Full text of DOI:10.1155/2001/798735

253
Experimental and numerical investigations on
deformation of cylindrical shell panels to
underwater explosion
K. Ramajeyathilagama, C.P. Vendhanb,∗ and
V. Bhujanga Raoa
sion energy, the structural response is either elastic or
plastic. The sequence of failures associated with un-
derwater explosion is elastic deformation, large defor-
mation, tensile tearing, shear failure etc. Keil [1,2] has
briefly dealt about underwater explosion, failure and
damage mechanisms. A number of investigations on
stiffened and unstiffened plates and beams subjected to
air blast loading [3–18] have been reported in the liter-
ature. There does not appear to be any published work
dealing with underwater explosion effects on cylindri-
cal panels.
aShock and Vibration Center, Naval Science and
Technological Laboratory, Visakhapatnam 530027,
India
bOcean Engineering Department, Indian Institute of
Technology Madras, Chennai, 600036, India
Received 5 May 2000
Revised 24 April 2001
Menkes and Opat [18] reported three failure modes,
namely, a) large deformation, b) tensile tearing and c)
shear failure for explosively loaded clamped beams.
Similar failure modes have been found in the case of
unstiffened and stiffened plates to air blast loading [11,
12]. Ramajeyathilagamet al. [19] have also established
the first two failure modes for rectangular plates under
air backed condition subjected to underwater explosion
on the basis of shock tank tests and numerical analysis.
Cylindrical shell panels find wide use in ship and
offshore structures and hence there is a need to under-
stand their behaviorundershock loading. In the present
paper an experimental investigation on the large de-
formation of cylindrical shell panels subjected to un-
derwater explosion is presented followed by numerical
modelling. The experiments were carried out on air-
backed cylindrical shell panels with rise-to-span ratio
h/l = (0.0, 0.05, 0.1), to study the curvature effects on
the shock response. The experiments were conducted
using small explosive charges of PEK I in a shock
tank. Numerical analysis has been carried out using
the CSA/GENSA [DYNA3D] [20] non-linear finite el-
ement code and the results comparedwith experimental
data.
Experimental and numerical investigations on cylindrical
shell panels subjected to underwater explosion loading are
presented. Experiments were conducted on panels of size
0.8 × 0.6 × 0.00314 m and shell rise-to-span ratios h/l =
0.0, 0.05, 0.1, using a box model set-up under air backed
conditions in a shock tank. Small charges of PEK I explo-
sive were employed. The plastic deformation of the panels
was measured for three loading conditions. Finite element
analysis was carried out using the CSA/GENSA [DYNA3D]
software to predict the plastic deformation for various load-
ing conditions. The analysis included material and geomet-
ric non-linearities, with strain rate effects incorporated based
on the Cowper-Symonds relation. The numerical results for
plastic deformation are compared with those from experi-
ments.
1. Introduction
Non-contact underwater explosions are widely used
to cause damage to ship structures as a part of naval
warfare strategy. The response of structures subjected
to the resultant shock loading is quite complex involv-
ing fluid – structure interaction, high strain rates, ma-
terial and geometric non-linearities, large deformation,
tensile tearing and rupture. Depending on the explo-
2. Experimental work
∗Corresponding author: Fax: +91 044 2350509; E-mail: vend-
han@iitm.ac.in.
Experiments were conducted in a shock tank of di-
mensions 15×12×10 (depth) m. The test fixture used
Shock and Vibration 8 (2001) 253–270
ISSN 1070-9622 / $8.00 2001, IOS Press. All rights reserved

254
K. Ramajeyathilagam et al. / Experimental and numerical investigations on deformation
to study the large deformation behavior of curved shell
panels is shown in Fig. 1. The test set-up consists of
a heavy steel box to simulate air- backed condition, a
top hollow cover plate and two end plates, which have
been machined to match the curvature of the test panels
corresponding to h/l = 0.05 and 0.1. All the fixture
plates were made of 0.025 m thick mild steel plates.
The test panels were fabricated from cold rolled high
strength, low alloy, mild steel plates of dimensions
0.8 × 0.6 × 0.00314 m. The cylindrical panels were
formed by setting the required radius on a plate bend-
ing machine. A typical shell panel is shown in Fig. 2.
The exposed (base) area of the panel to shock loading
is 0.6×0.4 m2, with the load acting on the convex side.
The experimentswere carried out using small explosive
charges of PEK I (1.17* TNT) of 20 gm, 30 gm and
50 gm weights. The explosive was packed in a small
spherical plastic container, appropriately designed for
each charge weight, and detonated electronically. In
the experimental set-up, the test panel was placed be-
tween the bottom box and the top cover plate. End
plates matching the curvature were used at the curved
ends to prevent water entering the box. The complete
set-up was bolted using 30 M12 bolts along the edges
with 0.1 m of the test panel held between the fixture
plates.
The test fixture along with the test panel was im-
mersed in the shock tank (at the NSTL, Visakhapat-
nam) to a water depth of 2.0 m to eliminate the effect of
surface reflections on the panel during early response.
The side walls and the bottom of the tank remained at
6 m and 8 m, respectively, away from the explosion.
Hence, the influence of reflections from the bottom and
side walls on the early response is considered negli-
gible. The explosive charge was attached at a known
distance by the use of a channel section fixed to the
box model (see Fig. 1). In all the experiments, the ex-
plosive was placed on the normal line passing through
the center of the panel at a stand-off distance of 0.2 m,
measured from the base of the panel.
Fig. 1. Panel test fixture.
panel was obtained by deducting the depth of the beam
and the thickness of the curved top cover plate from the
originally measured displacement.
3. Finite element analysis
3.1. Finite element formulation
The finite element analysis of the test panel was per-
formedusing CSA/GENSA [20], a commerciallyavail-
able version of the DYNA3D non-linear finite element
analysis software. Hughes-Liu shell elements [21]
available in the GENSA code were employed for mod-
elling the test panel. In the above code, the material
non-linearitiesare modeledby the Von- Mises yield cri-
terion and its associated flow rule with a bilinear stress
– strain law. The geometric non-linearities are based
on large deformation finite strain formulation. The fi-
nite element equations of motions for the assemblage
of elements, derived based on the principle of virtual
work, may be written in the form [20]
The permanent deformation of the panel was mea-
sured after each test, with the test panels secured to
the fixture, as follows: Grids were marked before each
test on a quadrant of the test panel corresponding to
the finite element mesh, employed in the numerical
analysis. Also markings were made on the top cover
plate to match with the grid lines. A thin rectangular
beam was placed over the curved top cover plate at
each grid line, one after another. The depth of dishing
was measured from the top of the beam with the help
of a vernier. Subsequently the final deformation of the
¨
[M]{X} = {P} − {F} + {H}
(1)
where [M] denotes the diagonal mass matrix, {P} the
sum of external and body force vectors, {F} the stress

K. Ramajeyathilagam et al. / Experimental and numerical investigations on deformation
255
Table 1
Maximum shock pressure at the center of the plate (standoff distance = 0.2 m)
S. No.
Test No.
Shell
rise ratio
(h/l)
Charge
weight
Effective
standoff
distance (m)
Shock factor
(0.45 × W1/2/R)
(kg1/2/m)
Peak
pressure
(MPa)
(× 10−3 kg)
1
2
3
4
5
6
7
8
9
HL001
HL002
HL003
HL051
HL052
HL053
HL11
0.00
0.00
0.00
0.05
0.05
0.05
0.10
0.10
0.10
20
30
50
20
30
50
20
30
50
0.2
0.2
0.2
0.18
0.18
0.18
0.16
0.16
0.16
0.318
0.390
0.503
0.354
0.433
0.559
0.398
0.487
0.629
156.0
181.0
220.0
175.0
204.0
247.0
199.0
232.0
282.0
HL12
HL13
divergence vector and {H} the hourglass resistance.
The stress divergence vector and hourglass resistance
vector are given by [20]
ꢀ
{F} = [B]T {σ}dv
(2)
(3)
v
ꢁ
{H} = −an
hijrjk (i = 1, 3)
j
where,
hij
ꢁ
=
Vikrjk
(4)
(5)
k
and
Fig. 2. Typical test panel (h/l = 0.1).
an = Qngρve2/3c/4
ements for some representative cases. Both the meshes
converged to virtually the same results. Therefore, for
all the subsequent analysis, the mesh of lower density
(10 × 10 grid ) as shown in Fig. 3 was used to achieve
computational economy.
Here [B] is the strain -displacement matrix (with t
denoting the transpose), {σ} the stress vector, rjk the
hourglass base vector, hij the magnitude of hourglass
mode, v the element volume, c the sound speed in the
material, Qng a constant with value between 0.05 and
0.15 and Vik the nodal velocity of the kth node in the
ith direction. The set of non-linear equations given in
Eq. (1) are solved using the central difference scheme.
3.3. Material model
The uniaxial stress strain properties obtained for high
strength, low alloy, mild steel have been used as the
primary material data:
3.2. Modeling of the test panels
Elastic modulus
Poisson’s ratio
Mass density
Tangent modulus Et = 250.0 MPa
Static yield stress σy = 400.0 MPa
E = 2.1 ∗ 105 MPa
Considering the experimental fixing conditions with
0.025 m thick plates, the 0.00314m thick test panel was
assumed to be fixed at the edges of the exposed area
(0.6 × 0.4 m2); i.e. the three displacements and two
bending slopes were assumed to be zero at the edges.
Because of symmetryin loading and structure, only one
quadrant of the exposed area (i.e. 0.3 × 0.2 m2) of the
test panel was consideredfor the analysis. The Hughes-
Liu shell element was selected to model the test panel
because of its robustness and greater accuracy when
encountering finite strain [20]. The quarter panel was
modeled using two grid sizes of 10×10 and 20×20 el-
γ = 0.3
ρ = 7860.0 kg/m3
Rupture strain
εrup = 0.23
The constitutive model selected for the non-linear
analysis of the panels using the GENSA code is the
elasto – plastic material model with isotropic harden-
ing. In order to include the effect of strain rate in the
finite element analysis, the elasto-plastic analysis has
been first carried out ignoring this effect, i.e. the static

256
K. Ramajeyathilagam et al. / Experimental and numerical investigations on deformation
Table 2
Permanent central deflection from experiments
S. No. Test No.
Shell
rise ratio
(h/l)
Charge
weight
Effective
standoff
distance (m)
Shock factor
(0.45 × W1/2/R)
(kg1/2/m)
Permanent
deformation
(m)∗∗
(× 10−3 kg)
1
2
3
4
5
6
7
8
9
HL001
HL002
HL003
HL051
HL052
HL053
HL11
0.00
0.00
0.00
0.05
0.05
0.05
0.10
0.10
0.10
20
30
50
20
30
50
20
30
50
0.2
0.2
0.2
0.18
0.18
0.18
0.16
0.16
0.16
0.318
0.390
0.503
0.354
0.433
0.559
0.398
0.487
0.629
0.0446
0.0554
0.0736
0.0687
0.0816
0.1079
0.0854
0.0988
0.1247
HL12
HL13
∗∗Measured positive towards the base from the top.
Fig. 3. Finite element modelling of the shell panel.
yield stress has been employed in the analysis. The
average strain rate has been calculated from the max-
imum effective plastic strain curve obtained from the
where σy is the static yield stress and D and n are other
material parameters, and for the strain rate ε˙, which
strictly speaking is a function of time, the average value
mentioned above has been employed. In the present
calculations, D = 40/sec and n = 5 [10], being the
commonly accepted values for mild steel, have been
used. Thermal effects were not includedin the analysis.
ꢂ
ꢃ
analysis. The effective platic strain εpeff is given
by [20]
ꢄ
ꢅ
ꢀ
1/2
t
2
3
εpeff
=
ε˙pij ε˙pij
dt
(6)
0
3.4. Shock pressure loading
where the plastic strain rate is given by the difference
between the total strain rate (ε˙ij) and elastic strain rate
(ε˙eij) as
The empirical pressure time history at any location
with an instantaneous pressure increase followed by
a decay, approximated by an exponential function, is
given by [23]
ε˙pij = ε˙ij − ε˙eij
(7)
P(t) = P0e−(t−t )/θ 0 ꢀ t ꢀ θ
(9)
d
Subsequently, the dynamic plastic effects on the de-
formation of the panel have been incorporated by ad-
justing the dynamic yield stress (σdy) using the Cowper
– Symonds relation [22] given by
For trinitrotoluene(TNT), the peak pressure(P0) and
the decay constant (θ) are given by
ꢄ
ꢅ1.13
ꢆ
W1/3
R
1
n
ꢇ
ꢇ
ꢇ
ꢇ
ꢇ
ꢇ
ꢇ
ꢇ
P0 = 52.16 ∗ 106
(10)
ε˙
D
σdy = σy 1 +
(8)

K. Ramajeyathilagam et al. / Experimental and numerical investigations on deformation
257
300
250
200
150
100
50
Charge W eight
Standoff Distance = 0.2 m
Shell rise ratio (h/l) = 0.1
= 50 gm
Free Field Pressure
Total Pressure
0
a) h/l=0.0
0
20
40
60
80
100
Time (microsec)
Fig. 4. Typical pressure time history at the center of the panel.
ꢄ
ꢅ−0.22
W1/3
R
θ = 92.5 ∗ W1/3
(11)
where P(t) is the pressure (Pa) at any instant of time
t, P0 the peak pressure of the shock front, θ the de-
cay constant (µsec), t the time variable (µsec), W the
charge weight (kg), td = (R − R0)/c the time delay
(µsec), R the stand off distance (m), R0 the shortest
radial distance and c the sound speed in water (m/sec).
When the free field pressure given in Eq. (9) im-
pinges on the panel, it undergoes diffraction and scat-
tering and hence the net pressure loading on the curved
panel is modified by the interaction between the fluid
and the structure. For an infinite plate submerged in
water, Taylor’s plate theory [24] provides the resultant
over pressure accounting for the fluid-structure interac-
tion effect. In the present study this theory is assumed
to be valid for finite curved panels also. Thus, based
on Taylor’s plate theory the total pressure on the panel
considering fluid – structure interaction is given by
b) h/l=0.05
Pt = 2 ∗ P(t) − (ρcv(t)/ sin φ)
where
v(t) =
(12)
ꢈ
ꢉ
c) h/l=0.1
2P0
ρc z − 1
1
et/zθ − et/θ
(13)
Fig. 5. Dishing of shell panels (3 in each category, see Table 1)
(Panels shown concave up).
m
z =
(14)
(15)
ρcθ
is noted that the shock wave propagation in the above
theory uses the plane wave acoustic approximation.
For the purpose of non-linear finite element analysis,
the pressure on each finite element is computed using
Eq. (12) at the center of the element. A spherical
spreadingassumption has been used to approximatethe
pressure variation over the panel. A typical pressure
φ = sin−1(R0/R)
Here, Pt is the total pressure (Pa) on the plate at time
t, ρ the fluid density (kg/m3), m the mass per unit area,
z the characteristic mass ratio, v(t) the plate velocity
(m/sec) and φ the angle of attack of the shock wave. It

K. Ramajeyathilagam et al. / Experimental and numerical investigations on deformation
259
0.14
h/l=0.00
h/l=0.05
0.12
h/l=0.10
0.10
0.08
0.06
0.04
0.30
0.40
0.50
0.60
0.70
1/2
Shock Factor (Kg
/m)
Fig. 7. Shock factor Vs permanent deformation (Experimental).
0.25
0.20
0.15
0.10
0.05
0.00
0
400
800
1200
1600
2000
Time (microsec)
Fig. 8. Typical effective plastic strain time history (Numerical).
time history (both total and free field using Eqs (9)
and (12) at the center of the panel (with h/l = 0.1)
correspondingto a 50 gm charge weight at 0.2 m stand-
off is shown in Fig. 4. The peak pressure estimated for
the various test conditions at the center of the panel are
given in Table 1.
(i.e. deflections normal to the base) for various loading
conditions are presented in Table 2. The ‘dished’ shell
panels are shown in Fig. 5. The permanent deformation
of the shell panels measured from the original position
along the longitudinal and transverse center lines (DC
and BC in Fig. 3) are shown in Fig. 6. Although the de-
formation pattern for h/l = 0.05 and 0.1 are somewhat
similar to that of the flat panel (i.e. h/l = 0.0), it is
essential to note that the shallow shell configurationun-
dergoes snap through. For computing the shock factor
(SF = 0.45 ∗ W1/2/R; W the charge weight (kg), R
the effective standoff distance (m)), the effective stand-
off distance is taken to be R = (standoff distance from
4. Results and discussions
4.1. Experimental
The permanent central deflection of the shell pan-
els, measured positive from the panel towards the base

260
K. Ramajeyathilagam et al. / Experimental and numerical investigations on deformation
0.10
Without Strain Rate Effects
With Strain Rate Effects
0.08
0.06
0.04
0.02
0.00
0
0
0
400
400
400
800 1200
Time (microsec)
1600
2000
a )SF = 0.318
Without Strain Rate Effects
With Strain Rate Effects
0.12
0.08
0.04
0.00
800 1200
Time (microsec)
1600
2000
b) SF = 0.390
Without Strain Rate Effects
With Strain Rate Effects
0.16
0.12
0.08
0.04
0.00
800 1200
Time (microsec)
1600
2000
c) SF =0.503
Fig. 9. Displacement time history for h/l = 0.0 (Numerical).
the base – the shell rise at the center). SF Vs the max-
imum permanent deformation (at the center, point C in
Fig. 3) for various shell panels is plotted in Fig. 7. The
permanentdeformationof the panel evidentlyincreases
with increase in the shock load. It is interesting to note
that the permanent set increases with rise ratio (for the

K. Ramajeyathilagam et al. / Experimental and numerical investigations on deformation
261
Without Strain Rate Effects
0.12
With Strain Rate Effects
0.08
0.04
0.00
0
0
0
400
400
400
800 1200
Time (microsec)
1600
2000
2000
2000
a) SF =0.354
0.16
0.12
0.08
0.04
0.00
W ithout Strain Rate Effects
W ith Strain Rate Effects
800 1200
Time (microsec)
1600
b) SF = 0.433
Without Strain Rate Effects
With Strain Rate Effects
0.20
0.16
0.12
0.08
0.04
0.00
800 1200
Time (microsec)
1600
c) SF =0.559
Fig. 10. Displacement time history for h/l = 0.05 (Numerical).
two cases tested), which may be ascribed to the snap
4.2. Numerical
through behavior. All the experiments were aimed at
causing mode I failure and in none of the experiments
tensile tearing was observed.
The numerical analysis for all the test condi-
tions have been carried out using the CSA/GENSA

262
K. Ramajeyathilagam et al. / Experimental and numerical investigations on deformation
Without Strain Rate Effects
With Strain Rate Effects
0.12
0.08
0.04
0.00
0
400
800 1200
Time (microsec)
1600
2000
a) SF =0.398
Without Strain Rate Effects
With Strain Rate Effects
0.16
0.12
0.08
0.04
0.00
0
400
800 1200
Time (microsec)
1600
2000
b) SF =0.487
W ithout Strain R ate Effects
W ith Strain R ate E ffects
0.20
0.16
0.12
0.08
0.04
0.00
0
400
800 1200
T im e (m icro sec)
1600
2000
c) SF =0.629
Fig. 11. Displacement time history for h/l = 0.1 (Numerical).
[DYNA3D] explicit integration code. Because the
method employed is conditionally stable, the choice
of time step is very important in the analysis. Three
different incremental time steps were used to study the
numerical convergence for a typical case. The selected
time steps of 0.5 µsec, 1.0 µsec and 2.0 µsec were
found to meet the stability requirements of the explicit
integration scheme. Since the results for the three time

K. Ramajeyathilagam et al. / Experimental and numerical investigations on deformation
263
Table 3
Strain rate and dynamic yield stress
S. No. Test No. Shell rise
Shock
factor
Strain
rate
(1/sec)
Dynamic
yield stress
(MPa)∗∗
Yield
stress
ratio
(h/l)
factor∗
1
2
3
4
5
6
7
8
9
HL001
HL002
HL003
HL051
HL052
HL053
HL11
0.00
0.00
0.00
0.05
0.05
0.05
0.10
0.10
0.10
0.318
0.390
0.503
0.354
0.433
0.559
0.398
0.487
0.629
504.0
832.0
1071.0
281.6
520.0
1023.0
176.0
460.0
772.0
1064.0
1134.0
1172.0
991.0
1068.0
1165.0
938.0
2.66
2.835
2.93
2.478
2.67
2.913
2.345
2.63
HL12
HL13
1052.0
1123.0
2.801
∗Yield stress factor = Dynamic yield stress/ Static yield stress.
∗∗Static yield stress = 400 MPa.
steps did not differ much, 1.0 µsec time step was se-
lected for subsequent analysis.
The strain rate and the corresponding dynamic yield
stress based on the numerical analysis are presented in
Table 3. The average strain rate varies considerably so
does the dynamic yield stress (Fig. 12) similar to that
reported for plates by Olson et al. [10] under air blast
loading and Ramajeyathilagamet al. [19], for underwa-
ter shock. The yield stress factor (dynamic yield stress
/ static yield stress) is found to vary between 2 and 3
for the test cases considered. At this juncture, it may
be noted that because the stand-off distance was kept at
0.2 m in all the tests, the three test panels experienced
different pressure loading, with the intensity increas-
ing with the rise ratio. In spite of this, the strain rate
is found to be less for the curved panels compared to
the flat plate, which may be ascribed to the shell mem-
brane effects. Also it is noted that smaller dynamic
yield stresses associated with reduced strain rates for
the curved panel may also be partly responsible for the
higher permanent set in the case of curved panels com-
pared to a flat plate (see Fig. 7). The influence of ge-
ometric imperfections as well as residual stresses that
might have been caused due to fabrication of the shell
panels, could not be investigated.
The analysis has been performed first without strain
rate effects and with strain rate effects later, by adjust-
ing the dynamic yield stress in the analysis based on
the Cowper – Symonds relation as discussed earlier.
The average strain rate has been obtained from the ef-
fective plastic strain time history curve obtained from
the numerical analysis. The maximum effective plas-
tic strain time history for the shell panel (h/l = 0.05)
corresponding to a charge weight of 20 gm is shown in
Fig. 8. The effective plastic strain is found to increase
until about 800 µsec and then reach a plateau. The
average strain rate used for the analysis has been com-
puted from the maximum plastic strain and the time
taken to reach the maximum.
The displacement time history for various load-
ing conditions for h/l = (0.0, 0.05, 0.1) is shown in
Figs 9–11. From the curves it can be seen that the dis-
placement time history at the center of the panel shows
a nearly linear trend upto the maximum and found to
have small oscillations thereafter as reported in other
studies [11,12,20]. This means that most of the kinetic
energy imparted to the panel by the shock loading has
been dissipated as plastic work. As a result, the dif-
ference between the maximum displacement and the
permanent deformation was quite small.
4.3. Comparisons
The central permanent deformations obtained from
experiments and numerical analysis, with and without
strain rate effects, are compared in Fig. 13. The nu-
merical results show the same trend as the experimen-
tal results, namely increase in permanent set with both
shock factor and shell rise ratio. It is seen that the com-
puted permanent set values without strain rate effect
are about 55% to 65% more than that of experimental
values. The numerical model using average strain rate
effects predicts values close to the experimental values
within 10% accuracy. The measured and computed
It is also observed from the curves that the total dis-
placement and the permanent deformation decrease by
about 44% for flat panels and 36% for curved panels
because of strain rate effects. The time of occurrence
of the maximum displacement was reduced consider-
ably (see Figs 9–11) when the strain rate effects were
introduced. The reduction in the permanent deforma-
tion and the occurrence of the maximum displacement
are similar to that reported for plates subjected to air
blast [10] or underwater shock [19].

264
K. Ramajeyathilagam et al. / Experimental and numerical investigations on deformation
1200
800
400
h/l=0.0
h/l=0.05
h/l=0.10
0
0.30
0.40
0.50
0.60
0.70
1/2
Shock Factor (kg
/m )
a) Shock factor Vs Strain rate
1200
1100
1000
900
h/l=0.0
h/l=0.05
h/l=0.10
0.30
0.40
0.50
0.60
/m)
0.70
1/2
Shock Factor ( Kg
b) Shock factor Vs Dynamic yield
Fig. 12. Strain rate and dynamic yield stress.
permanent deformations of the panels (with strain rate
effects) along the center lines (DC and BC in Fig. 3)
are compared in Figs 14–16. The numerical prediction
was found to be lower than the experimental values
near the edges and crossed to the other side roughly
at a distance of one fourth length of the panel for the
flat plate. In the center of the panel, the predicted
permanent deformation was found to be on the higher
side compared to the experimental values. This trend
for the flat panel is similar to that reported for square
plates [10]. For the cylindrical shell panels, the cross-
ing over has progressed towards the center of the panel
(Figs 15 and 16). The comparison of permanent defor-
mation pattern over a quadrant for a typical test panel
(h/l = 0.1) corresponding to a charge weight of 30 gm
is shown in Fig. 17. Overall, numerically predicted de-
formation shows a reasonable correlation with that of
experiments, the comparison being better in the central
region. The numerical analysis is not able to predict
the permanent set near the edges accurately because of
the plastic hinge formed near the edges, which could
not be considered in the numerical model.
5. Conclusions
The results of the experimental and numerical in-
vestigations carried out on cylindrical shell panels
(h/l = 0.0, 0.05, 0.1) subjected to underwater explo-
sion are presented. The numerical study shows that the

K. Ramajeyathilagam et al. / Experimental and numerical investigations on deformation
265
Without Strain Rate Effects
With Strain Rate Effects
Experimental
0.16
0.12
0.08
0.04
0.30
0.35
0.40
0.45
0.50
1/2
Shock Factor (Kg
/m)
a) h/l=0.00
0.20
0.16
0.12
0.08
0.04
Without Strain Rate Effects
With Strain Rate Effects
Experimental
0.30
0.40
0.50
0.60
1/2
Shock Factor (Kg
/m)
b) h/l=0.05
Without Strain Rate Effects
With Strain Rate Effects
Experimental
0.20
0.16
0.12
0.08
0.04
0.30
0.40
0.50
0.60
0.70
1/2
Shock Factor (Kg
/m)
c) h/l=0.10
Fig. 13. Comparison of central (Point C in Fig. 3) permanent deformation for various shell panels.
strain rate considerably affects the dynamic response.
dicts the permanent set of the panel with engineering
Use of average strain rate in finite element analysis in
conjunction with the Cowper – Symonds relation pre-
accuracy. Such an analysis will evidently be more eco-
nomical than the one using instantaneous strain rates.

K. Ramajeyathilagam et al. / Experimental and numerical investigations on deformation
269
C
____ Experimental
------ Numeri
0.1
0.05
0
0.2
D
B
0.16
0.3
0.12
0.24
0.08
0.18
0.12
0.04
0.06
0
0
A
Fig. 17. Comparison of permanent deformation on a quadrant of the test panel (see Fig. 3) (Permanent deformation measured from initial panel
surface towards base) (h/l = 0.1, Charge weight = 30 gm).
The membrane action of the shell panel considerably
reduces the strain rate and hence, the dynamic yield
compared to that of flat plates. Because of this as well
as due to snap through deformation, the curved panels
show larger permanent set than the flat plate. The influ-
ence of geometric imperfections and residual stresses
in the curved panels needs to be investigated. The
difference between numerical and experimental results
may partly be due to the use of Taylor’s plate theory
to approximate fluid-structure interaction in the case of
curved panels.
[2] A.H. Keil, The response of ships to underwater explosion,
Transactions of Society of Naval Architects and Marine Engi-
neers 69 (1961), 360–410.
[3] A.D. Gupta, F.H. Gregory, R.L. Bitting and S. Bhattacharya,
Dynamic analysis of an explosively loaded hinged rectangular
plate, Computers and Structures 26 (1987), 339–344.
[4] R. Houlston and J.E. Slater, Damage analysis with ADINA of
Naval panels subjected to a confined air blast wave, Computers
and Structures 47 (1993), 629–639.
[5] R. Houlston, J.E. Slater, N. Pegg and C.G. DesRachers, On
analysis of structural response of ship panels to air blast load-
ing, Computers and Structures 21 (1985), 273–289.
[6] J. Jiang and M.D. Olson, New Design – Analysis Techniques
for Blast Loaded Stiffened Box and Cylindrical Shell Struc-
tures, International Journal of Impact Engineering 13 (1993),
189–202.
[7] M.R. Khalil, M.D. Olson and D.L. Anderson, Non-linear Dy-
namic Analysis of Stiffened Plates, Computers and Structures
29 (1988), 929–941.
Acknowledgment
[8] T.S. Koko and M.D. Olson, Non-linear Analysis of Stiffened
plates using super elements, International Journal for Numer-
ical Methods in Engineering 31 (1991), 319–343.
[9] T.S. Koko and M.D. Olson, Non-linear Transient Response
of Stiffened plates to Air Blast Loading by a super element
Approach, Computer Methods in Applied Mechanics and En-
gineering (1991), 737–760.
The authors are grateful to RAdm S.Mohapatra, Di-
rector, NSTL for his permission to avail the experi-
mental facilities and publication of the work. The ex-
perimental support rendered by members of the shock
studies division of NSTL is gratefully acknowledged.
[10] M.D. Olson, G.N. Nurick and J.R. Fagnan, Deformation and
rupture of blast loaded square plates – predictions and Ex-
periments, International Journal of Impact Engineering 13
(1993), 279–291.
References
[11] G.N. Nurick, M.D. Olson, J.R. Fagnan and A. Levin, Defor-
mation and tearing of blast loaded stiffened square plates, In-
ternational Journal of Impact Engineering 15 (1995), 273–
291.
[1] A.H. Keil, Introduction to underwater explosion research,
UERD report 19–56, Norfolk naval ship yard, Portsmouth,
Virginia, 1956.

270
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[12] M.H. Klaus, Response of a Panel Wall Subjected to Blast
Loading, Computers and Structures 21 (1985), 129–135.
[13] L.A. Louca and J.E. Harding, Non-linear analysis of imperfect
plates under transient lateral pressure loading, Computers and
Structures 63 (1997), 27–37.
[14] Z. Qingjle, L. Shiqi and J. Zheng, Nonlinear dynamic behavior
of stiffened plate under instantaneous loading, Computers and
Structures 40 (1991), 1351–1356.
[15] T. Wierzbicki and A.L. Florence, A theoretical and experimen-
tal investigation of impulsively loaded clamped circular vis-
coplastic plates, International Journal of solids and structures
6 (1970), 553–568.
[16] G. Sinha and M. Mukhopadyay, Linear transient dynamic re-
sponse of stiffened plates by the finite element method, Report
No. MST/10, Department of naval Architecture, IIT, Kharag-
pur, India, 1992.
[18] S.B. Menkes and H.J. Opat, Tearing and shear failures in
explosively loaded clamped beams, Explosion Mechanics 13
(1973), 480–486.
[19] K. Ramajeyathilagam, C.P. Vendhan and V. Bhujanga Rao,
Non-linear transient dynamic response of rectangular plates
under shock loading, International Journal of Impact Engi-
neering 24(10) (2000), 999–1015.
[20] CSA/GENSA user manual, CSA Research Corporation, Cali-
fornia, USA, 1996.
[21] T.J.R. Hughes and W.K. Liu, Nonlinear finite element analysis
of shells: Part I. Three dimensional shells, Computer methods
in Applied Mechanical Engineering 26 331–362.
[22] N. Jones, Structural Impact, Cambridge University press,
Cambridge, UK, 1989.
[23] R.H. Cole, Underwater Explosions, Dover Publications Inc,
Newyork, USA, 1948.
[17] M.V. Dharaneepathy and C.K. Madheswaran, Influence of
stiffener patterns on impact response of steel panels, IE(I)
Journal-CV (1997), 44–48.
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sion waves on plates, Compendium of Underwater Explosion
Research, ONR 1 (1950), 1155–1174.

258
K. Ramajeyathilagam et al. / Experimental and numerical investigations on deformation
SF=0.318
0.08
0.08
0.06
0.04
0.02
0.00
SF=0.318
SF=0.390
SF=0.390
SF=0.503
SF=0.503
0.06
0.04
0.02
0.00
0.00
0.05
0.10
Distance from edge (m)
0.15
0.20
0.00
0.10 0.20
Distance from edge (m)
0.30
h/l=0.0
0.12
0.08
0.04
0.00
SF=0.354
SF=0.433
SF=0.559
SF=0.354
SF=0.433
SF=0.559
0.12
0.08
0.04
0.00
0.00
0.05
0.10
0.15
Distance from edge (m)
0.20
0.00
0.10 0.20
Distance from edge (m)
0.30
h/l=0.05
0.16
0.12
0.08
0.04
0.00
0.16
0.12
0.08
0.04
0.00
SF=0.398
SF=0.487
SF=0.629
SF=0.398
SF=0.487
SF=0.629
0.00
0.10
0.20
0.30
0.00
0.05
0.10
0.15
0.20
Distance from edge (m)
D istan ce from edge (m )
h/l=0.1
a) Longitudinal ( line DC in Fig.3)
b) Transverse ( line BC in Fig.3)
Fig. 6. Permanent deformation of the shell panel measured from the original position (Experimental) (SF = Shock Factor).

266
K. Ramajeyathilagam et al. / Experimental and numerical investigations on deformation
0.05
0.05
0.04
0.03
0.02
0.01
0.00
Experimental
Experimental
Numerical
Numerical
0.04
0.03
0.02
0.01
0.00
SF = 0.318
SF = 0.318
0.00
0.10
0.20
0.30
0.00
0.05
0.10
0.15
0.20
Distance from edge (m)
Distance from edge (m)
0.08
0.06
0.04
0.02
0.00
0.08
0.06
0.04
0.02
0.00
Experimental
Num erical
Experimental
Num erical
SF = 0.390
SF = 0.390
0.00
0.10
0.20
0.30
0.00
0.05
0.10
0.15
0.20
Distance from edge (m )
Distance from edge (m)
0.10
0.08
0.06
0.04
0.02
0.00
0.10
0.08
0.06
0.04
0.02
0.00
Experimental
Numerical
Experimental
Numerical
SF = 0.503
SF = 0.503
0.00
0.10
0.20
0.30
0.00
0.05
0.10
0.15
0.20
Distance from edge (m)
Distance from edge (m )
a) Longitudinal (DC in Fig. 3)
b) Transverse ( BC in Fig. 3)
Fig. 14. Comparison of permanent deformation along center line (h/l = 0.0).

K. Ramajeyathilagam et al. / Experimental and numerical investigations on deformation
267
0.08
0.06
0.04
0.02
0.00
0.08
Experimental
Experimental
Numerical
Numerical
0.06
0.04
SF = 0.354
SF = 0.354
0.02
0.00
0.00
0.10
0.20
0.30
0.00
0.05
0.10
0.15
0.20
Distance from edge (m)
Distance from edge (m)
0.10
0.08
0.06
0.04
0.02
0.00
0.10
0.08
0.06
0.04
0.02
0.00
Experimental
Numerical
Experimental
Numerical
SF = 0.433
SF = 0.433
0.00
0.10
0.20
0.30
0.00
0.05
0.10
0.15
0.20
Distance from edge (m)
Distance from edge (m)
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0.12
0.10
0.08
0.06
0.04
0.02
0.00
Experimental
Numerical
Experimental
Numerical
SF = 0.559
SF = 0.559
0.00
0.05
0.10
0.15
0.20
0.00
0.10
0.20
0.30
Distance from edge (m)
Distance from edge (m)
a) Longitudinal (DC in Fig. 3)
b) Transverse ( BC in Fig. 3)
Fig. 15. Comparison of permanent deformation along center line (h/l = 0.05).

268
K. Ramajeyathilagam et al. / Experimental and numerical investigations on deformation
0.08
0.08
0.06
0.04
0.02
0.00
Experimental
Numerical
Experimental
Numerical
0.06
0.04
0.02
0.00
SF = 0.398
SF = 0.398
0.00
0.10
0.20
0.30
0.00
0.05
0.10
0.15
0.20
Distance from edge (m )
Distance from edge (m)
0.10
0.08
0.06
0.04
0.02
0.00
0.10
0.08
0.06
0.04
0.02
0.00
Experimental
Numerical
Experimental
Numerical
SF = 0.487
SF = 0.487
0.00
0.10
0.20
0.30
0.00
0.05
0.10
0.15
0.20
Distance from edge (m)
Distance from edge (m)
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
Experimental
Numerical
Experimental
Numerical
SF = 0.629
SF = 0.629
0.00
0.10
0.20
0.30
0.00
0.05
0.10
0.15
0.20
Distance from edge (m)
Distance from edge (m)
(a) Longitudinal (DC in Fig. 3)
(b) Transverse ( BC in Fig. 3)
Fig. 16. Comparison of permanent deformation along center line (h/l = 0.1).

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