr'

h = tune duration of pressure

t* = natural pitching period U = scale ratio in conformal

mapping

V = traveling velocity of

pres-sure in longitudinal di-rection

Wa = maximum permanent

de-flection at panel center

X = random variable

Z = X + iy, coordinate for

sec-tion in conformal map-ping

z = vertical displacement

a = constant

/3 = ap/bp

y — angle of rotation of a

sec-tion about a neutral transvei-se axis

fl, 5 = constants associated with

deflection and stress of ship bottom plate, re-spectively

tx = phase between vertical

motion at point x and . wave at center of

grav-ity (CG)

e, = phase between heaving

and wave crest at CG, positive if heave leads wave

te = phase between pitching

and wave crest at CG, positive if pitch leads wave

f = ^+ir], coordinate for circle in conformal mapping

V = average number of

occur-rences in a specified time (o, i.e., N,U

X = l/jkRf'^ = 2/{pkiRf')

^ 0 = /3(V3

+

p = density of water

<r, = yield stress in pure ten-sion, 40 X 10' psi for steel ff» = hog-increasing whipping stress cr+* = sag-increasing whipping stress CO = frequency, cps

120 240 « \ \ \

i\

P n \ \ \ \ in-** 4 0 " ^ H 3URS A Pn \ ( 0 . 0 1 ) 2 0 0 4 0 ^ I \ ||MP/ VCT) \ 1 0.15-p t 0.20 — 0.25

Fig. 9 Distribution of extreme pressures along the ship length; Mariner, Sea State 7, significant wave height 25 f t ,

ship speed 10 knots, light draft

\

\ \

/ \ \ \ \ j _{>PEE[ 1} \ * / \ \ 1,—X 5 K N O T S /

^ ^

\

j

7. i K N ( T s / / A , \

/ \

y

I

\ 1 1 K N ( T s / ' 12 i K N ( T s / p, IMPA

y^

: T |V -\ \ \ IMPA

y^

: T |V -\ \ \ 0.05 0.15 H z 0.20 —

s

cc

Fig. 10 Distribution of extreme pressure and frequency of impacts along the ship length; Mariner, Sea State 7, significant wave height 25 f t , ship operation time '35 hr,

light draft

increasing distance from the forward perpendicular (FP). For example, the magnitude of extreme pressures at Stations 2 aud 3 is nearly equal for a given ship operation time, but the frequency of impacts at Station 2 is much more frequent; 2.2 times that at Station 3. Hence, the most probable extreme pressure expected to occur in 10 hr is 117 psi at both Stations 2 and 3; however, this same magnitude of extreme pressure occurs once in 723 impacts in 10 hr at Station 2, while i t occurs once in 337 impacts at Station 3. Similarly, the most prob-able e.xtreme pressure of 52 psi is applied at Station 5, but i t is one of only five impacts at this station during the 10-hr ship operation time.

Figure 10 shows the distribution of the most probable e.x-treme pressures along the ship length expected to occur in 35 hr of ship operation for various ship speeds. The probabili-ties of occurrence of impacts are also included in the figure. The justification of the 35-hr period used in the computation is that i t can be considered as the maximum ship operation time in this sea, since a sea of significant wave height of 25 f t would not persist for more than 35 hr in the North Atlantic Ocean, as will be discussed in Appendix 1. As can be seen in the figure, the pressure magnitude as well as frequency of impact significantly increase with increase in ship speed, and the location where the peak pressure will occur moves aft with increasing ship speed; the trend agrees well with experi-mentally obtained results [29].

An example of the effect of ship operation time on the most probable extreme pressure is shown in Fig. 11 for various shi]j speeds in the same sea used in the previous examples. Although the figure pertains to Station 3 of the Mariner, the result shown in the figure is equally apphcable to any location along the shi)5 length. That is, the extreme value increases significantly during the first 10 to 15 hr and then increases very slowly with time. This same trend can be observed for all sea states considered in the present study.

Figure 12 shows an example of a comparison between the predicted and observed ex-treme pressures. The observed values are the maximum pressures appearing in 30 min

(full-scale time) observation from model experiments carried out in irregular seas generated i n the towing tank [29]. The com-putation is made i n this case using the wave spectrum ob-tained from the experiments, and the predicted values are the most probable maximum pressures expected i n 30 min. As can be seen i n the figure, there is some discrepancy be-tween the predicted and observed values for a 5-knot ship speed. However, this discrepancy is not surprising since the observation time is short and hence the compaiisoii is made between the largest values in only eight samples at Station 3. For a 10-knot speed, on the other hand, the sample size is in-creased to 36 at Station 3, and therefore the agreement between predicted and observed extreme values becomas satisfactory.

Bottom plate response to slamming impact

I n evaluating the ship bottom plate response to impact pressure for design consideration, the following two simplifi-cations will be introduced:

1 The ship bottom plating is considered as a rectangular panel consisting of floors and longitudinal girders with all edges clamped, and is subject to a uniformly distributed load. 2 The response of a panel is evaluated as if the slamming pressures are apphed statically.

I t may be weh to give a more detailed explanation to the second statement. Many examples of time histories of slamming impact pressures observed during full-scale trials as well as in model experiments have shown that the impact pressure is approximately of the triangular type, aud the time duration varies considerably from one impact to another. For example, duration within the range of 0.025 to 0.25 sec was observed on the Wolverine Slate (length = 496 f t ) oper-ated in the North Atlantic Ocean [18]. Model e.xperiments in irregular waves also show the same order of variation, but the time duration for most of the impacts falls into the range of 0.08 to 0.12 sec (converted to a 520-ft vessel) [29].

Thus, the time duration of pressure may be assumed as 0.1 sec as an average for a 520-ft vessel. On the other hand, the Prediction of S l a m m i n g Characteristics and Hull R e s p o n s e s for Ship Design 153

200 160 : 120 so SPEE I 1 2. 5 K 1 lOTS 10 7.5 1 5 J ~ 10 15 20 25 30 SHIP OPERATION TIME IN HOURS

Fig. 11 Effect of ship operation time on most probable extreme pressm'e; Mariner, Station 3, Sea State 7,

signifi-cant wave height 25 f t , light draft

natural period of bottom panel for this size of ship is very smaU in comparison with the time duration of pressure. As an example, consider a pauel having dimensions of 30 X 90 X 0.8 in. (steel). These dimensions are equivalent to those of a panel surrounded by a floor, center girder, aud side girder near Stations 3 and 4 of the Mariner (520 f t long). The natural frequency of the first mode of vibration of this i)anel then becomes about 200 cps. Thus, the vibration period of an ordinary ship bottom panel is much shorter than the time duration of pressure, and hence the dynamic load factor is close to unity [30]. I n other words, it may safely be assumed that the pressures are applied statically to the plating in evaluating the plate response to slam impact.

The maximum deflection of the panel occurs at the center of the panel, and the maximum stress occm's at the middle of the long side for a panel with ah edges fixed. Their respective magnitudes are obtained as follows:

Maximum deflection = fx Maximum stress = 5 (19) where K E slamming jjressure width of plate (short side) thickness of plate

modulus of elasticity, 30 X 10^ psi for steel constants given in Table 3

I n equation (19), i t is assumed that the panel is in the elastic small-deflection region. I f the loads are large, however, the panel may enter the large deflection region, and the load is resisted by both bending and tension in the middle plane of the panel. I f the load is extremely large, the tensile stress will become predominaut and ultimately the plate may be con-sidered to act as a membrane. Greenspon [31] and Nagai

[32] have studied the efïect of elongation in the middle plane of the panel on plate response.

Perhaps of much more importance in practice is the evalua-tion of magnitude of pressure for which permanent

deforma-Table 3 n and 6-values for a clamped rectangular plate under uniform load (Poisson ratio = 0.3)

1.0 0.014 0.043 1.5 0.024 0.071 2.0 0.028 0.081 2.5 0.028 0.083 3.0 0.029 0.084 4.0 5.0 0.029 0.030 0.085 0.086 25 100 125

\

1 \ V \ ^ 6 Kl lOTS^ • p REDIC r E O ^ 'A ) /

/

QBSER VED^ 10 K 10 K MOTS

Fig. 12 Comparison between predicted and observed ex-treme pressures; Mariner, Sea State severe 7, significant wave height 35 f t , light draft, observed data from reference

[29]

tion (damage) of the i)anel takes place and the magnitude of the associated deflection. For this, formulas based on Clark-sou's allowable pr&ssure criteria [33] aud Wood's formula [34] wiU be introduced among others, and these formulas will be used later in conjunction with the e.xtreme pressure magiutude so that the attainable (maximum) ship speed below which no damage of bottom plating occm-s will be estimated.

Clarkson gives the following approximate formula for the allowable pressure, pa, foUowing his criteria required for the permanent set of a panel [33]:

Pa = G 1 (20)

whei'e

r = l'^-^'^ bp/üp = CD

\6.46 for Ip/ap = 1

bp = length of plate (long side)

O's = yield stress in pure tension, 40 X 10' psi for steel As can be seen in the above equation, Clarkson gives two values of allowable pressure—one for a square plate with fixed edges, the other for an infinitely long plate. Equation (20), therefore, gives the lower and upper bomids of the allowable pressure of a panel having an arbitrary length/beam ratio. Using the formula for the allowable pressure for an infinitely long plate, Greenspoii gives a curve to evaluate the permanent set at the center for the onset of plasticity [31 ].

Recently, Jones has developed foïmulas to evaluate the permanent deflection of a uniformly loaded, fully clamped rectangular plate from the plasticity viewpoint [35, 36]. His formulas for the static behavior of a plate for which the time duration of impact is much longer than the natural period of the plate, as is the case for ship slamming, are given by Wo _ 3(3 - go) + (3 - 2fo) for 1 ^ ^ ^ ' - ^ 7^0 + 2^ 0 ^ ) 3(3 - go) (21)

300 250 u 200 OC 3 160 Pa • V v a , = 1 bp/ap Vhp

Fig. 13 Comparison between Clarkson's allowable pressure, Pa, and Wood's collapse pressure, Pc for ip/Op = 3

280 240 200 STATION / / _ è / / — p . .

/

15 SHIP SPEED IN KNOTS

0.05 0.10 0.15 FROUDE NUMBER

Fig. 14 Most probable extreme pressm-es, p„, as a function of ship speed; Mariner, Sea State 7, significant wave height

25 f t , ship operation time 35 hr

Wo _ 2p

P. \ { i y _ 4go(2 - fo) j _ go(2 - f o ) l T / ! p o " ^ L \ 2 p J 3 ( 3 - lo) 1 3 - f o J , , p ^ 2(9 - 7fo + 2fo') p - ; - 3 ( 3 - f „ ) (21) (cont'd) where

Pc = static coUapse pressure for a clamped perfectly plastic rectangular plate given by Wood [34]

12cr,V

(3 - 2fo)op (22)

Wo = maximum permanent deflection (damage) at plate center

p = slam impact pressure 0 = ap/bp ^ 1

fo = KVs + P'- fi)

Wood's collapse pressure, Pc, is equivalent to Clarkson's allowable pressure, Pa- Hence, comparisons between them are made for various plate thicknesses, ap/hp, of steel plate having the ratio bp/up = 3, and the results are shown i n Fig. 13. As can be seen in the figure, the lower band of Clark-son's ahowable pressure for bp/up = co is very close to Wood's collapse pressure in this example. Since Wood's formula can be used for any èp/öp-ratio, i t will be used in the present study for evaluating the relationship between ship speed and slamming damage, a subject which wih be discussed in the subsequent section.

Ship speed and slamming damage

I t may be of considerable interest i n practice to estimate (o) the ship speed at which damage to bottom plating due to slam impact wiU occur; (6) the attainable (maximum) ship speed free from slam damage; and (c) the limiting speed for which slamming impact is tolerable.

The first two subjects can be estimated by equating the collapse pres.sure for bottom plate discussed in the previous section to the extreme impact, pressures which are a function of ship- speed and ship operation time. I n particular, the second subject is of importance to ship design, since i t can be estimated with probability of assurance as was discussed earlier in connection with the extreme values of impact pres-sures.

The third subject, hereafter referred to as tolerable speed for slam impact, may be estimated by using criteria provided by Aertssen obtained from analysis of many full-scale trial results. The significance of eistimating this tolerable speed for slam impact hes in a comparison of the tolerable speed with those associated with slamming damage, as will be dis-cussed later in this section.

The procedure of estimating these three speeds will be dis-cussed in the following, together m t h a practical example.

Estunation of ship speed for which bottom plate damage is most likely to occur

(i) I n the desired sea state, evaluate the most probable extreme pressure, p,„ at several locations along the ship's forward sections as a function of ship speed. The ship opera-tion time necessary to evaluate the extreme value is a func-tiou of sea state and i t may be obtained from Fig. 30 in Ap-pendix 1.

(ii) Evaluate the collapse pressure of the bottom pauel by equation (22) for the given panel dimensions.

(iii) From (i) and (ii), determme the ship speed at which the most probable e.xtreme pressure, p„, reaches the collapse pressure. This speed may be called the critical speed for slam damage.

(iv) Plot the critical speeds for various locations along the ship length. Then, the lowest critical speed i n the curve will provide the speed at which the plastic deformation (damage) is most hkely to occur and the location where the damage wih take place.

An e.xample of the estimation will be given for the Mariner operating in head seas of State 7 (significant wave height 25 f t ) . Ship operation time is chosen as 35 hr (see Appendix 1). The most probable extreme pressures, p„, are calculated for the j\Iariner at each forward station by equation (16) for vari-Predictlon of S l a m m i n g Cfiaracteristics and Hull R e s p o n s e s for Ship Design 155

ous ship speeds, and the results are shown in Fig. 14. The collapse pressure is then evaluated by equation (22) for a bot-tom panel having dimensions of 30 X 90 X 0.8 in. (steel). These dimensions are equivalent to those of a panel near Stations 2 | through 5 of the Mariner, but i t is assumed that the dimensions are also applicable for stations farther for-ward.

By equating the coUapse pressure of 166.7 psi thus evalu-ated to the most probable extreme pressure, the critical speeds at which damage may occur are determined from Fig. 14 for each station. The critical speeds are next plotted against stations as shown in Fig. 15. I t can be seen from the figure that the damage is most hkely to occur near Station 3 at 11.4 knots (the lowest critical speed) in this example.

Estimation of the attainable (maximum) speed free from slam damage

The attainable (maximum) ship speed below which no damage to a panel would occur at any location along the ship length can be obtained by the same procedure as for the speed at which damage is likely to occur. I n this case, however, the extreme pressure for which the probabihty of being exceeded is a preassigned value a, pja), should be used instead of the most probable extreme pressure, p„. Here, the probabihty a may be taken as 0.01 so that the jirediction is made with 99-percent assurance as mentioned earlier.

I t is noted that the computations cau be carried out for various different plate thicknesses in order to estimate the minimum plate thickness for which no damage will oecxu at a desired ship speed.

Estimation of attainable speed free from slam damage is carried out by choosing a = 0.01 ou the same example used previously, and the result is shown in the lower curve in Fig. 15. As can be seen, the maximum attainable speed free from slam damage with 99 percent assurance is 7.7 knots for the given plate thickness (0.8 in.). I n other words, if the .ship is operated below 7.7 knots in this sea, bottom plate damage would not occur anywhere along the ship length; however, the chance of damage occurring will gradually increase with increasing speed, and the damage is most hkely to occur when the ship speed reaches 11.4 knots.

Estimation of tolerable limit speed for slam impacts

The above-estimated two limiting speeds are from hydro-dynamic and structural viewpoints, and the human element is not considered. I n the practical operation of a ship in rough seas at hght draft condition, however, speed is deliber-ately reduced in an attempt to ease severe motions. This speed reduction is called "voluntary .speed reduction," and the speed whioh provides an acceptable level of slam im-pacts for the crew depends on the personal judgment of the ship captain.

If the tolerable speed for slam impact is below that free from slam damage, then the ship will be completely safe as far as slamming is concerned. I f the situation is reversed, then there is a possibility that the .ship will be operated at si)eeds where damage may take place, and consideration must be given to increasing the bottom plate thickness to the extent such that speed free from slam damage will be above the tolerable speed.

The criterion for estimation of the tolerable speed for slam impact can best be established from an analysis of results of data obtained from full-scale trials, and the criterion will be expressed in terms of probability of acceptance. For this purpose, Aertssen's criterion suggested through his series of many full-scale trials will be used. That is, voluntary reduc-tion in speed is exjiected for light draft condireduc-tion when either the number of occurrences of slamming reaches a certain limit

18

16

FREE FROM DAMAGE I WUH 1 - a = 0. 9 9

5 4 3 2 1 FP STATION

Fig. 15 Critical speeds for damage most likely to occur and for free-from-slam damage; Mariner, Sea State 7, significant

wave height 25 f t

(three times in every 100 pitch oscillations) and/or the signifi-cant amphtude of acceleration at the bow reaches a level of 0.4 g [17, 37, 38, 39]. I t should be noted that slamming de-fined in the Aertssen's study is not for every impact dehvered on the bottom but for a severe impact that may subsequently cause the captain to reduce speed, as was mentioned earher in connection with the threshold velocity.

Taking Aertssen's definition of slamming into considera-tion, the frequency of slam impact at Station 3 should be con-sidered. This is because results of model experiments have shown that an impact near Station 2 and forward does not cause any appreciable huU response, but whenever an impact pressure is applied near Station 3, impact pressures are also delivered to locations farther forward and this produces an

appreciable hull response. Thus, the tolerable speed may be

estimated by taking the lower of the two speeds: one asso-ciated with the probability of slam imjiact at Station 3 reach-ing a level of 0.03 [by equation (1)], and the other associated with the significant amplitude of acceleration at the bow becoming 0.4 g.

The results of numerical calculations made on the same example previously used show that the probability of impact at Station 3 reaches the critical level at 7.5 knots and the significant value of the bow acceleration reaches the critical level at 7.4 knots; hence the tolerable speed is 7.4 knots in this sea. By comparing tliis speed with the speed free from slam damage, i t is surmised that the Mariner would be un-likely to suffer from slam damage in this sea if these criteria were followed.

Additional computations on the estimation of speeds asso-ciated with damage and tolerable siieeds for the crew are carried out on the Mariuer for various sea states, and the re-sults are shown in Fig. 16. Dimensions of the bottom panel used in tliis computation are the same as those used in the previous calculation (thickness 0.8 in.), and details of sea states and ship operation time in each sea are given in Appen-dix 1. The foUowiug conclusions may be drawn from the figure:

• 20 i2 i i 2 z ' 1 D AMABE KELYl MOST 1

1

0 O C C U R

\

F R E : F R O M DAMAC 0.99 E T OLEflA iLESPf ED 0 ,FEZO *^ 10 20 30 40 SIGNIFICANT WAVE HEIGHT IN FT.

SEA STATE

Fig. 16 Comparison between speeds for (i) damage most lilcely to occur, (ii) free-from-slam damage, and (iii)

toler-able limit for slam impact; Mariner, light draft

(i) There appears to be no problem of slamming damage in seas of State 5 or lower at any ship speed up to the design speed of 20 knots.

(ii) A considerable amount of reduction in speed is ex-pected in seas of State 6 or higher for navigation comfort as well as for safe operation to avoid possible structural damage on bottom plating.

(iii) The tolerable speeds are nearly equal to or some-what below the speeds free from slam damage in almost all seas having a significant wave height higher than 15 f t . This implies that for the given bottom panel dimensions, the chance of occurrence of slam damage in severe seas is quite small provided that the ship is operated foUowing the criteria de-scribed in this section. I t should be pointed out that the reduction of speed from the design value in Fig, 16 is the voluntary reduction, and this governs the speed loss in heavy seas. I n milder sea states having a significant wave height of about 15 f t or less, however, speed reduction associated with resistance increase in waves exists even though voluntary reduction is not attempted. Hence, further reduction in ship operation speed than that shown i n the figure may be ex-pected in mild seas.

Slamming impact force on a hull

In order to estimate the main hull girder resjwnse to slam impact, impact loads (force) applied to the ship bottom for-ward must be estimated. Since slamming impact pressure usually travels either forward or aft with changing magnitude, the spatial distribution of pressure on the bottom as a func-tion of time is necessary for predicting the loads. Prior to determining the domain where impact pressures are applied to the hull, the traveling time of the pressure will be discussed.

Impact pressure traveling time

As was mentioned earlier in connection with .bottom plate response to slam impact, i t is assumed that the inijiact pres-sure is of triangular shape, and that the time duration of pressure at any location, regardless of whether i t is on the flat bottom or on the bilge, is 0.1 seo as au average for a 520-ft vessel. Using Fronde's law, the duration time of ]3ressure U at any point for a shij) of length h may be given by

i 2 8 UJ > I'' LU z s SIGNIFIC RELVELC "S ANT / CITY ^ /NO.OF IMPACTS 1 IN 35 KOU RS STi 1= 1 AT \TL0N 6 1400 1200 Ï 10001 800 ! 800 i 200 6 4 STATION F P

Fig. 17 Distribution of the significant relative velocity and number of slam impacts along the ship length; Mariner, Sea State 7, significant wave height 25 f t , ship speed 7.7

knots, light draft where

i\ = time duration of pressure L = ship length, f t

Results of model experiments carried out on the Mariner have shown that the pressure traveling time i n the longitu-dinal direction is approximately 0.15 to 0.30 sec (full scale) from Stations 2 to 5, or vice versa [29]. This implies that the pressure traveling velocity for a 520-ft vessel is i n the order of 260 to 520 fps. Sometimes impact pressures are applied over an extended area of the bottom almost simultaneously. However, this does not imply that the severest impact pres-sures (extreme values) are applied simultaneously over the bottom. On the contrary, the traveling time for severe pact is relatively slow in comparison with that for mild im-pact; hence i t may be appropriate to take the slowest travel-ing velocity observed in the model e.xperiments; namely, 260 fps for a 520-ft vessel. This value appears to be reason-able since observed data obtained on the Wolverine State

(length 496 f t ) show that the average value of the traveling velocity for relatively severe impacts was 220 fps. Then, for a ship of length L (ft), the impact pressure travels along the ship length with changing magnitude with the following velocity:

(24)

(23)

where v is the travehng velocity of pressure in the longitudhial direction fps.

The travehng velocity of the pressure in the upward di-rection at a particular station may be determined from the magnitude of the vertical relative velocity between ship bot-tom and waves at that station, assuming that the velocity is constant during the impact process. Since the magnitude of relative velocity is largest at the forward perpendicular and reduces with increasing distance from the FP in general, the traveling velocity in the upward direction of the pressure at forward locations, such as Station 2, is greater than that of the aft locations.

As an example. Fig. 17 shows the distribution of the sig-nificant relative velocity along the ship length in a Sea State 7

(significant wave height 25.0 f t ) for a 7.7-knot speed, the speed that is free from slam damage in this sea. As can be seen in the figure, the relative velocity at Station 2 is about 1.7 times that at Station 5, and this difference will be con-sidered i n determining the pressure domain for impact force estimation.