Damage response of submerged imperfect cylindrical structures to underwater explosion

00457949(9!5)00441-6

Cmpurers & Srrucrures Vol. 60, No. 5. pp. 683-693. 1996 Pubhshed by Elsevier Science Ltd

Pnnted in Great Brimn 0045.7949/96 1615.00 + 0.00

DAMAGE RESPONSE OF SUBMERGED IMPERFECT CYLINDRICAL STRUCTURES TO UNDERWATER

EXPLOSION

Y. S. Shin and D. T. Hooker

Department of Mechanical Engineering, Naval Postgraduate School, Monterey, CA 93943, U.S.A

(Received 31 October 1995)

Abstract-The effect of the initial geometric imperfections to the damage response of submerged structures is investigated. The type of the submerged structure investigated is the ring-stiffened long circular cylinders submerged in the fluid. The strain hardening mild steel is used in the analysis. The type of the loading is the underwater shock induced by underwater explosions. The modal imperfection concept has been used to simulate the initial geometric imperfections. The numerical analyses were performed to look into the details of damage response of ring-stiffened cylindrical shells. The following models were considered: (i) three-dimensional infinite ring-stiffened cylinder model, and (ii) three-dimensional finite ring-stiffened cylinder model. A finite element hydrocode was used to numerically predict model responses.

INTRODUCTION

The response of marine structures subjected to underwater explosion is greatly complicated by free field problems (e.g. incident/reflected/refracted/rarefection wave propagations, gas bubble oscillation and mi- gration toward free surface, bulk cavitations), com- plex fluid-structure interaction phenomena, and nonlinear dynamic behavior of the structures. One of the significant parameters affecting the damage response of the structure is the initial geometric imperfection of structures. The effect of the initial geometric imperfections to the nonlinear response of submerged cylindrical structures is investigated. The structure is subjected to underwater shock pressure loading induced by underwater explosions. The type of the submerged structure investigated is the ring- stiffened long circular cylinderical shell submerged in the fluid. The material used in the analysis is the strain hardening mild steel. The initial circumferential geometric imperfections are introduced to the circular cylindrical shell and their effects on damage responses are evaluated.

The finite element code used for dynamic response analysis of the submerged cylinders was VEC/ DYNA3D (nonlinear dynamic analysis of structures in three dimensions) [l]. This explicit code has been widely used for a variety of nonlinear engineering problems since its introduction in 1976. Its wide array of material models and equations of state result in a high degree of versatility. This code is also accessible to many users in that it is operational on worksta- tions and mainframe computers. The accessibility of this code is further enhanced by its availability in the public domain. The LS-INGRID pre-processor [2]

was used for three-dimensional mesh generation for VEC/DYNA3D and the LS-TAURUS post- processor [3] was used for the output display of contours, fringe plots and time histories.

The modeling of the surrounding water media was accomplished by use of a boundary element code that eliminates the need to discritize the fluid media. Instead, equivalent fluid forces and masses are applied to the nodes of a two-dimensional mesh which is superimposed over the surface of the cylinder. The boundary element code used in this analysis was USA (underwater shock analysis) code [4, 51. The coupling of the USA code with VEC/DYNA3D code called USA/DYNA3D was accomplished in 1991 which introduces the capability to calculate shock-induced nonlinear response of submerged structures. The verification of the satisfactory performance of the USA/DYNA3D code was recently conducted using analytical problems for spherical and infinite cylinder models [6].

The use of USA/DYNA3D for analyzing structures subjected to underwater shock has been further shown to be an effective tool in predicting the response of these structures and the resulting damage [7]. In some cases the actual deformation of a cylinder subjected to an underwater shock is not accurately predicted. One explanation for this inac- curacy is that actual cylinders have many imperfections (e.g. out of roundness, thin sections, voids, etc.) and the finite element modeling of these cylinders often do not take into account of the initial imperfections that are present. By introducing imperfections in the position of the node points generated by a finite element mesh generator it is possible to more accurately model the geometry of an actual cylinder. With

683

684 Y. S. Shin and D. T. Hooker

Fig. I Three-dimensional ring-stiffened infinite circular cylinder model.

a more accurate model of the actual cylinder the finite element numerical analysis of the resulting damage due to underwater shock may be more accurate as well.

USA BOUNDARY ELEMENT CODE

The underwater shock analysis (USA) code is a boundary element code for the underwater-shock

(a) Front View at 0.2 ms

(Scaled up 50X)

i

(c) Front View at 5.0 ms

fluid-structure interaction problem [4, 51 based on doubly asymptotic approximation (DAA) [8,9]. The DAA approach models the acoustic medium surrounding the structure as a membrane covering the wet surface of the structure. The principal advantage of the DAA is that it models the interaction of the submerged structure with the surrounding acoustic medium in terms of wet-surface response variables only, eliminating the need for fluid volume elements around the outside of the structure. The DAA may be used to determine the fluid pressure generated by the scattered wave on the wet surface of the structure. The governing equation of motion for the structure and the DAA equation with the interface compatibility relation are used to solve the dynamic response of the system.

The discretized differential equation for the structure can be expressed as

M,%+C,%+K,x=f, (I)

where x is the structural displacement vector, M,, C, and K, are mass, damping and stiffness matrices, respectively, f is the external force vector and a dot denotes a temporal derivative. For excitation of a submerged structure by an acoustic wave, f is

given by

(b) End View at 0.2 ms (Scaled up 50X)

(d) End View at 5.0 ms

SHOCK WAVE

Fig. 2. Deformation of ring-stiffened infinite length perfect circular cylinder subjected to a square pressure pulse of 500 psi for 1 ms.

Submerged imperfect cylindrical structures 685

i i

(a) Mode 6 Initial Imperfection A6= 0.05h (Scaled up 100X)

(b) Front View at 0.2 ms (Scaled up 50X)

(c) End View at 0.2 ms (Scaled up 50X)

SHOCK WAVE

SHOCK WAVE

(d) Front View at 5.0 ms (e) End View at 5.0 ms

Fig. 3. Deformation of ring-stiffened infinite length imperfect circular cylinder with a mode 6 initial imperfection subjected to a square pressure pulse of 500 psi for 1 ms.

f = -GA& + p,] + f, (2)

where pi and ps are nodal incident (known) and scattered (unknown) wave pressure vectors for the wet-surface fluid mesh, respectively, fd is the applied force vector for the dry-structure, A, is the diagonal area matrix converting nodal pressures to nodal forces and G is the transformation matrix associated with fluid and structural nodal surface forces.

The first order doubly asymptotic approximation can be expressed as [S, 91

(3)

where Mr is the fluid mass matrix for the wet-surface fluid mesh, p and c are the density and acoustic velocity of the fluid, respectively, and II, is the vector of scattered-wave fluid particle velocities normal to the structure’s wet surface. The DAA eqn (3) is called ‘doubly asymptotic’ because it exhibits the correct asyptotic behavior in both the high-frequency (early- time) and low-frequency (late-time) limits. For high- frequency motion, 1 i, I>> jp, 1, so that the second term in the left-hand side of eqn (3) is negligible compared with the first term, resulting in the plane wave approximation, p, = pcu, which is accurate for sufficiently short acoustic wavelengths. For low- frequency motion, Ih 1 ec 1 p, I, and eqn (3) approaches


the virtual mass approximation A,p, = M&I, which is accurate for sufficiently long acoustic wavelengths.

The fluid-structure interface compatibility relation can be expressed as

GT~=u,+ucr (4)

where the superscript ‘T’ denotes matrix transposi-

tion. Equation (4) expresses that the fluid particle velocity and the structural velocity normal to the wet-surface of the structure are equal at the interface.

Substituting eqn (2) into eqn (l), eqn (4) into eqn (3) and assuming fd is zero,

MS% + C,ir + K,x = -GAJp, + p,], (5)

and

.

--

; L-1 (a) Initial Imperfection(Scaled up 200X)

(a) 3D Model

(b) End Plate Model

(b) Front View at 5.0 ms

Fig. 5. Three-dimensional ring-stiffened finite circular cylinder model.

M,P, + pcA,p, = pcM,[GTR - ti,]. (6)

Equation (5) and (6) can be solved simultaneously at each time step for x and pS. The improved second

SHOCK order DAA [9] is also available in USA code. The WAVE detailed description of USA code is comprehensively

e described in Refs [4, 51.

INITIAL IMPERFECTIONS

The location of the node points of a structure modeled by a finite element mesh generator are

(b) End View at 5.0 ms precisely located (to the numerical accuracy of the

Fig. 4. Ring-stiffened infinite length imperfect cylinder modeling code) at the positions specified in the inputs

subjected to a square pressure pulse of 500 psi for 1 ms, to the mesh generator algorithm. Fabricated struc-

initial imperfection first 10 modes with random phase shifts. tures usually have many imperfections that are not


accurately modeled by many finite element mesh generators. Surveys have been performed to measure the imperfections that naturally exist in cylindrical shell structures [IO]. This data has been collected into data banks, and Arbocz suggests that it be used to improve design criteria for the buckling of thin shells. The imperfection data banks show that imperfections have characteristic distributions that include decreasing modal amplitudes with increasing mode number ducing initial

Kirkpatrick [l l] found that by intro- modal imperfections in a cylindrical

structure subjected to air blast shock loading the resulting numerical analysis agreed more closely to the experimental data than the analysis without these initial imperfections.

The initial imperfections were introduced into the location of the node points of the finite element model using a summation of modal imperfections expressed as the cosine series as shown in eqn (7):

AR(B) = : A, cos(n8 + &), (7) n=?

SHOCK WAVE

(a) Front View 0.2 ms (scaled up 50X) (b) End View

SHOCK WAVE

(c) Front View 1 .O ms (scaled up 2X) (d) End View

SHOCK WAVE

(e) Front View 2.0 ms (scaled up 2X) (f) End View

SHOCK WAVE

(g) Front View 5.0 ms (scaled up 2X) (h) End View

Fig. 6. Deformation of ring-stiffened finite length perfect circular cylinder subjected to a square pressure pulse of 500 psi for 1 ms.


SHOCK WAVE

---. 0 -._A

(a) Initial Imperfection (scaled up 100X)

\! ! ! ! ! ! ! ! ! I I ! ! ! I

(b) Front View at 5.0 ms

(c) End View at 5.0 ms

Fig. 7. Ring-stiffened finite cylinder subjected to a square pressure pulse of 500.

where AR is the radial imperfection, 0 is the angular position, n is the mode number, N is the maximum modal contribution, A, is the n th modal imperfection amplitude and 4, is a n th random modal phase shift. The assumption that the modal phase shift is a random variable is reasonable for many shells. The form for the modal amplitude can be expressed as shown in eqn (8):

where X and r are coefficients used to fit the available data for shells of a given construction. The modal amplitude can be modeled as a constant as shown in eqn (9) at 1% of the shell thickness for all modes:

A, = O.Olh for all n, (9)

where h is the shell thickness. Kirkpatrick [I I] suggested the following modal amplitudes in eqns (IO) and (11) based on his air blast tests for shells:

A,, = 0.05h (n < 6), (10)

An=; (n 37). (11)

The modal imperfections introduced in this study are in circumferential direction and no initial imperfections are introduced in cylinder axis direction.

RlNG STIFFENED 1NFlNlTE CYLINDER MODELS

The three-dimensional ring stiffened infinite cylinder model shown in Fig. 1 was developed. This model has a total of 400 elements with 40 elements in the circumferential direction and 10 elements along the length. The symmetric boundary conditions were imposed on both ends of the cylinder to model an infinite cylinder. This three-dimensional model has stiffeners 0.12 in thick and 1 in deep located on 12 in spacing. The shell is 0.06 in thick. This model and all subsequent models (unless specifically noted) used Belytschko-Tsay [12] shell elements for their numerical efficiency and also to eliminate the problems of using a solid element to model a thin walled structure. The shell material is modeled as mild steel, an elastic-plastic material with a Young’s modulus of 2.9 x 10' psi, a Poisson’s ratio of 0.3, a hardening modulus of 5100 psi and a yield stress of 32,000 psi.

Perfect cylinder model

This cylinder was subjected to an underwater shock wave modelled as a plane wave with an amplitude of 500 psi and a duration of 1 ms resulting from an explosive charge located along the x-axis. The early deformation of the cylinder at 0.2 ms (Fig. 2a) shows the outer shell pinching in on either side of the stiffener located in the center of the cylinder.

At this time in the deformation history, there is no evidence of the formation of a locally raised area on the front of the cylinder facing the charge. The final deformation pattern shown in Fig. 2c shows pinching

\\ l/ _--- Fig. 8. Initial imperfection first 10 modes A, = O.Olh with

random phase shifts (scaled up 200 x ).


(a) Front View A,=0 ‘.OOOlh

SHOCK WAVE

SHOCK WAVE

(c) Front View A,, = O.OOlh

SHOCK WAVE

(e) Front View A, = 0.01h

(b) End View

(d) End View

(f) End View

Fig. 9. Deformations at 5.0 ms of ring-stiffened finite length imperfect cylinder subjected to a square pressure pulse of 500 psi for 1 ms, initial imperfection first IO modes with random phase shifts.

of the shell on either side of the stiffener. The final deformation pattern of the cylinder side facing the charge has a local protrusion of the shell material towards the explosive charge as shown in Fig. 2d. Again, it is hypothesized that this damage pattern is the result of the model cylinder being a perfect cylinder while fabricated cylinders will always have imperfections in the shell of the cylinder.

Imperfect cylinder models

The introduction of initial imperfections signifi- cantly changes both the shape and magnitude of the final deformations seen in this cylinder model. The 5% mode 6 imperfection shown in Fig. 3 was introduced into this model. The resulting deformation due to a 500 psi with 1 m plane wave pressure pulse is shown in Fig. 3b-e. The initial imperfection shape is clearly evident in the final damage pattern. However, the shape of the cylinder at 0.2 ms does not show any of the initial imperfection. There is insufficient time elapsed for the deformations to grow large enough to show this initial imperfection. However, the pinching

of the shell on either side of the stiffener has already begun at 0.2 ms. The final deformation still shows this pinching effect, however, the magnitude of this type of deformation is reduced by the initial imperfection.

With the introduction of an initial imperfection of the first 10 modes with modal imperfection amph- tudes of 1% of the shell thickness and with random phase shifts as shown in Fig. 4, the resulting damage pattern again shows that the deformation of the shell preferentially follows the initial imperfections. The resulting damage pattern is very different from the perfect cylinder case (Fig. 2). Most noteworthy is the elimination of the pinch in the outer shell on either side of the stiffener. In additionthe magnitude of the deformation in the outer shell is much greater than for the perfect cylinder case. The introduction of these imperfections results in sites where the cylinder preferentially deforms during the shock pressure. Thus when the cylinder deforms, the deformation follows the initial imperfections resulting in the final shape of the cylinder which resembles the initial imperfection shape.


RING STIFFENED FINITE LENGTH CYLINDER MODELS

From the three-dimensional infinite cylinder a three-dimensional finite length cylinder was developed. This cylinder is 3 ft long, 1 ft in diameter with two stiffeners evenly spaced 12 in apart. The shell of the cylinder is 0.06 in thick mild steel. The stiffeners are mild steel 0.12 in thick and 1 in deep. The end plate is HY-100 steel 0.25 in thick. The HY-100 steel is modeled as an elastic-plastic material with a Young’s modulus of 2.9 x 10’ psi, a Poisson’s ratio of 0.3, a hardening modulus of 5020 psi and a yield stress of 108,000 psi. This cylinder is modeled as a half cylinder with a plane of symmetry perpendicular to the axis of rotation. The model has 40 elements in the circumferential direction and 15 elements in the axial direction for a total of 600 elements and 921 nodes. The circular end plate is modeled using 260 quadrilateral elements. The use of a half symmetry model with a symmetric boundary condition results in a smaller number of elements and greater compu- tational efficiency for the finite element analysis. The finite element model of a half cylinder is shown in Fig. 5. The end plate is HY-100 high strength steel and thickness is over four time thicker than the cylindrical shell. The deformation of the end plate due to shock pressure loading is relatively very small compared with the cylindrical shell. The focus is on the damage response of the cylindrical shell, not the end plate.

Perfect cylinder models

This cylinder was subjected to the same planar shock wave as in the previous models. The resulting damage patterns for a perfect cylinder are shown in Fig. 6. Again the side of the cylinder facing the planar shock wave showed a local raised area that was also seen in the previous models. In addition there is a pinch in the shell near the end plates and on either side of the stiffeners. Most of the deformation of the cylinder shell occurs between 1.0 and 2.0 ms.

Imperfect cylinder models

The addition of initial imperfections greatly changes the resulting deformation pattern of the shell of the cylinder. The addition of a mode 6 imperfection (shown in Fig. 7a) with a modal amplitude of 5%

SHOCK WAVE

Fig. 10. Deformation of ring-stiffened finite length perfect cylinder subjected to a square pressure pulse of 500 psi for

1 ms, cylinder rotated 4.5”.

SHOCK WAVE

(a) Front View

(b) Offset View

(c) End View

Fig. 11. Deformation at 5.0 ms of ring-stiffened finite length perfect circular cylinder subjected to a shock pressure by

40 lbs PETN explosive charge, 30 ft standoff distance.

of the shell thickness and with no random phase shift results in the final damage pattern shown in Fig. 7b and c. The end view clearly shows the strong effect of the mode 6 initial imperfection causing the final deformation to follow this initial imperfection. The front view of the cylinder shows that the pinching of the shell near the end plates and stiffeners is very much reduced due to the introduction of this imperfection.

An investigation into the effect of the modal imperfection amplitude on the final deformation pattern showed that very small modal imperfection amplitudes have a significant effect on the response of the cylinder. An initial imperfection consisting of the first 10 mode shapes and random phase shifts is shown in Fig. 8.

The shell deformation again follows the initial imperfections present in the shell structure. As the modal imperfection amplitude increases, the deformation of the shell of the cylinder also increases. At a modal amplitude of 0.01% of the shell thickness (Fig. 9a and b), the front of the cylinder is distinctly


raised toward the explosive charge as in the perfect cylinder. Yet it can also be seen from the end view that the deformation pattern even at this magnitude of imperfection has changed from the pattern for the perfect cylinder. At a modal imperfection amplitude of 0.1% (Fig. 9c and d) this alteration of the pattern is quite obvious. The pinching of the shell of the cylinder persists until higher modal imperfection amplitudes are reached. Finally, at a modal imperfection amplitude of 1 .O% of the shell thickness (Fig. 9e and f) the pinching of the shell is eliminated, but dynamic buckling or dishing type of shell damage is pronounced. It should be noted that an amplitude of 0.1% of the shell thickness represents a maximum modal imperfection amplitude of 0.0006 in for this model. This magnitude of imperfections would almost certainly be present in a fabricated cylinder.

The development of a raised section on the cylinder facing the shock wave was unexpected. With the


(b) Front View

SHOCK WAVE

(c) End View

Fig. 12. Deformation at 5.0 ms of ring-stiffened finite length imperfect circular cylinder subjected to a shock pressure by 40 Ibs PETN, 30 ft standoff distance, initial mode 6 imper-

fection A, = 0.05/z.

SHOCK WAVE

f

,/----“\, ‘\

\

I /,” _. ”


(b) Front View

(c) End View

Fig. 13. Deformation at 5.0 ms of ring-stiffened finite length imperfect cylinder subjected to a shock pressure by 40 Ibs PETN, 30 ft standoff distance, initial imperfection first 10

modes A, = O.Olh with random phase shifts.

perfect cylinder the shock wave impacted the cylinder along the line of nodes at the front of the cylinder. The cylinder was then rotated so that the shock wave would impact the cylinder along a line of elements. The resulting deformation pattern is shown in Fig. 10. There is still a locally raised section on the cylinder facing the shock wave front. Thus it is felt that the shot geometry of the finite element mesh is not a contributing factor to the development of this raised section.

The use of Hughes-Liu shell element [13] vs the Belytschko-Tsay shell element was investigated. The Hughes-Liu shell element does not use many of the simplifying assumptions used in formulating the Belytschko-Tsay shell element. As a result, the use of the Hughes-Liu shell element requires longer compu- tational times. The resulting deformation to a model using Hughes-Liu shell elements is identical to the previous models with the exception of the use of the


Hughes-Liu shell element. This model has initial imperfections with the first 10 modes and modal imperfection amplitudes of 1% of the shell thickness and random phase shifts. The results show that the Hughes-Liu and the Belytschko-Tsay shell elements have the same final deformations. In this case the use of the numerically more complicated Hughes-Liu shell element is not warranted.

Exponentially decaying shock wave

The use of an explosive shock wave modeled as a square wave is representative of a very large explosion at a great standoff distance from the cylinder. In order to model a smaller explosion close to the cylinder, a pressure profile corresponding to that produced by 40 Ibs of PETN explosive charge at a standoff distance of 30 ft was used. This explosion produces a peak pressure of 1839 psi and an exponentially decaying pressure history. The resulting damage to the cylinder from this shock wave is shown in Fig. 11. From the offset view it can be seen that the response of the cylinder is different from that to the previous plane wave. The small standoff distance causes the shock wave to be a spherical wave at the cylinder. The mid-bay still has a protrusion in the shell toward the explosive charge. However, the end-bay shell has a dishing type shell deformation facing the charge. Also evident from the front view is the severe pinching of the shell near the end plate and on either side of the stiffener. This pinching was very severe near the end plate.

into the cylinder models. The introduction of these imperfections not only caused the shape of the shell of the cylinder (as viewed from the end) to follow the shape of the initial imperfection, but it has also changed the response of the shell near the end plates and stiffeners of the model cylinder. The pinching of the shell near these stiffeners and end plates was greatly reduced or eliminated by the introduction of initial imperfections in the model cylinders. This resulting response is closer to the response observed in test cylinders subjected to actual underwater shock loading. The dynamic buckling type or dishing type of shell deformation is very much pronounced in the imperfect model. In addition, the introduction of the initial imperfections increased the magnitude of the shell deformations compared to the deformations for cylinder modeled as perfect cylinders.

Changes in the cylinder geometry resulted in some changes in the response of the cylinder to underwater shock. However, in all cases the introduction of initial imperfections greatly affected the response of the cylinder as compared to the response of a perfect cylinder. If the initial imperfections of a test cylinder are known and introduced into a model cylinder used in finite element analysis, the results of the finite element analysis may more closely simulate the actual

response of the test cylinder.

REFERENCES

The introduction of a 5% mode 6 imperfection to the model results in the deformation shown in Fig. 12. Again, it is evident that the damage pattern of the cylinder due to the explosion shock clearly followed the initial imperfection. The pinching of the shell material near the end plate and stiffeners was reduced, but not eliminated.

Introducing an imperfection of the first 10 mode shapes with a modal imperfection amplitude of 1% of the shell thickness with random phase shifts results in the final damage pattern shown in Fig. 13. The deformation of the shell of the cylinder was again influenced by the initial imperfection pattern. The local pinching of the shell material on either side of the stiffener was eliminated and the pinching near the end plate was reduced in magnitude. The dynamic buckling type or dishing type of shell deformation is very much pronounced.

1. J. 0. Hallquist and D. W. Stillman, VEC/DYNA3D user’s manual-nonlinear dynamic analysis of structures in three dimensions. LSTC Reoort 1018, Liver- more Software Technology Corporation, June (1990).

2. D. W. Stillman and J. 0. Hallquist, LS-INGRID: a ore-processor and three-dimensional mesh generator for ihe program LS-DYNA3D, LS-NIKE3D and TOPAZ- 3D. version 3.0. LSTC Reoort 1009. LSTC Reoort 1019, Livermore Software -Technology Corporaiion, June (1991).

3. J. 0. Hallquist, LS-TAURUS: an interactive post-processor for the programs LS-DYNA3D, LS-NIKE3D and TOPAZ-3D. LSTC Report 1009, Livermore Software Technology Corporation, April (1990).

4. J. A. DeRuntz Jr, T. L. Geers and C. A. Flippa, The underwater shock analysis code (USA-version 3)-ref- erence manual. DNA Renort 5615F. September (1980).

5. J. A. DeRuntz Jr, The underwater shock analysis code and its applications, In: Proc. 60th Shock and Vibration Symp., Vol. I, pp. 89-107, Virginia Beach VA, 14-16 November (1989).

SUMMARY AND CONCLUSIONS

6. P. K. Fox, Y. W. Kwon and Y. S. Shin, Nonlinear response of cylindrical shells to underwater explosion: testing and numerical prediction using USA/DYNA3D. NPS Renort NPSME-92-002. Naval Postgraduate

The damage response of submerged cylindrical shell to underwater explosive shock is a complicated function of many factors. One of the important parameter affecting the response is the initial imperfections that are present in the fabricated cylinders. The response of model cylinders subjected to simu- lated underwater explosive shock has been shown to be very subject to the initial imperfections introduced

School. Monterey CA, March (1992). - 7. Y. S. Shin and J. E. Chisum, Damage response predic-

tions of double hull cylinders to underwater explosions: initial imperfection et?ects (U). In: Proc. 63rd Shock and Vibration Svmo.. Vol. III (unclassified/limited distri-

(1971).

bution), pp.‘895-913, Las Cruces NM, 27-29 October (1992).

8. T. L. Geers, Residual potential and approximate methods for three-dimensional fluid-structure interaction problems. J. Acoust. Sot. Am. 49, 1505-1510


9. T. L. Geers, Doubly asymptotic approximations for transient motions of submerged structures. J. Acoust. Sot. Am 64, 1500-1508 (1978).

10. J. Arbocz, The imperfection data bank, a means to obtain realistic buckling loads. In: &z/&g of Shells, Proc. ofa State-of-the-Art Colloquium, New York (1982).

11. S. W. Kirkpatrick and B. S. Holmes, Effect of initial imperfections on dynamic buckling of shells. J. Engng Mech. 115, 1075-1093 (1989).

12. T. B. Belytschko and C. S. Tsay, Explicit algorithms for nonlinear dynamics of shells. Comp. Meth. appl. Mech. Engng 43, 21-276 (1984).

13. T. J. R. Hughes, W. K. Liu and I. Levitt, Nonlinear dynamics finite element analysis of shells. In: Nonlinear Finite Element Analysis in Structural Mechanics (Edited by W. Wunderlich, E. Stein and K. J. Bathe), pp. 1.5-168. Springer-Verlag, Berlin (1981).

Documents

Damage response of submerged imperfect cylindrical structures to underwater explosion