Indian Journal of Engineering & Materials Sciences Vol. 6, August 1999, pp. 188- 197

Analytical investigation of wave slamming loads on horizontal circular cylinders

Gazi Md . Khalil

Department of Naval Architecture and Marine Engineering, Bangladesh University of Engineering and Technology Dhaka 1000, Bangladesh

Received 18 Jun e 1997; accepted 6 May 1999

Hori zo ntal cylinders are subjected to impact loads when suddenly submerged in water. Analys is of the wave forces acting on a fixed , slender, horizontal circular cylinder in the vic inity of the free surface is described here by taking into account the int ermittency of submergence and wave slamming by a new analytical approach for deriving an expression for th e slamming coefficient. A computer programme is developed on the basis of the aforesaid analysis. The programme is written in FORTRAN 77 and executed on the IBM 4331 L02 computer. The computational results are plotted and illlcrpreted to explain the salient features of wave slamming. These results are expected to be useful in the assessment of hydrod ynamic loads on offshore structu res like the braces between legs or hull s for jacket structures and semi-submersibles.

When a body enters a free water surface at speed, it experi ences large trans ient forces due to the acce leration fi e ld it sets up within the water. These forces are generally termed impact or slamming loads . information concern ing the forces acting on bluff bodies subj ec ted to wave slamming is of great imporlance 111 naval architecture and ocean engineering. The design of structures which must survi ve in a wave environment is dependent on the knowledge of forces which occur at impact as well as on the dynamic response of the system. Two typical examples include the structural members of offshore drilling platforms at the splash zone and the of ten­encountered s lammi ng of ships. Impulsive forces can arise due to severe pitching motion of high-speed ships in heavy seas. If the pitching is sufficiently heavy, the bow of the ship may leave the water and when it plunges into the water again, the slamming phenomenon occurs and a large instantaneous force is generated near the bow. A bar struck by a hammer, would vibrate in its natural frequency of vibration and would even tuall y come to rest due to damping. Similarly. a ship when struck by the slamming force , wo Id vibrate in its natural frequency and eventually cotTle to rest Hi gh stresses can result from this whipping action causing severe hull damage sometimes. Offshore structures often have appreciable numbers of main structural and bracing members near the still water leve l. As waves pass the structure, some of these members are alternately exposed to the air and then submerged in the water: ·Each time they enter the

water, there are slamming forces. Fai lures of some members have been partially attributed to slamming.

The history of research on slamming and impact loads is re lat ive ly recent mainly because it is, as far as naval archi tecture is concerned, predominantly a high speed phenomenon. The dynamics of water entry of projectiles has been extensively studied and includes a slamming phase which partially governs the underwater trajectory. The general problem of hydrodynami c impact has also been stud ied extensively motivated in part by its importance in ordinance and missile technology I . A large number of mathematical model s have been developed for cases of simple geometry such as spheres and wedges. These models have been well supported by experiments. But, the special cases of wave impacts have not been studied extensively .

Wave slamming on slender structural members has been discussed in several reviews and texts, e.g., Miller2 and Sarpkaya and IsaacsonJ

• The vertical force on a member may be considered to be made up of components corresponding to the slamming force, a ti;ne-vary ing buoyancy force resulting from the intermittent SUbmergence, and forces that may be described by the Morrison equation during complete submergence.

Dalton and Nash4 conducted slamming experiments with a 1.27 cm diameter cylinder with small amplitude waves generated in a laboratory tank. But, the data exhibited large scatter and showed no particular

-

KHALIL: ANALYSIS OF WAVE FORCES ON HORIZONTAL CIRCULAR CYLINDERS 189

correlation with either the predictions of the hydrodynamic theory or identifiable wave parameters.

Mille r5 presented the results of a series of wave-tank experiments to establish the magnitude of the wave­force slamming coeffi c ient for a horizontal c ircular cylinder. The average slammjng coefficient found in these experiments was 3.6 for the trials in which slamming was dominant.

Fa ltinsen e( a l6 in vestigated the load acting on rigid

hori zontal c ircul ar cylinders (with end plates and length-to-diameter ratios of about one) which were forced with constant ve loc ity th rough an initially calm free surface. T hey found that the slamming coeffi c ient ranged from 4 .1 to 6.4. They also carried out experiments with fl exible hori zontal cylinders and fo und that the analyticall y predicted values were always 50-90% lower than those found experimentally.

Khalil and Miyata7 conducted a seri es of experiments on the evolution of forces acting on shallow ly submerged horizontal cy linders subjected to impact by sinusoidal surface waves. T he principal concl usion of thi s ex perimental work is that the break ing of waves behi nd a shallowly submerged cylinder is primarily responsible for the generation of nonlinear wave forces. The negat ive drifting force, which acts on the cylinder, is a direct consequence of wave breaking. This force attains a peak value when the top su rface of the cy I i nder pierces through the free water surface . This has been discussed at length by Khalil ~ . Wave s lamming is one of the most significant phenomena assoc iated with the offshore platform horizontal members in the splash zone. It is an area of hydrodynami cs st i II poorly understood, and further studies are needed to unders tand it completely.

Theoretical Analysis The general case of hydrodynamic impact is usually

described by us ing incompressible potential flow theory. The particular problem considered here is treated by a two-dimensional analysis for a horizontal circular cylindrical sect ion, assuming that the wave system propagates in a direction normal to the hori zonta l cylinder. A pictorial illustration of the si tuation being analysed is shown in Fig. I where the height of the cy linder centre is H above the mean water

level, 11 is the wave elevat ion measured relative to the mean water level and:: is the extent of penetration of the cylinder in the water. The circular cy li nder has radius r, wit h the immersed area Ai defined by the angle 8.

The total vertical force acting on the immersed cylinder section is made up of a buoyant force and a hydrodynamic force of inertial nature due to the body­wave interaction . The force on the c ircular cylinder is represented by Eq. (I ).

· .. ( I )

where Pb is the buoyant force given by re lation (2):

· .. (2)

The inert ial hydrodynamic fo rce is obtained by generali zing the re lations given by Kaplan and Hu9 and Kaplanl o for body-wave interaction as per Eq. (3):

· .. (3)

where p is the density of water, m is the vertical added mass of the cy linder section, 1'1 is the vertical wave

ve locity and ii the vertical wave accelerati on . The first

H

HW L

Fig. I- Defin ition sketch of the horizontal circular cylinder subjected to wave slamming

190 INDIAN 1. ENG. MATER. SCI., AUGUST 1999

term in Eq. (3) arises due to the spatial variation of the wave characteristics and the second term arises from the lime rate of change of fluid momentum associated with the immersed portion of the cylinder section.

The expression for Pi can be simplified as:

P A·· .. . am

i = P iTl +m Tl+Tl-ar

or

P ( A ).. . am az

j= m+p j Tl+Tl~at

or

1>; = (m + pA)it ~71'12 . . . (4)

Thus, the total vertical force acting on the immersed cylinder section is represented by expression (5):

· .. (5)

This expr~ssion holds good only if there is water contact re~ulting immersion of a portion of the cylinder section. An evaluation of this force expression, which is a two-dimensional force per unit length of the cylinder, requires determination of the immersed area Ai, the vertical added mass frI, as well as its rate of

h . h ' . am . c ange Wit ImmerSIOn a z ,the vertical wave velocity

1'1 and the vertical wave acceleration it .

The immersed area, Ai is expressed as a function of the angle 0 (defined in Fig. 1) by relation (6):

· .. (6)

where r represents the radius of the circular cylinder. Thus, we can write Eq. (6) as Eq. (7)

· . . (7)

where II (0) = 0 - sin 0

The extent of penetration z of the cylinder in the water can be related to the variables rand 0 by Eq. (8):

. . . (8)

The problem of the vertical added mass of segments of a circle has been solved by Taylorll , as described by Eqs (9) or (10) :

[

21t3

(I-cosO) 1t 1 1 2 -- 2 +-(l-cosO) m=-pr 3 (21t-0) 3

2 +(sinO~O)

. . . (9)

. .. (1 0)

where

21t3 (1- cos 0) 1t . 12(0)=- 2 +-(l-cosO)+ (smO-O)

3 (21t- 0) 3

... (II)

At 0 = 2n:, h takes the indeterminate form 0/0. In such a case:

It may be mentioned here that if two functions <p(x) and \jI(x) vanish at x = a, the fraction <p(x)I\jI(x) is an indeterminate form of the type ala at x = Q . To find the li.mit of <p(x) I \jI(x) in the case of an indeterminacy of the form 01 0, the ratio of the functions can be replaced by the ratio of the derivatives, and the limit found of this new ratio. This rule was given by the French mathematician, L' Hopital, and is usually named after him (SmirnovI2

) .

Thus, Eq. (9) can be written as Eq. (12):

am _ 1 2[21t3 {sinO 2(l-COSO)} ---pr -- +---ao 2 3 (21t-0)2 (21t-0) 3

+ ;sino+(COsO-l)] ... (12)

From Eq. (8), we get:

az r . 0 -=-SIO-ao 2 2

. ~

KHALIL: ANALYSIS OF WAVE FORCES ON HORIZONTAL CIRCULAR CYLINDERS 191

But dm = dni d8 ' dz d8 dz

or

dm _ dm/d z -- - -dz d8 d8

Hence, we get Eg , (13) :

dm=~[2rr'{ sin8 +2(l-COS8)} dz . 8 3 (2rr-8)2 (2rr-9) 3

sm -2

+~ Sin8+ (COS 8-1 ) ] " , (13)

Eg, (13) can be presented as expression (14) also:

dm - = pr f l( 8) dz .

.. , (14)

where,

fl (8) = _I _'[ 2rr ' { sin 8 + 2(1 - cos 8)} . . 8 3 (2rr - 8) 2 (2rr _ 8)3

sm -2

+~ sin8+(COS8-1)] " ,(15)

or

d f J 8 f l (8) = - - /s in -

. d8 2 , , , ( 16)

According to L' Hopital's rule

Limfl (8) = IT . "

The moti on of the free surface is re lated to its maximum amplitude by ex press ion ( 17):

. (2m) 11 = As m T , . , ( 17)

where A and T are the amplitude and period of the free surface oscillations respectively ,

Therefore,

, 2M (2m) 1] =TcOS T , , . ( 18)

The maximum velocity of the free surface oscillations is given by Eg, (19):

v = 2rrA m T

From Egs ( 18) and ( 19), we get:

, (2 ITt ) 11 = V m cos T Therefore

,2 V 2 2 (2rrt ) 11 = m cos -T

or

or

or 2

.. Vm 11=--11

A2

.. , ( 19)

, , , (20)

, , , (21 )

The expression for the slamming coeffi c ient, C. , can be written as:

192 INDIAN 1. ENG. MATER. SCI., AUGUST 1999

or

F C =--

S pU;r · .. (22)

where rand D are the radius and diameter of the circular cylinder, respectively.

From Egs (5) and (22), we get Eg. (23):

A A ).. dm . 2

pg j + (m + Pill +-11

or

T. = grAj I U 2 2

m r

or

T. =(~pj I u 2 2

m r

C = dZ S pU;r

· . . (23) or

The well-known Froude number is evolved if a dimensionless relationship is set up between the inertia and the gravity forces. Normally; the Froude number is considered as an indicator of geometrically similar flow with respect to both velocity and pressure phenomena when gravity effects such as surface waves are involved. In the present case, it can be represented by Eg. (24):

F=~ '.rg;

We can write Eg. (23) as Eg. (25):

where

T. = pgA j

I pU~r

T _ (m+pA)ii

2 -pU~r

dm .2 - 11

T _ dZ

1 - ---:--. pU~r

TI , T2 and TJ can be simplified as:

T. = gAj I U 2

m r

· .. (24)

· .. (25)

T. __ 1_,2 II (0) I - F2 2' 2 , r

or

1 ~ =--2 11(8)

2F,

T _ (m+ pAj)tj

2-pU~r

or

or

or

T2 = _ 1 pr2I2 (8)11 _! pr2 II (8)11

2 pA 2 r 2 pA 2 r

or

or 1 r 11

T2 = --[12 (8) + II (8)]--2 AA

... (26)

... (27)

KHALIL: ANALYSIS OF WAVE FORCES ON HORIZONTAL CIRCULAR CYLINDERS 193

or

or

T, = f,(e{ 1- ~: ) . .. (28)

The Eqs (25) to (28) finally lead to Eq. (29) relating the coefficient of slamming as:

I I r 7J C,. = -2 f, (8 ) - -[I, (8) + 12 (8)]--

. 2F " 2 A A r

where 1'\0 is the height of the bottom surface of the circular cylinder above the mean water level. This relation can be written as:

1'\ 1'\0 + Z -=--A A

or

1'\ 110 Z r -=-+--A A r A

.. . (30)

Eq. (8) can be written as Eq. (31):

+ /, (e{ 1 - ~: ) .. . (29) Z a - = l-cos­r 2

. . , (31)

From Fig. I, the wave elevation may be expressed as:

11 = 110 + z

..... ~

'::::: ~,

<> ",,'

'" N

0

.....

'"

.n, o

:~----~-----.-----,------~~ 0.0 0.5 1.0 1.5 2.0

z/r

For 0 ~ e ~ 2x, Eq. (31) gives 0 ~ ~ ~ 2. r

..... ~ Cl..

E

~~----------------------~~~ N

"!

<>

~+------r-----.----~~----~-J 0.0 0.5 1.0

zlr 1.5 2.0

Fig, 2-Y ariation of the area as a function of immersion of the Fig. 3-Variation of the added mass as a function of immersion of circular cylinder the circular cylinder

194 INDIAN 1. ENG. MATER. SCI., AUGUST 1999

Results and Discussion

A computer program has been developed on the basis of the theoretical analysis of the problem presented above. The program is written in FORTRAN 77 and executed on the mM 4331 L02 computer. The computational results are plotted in terms of non­dimensional parameters and interpreted with a view to explain the salient features of wave slamming on horizontal circular cylinders.

Results for Cs with respect to ~ are generated from r

Eqs (29) to (31) by varying e from 0 to 21t, for the

specified values of Fr, ~ and 2l2.. A A

Fig. 2 shows the variation of the non­dimensionalized immersed sectional area (A/rl) of the horizontal circular cylinder as a function of relative submergence (zlr). The non-dimensionalized immersed sectional area is found to increase almost linearly with the relative submergence.

Fig. 3 shows the variation of the non-

EIN "0"0

'" ....

<> ..

o+------,r-----~------_r------~ o O.~ to

z T

1.~ 20

Fig. 4--Variation of the rate of change of added mass with immersion of the circular cylinder

dimensionalired added mass (p~2 ) of 'he circular

cylinder as a function of relative submergence (~)­The non-dimensionalized added mass is observed to increase continuously with relative submergence.

Fig. 4 shows that the non-dimensionalized rate of

change of added mass am ~pr) decreases az I' continuously up to the value of zlr = 1.5, but then onwards it increases sharply.

Fig. 5 shows the variation of the slamming cbefficient (Cs) with the relative submergence (zlr) for different ratios of cylinder radius to wave amplitude (riA). The ratios of riA are taken as 0.0125, 0.025, 0.0375,0.05,0.0625,0.10,0.125,0.15 and 0.20. For all the nine ratios, the cylinder is located at llJA = 0, i.e., the bottom of the cylinder just touches the mean water level (MWL). At the instance of impact, the slamming coefficient is approximately equal to 3.14 for all the nine cases. As the ratio of riA increases (i.e., the radius

'" ",.

0 .ri

'" ..j

~-

'" ,..;

0 ,..;

VI U

'" N

C> N

"!

!3

'" d

0

0

~ =0.00

aGee rIA- 0.0125 x xx x riA = 0.025 uu riA ~ 0.0375

oc:1I:l a r fA = 0.05 ~*"r/A ~ 0.0625 1111111111 r/Aa 0.10 _. riA- 0.125 ~r/A~O.15 + + ++ riA - 0.20

0.5 1.0

.1.... r

15 2.0

Fig. 5-Variation of the slamming coefficient with relative submergence and position of the circular cylinder

')

~

KHALIL: ANALYSIS OF WAVE FORCES ON HORIZONTAL CIRCULAR CYLINDERS 195

III U

o .,.(

o N

r = 0.0125 A

lJQGl31!1 "lolA- 0.00 )(XXXX l)JA_O.2~

UAU "l<IA,. O.~O uooo 'l\JA-0.7~

O+-------r-----~r_----~------_r--~

o O.S 1.0

.1.­r

1.S 2.0

Fig. 6-Variation of the slamming coefficIent with relative submergence and position of the circular cylinder

co to.;

f = 0.025

~ alUU11D 1\.1 A- 0 .00 •••• x 1')./A-0 . 25

.,' " I)./A- 0 . 50 ~ N

•• 0 .... 1J./A:O . 7~

III U

'"

~

"I <>

co

0 OoS 1.0 1.5 2.0

1:-Fig. 7-Variation of the slamming coefficient with relative submergence and position of the circular cylinder

<> t.;

~ = 0.0375 .,.. N

""GIllE! 'I\. IA- 0.00 x xx. x 11./A= 0.25 '. , " 'II./A-0 .50

<> flfle •• 1\./A-0.75 III N U

~ .

~

.,..

.,;

<> +-----,------r-----.r_----~-J o 0.5 1.0

Z T

1.5 2.0

Fig. 8-Variation of the slamming coefficient with relative submergence and position of the circular cylinder

III U

o rri

'" N

0 N

U"I

0 '.

"1 0

0

0

Fig. 9-Variation

0.5

of the

l= 0.05

EI EI EI E113 l)./A=O.OO xx x xx 1'\.1 A :: 0.25 AAA&& 1\01 A = 0.50 (l)GGlGO 1'}./A = 0.7~

1.0 1.5

z -r-

slamming coefficient submergence and position of the circular cylinder

2.0

with relative

196 INDIAN 1. ENG. MATER. SCI., AUGUST 1999

VI U

0 ,..;

L A = 0.0625

~ N

81H:! 8 Q n./A = 0 .00 xxxxx l)./A = 0. 25 AAAAA 'l./A = UO

C> 00000 T\./A = 0·75 N

'"

o+------,.------.-------.------.-~

o 0.5 1.0

Z T

1.S 20

Fig. IO---Variation of the slamming coefficient with relative submergellce and position of the circular cylinder

of the cylinder increases), the value of the slamming coefficient Increases sharply with relative submergence. One of the important assumptions made in the ana lysis of the present problem is that the cylinder diameter is much smaller than the wave ampl itude. As the diameter of the cylinder increases, this assumption is violated, and consequently, the value of the slamming coefficient increases drastically . But at the instant of impact. the value of slamming coeffic ient is the ~amL. ,equal to 3. 14) for all the above mentioned cases .

Figs 6-10 show the variation of the slamming coefficient with relati v'e submergence (Jr) and relative position (T\ jA) for the ratio of riA = 0.0 125, 0.025, 0 .0375, 0.05. and 0.0625 respectively. As the relative position of the cylinder fro m the MWL increases, the value of the slamming coeffi cient decreases considerably . Obviously, the intensity of the impact force decreases as the cy linder moves away from the MWL. The sharp rate of decline of the slamming coefficient curve with the tIlcrease of relative

submergence indicates the impulsive nature of · the slamming force . However, this rate of decline of the slamming coefficient curve with relative submergence increases as the ratio of riA decreases.

Conclusions The computational results and analyses on the

estimation of slamming loads on horizontal ci rcular cylinders show that: The max imum value of the slamming coefficient, at the instance of impact, is computed to be approx imate ly 3.14 for a circular cylinder; The value of the slamming coefficient, at the instance of impact, does not depend on either the diameter of the cylinder or the wave parameters like length, ampli tude and period; The value of the slamming coeffic ient depends on the relative submergence of the cylinder as well as the cylinder position with respect to the mean water level, and; The impulsive natu re of the slamming force is clearly exhibited by all the slamming coefficient curves.

The resulb of these numerical investigations are expected to be useful til the assessment of hydrodynamic loads on offshore structures like the braces between legs or hu lls for jacket structures and semi-submersibles.

Acknowledgement Sincere thanks are due to Prof. Hisashi Kaj itan i and

Shozo Kuzumi San of the Towing Tank Laboratory of the Univers ity of Tokyo for the useful di scussions leading to improved quality of the paper.

Nomenclature A, immersed sectIOnal area of the circular cyli nder C, coefficient of slamming D diameter of the circular cylinder F total vel1ical force acting on the circular cylinder due to wave

slamming F, Froude number g acceleration due to gravity H height of the cylinder centre above the mean water level III added mas~ per unit lengt h of the ci rcular cylinder Ph buoyant force acting on the circular cylinder P, hydrodynamic inertial force acting on the circular cylinder ,. rad lu~ of the circular cylinder T wave period

time Um maximum veloc it y of the free surface oscillations

~ eXlent of penetration of the circular cylinder In the water

11 wave ele vation measured rel at ive to the mean water level

11 .. heigh t of the bottom surface of the c ircular cylinder above the mean water level

p mass dens ity of water

KHALIL: ANALYSIS OF WAVE FORCES ON HORIZONTAL CIRCULAR CYLINDERS 197

. Re(frences I . Szebehely V G, Appl Mech Rev, 12 (5)( 1959) 297-300. 2 MiJler B L. Wave Slamming on Offshore Strnctures, Report

No . NM I-R81 (National Maritime Institute, Feltharn, Midd lesex, U. K.), 1980.

3 S~lfpkaya. T & Isaacson M, Mechanics of Wave Forces on Offshore Structures (Van Nostrand Reinhold, New York), 198 1.

4 Dalton C & Nash J M, Offshore Technol Con/. Houston, Texas, Paper No. OTC 2500, 1976.

5 Miller B L, Paper presented at the Spring Meet Royal [nst Naval Architects, Lolldon , Paper NO.5 (1977), 81-98.

6 Faltinsen 0, Kjaerland 0, Nottveit A & Vinje T, Offshore Tee/lIlol Con/. Houston, Texas, Paper No. OTC 2741 (1977).

7 Khalil G M & Mi yata H, 1 Kansai Soc Naval Architects. lap, 209(1988) 11 -23.

8 Khalil G M, Fellowship Res Bull (The Matsurnae International Foundation, Japan), II (1991 ) 37-54.

9 Kaplan P & Hu P N, Proc Sixth Ann Conf Fluid Mech, University of Texas, (1959).

10 Kaplan P, 1 Ship Res. I (No.3) (1957). II Taylor J L, Phi[os Mag , Ser 7, 9 (No. 55) (1930) 161 -183. 12 Smimov V I, A Course of Higher Mathemntics (Pergamon

Press, London), 1(1964) 153-155.