Loads on fender structures and dolphins by sailing ships

(1)

pi: 2303$3

rijkswaterstaat

communications

loads on fender structures

and dolphins by sailing ships

by

(2)

(3)

rijkswaterstaat communications

loads on fender structures and dolphins by

sailing ships

by

dr.ir. a. vrijburcht, delft hydraulics

R U K S W A T E R S T A j ^

Qtenst 3innen\*ateren RIZA Maerlant 4-6

8224 AC Postbus 17 8200 AA Lelystad

(4)

(5)

all correspondence and applications should be a d d r e s s e d to: rijkswaterstaat dienst getijdewateren koningskade 4 postbus 20907 2500 ex the hague netherlands

the views in this article are the authors' own

recommended catalogue entry:

Vrijburcht, A.

L o a d s on fender structures and dolphins by sailing ships /

(6)

Introduction

Fender structures and dolphins form an important part of the navigable canal. They have many functions: they must be able to guide and slow down ships, they must protect structures aft of ships against impacts from ships, and must protect ships from these structures. In addition, they must act as berthing or waiting places for ships. Fender structures and dolphins are installed where one or more of these functions are required, generally when passing nearby sluices, bridges, harbour entrances, etc. The costs of constructing and maintaining fender structures and dolphins may be con-siderable, thus savings in construction or maintenance costs, together with efficient operation, are very important.

U p until recently schematised calculation methods, with dimensioning approach vel-ocities and angles of the ships, obtained from practical observation, were used by the Netherlands Ministry of Public Works, Locks and Weirs Directorate, for designing fender structures and dolphins. The fender structures built meet the requirements previously laid down.

However, at a time when fender structures and dolphins must be designed as econom-ically as possible, and when ships are increasing in both size (e.g. push tows) and speed, there is a need to provide better calculation methods and more broadly based general conditions.

For these reasons a study has been carried out to determine the loading of fender structures and dolphins by sailing ships. This Rijkswaterstaat Communication gives a survey of the study carried out.

The investigation into the loading of fender structures and dolphins forms part of the Applied Research Programme - Hydraulic Structures of the Netherlands Ministry of Public Works, and was carried out in the years 1974-1983 by Delft Hydraulics, in close collaboration with the Locks and Weirs Directorate of Rijkswaterstaat. D r . H . L . F O N -T I J N , from the Delft University of -Technology, was consultant for this study. This Rijkswaterstaat Communication is drawn up in two parts. The first part contains an introduction to, and summary of, the study (Chapt. 1), and a determination of the data required for determining the impact forces (Chapters 2 to 5 inclusive).

The second part contains the determination of the impact forces by calculations and model investigation (Chapters 6 to 9 inclusive).

(10)

Part I: Summary and Behaviour of Fender Structures and Ships

(11)

1 Purpose, summary and conclusions of the study

1.1 Purpose

The purpose of the study is to determine the loads on fender structures and dolphins by sailing ships under actual prototype conditions.

This knowledge will have to be made available in the form of a mathematical model. This must enable the fender structure or dolphin to be designed as economically as possible.

This means that the following must be established from the study: - the dynamic properties of fender structures and dolphins

- the approach velocities and angles of the fender structures and dolphins

- a mathematical model for determining the loads on fender structures and dolphins at a given combination of ship/approach velocity and angle, and with given proper-ties of the fender structure or dolphin.

The following may be mentioned as limitations of the study:

- particular consideration is given to ships used for inland navigation, although the possibility of sea-going ships is also examined.

- emphasis is placed on the hydraulics and theoretical mechanics of events; the applied mechanics for stresses and deflections of the structure and soil mechanics have not been considered.

- the impacts must not have any influence on closed, vertical walls (quay walls, etc.) - deformations of the ships induced by the impacts are absent.

- loads applied by berthed ships are not taken into consideration.

The purpose of this R W S Communication is to provide a summary of the study con-ducted over the past few years. No attempt is made here to give full details of the study.

1.2 Study procedure

The procedure used for the entire study is described here with reference to Delft Hydraulics reports which have already been published (see References).

Attention is first directed to the dynamic properties of dolphins (ref. [21]) and to the approach velocities and angles of ships near fender structures [20].

(12)

A model investigation into impact forces was then carried out, with the first develop-ment of a mathematical model of the impact forces [22]. It was then felt necessary, however, to develop a complete mathematical model of the impact forces, and for this purpose use was made of the knowledge of D r . H . L . F O N T I J N (the Delft University of Technology) [3], [4] and [5] and D r . P. A . K O L K M A N (Delft Hydraulics) [13]. To establish this complete mathematical model it is necessary to know the dynamic properties of the ship in water. For this purpose a model investigation was carried out into hydrodynamic coefficients [23], and at the same time the complete mathematical model B O T S was developed [15] and [16].

A l l the mathematical models developed for impact forces were then compared with the model investigation into impact forces [13].

Finally a report was drawn up for the consultant and designer [24], of which this is an extended summary. [11] is a brief summary presented at a symposium.

In 1988 D r . H . L . F O N T I J N published a thesis [5A] describing the behaviour of a ship berthing to an open and a closed structure and the related fender forces. The results of this study are not presented here.

1.3 Summary

The study breaks down into two parts. The general data and general conditions are set out in part I, and the mathematical models for the impact forces, with the associated model investigation, are contained in part II.

Part I: Behaviour of fender structures and ships (Chapters 2 to 5 inclusive)

The function of a fender structure is to act as a guide for shipping, to protect the ship and the structure behind the fender structure and to act as a berthing or waiting place. The fender structure generally consists of a row of piles driven into the ground, con-nected to a horizontal beam lined with a wooden protection.

A dolphin is a detached element consisting of one or more piles, having a function corresponding to that of the fender structure, but located in a position which is often less important than that of the fender structure.

Prototype measurements have been carried out to determine the behaviour of a dolphin when a ship collides. The dolphin was a pile driven into the ground, and the load consisted of a ship impact and a fixed force.

In the area of measurement there appeared to be a linear spring stiffness which showed little variation in the dynamic and static situation. Moreover, there was considerable damping in the system (soil, structure and water).

(13)

A n insight was gained, as a result of the prototype measurements, into how ships normally behave in the vicinity of fender structures, and into the approach velocities and angles. Emphasis was placed on large ships and wind. The measurements were processed to give a forecast of extreme manoeuvres, with a certain probability of exceedance. Unladen ships, on average, hit the fender at a greater velocity than laden ships.

A n investigation was carried out with a mathematical manoeuvering model into the possible maximum approach velocities and angles of a push tow in a sudden deviating manoeuvre. A push tow is assumed, which initially sails parallel with the fender struc-ture and which gives its rudder maximum deflection in a short space of time. Part II: Determination of impact forces

Because the water is a very complicated factor in the calculations, the ship/fender system without the influence water is considered in the first instance. The ship sails at a certain angle and velocity up to the fender structure, collides against the fender, causes the fender to spring inwards, the ship turns away, thereby being released from the fender. On first contact between the ship and fender, the ship is decelerated in a very short time, and the fender structure is then accelerated. This gives rise to a very short impact force.

This is followed by the inward springing action resulting in a force with a sinus shape. In this case use can be made of a virtual ship mass, which acts in the transverse direction on the fender. Consideration is also given to the sliding of the fore part of the ship along the fender and to dolphins.

The ship/water system without fender is then considered. It is shown that the effect of the water on an accelerating ship can be expressed in a hydrodynamic force. This hydrodynamic force consists of an added water mass term (acceleration term), an added water damping term (velocity term) and a memory effect. This memory effect indicates that the previous movements also have an effect on the present movement. The hydrodynamic coefficients, added water mass and damping, are determined in both model and theory, for a harmonic oscillation, and are dependent on frequency, water depth, shape of the ship and direction of motion.

Subsequently these methods are introduced for the ship/water/fender system. The hyd-rodynamic force is introduced in the equation of motion of the ship/fender system. The virtual water mass method introduces as a hydrodynamic force a (constant) quan-tity of water mass multiplied by the acceleration.

This quantity of water, the virtual water mass, is determined from a model investigation at such a manner that the maximum force from the model is equal to that from the

(14)

method mentioned. The development of the force, after the time of maximum force calculated by the method mentioned, does not agree very well with the model results. The impulse response function method has introduced the full hydrodynamic force, in the form of an added mass and damping term, with memory effect, into the ship/fender system. The method is set up according to the system theory whereby the input signal (= external fender force) is transmitted by a black box (= ship/water system), resulting in the output signal (= ship velocity). This method fully covers the model results of the impact force throughout the impact time. The method is further used to develop the mathematical model ' B O T S ' (Ref. [24]).

1.4 Conclusions and recommendations

Part I: Summary and behaviour of fender structures and ships Conclusions:

- The dolphin, in the normal working area, must be considered as a linear spring/ damper, the static spring stiffness deviating little from the dynamic spring stiffness. The damping of the dolphin is considerable. The impact of a ship against a dolphin can be described by means of a one-mass spring system.

- Extreme combinations of approach velocity perpendicular to the fender structure can be indicated with a certain probability of exceedance for the Volkerak locks. Unladen ships, on average, have greater approach velocities and angles.

There is a certain interdependence between the velocity parallel with the fender and the approach angle: the approach angle decreases as the velocity increases. - The maximum possible approach velocity and angle for a push tow can be indicated

by means of a mathematical model for ship manoeuvres. Recommendations:

- For normal fender structures and dolphins (without special provisions, such as a rubber fender), a linear spring/damper combination can be used. The damping values may amount to 5 to 10% of the critical damping (y = 0.05 to 0.10).

- It is difficult to ascertain the design velocities and angles which must be assumed for a particular design, in view of differences in berthing, type of fender structure, navigation frequency, ship size and the like. However, the extreme combinations for the Volkerak locks and the results of the manoeuvering model may be taken as a guideline.

(15)

Part II Determination of impact forces (Chapters 6 to 9) Conclusions

- On initial contact between the ship and fender structure, the fender structure is taken along with the forward edge of the ship in perpendicular direction. If the fender mass is considerable in relation to the mass of the ship, the velocity perpen-dicular to the fender structure is directly adapted.

The quantity of ship mass which is applied in the transverse direction against the fender structure depends very much on numerous factors. In addition to the mass and moment of inertia of the ship, the angle of approach, the coefficient of friction between the ship and fender structure, and the position of the point of impact, are also important.

- In the case of ships which undergo accelerations or decelerations, use must be made of acceleration and velocity terms for calculating the hydrodynamic force. In addi-tion to added water mass, added water damping and the effect of previous move-ments (memory effects) are also taken into consideration.

The hydrodynamic coefficients (added water mass and damping) can be satisfactorily calculated with the 2-dimensional potential flow theory for normal inland navigation situations. The effect of water depth and frequency is important.

- The virtual water mass method is based on calculations in which the influence of the water must be introduced using mass terms determined on the basis of a model investigation or literature data.

Where there are sufficient model results, reliable maximum forces and displace-ments can be calculated. However, the complete development of the force in time cannot be clearly represented due to the absence of water damping and memory effects. The disadvantage of this method is that a model investigation is mostly required to determine the effect of the water.

The coefficients of the water given in the literature [9] and [10] may in many situa-tions give rise to impact forces which are too low.

- The results of the calculations with the impulse response function method (with hydrodynamic coefficients from the 2-dimensional potential flow) and the model investigation have been compared. It appears that the calculated impact force as function of time agrees very well with the measured impact force.

The major advantage of the impulse response function method is that a reliable impact force is obtained by calculations without requiring the model investigation.

(16)

- The calculation examples indicate that there are major differences between the effect of the water for fender structures (impact of bow of ship) and for dolphins (impact with side of ship). Also the effect of keelclearance is important.

Recommendations

- The impulse response function method, with hydrodynamic coefficients according to the 2-dimensional potential flow, is recommended for situations in inland naviga-tion. The mathematical model "BOTS' is based on this (Ref. [24]). The method can also be fully applied to seagoing navigation, but for short ships with large diameters hydrodynamic coefficients according to 3-dimensional theories must be used.

(17)

2 General definitions

2.1 Fender structures

The function of fender structures is more than merely that of slowing down ships: the fender structure must be able to guide and slow down ships, it must protect structures (locks, bridges, etc.) against ship impacts and must protect ships against structures, and it must also be able to act as a boundary for the berthing or waiting place for ships. A complete fender structure consists of waiting places for ships which have yet to pass through the lock or bridge, and consists of guiding structures for allowing ships to sail in smoothly via a funnel (Figure 1).

Figure 1: Example of layout

The contour of the fender structures is very important for the capacity of the lock complex. See photographs 1 and 2.The waiting place must be such that the ships can easily berth, so that they do not meet any obstructions from passing ships during berthing and so that they can easily be manoeuvred to the lock. Fender structures for the waiting place are therefore often installed at an inclination to the lock axis (e.g.

1:10 or 1:20).

Access to the lock takes the form of a funnel, the guiding structure being set at such an inclination (between 1:6 and 1:4), with rounding that even wider ships (push tows) can sail into the lock without becoming stuck. See Ref. [7].

The systems of fender structures may be distinguished on the basis of a number of facets:

- continuous or not.

Fender structures may form a continuous line for shipping (where there is heavy inland navigation traffic), or may consist of detached dolphins (little inland naviga-tion or sea locks)

- degree of openness.

Fender structures may be of the open type, due to the use of connected free-standing piles, or of the closed type, in the case of sheet piling

(18)

- floating or not.

Fender structures are fixed, with water levels which vary little, or floating, with variable water levels.

- elasticity.

The elastic structures are the piles connected with horizontal beams. The rigid structures are sheet piling, these can be combined with loose rubber fenders. Continuous, open, elastic fender structures are normally used for important Dutch locks.

The construction of the elastic fender structure is indicated in figure 2 and photos 3 and 4.

The fender structure consists of a row of piles driven into the ground, connected by a continuous horizontal beam. The piles and the beams are in most cases round steel tubes. Short vertical beams are secured to the horizontal beam, to which are suspended aprons consisting of horizontal wooden beams.

In the case of the fixed fender structure, the beam and the piles are rigidly connected. In the case of the floating fender structure, the construction consists of a floating beam

Photo 1: Fender structures at Kreekrak Locks (Aerophoto Schiphol)

(19)

Photo 2: Fender structures for Locks at Tiel ( K L M Aerocarto)

(20)

Q UJ x u. £ 3 " s! s 2 e CM E £ i

Figure 2: Different types of fender structures

(21)

Photo 3: Fixed fender structure. Volkerak Locks (RWS photo, June 1977)

(with or without floating boxes), sliding along the piles, or of a floating beam with float piles, the float pile sliding into a deep jacket pile.

Dolphins consist of one single or several connected piles protected by a wooden lining. The dynamic properties of the fender structure are defined as the reaction of the fender structure to an impact of a ship. Mass, spring stiffness and damping are important factors.

The mass of the fender structure in relation to the mass of the ship is important for the initial joint velocity of the ship and fender structure.

The spring stiffness (force/displacement relation) is important for the frequency of the system. It may also be stated that elastic structures make lower forces and greater displacements than more rigid structures.

In the case of the elastic fender structure with a continuous beam, the spring stiffness is a combination of the elastic elements, and is dependent on:

- point of application of the force, with regards to height and distance between the piles (Figure 3).

- soil mechanics aspects, depth of driving of piles, - stiffness (EI) of beam and piles

(22)

Photo 4: Floating fender structure, partly without aprons, Volkerak Locks

( R W S photo, January 1966)

- whether beam is continuous or not, or in end section - connections between beam and piles.

Above a certain value of the force there are present permanent deformations. In the case of the elastic fender structure with a continuous beam, the spring stiffness parallel with the fender is in most cases much higher than perpendicular to the fender, but in the case of the free standing dolphin, the spring stiffness is equal in all directions. The damping (force/velocity relation) of the fender structure must be regarded as internal friction, and is caused by the possibilities of motion between the parts of the structure, the material damping and the soil damping.

(23)

2.2 Ship in water

The dynamic properties of the ship are important during impact. The mass of the ship, how the mass is distributed across the ship, and the position of the centre of mass, must all be known. The rigidity of the ship is also important.

The effect of the water during the impact of the ship against the fender structure may be considerable. In the case of an open fender structure, the forces of the water act directly on the ship, and indirectly on the fender structure. The shape of the hull of the ship, the water depth, directions of movement of the ship, accelerations and vel-ocities of the ship are important.

2.3 Ship/fender structure contact

The approach up to the fender structure takes place differently for each ship. It is characterised by the sailing velocity (including angular velocity, angle of approach and propeller and rudder action), and is determined by the behaviour of the master, ship type, location, wind, current, etc.

The manoeuvres of the ships may in principle show the following pattern during impact:

Figure 4: Example of manoeuvre for continuous fender structure and freestanding dolphin

Position 1 is before contact, position 2 is the initial impact, position 3 is the springing in of the fender structure/dolphin, in position 4 the ship releases the fender dolphin, and position 5 relates to the time after impact.

The ship turns away during the impact, whilst the fender structure springs inwards. On the fender structure the point of contact is always the forward edge of the ship, and is displaced along the fender structure during impact. On the dolphin, the point of contact may be along the whole length of the hull, and is displaced along the hull during impact.

The force on the fender structure (or dolphin) has an extended semi sinusoidal shape when the fender structure (or dolphin) acts as a spring (without mass and damping). See figure 5.

(24)

Figure 5: Fender structure (or dolphin) force/time relation.

The deformation of the fender structure has almost the same shape as the force diag-ram, although this deformation is sometimes incomplete because of permanent defor-mation.

Point 2 is the initial impact, point 3 is the maximum deformation/force and point 4 is the release of the ship from the fender structure.

(25)

3 Dynamic and static behaviour of a dolphin

3.1 Prototype measurements

The purpose of the measurements was to describe the behaviour of a dolphin subjected to dynamic and static loads.

The dolphin consisted of a steel tube, 1.20 m in diameter, driven into the ground, with a wall thickness of 13 to 19 mm, and lined with wooden beams (See Figure 6). The soil consisted of a layer of clay, containing some peat, and underneath sand with a high load carrying capacity.

A push boat with a pontoon connected to it was used for the dynamic tests. This combination (550,000 kg) collided several times on the dolphin (See Photo 5). In the static tests the push boat was applied with cables so slowly that the groundwater stress did not vary around the pile.

The force on the dolphin, the displacements of the dolphin, the velocity of the ship and the groundwater stresses were measured.

The relation between the maximum force and displacement is indicated in figure 7 for the dynamic tests. The measuring points lie almost in a straight line. The relation is almost independent of the size and repetition of the load. The permanent displacements during the tests accumulated regularly up to a limited value (0.03 m).

The relation between the maximum force and approach velocity is indicated in figure 8. This relation is almost linear. The ratio between the approach velocity and the velocity of release was an average of 0.57, with little variation. The times of maximum force, maximum displacement and release showed little variation during the different tests.

The relation between the force and elastic displacement for the static tests is indicated in figure 9. The measuring points lie almost in a straight line. The permanent displace-ment accumulated to a limited value (0.05 m).

The ratio between the maximum force and maximum elastic displacement differed during the dynamic tests and static tests: 3.75 x 106 and 2.83 x 106 N/m respectively.

(26)

3.2 Calculation

The purpose of the calculations was to assess whether the impact of a ship against a dolphin can be schematised by means of a one-mass spring system.

(27)

800 Fy(103N) A 600 400 200 . r g a = 3.75 x 7 0s fV/m 0 O 0 5 0~7o 0.75 0.20 »- y(m)

Figure 7: Relation maximum force/maximum elastic displacement, dynamic tests

Fy(103N)

• 4 0 0

200

0.20 0.40 0.60

*• 9o(m/s)

Figure 8: Relation maximum force/initial velocity, dynamic tests

600 400 200 tga = 2.83 x 106 N/m 0 04 0.08 0.12 0.16 0.20 0.24 ~~ y(m)

(28)

Photo 5: Push boat near the measuring dolphin

First the initial velocity of a one mass spring system is determined by means of an impulse equation.

The ship sails to a dolphin at a uniform speed y„. A t time t() the first contact takes place

between the ship and the dolphin. It is assumed that this impact is fully elastic, i.e. in a very short time the ship and the dolphin move as one mass at an adapted initial velocity y,',. (Figure 10)

u

o

Figure 10: Before and after the first contact

(29)

The impulse balance is (where y = dy/dt = velocity): my,, = (m + M ) y,',

where m = mass of the ship (kg) M = mass of the dolphin (kg) The new initial joint velocity is then:

Yo = (m/(m + M))y„

(1)

(2) It is assumed that the movement of the ship with the dolphin in the water can be described as (Figure 11):

(m + M ) y + cy + ky = - ( a „ y + bn( y - ft))

where y, y, y = acceleration, velocity, displacement (m/s2, m/s, m)

c, k = damping, spring, rigidity (Ns/m, N/m) au, b|, = added water mass, damping (kg, kg/s)

(3)

IS

t

Figure 11: One-mass spring system

The terms in the left term are the inertia force of the ship and dolphin and the resistance force of the dolphin. It is assumed that this resistance can be described as a linear spring and damper.

The right term includes the influence of the water. It is assumed that this hydrodynamic force can be written as an added mass term and an added damping term, these terms being added to the original initial situation, with acceleration y0 = 0 and velocity ft.

Using these equations (2) and (3), the measured impacts can be recalculated.

Measurements have been carried out at Delft Hydraulics (Ref. [6]) relating to the added mass of water aH and damping bn of a short ship in shallow water.

The mass of the ship m and the dolphin M are known and also the initial velocity y0

of the ship.

In this case combinations of spring stiffness k and damping c are chosen so that both the force and displacement correspond as closely as possible, as a function of time in the calculation, to the measured values.

(30)

*> time (s)

Figure 12: Displacement/time relation, test 15

0.6

Figure 13: Force/time relation, test 15

(31)

In all tests, a spring stiffness k = 3.75 x IO6 N/m and a damping c = 0.4 x 10A Ns/m

were found to be the best combination. The damping/critical damping ratio y = c/cc r =

= c/(2\/(km)) = 0.17 is considerable. See figures 12 and 13 as a comparison between the measurements and the calculations for force and displacement.

The presence of a higher vibration frequency is caused by the initial impact between the ship and dolphin.

The quantity of energy in the different terms is represented in figure 14.

The kinetic energy of the ship decreases to 0 at maximum displacement, then increases to a limited degree up to the time of release.

The energy in the spring increases until the time of maximum displacement, then disappears completely after the release. The energy in the damping of the dolphin and the water increases gradually from the beginning and this energy does not decrease.

Energy from:

Energy (103 Nm)

f

70

I mass of ship II added mass of water III added damping of water IV mass of dolphin V spring shiffness dolphin VI damping dolphin 60 56.3 x 103 Nm 0.8 Tb 1.2 • TIME (s) IV 0 _0.2 _0.4 _0.6 1.4

(32)

3.3 Conclusions

In the measurement area, the ratios between maximum force/maximum elastic displace-ment were almost constant for the dynamic and static tests, regardless of how many times the tests were repeated. When converting to the same point of application, the ratios were also more or less the same for the dynamic and static tests. The permanent displacements remained limited and were regularly accumulated.

The maximum force/initial velocity ratios and initial velocity/release velocity ratio were almost constant.

The impact of a ship against a dolphin can be easily calculated by means of a one mass spring system, but here a combination of a spring and a damper must be taken as the resistance.

The damping of the dolphin, driven into the ground, is found to be considerable.

(33)

4 Approach velocities and angles of fender structures

4.1 Prototype measurements

The purpose of the prototype measurements was to gain an impression of the normal behaviour of ships in the vicinity of fender structures, and the extreme approach vel-ocities and angles which may occur. A n attempt is made to lay emphasis on conditions associated with wind and large ships.

The Volkerak locks (in the south-west part of the Netherlands) were selected for the prototype measurements. These locks possess waiting places with modern equipment for heavy shipping traffic.

Aerial photographs were taken of the lock harbour every 10 seconds by using a helicop-ter. The approach velocities and angles were determined from a comparison of succes-sive photographs for interesting manoeuvres. A calculation programme was developed to interpret these photographs.

Many manoeuvres were measured for a period of 7 days of measurement, see for example photographs 6 to 10a.

The most important manoeuvres were selected on the basis of approach velocities and angles and ship size. From the total numbers of ships normally passing through the Volkerak locks (approximately 200,000 every year) it appeared that approximately 20% fulfil the requirements of the selection criteria established.

Of the manoeuvres selected, 5 categories may be listed: category 1: all the ships selected

category 2: unladen, > 1000 tons loading capacity category 3: unladen, < 1000 tons loading capacity category 4: laden , < 1000 tons loading capacity category 5: laden , > 1000 tons loading capacity

The mean values m and standard deviations s of the approach velocities and angles are indicated in table 1 for these categories. Here the notations are:

vx = velocity of fore part of the ship, parellel to fender structure (m/s).

vy = velocity of fore part of the ship, perpendicular fender structure (m/s).

cj) = angle between ship and fender structure (rad). n = number of observations.

(34)

(35)

Photo 8: 30 s before impact Photo 9: 20 s before impact

(36)

Vc

Figure 15: Approach velocities and angle

In table 1 category 5' is the same as category 3, except that one value has been omitted.

Table I. Mean values (m) and standard deviations (s).

Category n (m/s) (m/s) mv, (m/s) (m/s) (rad) (rad) 1 55 0.99 0.11 0.09 0.12 0.20 2 15 0.90 0.45 0.14 0.11 0.12 0.11 3 19 1.16 0.61 0.12 0.09 0.18 0.32 4 11 0.62 0.28 0.05 0.05 0.08 0.09 5 10 1.20 0.32 0.09 0.06 0.06 0.03 5' 18 0.12 0.09 0.11 0.11

4.2 Analysis of measurements and extreme situations

The mean values and standard deviations of the observations in table 1 indicate a considerable variation in velocities and angles. The standard deviations lie in the region of the mean values.

According to statistical tests with a reliability of 95% it can be expected:

unladen > 1000 tons (2) greatervx and vy than for laden > 1000 tons (4)

unladen > 1000 tons (2) smallervx than for laden < 1000 tons (5)

unladen < 1000 tons (3) greatervx and vy than for laden > 1000 tons (4)

laden > 1000 tons (4) smallervx than for laden < 1000 tons (5)

The following is stated on the correlations between vx, vy and (f>. The coefficient of

correlation OV j V j between vx and vy is, by definition, zero if vx and vy are stochastically

independent, and is, by definition, one, if vx and vv are stochastically dependent.

It was shown by calculations for all the ships selected (category 1) that the expected value for the coefficient of correlation r is:

rV x V ? = 0.28 , rV i < p = 0.45 , rv > 4 = 0.04

(37)

According to statistical tests it was shown that with a reliability of 90% it can be expected that:

- there is a certain linear correlation between vx and 4>

- vy and 4> are uncorrelated (independent).

The effect of vx, vy and <f> on the impact force is highly different. The maximum impact

force Fy is expressed in the simplest form in Chapter 6 as:

This shows that the force is linear dependent on vy and also dependent on (f>, but much

less than linear. The velocity vx is not included.

The extreme velocities and angles of approach are determined by assuming a Gumbel distribution (Ref. [17]). This distribution function lies asymmetrically around the mean value, and is normally used for determining extreme values (water levels, discharges, etc.).

The following applies to a Gumbel distribution:

Fy = vyV ( k mb w) (4)

where Fy = impact force (N)

k = spring stiffness (N/m) mb w = m ( l + 3 cos2t]))"1(kg)

m = mass of the ship (kg)

Pr (x =s t) = e x p ( - e x p ( - a ( t - u))) = 1 - p (5) where Pr(x t) = chance that x s£ t with exceeding chance p

a = s/oN and oN as a function for n observations, (Ref. [17])

s = expected value of standard deviation

u = m — yN/ a and yN as a function for n observations, (Ref. [17])

m = expected mean value.

The following applies with regard to the reliability interval:

xp - ^ < xp < Xp + IjJ (6)

where y = 2 Ws

2 = normal distribution from table for reliability (1 — 2a) W = V ( ( . l + 1.14 K + 1.10 K2)/n)

(38)

Extreme combinations obtained by the above mentioned method are summarised in table 2.

Independence of the approach velocity perpendicular to the fender structure vy and the

approach angle <t) are assumed, so that the probability of simultaneous exceedance of vy p and (j>p is p2. Moreover, the effect of vy on the impact force Fy is much greater than

the effect of (j) on the final impact force Fy A s a result those combinations are selected where the probability of exceedance for vy is much smaller than for <t>.

Table 2. Extreme combinations.

category ships exceedances extreme combinations vy and tj) with

80% reliability interval 4)* 5)* 1)* 2)* 3)* vy (m/s) (}> (rad) 1 2 x IO"5 4 10' •4 _{0.49 < 0.59 < 0.68} _{0.33 < 0 . 4 1 < 0.49} 2 0.6 x 10"3 1.2 UT -4 0.55 < 0.80 < 1.05 0 . 2 K 0.30 < 0.39 3 _0.6_x_10"5 1.2 lo--4 0.49 < 0.68 < 0.87 0.47 < 0.69 < 0.92 4 0.4 x 10"5 0.8 ur -4 0.22 < 0.35 < 0.48 0.14 < 0.23 < 0.32 5 0.4 x 10"5 0.8 10" -4 0 . 3 K 0.49 < 0.68 0.08 < 0 . 1 K 0.14 3' _0.6_x_10"' 1.2 ur •4 0.47 < 0.66 < 0.85 0 . 2 K 0.30 < 0.38 1) * Chance of exceedance of the combination, in relation to total number of ships.

2) * Number of exceedances per year.

3) * Chance of exceedance of the combination, in relation to category. 4) * Separate probability of exceedance = 10~3.

5) * Separate probability of exceedance = 10"1.

4.3 Conclusions

Unladen ships generally have greater approach velocities and angles than laden ships. In this case large laden ships have an extra small approach velocity and angle. There is a certain dependence of the velocity parallel with the fender vx and the

ap-proach angle <}>: vx is reduced as (t> increases. The velocity perpendicular to the fender

vy is almost independent of the angle of approach cf>.

Extreme combinations of approach velocity perpendicular to the fender vy and angle

of approach <f> are presented in table 2 for the measured situation at the Volkerak locks. The approach velocities and angles which must be taken into account when designing a fender structure in a different situation cannot be indicated from this table because each layout has its own manoeuvres.

Which chance of exceedance must be taken into account is also dependent on the part of the fender structure. The chance of exceedance of the main components of the fender structure is higher than for the different parts.

(39)

5 Special approach situations

5.1 Occurrence of special approach situations

In the lock harbours at waiting places for push tows, a number of events frequently occur simultaneously:

- ships to be locked berth at the waiting place

- ships to be locked leave the waiting place, to sail to the lock - ships leave the lock.

' ' / / / ' '

y / / / / / / / .

Figure 16: Manoeuvres near lock harbour

In this case the ships will meet each other at a short distance, giving rise to unexpected situations where extra manoeuvres are required for the ships to avoid each other. Here it is possible for the fender structure to be approached at a too great velocity or angle of the ships.

The reasons for the unexpected situations may be poor visibility, wind, communication problems, inexperience, pleasure shipping, etc. In confined, busy lock harbours, these unexpected meetings can regularly occur, but the ship velocities are limited. In spacious lock harbours unexpected situations will occur less frequently, but here the ship vel-ocities are perhaps higher.

The purpose of the following calculations is to indicate the upper limit of the approach velocities and angles for fender structure.

5.2 Mathematical model for ship manoeuvres

The S H I P M A mathematical model has been developed at Delft Hydraulics for calculat-ing the course of a push tow. (Ref. [19]) Steercalculat-ing is effected by means of an automatic pilot, which anticipates the desired course. The number of propeller revolutions is predetermined.

(40)

Three horizontal equations of motion are assumed (surging, swaying and yawning) for the ship. These equations contain forces of inertia of the ship, resistance, propeller and rudder forces and the effect of wind and current.

The coefficients for the resistance, propeller and rudder forces are determined by model ship tests.

The steering of the ship by means of the rudder with an automatic pilot is such that there is a reaction to the angle between the desired course and the resultant ship velocity, and to the distance between the centre of gravity and the desired course. The mathematical model is extensively tested with prototype and model investigations. 5.3 Calculations carried out

The calculations were carried out on a push tow - 2 x 2 Europa II barges 153 x 22.80 n r 4 m 13,000 m3 38 x 10 n r 1.80 m 468 m3 horizontal dimensions draught water displacements push tug Vulcaan I horizontal dimensions draught

water displacements

2 propellers, blade diameter 1.95 m, max. capacity 2 x 1500 k W

Two calculations are presented with manoeuvres caused by a sudden change of the rudder angle:

1. waterdepth 5 m, no wind

before change of rudder angle a constant ship speed 2.5 m/s constant revolutions per second n = 4.59 rev/s

rudder angle (Fig. 17):

o 10

45°

time (s)

Figure 17: Rudder angle/time relation, calculation 1 and 2

2. waterdepth 5 m, no wind rudder angle as in 1. (Fig. 17) revolutions per second n (Fig. 18):

(41)

n — r 4,59 rev.Is

\

45 0 20 \ time (s) -4,59 rev.Is

Figure 18: Revolutions per second screw/time relation, calculation 2

The notations for the displacements and velocities are (Fig. 19) xv, xc = displacement V , C in x-direction (m)

yv, yc = displacement V , C in y-direction (m)

<p = angle between ship axis/x-axis (°)

time intervals 30 s

(42)

500 —• x(m) 750 250

I

time intervals 30 s y(m) 500

Figure 21: Course of push tow, calculation 2

xv, xc = velocity V , C in x-direction

yv, yc = velocity V , C in y-direction

4> = angular velocity of ship

(m/s) (m/s) (°/s)

The course of the ship is indicated in Figures 20 and 21, and the displacements and velocities of the 2 calculations are represented in Figures 22 and 23.

5.4 Processing of the results

The results of the calculation indicate the upper limit of the approach velocities and angles of the fender structure.

By projecting the layout of the lock harbour on the course of the push tow (Figures 20 and 21), the maximum approach velocities and angles can be directly determined. It is supposed, as an example, that the push tow sails parallel with the fender structure, at a distance of 30 m from the ship axis to the fender structure. The push tow must deviate in the manner indicated in calculations 1 and 2.

- calculation 1

From figure 22 with yv = 30 m the push tow reaches the fender structure after

t = 63 s.

In this case yv should be = 0.80 m/s and tt> = 13°

- calculation 2

From figure 23 with yv = 30 m the push tow reaches the fender structure after

t = 45 s.

In this case yv should be = 0.42 m/s and (j> = 10°

(43)

500 m 200 m 100° 400 m 160 m 80° 300 m 120 m 60 200 m 80 m 40° 100 m 40 m 20 2 mis 2 m/s 0.57s 1.6 mis 1.6 m/s 0.4°Is 1.2 mis 1.2 m/s 0.3°/s 0.8 mis 0.8 mis 0.27s 0.4 mis 0.4 mis 0.77s

(44)

500 m 200 m 100 400 m 160 m 80 300 m 120 m 60° 200 m 80 m 40 100 m 40 m 20° 2 mis 2 m/s 0.57s 1.6 mis 1.6 m/s 0.4°/s 1.2 mis 1.2 mis 0.3°ls 0.8 m:s 0.8 mis 0.2°Is 0.4 m/s 0.4 mis 0.77s

Figure 23: Displacements and velocities, calculation 2

(45)

The approach situations which must be assumed when designing fender structures is not generally indicated.

It may be stated that if the above mentioned examples are used, the fender structures are considerably loaded, but this relates to a physical upper limit.

(46)

6 Ship/fender system, without the influence of water

6.1 Introduction

The purpose of the following presentation is to give insight into the forces and the movements of the ship and the fender during their contact without the influence of water: it implies a theoretical situation.

1. before contact 2. impact

3. inward springing 4. release

5. after contact

Figure 24: Example of a manoeuvre

A ship/fender system is assumed without the influence of water. The fender may spring in the perpendicular direction (y-axis) as an undamped system, and is assumed to be infinitely rigid in the parallel direction (x-axis). The ship can perform all the horizontal movements during the impact process: surging, swaying and yawing, in addition to its original movement.

The ship sails at a uniform forward (= longitudinal) velocity, drift (= transverse) velocity and yaw (= angular) velocity at a certain angle to the fender, collides against the fender and causes the fender to spring inwards, causing the ship to turn away, after which the ship is released from the fender.

Before the first contact between ship and fender, the ship sails at a uniform velocity, and the fender is at rest. A t the moment of impact the fender will follow the movement of the fore part of the ship in the y-direction in a very short time: the impact stage. The fender then springs inwards, and the fore part of the ship and fender may move in the y-direction as a unit: the movement stage. After the ship is released from the fender, the ship and fender move separately.

(47)

The following is further assumed:

- ship: rectangular B x L , mass m, moment of inertia I, point of impact fore part of ship V , centre of mass C.

- fender: spring stiffness k, mass M . - friction: coefficient of friction tg p\ - distances a and b constant during contact:

a = 0.5(L cos dp - B sin dp) (7) b = 0.5(L sin dp + B cos dp) (8) - fore part of ship and fender associated by velocities:

xv = xc + bdp (9)

yv = yc - adp (10)

where xv, xc = velocities of V , C in x-direction

yv> Yc - velocities of V , C in y-direction dp = angular velocity

- the variations of dp and terms with dp2 are ignored.

y

Figure 25: Notations

6.2 Impact stage

During the impact stage the ship is decelerated in a very short time and the fender at rest is accelerated.

A t the beginning of the impact stage (t0) the ship is sailing and the fender is at rest,

whilst at the end of the impact stage (to), the fore part of the ship and fender are moving in the y-direction, together, at a new velocity. This means an inelastic impact. The

(48)

tg = time just before impact to = time just after impact

Figure 26: Situation during impact stage

duration of impact is so short that fender and ship are hardly displaced. The impact is defined thus:

S = / F d t

The impulse equations are:

ship: m(Xc(to) - xc(t0)) = S tg 6

m(yc(to) - yc(t„)) = s

I(4>(t6)-<i>(to)) = S ( b t g ( 3 - a ) fender: M(yv(t/))) = - S

From (10), (12), (13) and (14) it follows that yc(t0) - a<j>(to) S = -1/ ma \ 1 1 + ( a b t g | 3 ) + -m \ / M (11) (12) (13) (14) (15) Now let: CrH — Ml ma 1 + - - ( a - b t g | 3 ) + 1 m \ I (16)

Then the velocity of the fore part of the ship and fender in the y-direction, immediately after impact, from (14), (15) and (16), becomes:

yv(to) = cc dyv(to) (17)

(49)

1.0 (-) 0.5 M • 0.05 tgp = 0.3 BIL = 0.2 10 20 30 40 50 60 70 —*

Figure 27: Reduction of initial velocity as a function of approach angle

The relation ccd/(t» for various M/m and tg |3 values and for B / L = 0.2 is given in figure

27.

It will be seen that the reduction in the initial velocity is greater (or cc d smaller) as the

fender mass increases and the approach angle and friction decrease.

6.3 Movement stage

During the movement stage, the fore part of the ship and fender move together in the y-direction. The initial velocities of this movement stage are the final velocities of the impact stage. The end of the movement stage is reached when the force between the ship and the fender is negative. (Fig. 28)

(50)

T k

Figure 28: Situation during movement stage

The movement equations are:

ship: mxc = Fx (18)

myc = Fy (19)

I<|> = Fxb - Fya (20)

fender: tg P Fy = Fx (21)

M yv - kyv = - F , (22)

Connection between fore part of ship and fender from (9) and (10):

xc = xv - b$ (23)

yc = yv + a$ (24)

From (18) to (24) it follows that:

m(xv - b$) = - (Myv + kyv) tg |3 (25)

m(yv + ac^) = - ( M yv + kyv) (26)

1$ = ( M yv + k yv) ( a - b t g P ) (27)

From (26) and (27) it follows that:

/ M + - \ yv + kyv = 0 (28)

ma

1 1 + j ( a - b t g p ) l

(51)

It is assumed that there is a virtual ship mass mm d:

mm d = cm d • m (29)

( ma \->

where cm d = I 1 + —-(a - b tg (3) I (30)

The coefficient cm d indicates what proportion of the total mass of the ship contributes

to the movement of the fore part of the ship and fender in the y-direction. The relation cmd/ct> for different B / L values is presented in fig. 29. It is clear from this that the

coefficient cm d has a minimum of 0.25 for an angle of approach <b = 0, B / L = 0 and

tg (3 = 0 and may approach a maximum of 1 at an angle of approach of cp = 90°. According to (28), the system is reduced to a one-mass spring system, with a mass ( M + cm dm ) and a spring stiffness k.

(52)

A t t = t{) = 0 are the initial conditions yv = yv(to) and yv = 0. The solution of (28) is then:

yv = yv(to) cos tot (31)

where co2 = k / ( M + cm dm ) (32)

From (31) it follows that:

sin cot (33)

yv = co

The force between the ship and the fender is with (22), (31) and (32): Fy = - ( M yv + kyv)

= — (—Mco + k/co)yv(to) sin cot (34)

The maximum force and displacement are reached after a quarter of a period, i.e. at time t2 = JT,/(2CO).

The release of the ship and fender is reached after half a period, i.e. at time t3 = JT/CO.

6.4 Special situations

Standstill of the fore part of the ship

If the angle of approach and/or the coefficient of friction is so great that the fore part of the ship in the x-direction (= longitudinal direction of fender) comes to a standstill during contact, then the equations given in paragraph 6.3 can no longer be assumed. In a (theoretical) situation where B = 0 (width of the ship), M = 0 (fender mass) and with the initial velocity along the longitudinal axis of the ship, the fore part of the ship comes to a standstill if:

The resultant force of the fender coincides with the longitudinal axis of the ship with tg p = 1/tg cb.

The impact of a ship against a dolphin, in which the ship slides along the dolphin, is very similar to the situation described in paragraph 6.3. However, the point of impact is no longer the fore part of the ship, but is located somewhere along the hull of the ship.

tg p > 1/tg 4> (35)

Dolphin

(53)

The ship has a transverse velocity for coming into contact with the dolphin, and has a longitudinal velocity for sliding along the dolphin.

The dolphin is represented by a point, has no mass, has a spring stiffness k in all directions, and the coefficient of friction is tg p.

The dolphin will spring inwards in the direction of the resultant force, and since this force is at an angle (90° - P) to the axis of the ship, the dolphin will also be displaced in this direction (= z-direction). (Figure 30)

It can be demonstrated that here too the system can be reduced to a one-mass spring system. A coefficient cm d, according to equation (30), which indicates the proportion

of the mass of ship which contributes to the movement, may also be used. The displace-ment of the dolphin in the z-direction is:

ZvOo) .

sin tot (36)

zv =

to

where = initial velocity of the dolphin in z-direction = (yc - acj>)/cos P

= k/(cm dm)

b

a = distance from C to V in x-direction = B/2 = half width of the ship. The resultant force F2 between ship and dolphin is:

k zv(t0) sin cot (37) (T) c b x a

(54)

6.5 Example of a fender structure - data L = 150.30 m fender structure: M = 1.1 x 106kg B = 22.80 m k = 4 x 106 N/m m = 11.3 x 106kg I = 2.13 x 1 01 0k g m2 friction: tg (3 = 0.15 Xv(t0) = 0 y»(to) = 0 <Kto) = 30° Xv(t0) = -0.866 m/s y»(to) = - 0 . 5 m/s <t»(t0) = 0 - impact stage

From (7), (8) and (16) it follows that: cc d = 0.796

FV(N) Yv(m)

1 I 2 3

-Ka)

Figure 31: Impact force and displacement/time relation, example paragraph 6.5

(55)

- movement stage

From (30) it follows that: cm d = 0.378

From (32), (33) and (34) the angular frequency, displacement and force between the ship and fender, in the y-direction, are:

co =0.863 (1/s)

yv = -0.461 sin (0.863 t) (m)

Fy = 1.467 x 106 sin (0.8631) (N)

The impact force Fy and displacement are presented in figure 31 as a function of time.

6.6 Conclusions

In a theoretical situation without the influence of water the contact between the ship and fender structure can, according to some assumptions, be described by means of analytical expressions.

On first contact between ship and fender structure the fender structure follows the movement of the fore part of the ship. If the fender mass is not negligible in relation to the mass of the ship, the initial velocity of the ship/fender is adapted. The reduction in this initial velocity is all the greater if the fender mass increases, the angle of approach decreases and the friction decreases.

The further joint movement between the fore part of the ship and the fender structure can be described according to a one-mass spring system.

The proportion of the mass of the ship of this movement forms part of the actual mass of the ship. The reduction in the mass of the ship is all the smaller if the angle of approach increases, the friction increases or the distance from the point of impact to the centre of mass decreases.

The course of the impact force and displacement in time is sinusoidal, which confirms the assumption that there is no damping.

Attention must be given to the question of whether the fore part of the ship comes to a standstill in the longitudinal direction of the fender structure. If it does, then another calculation scheme must be used.

(56)

7 Ship/water system without fender

7.1 Introduction

In the case of an accelerating or decelerating ship in water, the equation of motion for the ship/water system is in principle:

mx + fw(t) = 0 (38)

where: m = mass of the ship x = acceleration of ship fw(t) = hydrodynamic force

In this chapter the effect of the water on the accelerating or decelerating ship is deter-mined, i.e. the hydrodynamic force is determined for the situation of shallow water and without the influence of walls.

The first question discussed is what sort of terms are required to describe the hydro-dynamic force, these are acceleration and velocity terms.

The size of these terms must then be determined. This is done in this chapter by means of calculations and a model investigation.

7.2 Terms of the hydrodynamic force

When a body accelerates (or decelerates) in water, hydrodynamic forces are exerted on that body. These forces are the result of a change in the pressure distribution around the body.

If a submerged body is assumed, in water of infinite depth (figure 32), where the water is also incompressible and frictionless, then a resistance force from the water is exerted

Figure 32: A n accelerating body in infinite water

(57)

on that body when it is accelerated. This force is proportional to the acceleration of the body. The proportionality constant is indicated by the added water mass. The following applies:

fw = ax (39) where fw = hydrodynamic force

a = added water mass x = acceleration

If the body is subjected to a prescribed acceleration, then the hydrodynamic force can be directly calculated as a function of time. The added water mass is in this case constant, and is dependent only on the geometry of the body.

Figure 33: A harmonic movement of a body near the water surface

When the body is located near the water surface, waves are generated (figure 33). These waves draw energy from the movement and radiate it. To describe this situation an acceleration term is insufficient and a velocity term must also be introduced. The following applies for a harmonic movement:

fw = a(co)x + b(to)x (40)

where fw = hydrodynamic force

a(co) = added water mass b(cu) = added water damping x = acceleration

x = velocity

Here the proportionality constants are called hydrodynamic coefficients, and consist of an added water mass for the acceleration term, and an added water damping for the velocity term. These hydrodynamic coefficients are dependent on the frequency of the harmonic movement, because these coefficients all contain information on the (fre-quency-dependent) wave pattern. The waterdepth is also important because it influ-ences the wave pattern.

(58)

Figure 34: A n arbitrary movement of a body near the water surface

If the body is subjected to an arbitrary movement varying in time (fig. 34), instead of a harmonic movement, equation (40) will not suffice. The sum of 2 convolution integ-rals must then be used: one for the acceleration term, and one for the velocity term. A part of the past is incorporated in these convolution integrals: the memory effect. By this is meant that not only momentary accelerations and velocities are included, but also previous accelerations and velocities (integration of - infinity to t) The following applies:

t t

fw(t) = | A ( t - x) x(x)dx + J B(t - x) x(x)dx (41)

— oo —oo

where fw(t) = hydrodynamic force

x(t) = acceleration x(t) = velocity

This memory effect must be introduced because a previous movement, with the as-sociated generated wave (located some distance from the body) also exerts an influence on the pressure distribution round the body (= hydrodynamic force) at the present moment. It is also stated that the arbitrary movement is a summation of all sorts of harmonic movements with its own added water mass and damping.

It can be demonstrated, that the terms A(t) and B(t) from equation (41) can be deter-mined as follows:

1 "

A(t) = - J"a(to)e""'dto (42)

2k 0

1

-B(t) = - J"b(to)e,lo,dco (43)

2 J T0

where a(co) = added mass of water b(to) = added damping of water

Thus the hydrodynamic coefficients a(co) and b(to) for the entire frequency range from 0 to infinity are included in these terms A(t) and B(t).

(59)

Thus if it is possible to determine the hydrodynamic coefficients a(co) and b(co) for the entire frequency range from 0 to infinity, then A(t) and B(t) can be calculated on the basis of equations (42) and (43). These terms A(t) and B(t) are substituted in equation (41) with the given function of x(t) (acceleration) and x(t) velocity, this gives the hydrodynamic force fw(t) as a function of time.

For subsequent determination of the impact forces in Chapter 9, use is made of the latter approach to the hydrodynamic force.

In the above presentation the most important conditions are connected: - there is no influence from viscosity and friction.

- all the hydrodynamic forces are based on deviations from the state of rest or uniform movement.

- the theory is intended for horizontal movements of the body. For vertical move-ments of floating bodies other terms are required.

- all the equations and coefficients related to one and the same direction of move-ment.

- the ship water system is linear, i.e. with variations in acceleration and velocity of the body, the hydrodynamic forces vary accordingly. The displacements of the body must therefore be small in relation to the other dimensions of the body.

7.3 Calculations of the hydrodynamic forces 7.3.1 Notations

A ship has 6 movement possibilities

horizontal 1 surging 2 swaying 3 yawing vertical 3 heaving 4 rolling 5 pitching

Figure 35: Movement possibilities

In this report only the horizontal movement possibilities surging (1), swaying (2) and yawing (6) are considered.

Harmonic movements are required for determining the hydrodynamic coefficients (equation 40). The equations of motion for the ship/water system are (with a symmet-rical ship):

(60)

(m + aH)X | + bux, = f, (44)

(m + a2 2) x2 + b2 2x2 = f2 (45)

(I + a(„,)x6 + b6 6x6 = f6 (46)

where an, a2 2 = added water mass in surge, sway direction

aA 6 = added moment of inertia in yaw direction

bH, b2 2 = added damping in surge, sway direction

b ^ = added rotation damping in yaw direction

x,, x2, i i , x2 = acceleration, velocity in surge, sway direction

x6, x6 = angular acceleration, velocity in yaw direction

f|, f2 = external force in surge, sway direction

f6 = external moment in yaw direction

m = mass of ship

I = moment of inertia of ship.

The added mass terms are called the in-phase terms, i.e. in phase with the acceleration of the ship, and the added damping terms are called the out of phase terms, i.e. 90° out of phase with the acceleration.

In addition to the added water mass and damping terms mentioned here, coupling terms are used for asymmetrical ships, i.e. for a pure swaying movement, yawing forces are also generated. However, this aspect is not discussed here.

The coefficient aH and b,, in the surge direction are ignored during further investigation

because they are many times smaller than the mass of the ship.

For more information on hydrodynamic coefficients, see references [2], [8] and [12].

7.3.2 Two-dimensional potential flow (Ref. [2])

Consideration is given to a transverse section of a horizontally oscillating, rectangular ship of infinite length:

Figure 36: Transverse section for two-dimensional potential flow

(61)

The two-dimensional character is expressed in the vertical and horizontal water vel-ocities. The notations are:

u = horizontal water velocity w = vertical water velocity

uk c = horizontal water velocity underneath the ship

T) = wave height B = width of ship D = draught

h = water depth (mean) p = pressure

g = specific mass of water

The following applies to a non-compressible, non-stationary and frictionless flow in plane x-z (Euler and continuity equations):

3u 9u 3u 1 3p — + u — + w = (47) 9t 3x 3z q 3x 3w 3w 3w 1 3p 3t 3x 3z q 3z 3u 3w 3x + 3 z = ° <49>

It is assumed that the flow is free of rotation, and that it is potential flow. Then a velocity potential (j> exists, for which the following applies:

3<tj

U = 3x <50>

3<t>

W = 3z <51>

Continuity equation (49) with equation (50) and (51) becomes the Laplace equation: 32d> 32cj>

3x2 + 3 z2 =° <52>

Equations of movement (47) and (48) are added together, with substitution of (50) and 3cj>\ / 3ch\2