Ijems 6(4) 188 197

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 June 1997; accepted 6 May 1999
Horizontal cylinders are subjected to impact loads when suddenly submerged in water. Analysis of the wave forces
acting on a fixed , slender, horizontal circular cylinder in the vicinity of the free surface is described here by taking into
account the intermittency of submergence and wave slamming by a new analytical approach for deriving an expression for
the 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
hydrodynamic loads on offshore structures like the braces between legs or hulls for jacket structures and semi-submersibles.
When a body enters a free water surface at speed, it
experiences large transient forces due to the
acceleration field it sets up within the water. These
forces are generally termed impact or slamming loads.
information concern ing the forces acting on bluff
bodies subjected 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 often-
encountered slamming 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 eventually 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 High 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 level. 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. Failures of some
members have been partially attributed to slamming.
The history of research on slamming and impact
loads is relatively recent mainly because it is, as far as
naval architecture 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
hydrodynamic impact has also been studied extensively
motivated in part by its importance in ordinance and
missile technology I . A large number of mathematical
models 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.,
Miller
2
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-varying 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.
Miller
5
presented the results of a series of wave-tank
experiments to establish the magnitude of the wave-
force slamming coefficient for a horizontal circular
cylinder. The average slammjng coefficient found in
these experiments was 3.6 for the trials in which
slamming was dominant.
Faltinsen e( al
6
investigated the load acting on rigid
horizontal circular cylinders (with end plates and
length-to-diameter ratios of about one) which were
forced with constant velocity through an initially calm
free surface. They found that the slamming coefficient
ranged from 4 .1 to 6.4. They also carried out
experiments with flexible horizontal cylinders and
found that the analytically predicted values were
always 50-90% lower than those found experimentally.
Khalil and Miyata
7
conducted a series of
experiments on the evolution of forces acting on
shallowly submerged horizontal cylinders subjected to
impact by sinusoidal surface waves. The principal
conclusion of this experimental work is that the
breaking of waves behi nd a shallowly submerged
cylinder is primarily responsible for the generation of
nonlinear wave forces. The negative 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 cyIinder pierces through the free
water surface. This has been discussed at length by
Khalil~. Wave slamming is one of the most significant
phenomena associated with the offshore platform
horizontal members in the splash zone. It is an area of
hydrodynamics stiII poorly understood, and further
studies are needed to understand it completely.
Theoretical Analysis
The general case of hydrodynamic impact is usually
described by using incompressible potential flow
theory. The particular problem considered here is
treated by a two-dimensional analysis for a horizontal
circular cylindrical section, assuming that the wave
system propagates in a direction normal to the
horizontal cylinder. A pictorial illustration of the
situation being analysed is shown in Fig. I where the
height of the cylinder centre is H above the mean water
level, 11 is the wave elevation measured relative to the
mean water level and:: is the extent of penetration of
the cylinder in the water. The circular cylinder has
radius r, with 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 body-
wave interaction. The force on the circular cylinder is
represented by Eq. (I ).
· .. ( I)
where Pb is the buoyant force given by relation (2):
· .. (2)
The inertial hydrodynamic force is obtained by
generalizing the relations given by Kaplan and Hu
9
and
Kaplanlo for body-wave interaction as per Eq. (3):
· .. (3)
where p is the density of water, m is the vertical added
mass of the cylinder section, 1'1 is the vertical wave
velocity and ii the vertical wave acceleration. The first
H
HW L
Fig. I- Definition 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 az ,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):
[
21t
3
(I-cosO) 1t 11 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)IjI(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[21t
3
{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
. ~

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
- = prf 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) =- - /sin -
. d8 2
, , , ( 16)
According to L' Hopital's rule
Limfl (8) = IT
."
The motion of the free surface is related to its
maximum amplitude by expression ( 17):
.(2m)11 = Asm 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:
, (2ITt )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 coefficient, C. , can
be written as:

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. =(~pjI u2 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. = pgAj
I pU~r
T
_ (m+pA)ii
2 -
pU~r
dm .2
- 11
T
_ dZ
1 - ---:--
. pU~r
TI , T2and 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
~ =--211(8)
2F,
T
_ (m+ pAj)tj
2-
pU~r
or
or
or
T2 =_1 pr2I2 (8)11 _!pr2II (8)11
2 pA2
r 2 pA2
r
or
or
1 r 11
T2 =--[12(8) + II (8)]--
2 AA
... (26)
... (27)

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 Ar
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-cos-
r 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-Yariation 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

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 non-
dimensional 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 non-
dimensionalized 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
xxx 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
')

~
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 'lJA-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
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:-
<>
t.;
~ =0.0375
.,..
N
""GIllE! 'I. IA- 0.00
xxx. x 11./A= 0.25
'.," 'II./A-0 .50
<> flfle •• 1./A-0.75III N
U
~ .
~
.,..
.,;
<> +-----,------r-----.r_----~-J
o 0.5 1.0
Z
T
1.5 2.0
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&& 101 A = 0.50
(l)GGlGO 1'}./A = 0.7~
1.0 1.5
z-r-
slamming coefficient
2.0
with relative

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 = 0·75N
'"
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 analysis of the present problem is that the
cylinder diameter is much smaller than the wave
amplitude. 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 coefficient
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 from the MWL increases, the
value of the slamming coefficient decreases
considerably. Obviously, the intensity of the impact
force decreases as the cylinder 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 circular
cylinders show that: The maximum value of the
slamming coefficient, at the instance of impact, is
computed to be approximately 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, amplitude and period; The value of the
slamming coefficient depends on the relative
submergence of the cylinder as well as the cylinder
position with respect to the mean water level, and; The
impulsive nature 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 hulls for jacket structures and
semi-submersibles.
Acknowledgement
Sincere thanks are due to Prof. Hisashi Kaj itani and
Shozo Kuzumi San of the Towing Tank Laboratory of
the University of Tokyo for the useful discussions
leading to improved quality of the paper.
Nomenclature
A, immersed sectIOnal area of the circular cylinder
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 length 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 velocity of the free surface oscillations
~ eXlent of penetration of the circular cylinder In the water
11 wave elevation measured relative to the mean water level
11.. height of the bottom surface of the circular cylinder above the
mean water level
p mass density of water

. 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,
Middlesex, 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 & Miyata 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.

Ijems 6(4) 188 197

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