Shock responses of a surface ship subjected
to noncontact underwater explosions
Cho-Chung Lianga, Yuh-Shiou Taib,*
aDepartment of Mechanical and Automation Engineering, Da-Yeh University, 112 Shan-Jeau Road, Dah-Tsuen,
Changhwa 515, Taiwan, ROCbDepartment of Civil Engineering, R.O.C. Military Academy, 1 Wei-Wu Road, Fengshan 830, Taiwan, ROC
Received 22 September 2004; accepted 27 March 2005
Available online 15 August 2005
Abstract
In combat operations, a warship can be subjected to air blast and underwater shock loading, which
if detonated close to the ship can damage the vessel form a dished for hull plating or more serious
holing of the hull. This investigation develops a procedure which couples the nonlinear finite
element method with doubly asymptotic approximation method, and which considers the effects of
transient dynamic, geometrical nonlinear, elastoplastic material behavior and fluid–structure
interaction. This work addresses the problem of transient responses of a 2000-ton patrol-boat
subjected to an underwater explosion. The KSFZ0.8 is adopted to describe the shock severity.
Additionally, the shock loading history along keel, the acceleration, velocity and displacement time
histories are presented. Furthermore, the study elucidates the plastic zone spread phenomena and
deformed diagram of the ship. Information on transient responses of the ship to underwater shock is
useful in designing ship hulls so as to enhance their resistance to underwater shock damage.
q 2005 Elsevier Ltd. All rights reserved.
Keywords: Underwater explosions; Fluid–structure interaction; Surface ship
1. Introduction
Underwater explosions are very important and complex problems for naval surface
ships or submarines, since detonations near a ship can damage the vessel form a dished for
Ocean Engineering 33 (2006) 748–772
www.elsevier.com/locate/oceaneng
0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.oceaneng.2005.03.011
* Corresponding author. Fax: C886 7 745 6290.
E-mail address: [email protected] (Y.-S. Tai).
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772 749
hull plating or causing more serious holing. Analyzing these problems requires
understanding many different areas, including the process of underwater explosions,
shock wave propagation, explosion gas bubble behavior, nonlinear structural dynamics
and fluid–structure interaction phenomena. Previous literature dealt with the effectiveness
of underwater explosions on marine structures was restricted in this area for reasons of
national security. This investigation develops a procedure to examine the transient
responses of a ship hull subjected to noncontact underwater explosions.
The dynamic responses of submerged structures impinged by underwater explosion
have received attention since the 1950s. Many investigators studied the transient response
of structures shocked by acoustic waves, and the interaction between structures and
acoustic waves, and these investigations have considered a variety of structural geometry
and boundary conditions. Carrier (1951) proposed a solution for an infinite, elastic,
circular cylindrical shell submerged in an infinite fluid medium and impacted by a
transverse transient, acoustic wave. The solution developed by Carrier resulted from
transforming the governing equation using a modal expansion of the shell displacements
and then solving the transformed equations. The shell displacements, fluid pressures and
shell velocities were expressed in terms of modal inversion integrals. Mindlin and Bleich
(1953) proposed the early time asymptotic solution for the first three modes of the series
solution for the case of a transverse step incident wave. Due to the series approach not
being able to calculate the exact value of initial redial acceleration of the shell at the first
point of impact. Payton (1960) applied double integral transform techniques and obtained
the asymptotic solution for the early time total responses of the shell and the fluid motion
by the method of steepest descent. Meanwhile, Haywood (1958) introduced an
approximate relation between the fluid pressure and the velocity of a cylindrical wave
and used it to obtain the approximate modal solutions for the first three modes of the shell
response.
Since 1969, Huang has published a series of investigations dealing with the transient
interaction of structures and acoustic waves. These studies eliminated some of the
assumptions made in earlier investigations and covered a variety of structural geometry
interacting with a point loading, plane, or spherical waves. Huang employed series
expansion methods to investigate the interaction of plane (Huang, 1969)/spherical (Huang
et al., 1971) acoustic waves with an elastic spherical shell. That investigation presented the
exact solution of the problem, and also provided illustrative examples for subsequent
investigations. Huang (1974) dealt with the transient response of a large elastic plane to the
impact of an incident spherical shock wave. His investigation employed Laplace and
Hankel transform techniques to investigate the fluid–structure interaction. The work
displays the bending effect of a plate impacted by a convex acoustic wave. Huang (1979)
used classical techniques for separating variable and Laplace transforms to solve the wave
equations governing the fluid motions and shell equations of motion. This work studied the
transient response of two fluid-coupled cylindrical elastic shells to an incident pressure
pulse was studied. Huang and Wang (1985) presented the ranges within which asymptotic
fluid–structure interaction theories predict acoustic radiation accurately. According to
their results, added mass and plane wave approximation (PWA) is appropriate for very
low- and high-frequency situations. Huang (1986) subsequently addressed the linear
interaction of pressure pulses with a submerged spherical shell. In his investigation Huang
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772750
applied the boundary element method, which is based upon the exact Kirchhoff Retarded
Potential integral solution to the linear wave equation, in conjunction with the finite
element method. Recently, Huang and Kiddy (1995) studied the transient interaction of a
spherical shell with an underwater explosion shock wave and subsequent pulsating bubble,
based on their approach on the finite element method (PISCES 2DELK) coupled with the
Eulerian–Lagrangian method. According to their results, the structural response, as well as
interactions among the initial shock wave, the structure, its surrounding media and the
explosion bubble must be considered. In the 1970s, Geers systematically developed many
theories in the technical area of transient interaction between a submerged spherical shell
and an acoustic wave. Most notably, Geers (1971) summarized the effects in the transient
fluid–structure interaction. The investigation concluded that retarded potential integral,
spatial domain mapping, and surface approximation methods offer the optimum means of
analyzing complex submerged structures. Geers (1978) sequentially studied the transient
motions of submerged spherical shell subjected to step waves, and simultaneously
examined the free vibration and forced response characteristics of first-order doubly
asymptotic approximation (DAA1) and second-order doubly asymptotic approximation
(DAA2). Geers and Felippa (1983) not only used the first- and second-order DAA2 for
steady-state vibration analysis of submerged spherical shells, but also examined the
accuracy of DAA1 forms. Furthermore, Tang and Yen (1970) used the Laplace transform
and Watson’s transform to study the interaction of a plane acoustic step wave with an
elastic spherical shell, while considering the effects of membrane, bending, rotators
inertia, and shear deformation. Kwon and Fox (1993) applied numerical and experimental
techniques to investigate the nonlinear dynamic response of a cylinder subjected to a side-
on, far-field underwater explosion. Comparisons between the strain gage measurements
and the numerical results at different locations revealed a good agreement. Bathe et al.
(1995) developed a new effective three-field mixed finite element formulation for
analyzing acoustic fluids and their interactions with structures. The discretization use
displacements, pressure and a vorticity moment were variables with appropriate boundary
conditions. Shin and Chisum (1997) employed a coupled Lagrangian–Eulerian finite
element analysis technique as a basis to investigate the response of an infinite cylindrical
and a spherical shell subjected to a plane acoustic step wave. Ergin (1997) presented
experimental measurements and theoretically calculations of a cylindrical shell subjected
to an impulse, and the dynamic response is predicted based on the DAA method. Kwon
and Cunningham (1998) studied submerged structures subjected to an underwater
explosion and developed a technique which investigated for smearing of stiffeners.
Meanwhile, Kwon’s paper represented a cylindrical shell by a beam with a surface of
revolution (SOR) and the interface of a cylindrical shell with a SOR beam. Liang et al.
(1998) presented a procedure based on the methodology of Hibbit and Karlsson to analyze
the elastoplastic response and critical regions of an entire pressure hull subjected to an
underwater explosion. Liang et al. (2000) investigated the response of a submerged
spherical shell to a strong shock wave based on DAA2.
Using full-scale trials to examine the response of marine structures subjected to
underwater explosions is very costly and is limited by environmental safety concerns.
Meanwhile, the physical phenomena involved in these explosions cannot be scaled in a
practical experimental setup. Additionally, for simple geometric closed-forms solutions
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772 751
can be found, but for practical structures, numerical simulation is unavoidable. The brief
review above reveals that in the past decades many investigations have provided
preliminary results concerning the fluid–structure interaction of submerged spherical
shells. However, the transient interaction of a surface ship hull with an underwater shock
wave, has received limited attention. Greenhorn (1988) described a computer code
(SSVUL) which can assess the vulnerability of a surface ship to underwater attack by blast
weapons. Meanwhile, Shin and Santiago (1998) used a coupled USA-NASTRAN-CFA as
a basis to investigate the effects of fluid–structure interaction and cavitation on the
response of a surface ship subjected to an underwater explosion. Finally, Hung et al.
(1999) presented a numerical simulation of a ship-like structure subjected to underwater
explosions in an infinite fluid domain.
This study aims to develop a procedure that considers factors such as transient dynamic
response, geometrically nonlinear, elastoplastic material behavior, and the fluid–structure
interaction effect, to investigate the shock responses of a surface ship subjected to
underwater explosion. The nonlinear finite element method based on the methodology of
Hibbit and Karlsson (1979) is employed to model the structure, and the boundary element
method based on doubly asymptotic approximation (DAA) is used to model the fluid
domain. Meanwhile, the incident pressure from the explosive charge is determined
according to the empirical equation of Cole (1948). Furthermore, the most widely used
keel shock factor (KSF) value is adopted for the shock intensity consideration. This study
selects a 2000-ton patrol-boat subjected to shock intensity KSFZ0.8 for numerical study.
Additionally, the surface pressure, acceleration, velocity, displacement time histories and
plastic zone progress of the ship are also presented. Information on the transient response
of the ship to underwater shock is useful for designing a ship hull to enhance its resistance
to underwater shock damage.
2. Theoretical background
To study the transient dynamic elastoplastic response of surface ship subjected to a
shock wave, this work initially applies an incremental update Lagrangian finite element
procedure based on Hibbit and Karlsson’s methodology. The procedure used the Newmark
time implicit integration scheme and Newton–Raphson method, which includes dynamic
equilibrium interaction considering the half-step residual convergence tolerance proposed
by Hibbit (1979). The plastic relations, based on the von-Mises yield criterion, assume the
isotropic-hardening rule for the elastoplastic material behavior of the material under study.
The governing equation of the fluid medium based on the doubly asymptotic
approximation (Geers, 1971, 1978; DeRuntz et al., 1980; DeRuntz, 1989) is advantageous
in that it models the surrounding fluid medium as a membrane on the wet surface of
structure actually in contact with the homogeneous fluid. The effect of cavitation on a
structure modeled with a surrounding fluid also included. The fluid motion is described
only in term of the response of the wet surface, which is then linked by compatible
relations to the structural response. In addition, the staggered solution procedure is
adopted herein to perform the doubly asymptotic approximation.
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772752
2.1. Governing equations
In this section, we present the structural response equation, which is based on the
dynamic virtual work equation. The fluid and coupled fluid–structure equation are based
on the doubly asymptotic approximation.
2.1.1. Structural response equation
For a fully or partially submerged structure subjected to an underwater shock wave, the
structure may exhibit material and geometrical nonlinear behavior. The formulation is
based on the dynamic virtual work equation. Let the body force at any point within the
volume V be fb, and the surface force at any point on surface S be fs. The governing
equation of structural response isðVe
rs €uedue dV C
ðVe
rsa _uedue dV C
ðVe
tijeij dV K
ðVe
f bi due dV K
ðSe
f si due dS Z 0 (1)
where ue, uKe and uKe are the displacement, velocity and acceleration, respectively, of the
nodal at the element. Additionally, tij and eij represent the stress and strain tensor,
respectively. Furthermore, rs and ac are the material density and the mass proportional
damping factor, respectively. Based on the theorem of virtual displacement, the governing
equation of the problem can be expressed in matrix form as
½Ms�f €ugC ½Cs�f _ugC ½Ks�fug Z ff g (2)
where
½Ms� Z
ðVe
rs½N�T½N� dV ; ½Cs� Z
ðVe
rsac½N�T½N� dV
½Ks� Z
ðVe
½B�T½D�½B�dV ; ff g Z
ðVe
½N�Tf dV
{u} and {f} are the structural displacement and the external force vector, respectively.
Additionally, [Ms], [Cs] and [Ks] represent the structural mass, damping and stiffness
matrices, respectively. [N], [B], [D] are the shape function, strain matrix and matrix of
elastic–plastic tangent stiffness, respectively. For excitation of a submerged structure by
an acoustic wave, {f} can be expressed as
ff g ZK½G�½Af�ðfPIgC fPSgÞ (3)
where {PI} and {PS} are the nodal pressure vector for wet-surface fluid mesh pertaining to
the incident wave and scattered wave, respectively. Where [Af] represents the diagonal
area matrix associated with an element in the fluid mesh, and [G] represents the
transformation matrix relating the structural and fluid nodal surface forces.
2.1.2. Fluid surface equation
For a structure submerged in an infinite acoustic medium, the governing equation of the
wet surface of the shell is based on the Doubly Asymptotic Approximation (Geers, 1971,
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772 753
1978; DeRuntz et al., 1980). The second-order approximation (DAA2) is given in matrix
notation by
½Mf�f €PsgCrfc½Af�f _PsgCrfc½Uf�½Af�fPsg Z rfcð½Mf�ðf _vsgÞC ½Uf�½Mf�ðvsÞ�Þ (4)
where
½Uf� Z hrfc½Af�½Mf�K1 (5)
[Mf] denotes the symmetric fluid mass matrix, h is scale parameter bounded as 0%h%1,
and rf and c are the fluid density and sound velocity, respectively. Additionally, {ns} is the
vector of scattered-wave fluid particle velocities normal to the structural surface.
2.1.3. Coupled fluid–structure interaction equation
The fluid surface Eq. (4) is coupled to the structural response by the following equation
fvsg Z ½G�Tf _ugKfvIg (6)
where {nI} is the fluid incident velocity. The coupled fluid–structure interaction equations
can then be obtained by introducing Eq. (3) into Eq. (2), as well as, introducing Eq. (6) and
its derivative into Eq. (4)
½Ms�f €ugC ½Cs�f _ugC ½Ks�fug ZK½G�½Af�ðfPIgC fPsgÞ (7)
½Mf�f €qsgCrfc½Af�f _qsgCrfc½Uf�½Af�fqsg
Z rfc½Mf�ð½G�Tf €ugKf _vIgÞC ½Uf�½Mf�ð½G�Tf _ugKfvIgÞ (8)
where
fqsg Z
ðt
0
fPsðtÞgdt (9)
Multiplying by [Af][Mf]K1, Eq. (8) can be rewritten as follows:
½Af�f €qsgCrfc½Df1�f _qsgCr2f c2½Df2�fqsg
Z rfc½Af�ð½G�Tf €ugKf _vIgÞCrfc½Df1�ð½G�Tf _ugKfvIgÞ (10)
where the [Df1]Z[Af][Mf]K1[Af] and ½Df2�Z ½Af�½Mf�
K1½Af�½Mf�K1½Af�.
2.2. Solution method and convergence tolerance
The Newton–Raphson method and the Newmark implicit time integration scheme were
used as numerical techniques for solving the structural equations. In addition, the fluid
equation is treated with the staggered solution procedure (Park et al., 1977). The
convergence tolerance of the dynamic equilibrium equation is based on the half-step
residual proposed by Hibbit (1979). Fig. 1 displays a flow chart of the analysis procedure.
Fig. 1. The definition of HSF and KSF.
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772754
3. Shock pressure of underwater explosions
The sudden energy release associated with the underwater explosions of a conventional
high explosive or nuclear weapon generates a shock wave and the forms a superheated,
highly compressed gas bubble in the surrounding water (Cole, 1948; Keil, 1961). Of the
total energy released from a 1500-lb TNT underwater explosion, approximately 53% goes
into the shock wave and 47% goes into the pulsation of the bubble. Most cases
demonstrate that the damage done to marine structures (such as the surface ship and
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772 755
submarine) occurs early on and is due to the strikes of the shock wave. This investigation
only considers the effects of the shock wave.
The underwater shock wave generated by the explosion is superimposed on the
hydrostatic pressure. The pressure history P(t) of the shock wave at a fixed location starts
with an instantaneous pressure increase to a peak Pmax (in less than 10K7 s) followed by a
decline which initially is usually approximated by an exponential function. Thus,
according to the empirical equation of Cole (1948)
PðtÞ Z Pmax eKt=l; tR t1 (11)
where Pmax is the peak pressure in the shock front, t is the time elapsed since the arrival of
the shock, and l is the exponential decay time constant. The peak pressure and the decay
constant depend upon the size of the explosive charge and the stand off distance from this
charge at which the pressure is measured. The peak pressure Pmax and decay constant l in
Eq. (11) are expressed by
Pmax Z K1
W1=3
R
� �A1
ðMPaÞ (12)
l Z K2W1=3 W1=3
R
� �A2
ðmillisecond; msÞ (13)
where K1, K2, A1 and A2 are constants which depend on explosive charge type when
different explosives are used the input constants are according to Table 1 (Cole, 1948;
Smith and Hetherington, 1994; Reid, 1996), W is the weight of the explosive charge in
kilograms and R is the distance between explosive charge and target in meters. Moreover,
Cole (1948) gave further information on the systematic presentation of the physical effects
associated with underwater explosions, and this should be consulted.
When the pressure from an underwater explosion impinges upon a flexible surface such
as the hull of surface ship, the reflected pressure on the fluid–structure interaction surface
can be predicted reasonable accurately, based on Taylor’s plate theory (Taylor, 1950). For
an air backed plate of mass per unit area (m) subjected to an incident plane shock wave
Pi(t), a reflection wave of pressure Pr(t) will depart from the plate. Let np(t) be the velocity
of the plate and applying Newton’s second law of motion
mdvp
dtZ Pi CPr (14)
Table 1
Shock wave parameters for various explosive charges
Constants Type
HBX-1 TNT PETN Nuclear
K1 53.51 52.12 56.21 1.06!104
A1 1.144 1.180 1.194 1.13
K2 0.092 0.0895 0.086 3.627
A2 K0.247 K0.185 K0.257 K0.22
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772756
The fluid particle velocities behind the incident and reflected shock wave ni(t) and nr(t),
respectively, the velocity of the plate becomes
vpðtÞ Z viðtÞKvrðtÞ (15)
Incidence and reflected shock wave pressures are defined as PiZrfcvi and PrZrfcvr.
The rf and c are the fluid density and sound velocity, respectively. Substituting the
pressure into Eq. (14) and utilizing Eq. (11) then the Pr(t) can be expressed as
PrðtÞ Z PiðtÞKrfcvp Z Pmax eKt=l Krfcvp (16)
Then the equation of motion can be rewritten as
mdvp
dtCrfcvp Z 2Pmax eKt=l (17)
Eq. (17) is a first-order linear differential equation. Solving the differential equation
obtain the velocity of the plate
vp Z2Pmaxl
mð1KbÞ½eKbt=lKeKt=l� (18)
where the bZ(rfcl/m), and tO0. The total pressure on the plate is given by
PtðtÞ Z 2PiðtÞKrfcvp Z2Pmax
1Kb½eKt=l Kb eKbt=l� (19)
In Eq. (19), as b becomes large (light weight plate), the total pressure will become
negative at a early time. In reality, the pressure cannot be negative in water since the water
cannot sustain the tension. As the pressure reduces to vapor pressure, local cavitation
occurs in front of the plate.
4. Shock factor
Since a ship can be subjected to a large variety of underwater explosion (variation in
charge weight, standoff distance, relative attack orientation), the relation between attack
severity and geometry must be determined. The attack severity for high explosive charges
such as mines is usually described by shock factor which is proportional to the energy
density of the shock wave arriving at the ship’s hull (Keil, 1961; Reid, 1996). Because the
shock energy flux density is given by
E Z1
rfc
ð6:7l
0
PðtÞ2 dt (20)
where the pressure time history can be obtained by Eq. (11), and then the energy density at
a distance R from the explosion of W of trinitrotoluene (TNT) is
E ZP2
maxl
2rfc(21)
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772 757
or approximately expressed as
E z94:34W
R2(22)
For charge weight (W) and standoff distance (R), various combinations can generate
various pressure–time curves. Nevertheless, for the observation that the energy released
from various pressure loadings for structures indicates is roughly equal. Higher shock
factors represent an increasing proportion of energy being imparted to the ship by the
underwater shock. Thus, the underwater shock resistance of a vessel can specify at the
design stage in terms of a shock factor. The factor may be chosen that theoretically ensures
that a vessel will withstand a particular threat or as a value that experience has shown
reasonable for the type of vessel.
For damage predictions for submarines, this factor is referred to as the Hull Shock
Factor (HSF) (Bishop, 1993; Reid, 1996; O’Hara and Cunniff, 1993). The HSF represents
the energy contained in a shock wave which may contribute to damaging hull plating on
the ship (see Fig. 2a). It has been found that
HSF ZffiffiffiffiffiW
p=R (23)
where
W
is the weight of explosive in TNT equivalence (kg).
R is the stand off distance from the charge to the target (m).
For a surface ship, where the response is largely vertical, it is necessary to correct
for the angle at which the shock wave strikes the target. When the charge position is
measured relative to the keel of the ship and the angle of incidence of the shock wave with
respect to the ship is also considered the value is referred to as the Keel Shock Factor
(KSF) (Bishop, 1993; Reid, 1996). In this situation the above equation must also be
multiplied by (1Ccos q)/2, then the KSF can be expressed as
KSF Z
ffiffiffiffiffiW
p
R!
ð1 Ccos qÞ
2(24)
where q is the angle between a vertical line and a line drawn from the charge to the keel of
the ship (see Fig. 2b). This work adopts the KSF to study the shock resistance of a surface
ship.
5. Numerical Example—2000-ton patrol-boat subjected to underwater shocks
This investigation selected a 2000-ton patrol-boat subjected to underwater explosions
to study the transient response. A simplified notional ship design was adopted due to the
complexity of a global ship model, and this design was derived from an earlier design
modeled by the United Ship Design and Development Center (USDDC) of Taiwan.
Transient Response of surface shipsubjected to underwater explosions
fluid wet-surface mesh mesh geometry
element definitions
material properties
structural equation
finite element methodboundary element method
constraints
element definitions
fluid properties
constraints
fluid equation
Establish the coupled fluid-structure interaction eqs.
estimate the structural force kSu att+∆t from the extrapolation ofcurrent values and past values
solve fluid equation and obtainpreliminary of pressure at t+∆t
transform fluid pressure intostructural nodal forces
solve the displacement andveloctiy at t+∆t
transform the structural force kSuinto fluid node also involves theknow incident pressure at t+∆t
compute structural restoringforces and transform into fluid
node and reform the fluidequation at t+∆t
re-solve the fluid equationand obtain refined pressure at
t+∆t
Force vector { f }
Data outputstructural responses and shock wave
pressure
Dou
ble
Asy
mpt
otic
App
roxi
mat
ion
Tim
e In
terg
ratio
n
Cal
cula
te th
e m
ass
and
stif
fnes
s m
atri
xD
ynam
ic A
naly
sis
Fig. 2. Schematic diagram of the analysis procedure.
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772758
The shock intensity consideration KSF is set at 0.8 and cavitation effects are considered.
Moreover, for the following results is also presented.
1. The shock loading history at different keel locations.
2. Acceleration, velocity and displacement responses.
3. The plastic zone progress.
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772 759
5.1. Description of problem
5.1.1. Model description
Fig. 3 schematically depicts the 2000-ton patrol-boat analyzed herein, which has a
length of 90.0 m, breadth of 13.2 m, depth of 7.6 m and draft of 3.8 m. Additionally, the
figure illustrates the locations of the major cabins. The shell thickness is modeled using
the average thickness technique. For the stiffener, the cross-sectional area is blended into
the plate cross-sectional area. The method and smear ratio are based on the method of
Kwon and Cunningham (1998), and Table 2 lists the relative plate thickness at various
locations. Due to the symmetry of the structure, only half of the patrol-boat must be
modeled. The problem is modeled for the structure using three-node thin shell elements
with five degrees of freedom pre node—ux, uy, ux, qx, qy (Fig. 4). The fluid medium
comprises three-node shell–fluid interface elements with three degrees of freedom pre
node—ux, uy, ux (Fig. 5). Fig. 6 depicts the finite element mesh diagram with 1828 nodes,
5074 shell elements and 1075 fluid interface elements, and the fluid for the draft as a
interface element covers the wet surface of the structures. The symmetry boundary
conditions are imposed on the y–z plane of the centerline (CL).
5.1.2. Material properties
The 2000-ton patrol-boat was constructed of steel (ASTM A106 grade C), and this
work adopts the hardening rule of elastic–perfectly plastic. Meanwhile, the superstructure
was constructed by aluminum (6061-T6 Alloy). The material properties of the patrol-boat
are described as follows:
(1) Steel:
Mass density (kg/m3): 7860.0
Poisson’s ratio: 0.3
Young’s modulus (GPa): 204.0
Yield stress (MPa): 351.7
Fig. 3. The geometrical configuration of the 2000-ton patrol-boat.
Table 2
Equivalent plate thickness
Structural location Plate thickness (mm)
Outer plate 12.4
Bottom plate 16.4
Bulkhead 9.6
Main deck 9.0
First deck 7.0
Tank top 10.0
Superstructure 8.3
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772760
(2) Aluminum:
Mass density (kg/m3): 2710.0
Poisson’s ratio: 0.33
Young’s modulus (GPa): 70.0
Yield stress (MPa): 300.0
The material properties of seawater are described as:
Mass density (kg/m3): 999.6
Sound speed (m/s): 1461.2
5.1.3. Shock loading
This study adopts the Keel Shock Factor (KSF) to describe the shock severity. The
work assumes the KSF value is 0.8 ðKSFZ ðffiffiffiffiffiW
p=DÞ!ðð1Ccos qÞ=2ÞÞ and the charge is
positioned directly underneath the keel (qZ0, Fig. 6). Eq. (5) becomes the relationship
between weight and distance ððKSFZ0:8Z ðffiffiffiffiffiW
p=DÞÞ, and then the shock pressure-time
Node 2
Node 3
Node 1
X
Y
Z
n
Integration pt.
S2
S1
j
i
Fig. 4. The 3-node doubly curved thin shell element.
Node 2
Node 3
Node 1 Pressure
Interface element
ship-likestructure
Fluid
X
Y
Z
Fig. 5. The 3-node shell-fluid interface elements.
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772 761
curves can be generated by empirical equation, namely Eqs. (1)–(3). According to Table 1,
for the TNT explosive material, the constants K1, K2, A1 and A2 are 52.1, 1.18, 0.0895, and
K0.185, respectively. For charge weight (W) and standoff distance (R), various
combinations can generate various pressure–time curves ðPðtÞZPmax eKðt=lÞÞ.
Fig. 6. Finite element model of the 2000-ton patrol-boat.
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772762
For example, when KSFZ0.8, then if the weight WZ64.0 kg, the standoff distance R is
10.0 m, if WZ576.0 kg, then R is 30.0 m and if WZ1600.0 kg, then R is 50.0 m, and so
on. Nevertheless, for the observation that the total energy released from various pressure
loadings for structures indicates is roughly equal (60.38 m kPa), the relationship can be
calculated by Eq. (22). This study, assumes that WZ576.0 kg, and RZ30.0 m.
5.2. Results and discussion
Figs. 7–13 present the numerical results for the transient response of a patrol-boat
subjected to an underwater explosion shock wave. These results reveal the following:
5.2.1. The shock loading history at different locations along keel
For charges close to the ship, the shock waves propagate as spherical waves moving
towards the structure. It is apparent that different portions of the ship will encounter
different peak responses, depending on distance from the explosion and angle of attack.
Fig. 7 displays the fluid pressure history at locations A, B, C, D, E along the keel. The
greatest peak pressure occurred at location C, since the charge was positioned directly
beneath here, followed by locations D and B, and with locations E and A experiencing the
lowest peak pressure, a result that meets expectations.
The peak pressure at location C (the standoff point) instantly rises to 10.46 MPa at
tZ0.25 ms (Fig. 7(a)). Meanwhile, for locations D and B, the peak pressures, of 5.86 and
6.48 MPa arrive at tZ4.75 ms (Fig. 7(b)) and tZ5.0 ms (Fig. 7(c)), respectively. These
values are roughly 38.05–43.98% below the location C response. Finally, for locations E
and A, the peak pressures 4.25 and 4.25 MPa arrive at tZ11.25 ms (Fig. 7(d)) and
tZ17.0 ms (Fig. 7(e)), respectively. These two values are roughly 59.0% lower than the
location C response.
5.2.2. Acceleration, velocity and displacement response at different locations
through the ship
To examinate the responses for different locations in the ship, several important
locations (such as main engine room (location B2), steering gear room (location B3), bow
thruster room (location B1), main deck (location M) and combat direct tower (location S)
throughout the ship were chosen. Figs. 8–10 display the acceleration, velocity and
displacement responses at these different locations.
Fig. 8 indicates that the acceleration responses in the vertical, athwartships, and fore-aft
directions. Due to the charge being located below the main engine room (location B2), the
peak acceleration 2662.0 g at tZ0.25 ms in the vertical direction suddenly rises. The
athwartships and fore-aft peak acceleration responses are 1413.0 and 946.0 g, that is
approximately 53.1% of the vertical acceleration response. Following the peak response,
the response rapidly decays, after about 5.0–7.0 ms, and then the response tends to
steadies. Successively, in the steering gear room (location B3) the peak acceleration
response at tZ11.75 ms is 1054.0 g in the vertical direction. Meanwhile, in the bow
thruster room (location B1) the peak acceleration response occurs at tZ19.5 ms and is
398.0 g in the fore-aft direction. These values are about 39.59 and 14.95% of the location
B2 vertical acceleration response. The results for the main deck (location M) are different
–4.0
0.0
4.0
8.0
12.0
Pres
sure
(M
Pa)
–4.0
0.0
4.0
8.0
12.0
Pres
sure
(M
Pa)
–4.0
0.0
4.0
8.0
12.0
Pres
sure
(M
Pa)
–4.0
0.0
4.0
8.0
12.0
Pres
sure
(M
Pa)
0 5 10 15 20 25 30 35 40 45Time (ms)
–4.0
0.0
4.0
8.0
12.0
Pres
sure
(M
Pa)
0 5 10 15 20 25 30 35 40 45Time (ms)
(a) Location C
(b) Location D
0 5 10 15 20 25 30 35 40 45Time (ms)
(c) Location B
0 5 10 15 20 25 30 35 40 45Time (ms)
(d) Location E
0 5 10 15 20 25 30 35 40 45Time (ms)
(e) Location A
Fig. 7. Shock pressure history at different keel locations.
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772 763
Direction
LocationVertical direction Athwartships direction Fore and aft direction
B2
–2.0
0.0
2.0
4.0
–2.0
0.0
2.0
4.0
–2.0
0.0
2.0
4.0
–2.0
0.0
2.0
4.0
–2.0
0.0
2.0
4.0
2.6621.413 0.946
B31.054 0.452
0.471
B10.359 0.174 0.398
M0.597
1.183 0.770
S
Acc
eler
atio
n (
×103
), g
–2.0
0.0
2.0
4.0
–2.0
0.0
2.0
4.0
–2.0
0.0
2.0
4.0
–2.0
0.0
2.0
4.0
–2.0
0.0
2.0
4.0
Acc
eler
atio
n (
×103
), g
–2.0
0.0
2.0
4.0
–2.0
0.0
2.0
4.0
–2.0
0.0
2.0
4.0
–2.0
0.0
2.0
4.0
–2.0
0.0
2.0
4.0
Acc
eler
atio
n (
×103
), g
0.525
0 5 10 15 20 25 30 35 40 45
Time (ms)0 5 10 15 20 25 30 35 40 45
Time (ms)0 5 10 15 20 25 30 35 40 45
Time (ms)
0.464 0.439
Note: Athwartships (x-direction), Vertical (y-direction), Fore and aft (z-direction)
Fig. 8. Acceleration response underwater shock loading.
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772764
from the response of location B2. Unlike location B2, the peak acceleration responses for
the main deck are 1183.0 and 770.0 g and occur in the athwartships and fore-aft directions.
The peak acceleration response in the vertical direction is 597.0 g, just 50.46% of that in
the athwartships direction. The main reason for this phenomena is the hull hogging
response induced by the underwater explosion. In the combat direct tower (location S), the
peak acceleration response in the vertical direction is at 525.0 g. Meanwhile, the
athwartships direction response is 464.0 g and the fore-aft direction response is 439.0 g.
These values are roughly 19.72 and 16.5% lower than those of the vertical acceleration
response at location B2. Besides, due to the shock waves propagating spherically, they
9.674
2.462 2.207
5.4510.822 2.223
2.774
0.521 2.814
4.459 2.781 1.983
6.052
1.275 2.268
Direction
LocationVertical direction Athwartships direction Fore and aft direction
B2
–4.0
4.00.0
8.012.0
–4.0
4.00.0
8.012.0
–4.0
4.00.0
8.012.0
–4.0
4.00.0
8.012.0
–4.0
4.00.0
8.012.0
B3
B1
M
S
Vel
ocity
(m
ps)
–4.0
4.00.0
8.012.0
–4.0
4.00.0
8.012.0
–4.0
4.00.0
8.012.0
–4.0
4.00.0
8.012.0
–4.0
4.00.0
8.012.0
Vel
ocity
(m
ps)
–4.0
4.00.0
8.012.0
–4.0
4.00.0
8.012.0
–4.0
4.00.0
8.012.0
–4.0
4.00.0
8.012.0
–4.0
4.00.0
8.012.0
Vel
ocity
(m
ps)
0 5 10 15 20 25 30 35 40 45
Time (ms)0 5 10 15 20 25 30 35 40
Time (ms)0 5 10 15 20 25 30 35 40 45
Time (ms)
Note: Athwartships (x-direction), Vertical (y-direction), Fore and aft (z-direction)
Fig. 9. Velocity response underwater shock loading.
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772 765
arrive first in locations B2 and M, at 0.25 and 6.5 ms, then following arrival in location S,
at 7.25 ms, and finally reach locations B3 and B1 at 11.75 and 15.0 ms.
Fig. 9 represents the velocity response in the vertical, athwartships, fore-aft directions.
For the locations B2, B3, M and S, the greatest velocities all occurred in the vertical
direction, followed by the athwartships and fore-aft directions. Meanwhile, the greatest
velocity at location B1 was in the fore-aft direction, followed by the vertical direction, and
finally the athwartships direction.
Fig. 10 displays the displacement responses. For all locations, the greatest displacement
was in the vertical direction, followed by the fore-aft direction, while the athwartships
direction consistently had the lowest displacement. Such behavior displays an evident
–202468
10
–202468
10
–202468
10
–202468
10
–202468
10
6.05
–0.06–0.12–0.18
0.000.060.12
–0.06–0.12–0.18
0.000.060.12
–0.06–0.12–0.18
0.000.060.12
–0.06–0.12–0.18
0.000.060.12
–0.06–0.12–0.18
0.000.060.12
0.088
–0.4–0.8–1.2
0.00.40.8
–0.4–0.8–1.2
0.00.40.8
–0.4–0.8–1.2
0.00.40.8
–0.4–0.8–1.2
0.00.40.8
–0.4–0.8–1.2
0.00.40.8
0.20
2.26
0.029 0.34
1.48
0.036
1.08
7.71
0.1730.28
Dis
plac
emen
t (cm
)
Dis
plac
emen
t (cm
)
Dis
plac
emen
t (cm
)
5.49
0.095 0.86
Direction
LocationVertical direction Athwartships direction Fore and aft direction
B2
B3
B1
M
S
0 5 10 15 20 25 30 35 40 45
Time (ms)
0 5 10 15 20 25 30 35 40 45
Time (ms)
0 5 10 15 20 25 30 35 40 45
Time (ms)
Note: Athwartships (x-direction), Vertical (y-direction), Fore and aft (z-direction)
Fig. 10. Displacement response underwater shock loading.
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772766
difference in the order of displacement in the three directions. The peak vertical direction
displacements at tZ45.0 ms were observed at locations B2, M, S, B3, B1 and were 6.05,
7.71, and 5.49, 2.26 and 1.48 cm. These results reveal that during the initial shock, the
middle locations (B2, M, S) experience upward motion and cavities/hollows/holes form.
This behavior presents the hogging response. Fig. 11 illustrates a sequence of deformed
configurations during the analysis process.
The hull of the patrol-boat was constructed from the steel, and the superstructure was
constructed with aluminum. Fig. 12 focuses on the A–A section to investigate the dynamic
response of the amidships cross-section. At the keel the peak vertical acceleration 2520.0 g
Fig. 11. Deformation of patrol-boat subjected to underwater shock (displacement magnification factor is 100).
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772 767
and velocity 7.70 mps are suddenly rising, and the following responses rapidly decline.
The high frequency motion of the keel is very evident. On the main deck, the peak
acceleration is 533.0 g and velocity is 6.252 mps. The acceleration response is more
alleviative than the keel response, and is does not display the rapid initial rise to a peak
value. Physically, the keel is struck directly by the incident shock wave, and the evidently
y
xo
02 deck
Main deck
Keel
Aluminum
Steel
A-A section
Acceleration time history Velocity timehistory Displacement time history
–4.0
–2.0
0.0
2.0
4.0
Acc
eler
atio
n (1
03 ) g
–4.0
–2.0
0.0
2.0
4.0
Acc
eler
atio
n (1
03 ) g
–4.0
–2.0
0.0
2.0
4.0
Acc
eler
atio
n (1
03 ) g
2.520
–8.0
–4.0
0.0
4.0
8.0
Vel
ocit
y (m
ps)
–8.0
–4.0
0.0
4.0
8.0
Vel
ocit
y (m
ps)
–8.0
–4.0
0.0
4.0
8.0
Vel
ocit
y (m
ps)
7.70
–2.0
0.0
2.0
4.0
6.0
Dis
plac
emen
t (cm
)
–2.0
0.0
2.0
4.0
6.0
Dis
plac
emen
t (cm
)
–2.0
0.0
2.0
4.0
6.0
Dis
plac
emen
t (cm
)
6.10
Keel
0.533
6.252 5.68
Main deck
0.730
7.315
1.80
02 deck
0 5 10 15 20 25 30 35 40 45
Time (ms)
0 5 10 15 20 25 30 35 40 45
Time (ms)
0 5 10 15 20 25 30 35 40 45
Time (ms)
0 5 10 15 20 25 30 35 40 45
Time (ms)
0 5 10 15 20 25 30 35 40 45
Time (ms)
0 5 10 15 20 25 30 35 40 45
Time (ms)
0 5 10 15 20 25 30 35 40 45
Time (ms)
0 5 10 15 20 25 30 35 40 45
Time (ms)
0 5 10 15 20 25 30 35 40 45
Time (ms)
Fig. 12. The amidships section of the ship structure for acceleration and velocity response in the vertical direction.
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772768
high frequency motion of keel is predictable. When the shock energy was propagated
upward to the main deck, higher frequency motion was attenuated by structural damping
of the ship, and lower frequency responses became more prominent. However, at 02 deck
the peak acceleration of 730.0 g occurred at tZ27 ms. This response is 36.9% higher than
Fig. 13. The plastic zone progress of the patrol-boat subjected to underwater shock.
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772 769
the main deck and the time is obviously delayed. The main reason lies in the fact that the
superstructure is aluminum, which has a stiffness approximately 34.3% that of steel.
Consequently, when the shock wave propagated upward via the main deck the
superstructure responded excessively. At tZ45 ms, the peak displacement at the keel is
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772770
6.10 cm, which is a direct linear increase, and at the main deck, the peak displacement is
5.68 cm, which indicates that the patrol boat was subjected to an underwater explosion
with light upward rigid body motion. The upward motion of the main deck is obviously
delayed, which upward motion beginning at tZ31.5 ms. Finally, Fig. 12 shows that the
displacement responses at 02 deck are smaller than the keel and main deck.
5.2.3. The ship plastic zone progress
When the ship is subjected to the serious underwater explosion KSFZ0.8, the structure
will respond into the plastic range. Fig. 13 depicts the plastic zone spread diagram. This
work selects WZ576.0 kg and RZ30.0 m, with the shock wave propagating towards the
hull as spherical waves. Initially, the stress of the bottom structure at tZ2.0–3.0 ms
immediately exceeds the yielding stress and moves into the plastic range, with the
effective plastic strain being approximately 7.0!10K3. Then the following yield region
spreads through the fore-aft direction at tZ4.0–16.0 ms. Meanwhile, the patrol boat
oscillates between elastic and plastic states. Finally, the whole patrol boat recovers toward
completely elastic state at tZ20.0 ms.
6. Conclusion
This investigation developed a procedure to exanimate a surface ship under a shock
environment. It employed the finite element method coupled with the DAA2 to study the
transient dynamic response of a 2000-ton patrol-boat subjected to an underwater
explosion. It adopted the Keel Shock Factor (KSFZ0.8) to describe the shock severity.
Consequently, the shock loading history at the keel, the acceleration, velocity and
displacement time history at different locations, and the ship plastic zone progress are
presented in detail.
Based on the results, we can conclude the following:
1. The keel shock factor ðKSFZ ðffiffiffiffiffiW
p=RÞ!ðð1Ccos qÞ=2ÞÞ is used for describing shock
severity. An identical KSF can create various pressure shock loadings, and the total energy
generated by each pressure loading is approximately equal. For total energy equality, then
when q is constant, selecting a smaller charge W will reduce distance R, which means a
spherical shock wave and local damage to the hull of the ship occurring early. But, if a
larger charge W is selected, then the distance R will be longer, which means a plane shock
wave and a longer impulse duration on the ship hull, thus increasing the damage to
equipment. Therefore, the KSF is used to describe shock severity for the real shock threat,
and in addition various charge weights, distance and incident angle should be carefully
considered.
2. In most cases, the equipment is more sensitive then the structure to shock, and
damage may be caused by higher acceleration or displacement. This investigation
developed a procedure to analyses the shock response at different locations. When the ship
was subjected to underwater shock, typical acceleration, velocity and displacement time
histories were obtained. These results should confirm whether the specification
requirements were satisfied or not.
C.-C. Liang, Y.-S. Tai / Ocean Engineering 33 (2006) 748–772 771
This work represents a preliminary study of the transient responses of a surface ship
under shock loading. It aims to assist in the choice of structure and equipment to ensure
durability in a shock environment. The gas bubble effect and shock resistant design of ship
structure and attached equipment are merit further study.
Acknowledgements
The authors would like to thank the United Ship Design and Development Center of
Republic of China for financially supporting this work under contract No. USDDC-RD-
461.
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