shock responshock responses of a surface ship

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).

http://www.elsevier.com/locate/oceaneng

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


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.


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,


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.


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


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


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)


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.


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.


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


(2) Aluminum:


Poisson’s ratio: 0.33

Young’s modulus (GPa): 70.0

Yield stress (MPa): 300.0

The material properties of seawater are described as:


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.


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.


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.


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.


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


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)


Fig. 9. Velocity response underwater shock loading.


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


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)


Fig. 10. Displacement response underwater shock loading.


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).


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.


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.


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


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.


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.

References

Bathe, K.J., Nitikitpaiboon, C., Wang, X., 1995. A mixed displacement-based finite element formulation for

acoustic fluid-structure interaction. Computers and Structures 56 (2–3), 193–213.

Bishop, J.H., 1993. Underwater shock standards and tests for naval vessels. DSTO, Materials Research

Laboratory.

Carrier, G.F., 1951. The interaction of an acoustic wave and elastic cylindrical shell. Tech. Rept. No.4, Brown

University, Providence, R.I.

Cole, R.H., 1948. Underwater Explosions. Princeton University Press, Princeton.

DeRuntz Jr.., J.A., 1989. The Underwater Shock Analysis Code and its Applications 60th Shock and Vibration

Symposium Proceedings, vol. 1 1989 pp. 89–107.

DeRuntz, J.A., Jr., Geers, T.L. and Felippa, C.A., 1980. The underwater shock analysis (USA-Version 3) code, A

Reference Manual. DNA 5615F, Defense Nuclear Agency, Washington, D.C.

Ergin, A., 1997. The response behaviour of a submerged cylindrical shell using the doubly asymptotic

approximation method (DAA). Computers and Structures 62 (6), 1025–1034.

Geers, T.L., 1971. Residual potential and approximate methods for three-dimensional fluid-structure interaction

problems. The Journal of the Acoustical Society of America 49, 1505–1510.

Geers, T.L., 1978. Doubly asymptotic approximations for transient motions of submerged structures. The Journal

of the Acoustical Society of America 64, 1500–1508.

Geers, T.L., Felippa, C.A., 1983. Doubly asymptotic approximations for vibration analysis of submerged

structures. The Journal of the Acoustical Society of America 73, 1152–1159.

Greenhorn, J., 1988. The assessment of surface ship vulnerability to underwater attack, The Royal Institution of

naval Architects 1988 pp. 233–243.

Haywood, J.H., 1958. Response of an elastic cylindrical shell to a pressure pulse. Quart. J. Mech. Appl. Math. 11,

129–141.

Hibbit, H.D., Karlsson, B.I., 1979. Analysis of pipe whip. EPRI, Report, NP-1208.

Huang, H., 1969. Transient interaction of plane acoustic waves with a spherical elastic shell. The Journal of the

Acoustical Society of America 45, 661–670.

Huang, H., 1974. Transient bending of a large elastic plate by an incident spherical pressure wave. Journal of

Applied Mechanics, 772–776.

Huang, H., 1979. Transient response of two fluid-coupler cylindrical elastic shell to an incident pressure pulse.

Journal of Applied Mechanics 46, 513–518.

Huang, H., 1986. Numerical analysis of the linear interaction of pressure pulses with submerged structures. In:

Smith, C.S., Clarke, J.D. (Eds.), Advance in Marines Structures.

Huang, H., Wang, Y.F., 1985. Asymptotic fluid-structure interaction theories for acoustic radiation prediction.

The Journal of the Acoustical Society of America 77, 1389–1394.


Huang, H., Kiddy, K.C., 1995. Transient interaction of a spherical shell with an underwater explosion shock wave

and subsequent pulsating bubble. Shock and Vibration 2, 451–460.

Huang, H., Lu, Y.P., Wang, Y.F., 1971. Transient interaction of spherical acoustic waves and a spherical elastic

shell. Journal of Applied Mechanics 38, 71–74.

Hung, C.F., Hsu, P.Y., Chiang, L.C., Chu, L.N., 1999. Numerical simulation of structural dynamics to underwater

explosion, Proceeding of The 13th Asian Tech. Exchange and Advisory Meeting on Marine Structures 1999

pp. 155–164.

Keil, A.H., 1961. The response of ships to underwater explosions. Annual Meeting of The SNAME, New York.

Kwon, Y.W., Fox, P.K., 1993. Underwater shock response of a cylinder subjected to a side-on explosion.

Computers and Structures 48 (4), 637–646.

Kwon, Y.W., Cunningham, R.E., 1998. Comparison of USA-DYNA finite element models for a stiffened shell

subject to underwater shock. Computers and Structures 66 (1), 127–144.

Liang, C.C., Lai, W.H., Hsu, C.Y., 1998. Study of the nonlinear response of a submersible pressure hull.

International Journal of Pressure Vessels and Piping 75, 131–149.

Liang, C.C., Hsu, C.Y., Lai, W.H., 2000. A study of transient responses of a submerged spherical shell under

shock waves. Ocean Engineering 28, 71–94.

Mindlin, R.D., Bleich, H.H., 1953. Response of an elastic cylindrical shell to a transverse step shock wave.

Journal of Applied Mechanics 20, 189–195.

O’Hara, G.J., Cunniff, P.F., 1993. Scaling for shock response of equipment in different submarines. Shock and

vibration 1 (2), 161–170.

Park, K.C., Felippa, C.A., DeRuntz, J.A., 1977. Stabilization of staggered solution procedures for fluid-structure

interaction analysis. Computational Methods for Fluid-Structure Interaction Problems, Ed. T. Belytschko,

AMD 26, 94-124.

Payton, R.G., 1960. Transient interaction of an acoustic wave with a circular cylindrical shell. Journal of the

Acoustical Social of America 32, 722–729.

Reid, W.D., 1996. The response of surface ship’s to underwater explosions. DSTO, Aeronautical and Maritime

Research Laboratory.

Shin, Y.S., Chisum, J.E., 1997. Modeling and simulation of underwater shock problems using a coupled

lagrangian-eulerian analysis approach. Shock and Vibration 4, 1–10.

Shin, Y.S., Santiago, L.D., 1998. Surface ship shock modeling and simulation: two-dimensional analysis. Shock

and Vibration 5, 129–137.

Smith, P.D., Hetherington, J.G., 1994. Blast and Ballistic Loading of Structures. Butterworth-Heinemann Ltd,

London.

Tang, S.C., Yen, D.H.Y., 1970. Interaction of a plane acoustic wave with an elastic spherical shell. The Journal of

the Acoustical Society of America 47, 1325–1333.

Taylor, G.I., 1950. The pressure and impulse of submarine explosion waves on plates. Compendium Underwater

Explosion Res ONR 1, 1155–1174.

shock responshock responses of a surface ship

Documents

underwater shock damage

underwater shock loading

underwater shock isuseful

underwater explosionhave

shock responses

shock severity

ship hulls

shock loading history