luis a. boticario- dynamic response of cylindrical shells to underwater end-on explosion

8/3/2019 Luis A. Boticario- Dynamic Response of Cylindrical Shells to Underwater End-On Explosion

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NAVAL POSTGRADUATE SCHOOLMonterey, CaliforniaAD-A245 804 S&TATS.

THESIS IFB1i

DYNAMIC RESPONSE OF CYLINDRICAL SHELLS TOUNDERWATER END-ON EXPLOSION

byLuis A. Boticario

DECEMBER 1991

Thesis Advisor: Young W. Kwon

Approved for public release: Distribution is unlimited


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64 NAME OF PERFORMING ORGANIZATION 6b. OFFICE SYMBOL 7a NAME OF MONITORING ORGANIZATION(it applicable)Naval Postgraduate School ME Naval Postgraduate SchoolSe.ADDRESS (City, State and ZIP Code) 7b . ADDRESS (Cy, State, and ZIP Code)

Monterey, CA 93943-5000 Monterey, CA 93943-5000ft . NAME OF FUNDING/SPONSORING 8b OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBERORGANIZATION (I applicable)S. ADDRESS (Cit, State, and ZIP Code) 10 SOURCE OF FUNDING NUMBER

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11.TITLE (Indlude Seamny Classification)DYNAMIC RESPONSE OF CYLINDRICAL SHELLS TO UNDERWATER END-ON EXPLOSION

12 . PERSONAL AUTHORSLUIS A. BOTICARIO1.. TYPE OF REPORT 13b TIME COVERED 14. DATE O F REPORT (Year, Month, Day) 15. PAGE COUNTMaster's Thesis FROM _ TO DECEMBER 199116. SUPPLEMENTARY NOTATIONThe views expressed are those of the author and do not reflect the official policor position of the Department of Defense or the U.S. Government

17 COSATI CODES 18 SUBJECT TERMS (Conbnue on reverse t necessary and identityby blA numbers)FIELD GROUP SUB-GROUP underwater shock

19 . ABSTRACT (Continue on reverse it nec'ssary and identify b) bock numbers)Both numerical and experimental analyses were performed to investigate underwatshock propagation and the induced nonlinear response of cylindrical shells with end capThe cylinders were subjected to shocks from explosive charges at 12 inches (near-fieland 28 feet (far-field) from the cylinder. An underwater shock test was also performed withe far-field explosion. The numerical results were compared with the experimental datStresses and strains occurring in the structure as well as the pressure in the water westudied. The far-field explosion caused the largest circumferential deformations closeboth end plates and an accordion oscillatory motion of the cylindrical shell. The neafield explosion caused severe plastic deformation in the neighborhood of the closest eplate to the charge. The stiffeners had, as expected, a larger effect on tcircumferential stresses than on the longitudinal stresses. The measured and calculatstrains agreed well qualitatively near the remote end plate from the charge.

20 DISTRIBUTION AVAILABILITY OF ABSTPACT .1 ABST RACT SEPURITY CLASSIFICATIONXX UNCLASSIFIEDUNLIMITED _ SAME AS RPT _ DTIC USERS unclassified22a NAME OF RESPONSIBLE INDIVIDUAL no TELEPHONE (InciLde Area Code) 22c OFFICE SYMBOLYoung W. Kwon (408) 646-3385 ME/KwDD Form 1473, JUN 86 Previous editions are obsolete. SECUR YCLASSI:ICATON OF THIS PAG


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Approved foi public release: Distribution is unlimitedDynamic Response of Cylindrical Shells to Underwater End-OnExplosion

byLuis A. BoticarioLieutenant, United States NavyB.S., University of Michigan, 1986

Submitted in partial fulfillment of therequirements for the degree ofMASTER OF SCIENCE

IN MECHANICAL ENGINEERINGfrom the

NAVAL POSTGRADUATE SCHOOLDECEMBER 1991

Authcr: '6 - 'Luls A. BoticarioApproved by: _ _ _ _ __/C/ Youn W. Kwon, Thesis Advisor

Young S. Shin, Second Reader

A.Ji ealey, ChaiDepartment of Mechanical 4-ineering


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ABSTRACT

Both numerical and experimental analyses were performedto investigate underwater shock propagation and the inducednonlinear response of cylindrical shells with end caps. Thecylinders were subjected to shocks from explosive charges at12 inches (near-field) and 28 feet (far-field) from thecylinder. An underwater shock test was also performed with thefar-tield explosion. The numerical results were compared withthe experimental data. Stresses and strains occurring in thestructure as well as the pressure in the water were studied.The far-field explosion caused the largest circumferentialdeformations close to both end plates and an accordionoscillatory motion of the cylindrical shell. The near-fieldexplosion caused severe plastic deformation in theneighborhood of the closest end plate to the charge. Thestiffeners had, as expected, a larger effect on thecircumferential stresses than on the longitudinal stresses.The measured and calculated strains agreed well qualitativelynear the remote end plate from the charge.

Aoesseon Fo rXTIS GRA&IDTIC TAB 0Unannounoo tJustifLoat1oBy ....--...., Distributim/4AvadLa td1Wwt

Det SPOOL41


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TABLE OF CONTENTS

I. INTRODUCTION ......... .................. 1II. ANALYSIS AND EXPERIMENT ...... ............ 3

A. NUMERICAL ANALYSIS ....... ............. 3B. NUMERICAL MODELLING ....... ............. 6C. UNDERWATER EXPLOSION TEST .. .......... . 14

III. RESULTS AND DISCUSSIONS . ........... 23A. NUMERICAL RESULTS OF FAR-FIELD EXPLOSION . . 23B. COMPARISON BETWEEN NUMERICAL AND

EXPERIMENTAL RESULTS .... ........... . 36C. NUMERICAL RESULTS OF NEAR-FIELD EXPLOSION . . 45

1. Pressure Wave Study .......... 512. Ring Stiffener Study .. .......... . 51

IV. CONCLUSIONS . . .................... 65APPENDIX: UNIAXIAL TENSION TEST DATA FOR 6061-T6

ALUMINUM ........ .................... 68LIST OF REFERENCES ....... .................. 72INITIAL DISTRIBUTION LIST ................. . . 74

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LIST OF FIGURESFigure 2.1. Finite Element Meshes for Near-Field

Explosion ....... ............... 9Figure 2.2. Stiffened Cylinders with End Plates . . 10Figure 2.3. Finite element meshes of a cylinder

subject to a far-field explosion . ... 12Figure 2.4. Finite element discretization over 4

inch span at one end of cylinder . ... 13Figure 2.5. Cylinder and Crane Rigging ........ . 16Figure 2.6. Test Geometry at the Surface ......... 17Figure 2.7. Test Geometry (Ranges in Feet . . ... 18Figure 2.8. Water Plume from Shot ... ......... 19Figure 2.9. Cylinder Geometry .... ........... 20Figure 3.1. Deformation Time Histories ....... . 24Figure 3.1. (cont). Deformation Time Histories . . . 25Figure 3.1. (cont.) Deformation Time Histories . . . 26Figure 3.2. Axial Velocities of Nodes at Both End

Plates of Cylinder .... ........... 28Figure 3.3 Locations for Strain Computation . ... 30Figure 3.4 Circumferential Strains Close to EndPlates ................... 31Figure 3.5. Circumferential Strain 20.75 Inches

from the Closest End Plate to theExplosive Charge ... ............ ... 33

Figure 3.6. Longitudinal Strains in the Vicinity ofthe Closest End Plate to the ExplosiveCharge ..... ................. ... 35

Figure 3.7. Longitudinal Strains 37.5 Inches fromthe Closest End Plate, of DifferentDensities, to the Explosive Charge . . . 37

Figure 3.8. Circumferential Strains 20.75 Inchesfrom the Closest End Plate, ofDifferent Densities, to the ExplosiveCharge ..... ................. ... 38

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Figure 3.8. Ci :'umfe:ential Strains 20.75 Inchesfrom the Closest End Plate, ofDifferent Densities, to the ExplosiveCharge ..... ................. ... 38

Figure 3.9. Circumferential Strains 37.5 Inchesfrom the Closest End Plate, ofDifferent Stiffness, to the ExplosiveCharge ..... ................. ... 39

Figure 3.10. Longitudinal Strains 37.5 Inches fromthe Closest End Plate, of DifferentStiffness, to the Explosive Charge . . . 40

Figure 3.11. Nodal Velocities of Both End Plates ofStiffness Ten Times Lower Than theNominal Value for 6061-T6 Aluminum . . . 41

Figure 3.12. Measured and Computed LongitudinalStrains at 37.5 Inches from Closest EndPlate to Explosive . .......... .. 43

Figure 3.12. (cont.) Measured and ComputedLongitudinal Strains at 37.5 Inchesfrom Closest End Plate to Explosive . . 44

Figure 3.13. Measured and computed longitudinalstrains at 38.5 inches from closest endplate to explosive . . ......... 46

Figure 3.14. Measured and Computed LongitudinalStrains at 4.5 Inches from Closest EndPlate to Explosive . . . . . . . . ... 47

Figure 3.14. (cont.) Measured and ComputedLongitudinal Strains at 4.5 Inches fromClosest End Plate to Explosive ..... . 48

Figure 3.15. Measured and Computed LongitudinalStrains at 3.5 Inches from Closest EndPlate to Explosive . . ........... 49

Figure 3.16. Measured and Computed LongitudinalStrains at 20.75 Inches from ClosestEnd Plate to Explosive ......... . 50

Figure 3.17. Pressure Wave Propagation History inNear-Field Explosion Model ........ . 52

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Figure 3.18. Cavitation at End Plate Closest toNear-Field Explosion .. ............. 53

Figure 3.19. Near-Field Model Locations Chosen forAnalysis ..... ................ ... 55

Figure 3.20. Local Deformations for Near-FieldModels ......................... 57

Figure 3.21. Circumferential Strains at Both Sidesof Stiffener Closest to ExplosiveCharge ...... ................. ... 58

Figure 3.22. Circumferential Strains at Both Sidesof Stiffener Farthest from ExplosiveCharge ...... ................. ... 59

Figure 3.23. Circumferential Strains at BothStiffeners of Two-Stiffener Model . . . 60

Figure 3.24. Circumferential Strain Comparison forUnstiffened ar.d One-Stiffener Models . . 62

Figure 3.25. Longitudinal Strains (Unstiffened andTwo-Stiffened) at Stiffener LocationClosest to Charge ................ 63

Figure 3.26. Longitudinal Strains (Unstiffened andTwo-Stiffened) at Stiffener LocationFar from Charge .... ............ . 64

Figure A.I. Load-Dispiacement Curve for 6061-T6Aluminum ..... ................ . 69

Figure A.2. Stress-Strain Curve for 6061-T6Aluminum ..... ............. . . 70

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ACKNOWLEDGEMENTS

I would like to express my gratitude to Professors YoungW. Kwon and Young S. Shin for their guidance, encouragementand support in carrying out this research. In appreciation toDr. Thomas T. Tsai and Dr. Kent Goering, from the DefenseNuclear Agency, for supporting this research. I would alsolike to thank LCDR Padraic Fox and LT James Chisum for their

assistance in this research. Finally, I am most grateful to mymother Sylvia for her moral support and encouragement duringmy two years at the Naval Postgraduate School.

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I. INTRODUCTION

Because of the U.S. Navy's high interest in the underwatershock hardening effects on surface ships and submarines, thisresearch intends to provide more insight into the response ofa submerged vessel subjected to end-on underwater shock.

A continuous research has been taken at the NavalPostgraduate School to provide more insight into thedeformation and catastrophic failure of surface and subsurfacehulls. With simple cylindrical shells as a starting point, thestudy will then be extended to structures with more complexmaterial and geometric properties as the methods andpredictions improve.

Some of the previous studies in this subject are listed inreferences 1-3. These studies have served as building blocksfor the current research into dynamic response of cylindricalshells to underwater shock. The objective of this study is toprovide insight into the end-on shock dynamic response ofcylindrical shells by using numerical and experimentaltechniques.

An unstiffened cylinder subjected to a far-field explosionwas investigated using both numerical and experimentaltechniques. In addition, both unstiffened and ring-stiffenedcylinders subjected to a near-field explosion were studied


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numerically. Both the finite element and the boundary elementmethods were utilized for the numerical study.

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II. ANALYSIS AND EXPERIMENT

A. NUMERICAL ANALYSISThe numerical study of the behavior of cylindrical shells

loaded by underwater explosion was carried out by using thefinite element and boundary element methods.

For a problem with a three dimensional domain, the finiteelement method generates a three dimensional discretization ofthe entire domain whereas the boundary element methoddiscretizes the surface boundary of the domain with a twodimensional grid. The boundary element method reducessignificantly the number of elements required to model theproblem by using a two dimensional mesh. Furthermore, theboundary element method surpasses the finite element method incomputing tractions because these tractions are treated asprimary but not secondary unknowns [Ref. 4]. The matrixgenerated ty the boundary element method is generally fullypopulated while the matrix generated by the finite elementmethod is usually narrowly banded.

To study the propagation of the explosive pressure wavethrough the acoustic medium and its subsequent interactionwith the cylindrical shell, a finite element analysis programcalled VEC/DYNA3D [Ref. 5] was used. This program analyzes thedeformations of solids which usually are inelastic in nature.

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An alternative for modelling the acoustic medium is by usingthe boundary element method instead of using the finiteelement method. In doing so, the number of elements is greatlyreduced. The boundary element analysis method program calledUSA (Underwater Shock Analyzer) [Ref. 6], was used to computetransient responses of submerged structures to acoustic shockwaves.

In USA, the fluid-structure interaction was handled usingthe Doubly Asymptotic Approximation (DAA). The differentialequation of motion describing the structure response is givenbelow.

M5 + C5 + Ksxf ()where M,, C., and K. are the mass, damping and stiffnessmatrices respectively and x, k, x are the displacement,velocity and acceleration vectors of the structurerespectively. The excitation for a submerged structuresubjected to an acoustic wave is given below.

f= -GA,(P + P) + f. (2)where -GA, (P- + P,) is the force vector due to the fluid-structure interaction, f, is the force vector applied to thedry structure, P is the incident pressure, P, is the scatteredpressure, G is the transformation matrix relating fluid andstructure nodal forces and Af is the diagonal area matrixassociated with the fluid mesh.

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To further study the fluid-structure interaction andespecially to relate the scattered wave pressure to velocityover the wet surface, DAA was introduced. The DAA solutionapproaches the exact solutions for both early time and latetime responses. The early time response is the high frequencyresponse and the late time response is the low frequencyresponse. The DAA is given by the following equation.

Mfi* + pcAfPs = pcM).s (3)where M, is the symmetric fluid mass matrix for the wet-surface fluid mesh, u, is the vector of scattered fluidparticle velocities normal to the structure's wet surface, pis the density of the fluid, c is the sonic speed in the fluidand A, is the symmetric fluid mass matrix for the wet surfacefluid mesh. This matrix is created using the boundary elementmethod.

The high frequency approximation, i.e. plane waveapproximation implies that 115, >>IPsl where 1, is the timederivative of the acoustic pressure. Therefore, Equation (3)is converted into the following equation.

P, = pcu, (4)The low frequency approximation, i.e. virtual mass

approximation implies that IPj


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surrounding that structure at the low frequency motion of thestructure. Hence, Equation (3) is modified as seen below.

Af PS = Mf0. (5)Equation (3) denotes the first order of the Doubly

Asymptotic Approximations (DAAI). The second order of theDoubly Asymptotic Approximations, DAA2, was created to improvethe solution of DAAl for intermediate times and to correct forcurvature of the surface of the structure [Ref. 7].

The finite element model of both structure and fluid hasthe advantage of presenting the pressure wave propagation inthe fluid and the fluid-structure interaction. On the otherhand, it also requires a large number of elements. Using thefinite element model for the structure and the boundaryelement model for the fluid reduces the number of degrees offreedom in the system because the fluid domain is usually muchlarger than the structural dimension. However, it cannotpresent the propagation of the pressure wave across the fluid.

The post-processing of the VEC/DYNA3D and USA programs wasdone using LS-TAURUS [Ref. 8]. This post-processor generatespressure, strain and stress contours superposed on the mesh oralso generates element and node time history responses.

B. NUMERICAL MODELLINGTo study the deformation of a cylindrical shell subjected

to an end-on shock, two numerical models were created. The

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first model was designed to study the propagation of thepressure wave from the explosive to the cylinder and its earlyinteraction with the cylindrical shell. The second model wasdesigned to study stresses and strains of the cylindricalshell without analyzing the fluid or explosive around it.

To generate the first model, the pre-processor, INGRID(Ref. 9), generated the finite element meshes for theexplosive, fluid and cylindrical shell and VEC/DYNA3D computedtheir dynamic response.

The computational effort was minimized by creating aquarter model, possible only because of the symmetric geometryof the charge and cylindrical shell locations. Appropriatesymmetric boundary conditions were applied to the problem.

The explosive was modeled with a fine mesh of 416 elementstc avoid non-spherical propagation and severe distortion inthe finite element mesh as a result of the expansion of theexplosive in the fluid.

To model the explosion, the Jones-Wilkins-Lee (JWL)equation of state was invoked to describe the pressure-volume-energy behavior of high explosives. [Ref. 10]

P= A W jfR, V + Bfj W P, + wE(6(lA -(where A, B, and C are linear coefficients given in units ofMbar. RI, R2 and w are nonlinear coefficients, V is the volumeof detonation products divided by the volume of undetonated

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high explosive, P is the pressure, and E is the detonationenergy per unit volume in (Mbar cm3)/cm3 .

The Gruneisen equation of state was used to define thepressure for compressed materials, in this case water. Thisequation of state is provided in [Ref. 10].

The model including the charge, water and cylinder had11808 elements which consequently required large amounts ofstorage. Finite element meshes are given in Figure 2.1.Therefore, the charge was constrained to be close to thetarget cylinder to minimize the elements between explosive andcylinder and still be able to store the shock wave propagationand stress wave effect information. The cylindrical shell ofthis finite element model was modified to add ring stiffeners.One model had one ring stiffener located halfway between thecylinder end plates. The second model had two ring stiffeners

equidistant between the two end plates. A view of these twomodels is shown in Figure 2.2. The additional stiffeners onlyadded an additional 16 elements per stiffener to the total.The ring stiffeners were added to study their effect in stresswave propagation and reinforcement properties.

The other model used in this research was designed tostudy the stresses and strains of the cylinder for comparisonwith experimental results of an actual underwater shock test.The model dynamic response was analyzed using both VEC/DYNA3Dand USA. The finite element method was used for thecylindrical shell and the boundary element method was used for

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(a) Quarter Sphere

(b) Quarter Stiffener

I I I I ? I I I I I , I I I z! I

(c) Quarter Cylindrical Shell with End Plates

(d) Quarter Fluid MeshFigure 2.1. Finite Element Meshes for Near-Field Explosion

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(a) One-Stiffener Model

(b) Two-Stiffener ModelFigure 2.2. Stiffened Cylinders with End Plates

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the fluid-structure interaction. This model consisted of acylindrical shell capped at both ends by two end plates. Thecylindrical shell had 512 elements and the endplates had 48elements each. The full model had 608 total elements. Adiagram of this model is presented in Figure 2.3. Thedifferent size of the cylindrical shell elements was generatedto get a better solution in the strain gage locations of theexperiment. These locations will be presented when theexperimental procedure is explained.

A study of grid independence and time step instability wasperformed to verify that the choice of the element sizes andtime steps chosen were not affecting the solution. To analyzegrid independence, three different discretizations were madeover a four inch segment of the cylinder. The discretizationwas made in the longitudinal direction due to the symmetry ofthe problem. The coarse model had two elements of two incheseach in the axial direction. The finer model had six elementsof 0.66 inches each in the same direction. The finest meshmodel had eighteen elements of 0.22 inches each. The threediscretizations are shown in Figure 2.4. Comparison of thethree models revealed considerable differences between thecoarse model and the two finer mesh models. The six and 18element models had almost identical results. As a result ofthis comparison, the six element model was chosen as a goodestimate of the size of the elements to use for the actualmodel.

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(a) End plate

L LI L

(b) Cylindrical shellfigure 2.3. Finite element meshes of a cylinder subject to afar-field explosion

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(a) oarse model (2 inch elements)(2 elements in 4 inch span)

(b) iner model (0.66 inch elements)(6 elements in 4 inch span)

(c) Finest model (0.22 inch elements)(18 elements in 4 inch s-.an)Fi~gure 2.4. Finite element discretization over 4 inch span atone end of cylinder

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Another area of concern was the effect the time step wouldhave on the solution. An appropriate method to choose a stabletime step size was the Courant-Friederichs-Lewy Criterion. Theequation for this criterion is given below.

At I/ (7)where At is the time step size, I is the length of theelement, and c is the sonic speed in 6061-T6 Aluminum.

Two different time step sizes were chosen, 2xl0e "6 and2xl0e "' seconds. These time steps satisfied the above mentionedcriterion. The resulting solutions for the two time steps werealmost identical. Therefore, 4xl0e-7 seconds was chosen as thetime step for this problem since this value was between thetwo closen for the criterion verification.

C. UNDERWATER EXPLOSION TESTThe underwater explosion test was performed at Dynamic

Testing Inc. (DTI) facilities in Rustburg, Virgina. Thisfacility had a quarry that had been filled with water for usein underwater shock tests. The water depth was approximately130 feet at the location of the test which was deep enough toallow the study of the dynamic response of the cylinder priorto the arrival of the reflected shock wave from the bottom.

The cylinder was placed 12 feet below the water's surfaceand held in place by a crane with pendants attached to bothfront and rear end plates. The rig attachment can be observed

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in Figure 2.5. The 60 pound charge of HBXl was also placed 12feet below the surface and aligned with the cylinder with aspan wire from the charge float to the crane rig. Figure 2.6.

9gives a good view of the arrangement. The 12 foot depth waschosen so that the bubble generated by the explosion wouldvent to the surface prior to encountering the cylinder. Thetest geometry is shown in Figure 2.7. The explosive charge wasactivated by a radio device and the plume is pictured inFigure 2.8.

The strain gages used for this test were of type CEA-06-250UW-350. These are general purpose strain gages with anoptimum operating range of 1500 micro strain and are usedfor both static and dynamic test measurements. They wereattached to the cylinder using a M bond 200. There were atotal of seven strain gages placed at locations A, B and C, asseen in Figure 2.9, per axis for a total of 14 strain gages.The cylinder was oriented so that the gages at C would beclosest to the explosive charge. A pressure probe was alsoplaced 28 feet from the cylinder to measure the free fieldpressure.

The cylinder used for the underwater shock test wasmanufactured from 6061-T6 Aluminum. This alloy is primarily anAluminum-Magnesium-Silicon alloy. The T6 denominationindicates it was solution treated and artificially aged (Ref.11]. The cylinder consisted of a quarter inch thickcylindrical shell and two one-inch thick circular end plates

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Figure 2.5. Cylinder and Crane Rigging


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Figure 2.6. Test Geometry at the Surface17


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To croneCharge flot

12 " 12 '

As ACylinder Charge

figure 2.7. Test Geometry (Ranges in root)


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Figure 2. 8. Water Plume from Shot

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42"A .

I31 21 - -- -- - -- - - - - - --- - -- -- - - - - - - ---- I.-

,i I1" 16.S5"bU 16.7 .5

21"

Figure 2.9. Cylinder Geometry20


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as seen in Figure 2.9. The welding of the two end plates tothe cylindrical shell was done at the Naval PostgraduateSchool facilities. There was a concern in the welding processof the two circular end plates to the shell because of theheat generated at the weld and its effect on the alloymorphology close to the weld. However, the strain readingswere taken far enough from the heat affected zone to have anymeasurable effect on the readings. The welding was done usingtungsten inert gas (TIG) This procedure has been recommendedby most expert welders [Ref. 12].

The cylinder weighed 60.5 pounds and tensile tests weredone to verify that the material properties were close to thenominal properties of 6061-T6 Aluminum. This test can be seenin the Appendix. The value of Young's Modulus was 10800 ksiand the yield strength was approximately 43 ksi.

Post-shot observation showed no visible deformations onthe cylinder as seen in Figure 2.10. All strain gages werewell fixed when uncovering the bonding material. Only straingage Cl in Figure 2.9. was found wet when uncovered suggestingthe possibility of water insertion.

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.41

.P4

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III. RESULTS AND DISCUSSIONS

A. NUMERICAL RESULTS OF FAR-FIELD EXPLOSIONA cylinder was subjected to an end-on explosion with a

standoff distance. The sketch of the configuration waspresented in previous Figure 2.6. The deformations of thecylinder are shown in Figure 3.1 at different times as theshock wave propagates along the axial direction of thecylinder. Becaase of the symmetric loading, the deformationwas axially symmetric. In order to visualize the deformationsmore clearly, the actual deformations were magnified by thescale factor shown in Figure 3.1.

As the shock pressure wave hit one end plate, which is atthe nearest location from the charge, the compressive stresswave propagated from the end plate to the other end platethrough the aluminum alloy cylinder with a faster speed ofsound than the shock pressure wave propagating through thesurrounding water medium. The nearer end plate will be calledthe near plate while the other end plate will be called theremote end plate in the following discussion. The shock loadcaused initially a localized circumferential deformation closeto the near end plate as shown in Figure 3.1c. This localizeddeformation remained as a permanent deformation. As the shockpressure wave propagated toward the remote end plate, it

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(a) ime = OOOOOEOO sec

(b) time =.7960CE-04 sec

(c) time =.11960E-03 seccusp. scale factor = l.OOE+OlFigure 3.1. Deformation Time Ristoris

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E III i I

(d) time = .15960E-03 sec

9 I Ill(e) ime = .23960E-03 sec

(f) time = .47960E-03 secdisp. scale factor = 1.OOOE+01

Figure 3.1. (cont). Deformation Time Histories

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t IU-h -T- L\\\11m,m Il inii////n

(g) time = .51960E-03 sec

1777 LV I llI I -j-T1! L .

(h) time = .11596E-02 secdisp. scale factor = 1.OOOE+l

Figure 3.1 (cont.) Deformation Time Histories

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induced a radial contraction of the cylinder wall andsubsequently a recovery of it. After the shock pressure wavepassed the remote end plate, there was also a localizedcircumferential deformation near the remote end plate as shownin Figure 3.1h. The overall steady state deformation of thecylinder was nearly symmetric about the center plane which islocated at an equal distance between the two end plates. Atthis stage, the cylinder had the deformation near the two endplates, and the rest of the cylinder had very little radialdeformation. The axial deformation of the cylinder had anaccordion mode as expected. Figure 3.2 is the plot of axialvelocities at the centers of the two end plates. The twovelocities had a phase difference of 180 degrees. The phasedifference indicated the accordion mode.

The initial localized circumferential deformatior near theend plates may be explained as follows. The axisymmetricdeformation of a shell has the following governing equation.

Dd W+Fd2W+Eh w=p ()where w is the radial deflection, x is the axial direction, Dis the flexural rigidity of the shell, E is the elasticmodulus, h is the wall thickness of the shell, a is the shellradius, and F and p are the axial load and the radial load,respectively. Therefore, the effect of the inertia term on thedeformation was negligible. The axial load applied on thecylinder due to the shock pressure was lower than the critical

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A: Node at plate closest to charge3: Nods at plate farthest from' harge

-4.K+62

nods A 46Pes3.t Ime

Fiue32 xa elcte fNdsatBt n ltsoCylinde

U2


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buckling load of that cylinder from the linear bucklingtheory. In this case, the deformation of the cylinder was ofa decaying exponential form and it damped out rapidly if thecylinder was not short [Ref. 13]. The very stiff end plattessuppressed the deformation very near the end plates. Thisresulted in a shift of the locations of the circumferentialdeformation somewhat away from the end plates. The inertialforce was not included in the equation. When the density ofthe cylinder was varied from its nominal value to a value ofone order of less magnitude, the same kind of initiallocalized deformation was observed. However, the low densitycaused less permanent localized deformation.

Two normal strain components, i.e., hoop (orcircumferential) and axial (or longitudinal) strains, werecomputed at some selected locations. The selected points werelocated at the locally deformed zones as well as at the centerof the cylinder. The locations where the strains were computedare illustrated in Figure 3.3. Strain gages were also attachedto the same locations on the cylinder in the experimentalstudy for the comparison with the numerical study. All thestrains were computed and measured at the outside surface ofthe cylinder. The comparison between the numerical andexperimental results will be provided in the next section. Asshown in the strain plots of Figure 3.4, the circumferentialstrains at locations A and C of Figire 3.3 were very ,,],se tceach other. The strains had initially large compressive peaks

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169 (C) 191(B) 215 (A)

I!1,-ILu llLExplosion

side

1. Element 169: 4.5" from closest end toexplosion.2. Element 191: 20.75" from closest end toexplosion.3. Element 215: 37.5" from closest end toexplosion.

Figure 3.3 Locations for Strain Computation30


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, rs40.

MIwMR."1641641

gel W-0

aalExloiv Charg

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and returned to the steady state values which were about halfthe initial peak strain values. Location A had a smallcircumferential tensile strain before arrival of the shockpressure wave. The small tensile strain was caused by thelongitudinal compressive stress wave which arrived at thelocation through the cylinder. The longitudinal compressivestrain caused by the longitudinal stress wave resulted in thecircumferential tensile strain because of Poisson's effect.After the shock pressure wave arrived at the location, thecircumferential strain became compressive. The circumferentialstrain at the middle of the cylinder, i.e, location B inFigure 3.3, was much less that those at locations A and C asshown in Figure 3.5 because the localized deformations atlocations A and C induced the larger circumferential strains.

The initial peak of longitudinal strain was compressive atall locations due to the propagation of the longitudinalcompressive stress wave after the shock pressure hit the nearend plate. The early arrival of the shock pressure wave tolocation C in Figure 3.3, caused the immediate jump of thelongitudinal strain into tension after the initialcompression. On the other hand, the subsequent compressivestress wave resulted in more compression at locations A and Bthan at location C until the shock pressure wave arrived atthe locations. At later times the longitudinal strain werequite different at locations A and C. The longitudinal strainat location A stayed in tension just after the initial

32


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C00

U

-1.,$K-63m!m m

'I-1.6t1!me

Fiue35 irufrnilStan2.5Inhsfo-hClsetZ.5laetoth x-oiv hag

'33


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compressive peak while the longitudinal strain at location Calternated between tension and compression. However, thelongitudinal strain was quite sensitive with the locationsnear A and C because of the localized deformation.Longitudinal strains at various locations near position C wereplotted in Figure 3.6. The approximate frequency of thelongitudinal strains after the initial compression was about750 Hz at all locations.

The present study showed that the end plates played animportant role in the deformation of the cylinder. Someparametric study was performed to find the effect of the endplates on the dynamic response of the cylindrical shell. Thefirst study was to find out the effect of the inertial forceof the end plates. Therefore, the density of the end plateswas reduced tenfold with the stiffness of the plates remainingthe same. The light end plates had a larger effect on thelongitudinal strains than on the circumferential strains. Thefrequency of the longitudinal strains was approximately 840 Hzfor the light end plates. This frequency was higher than thatfor the heavy plates as expected. The longitudinal straindamped more quickly with the light end plates than with heavyend plates as shown in Figure 3.7. The light end platesresulted in an approximately 30% decrease of thecircumferential strain at location C while its effect wasnegligible on the circumferential strain at location A.Moreover, the circumferential strain at location B increased

34


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I. 4K- 31CI I 2-II63

C0 OKE-83

. 6.$K-644.)61

iv'4.L -2. IK-I4

-4.,11-14

-N, 2K '1(-O

*elmtsI Fi- 165 a- 169 C.* 173t 1me kme l 16

A. (165) 4.0 inches from end plateB. (169) 4.5 inches from end plateC. (173) 5.0 inches from end plate

Figure 3.6. Longitudinal Strains in the Vicinity of theClosest End Plate to the Explosive Charge

35


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about 20 percent for the light end plates as shown in Figure3.8.

The next parametric study was performed to investigate theeffect of stiffness of the end plates on the deformation ofthe cylinder. The elastic modulus of the plate was reducedtenfold without change of the rest of the material data. Theless stiff end plates made an effect on not only longitudinalstrain but also circumferential strain as shown in Figures 3.9and 3.10. The less stiff support of the cylinder at the endplates caused more fluctuation in the circumferential strainas shown in Figure 3.9. The wave pattern in the longitudinalstrain-time history plot was also severely altered by the lessstiff end plates as shown in Figure 3.10. The longitudinalvelocities of the centers of the less stiff end plates wereplotted in Figure 3.11 and showed a very different patternthan the accordion oscillation of the stiff plates in Figure3.2. The density change in the previous parametric study didno show a different velocity pattern of the end plates.

B. COMPARISON BETWEEN NUMERICAL AND EXPERIMENTAL RESULTSAn underwater explosion test was performed at Dynamic

Testing Inc. facilities in Rustburg, Virginia as described inChapter II. Unfortunately, a pressure gage, placed to measurea free-field pressure at the same stand-off distance as thecylinder, failed. Therefore, no information was available forthe shock pressure due to the charge. The previous numerical

36


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CL

t ffi $6Lon t d a l Sta1W7

n h s r m t e l s sExplosive 'W Ofeif

Ur er n S ~ t . , t h*3


47/84

c0

UL

M44

'rigrure~(a.8omb najensity end Patee ftnC Cr u f r ~ i l S r i s 2 .7 n h s f oC *8 si n d.,. f D ff' o t D n it e , t hx s v C a g h

03


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0-eC m

CLQ's rn44

r-a

C =.

0.1

L-4.r-

*I~%$ lit ie(a) nooiatiffness end plate e ftn

Figure 3.9~~~~. CrufrnilSris3. nhsfo hClss En PltofDfern-uifest hZxlsveCagC3


49/84

a,*

4L4

&..f4.~.

."

(a)om nal st'ffn ss end plate

(b) 10w r t fflp ll

Jp'ur 1. iffes ed lae (rdr f en

4-a0


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A: Nods at closest end plate to explosiveI.M*8e.8: Node at farthest end plate to explosive

6. SK of

4 Gee,71 IME #0

U 9.91C-400l,)o-2. c#szN

-4.8UC.5

-6.@K-62

modes A. 46 3. sot !me

figure 3.11. Nodal Velocities Of Both End Plates ofStiffness Ton Times Lower Than the NominalValue for 6061-T6 Aluminum41


51/84

simulation was carried out using the empirical equation forpressure with the nominal weight of the charge. As a result,the following comparison was made in a qualitative sense. Inaddition, several strain gages failed before and during thetest. All the strain gages to measure circumferential strainsfailed. Only the axial strains were compared here.

The speed of sound in the water was computed based on thearrival time of the pressure wave to the strain gage from theexplosion and the stand-off distance. The computed speed ofsound was close to a nominal value of 5000 ft/sec. Thenlocations of the strain gages were shown in Figure 3.3. Thelongitudinal strain at gage location A was compared in Figure3.12. All experimental data was filtered out at 2000 Hz lowpass. Therefore, there was no peak strain with a higherfrequency than 2000 Hz in the experimental data. Bothnumerical and experimental solutions indicated the initialcompression and tension at later times. They agreed wellqualitatively even if there was a mismatch in magnitude. Thelongitudinal strain was quite sensitive to the location. Thelongitudinal strain computed at just one inch away from thegage location was compared with the experimental measurementin Figure 3.13. Two strain gages, both of which were locatedat location A but were separated in the circumferentialdirection with an angle of 180 degrees, gave a similarlongitudinal measurement as shown in Figure 3.12. Thismeasurement indic-ated the axisymmetric nature of the

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TEST 1, LOCATION AIL

z %,,1111,11111,11

Iv

0 a1TIMME

AXA SRIEXEIMNA DAT

(a Stai Gag Loato (Se Fiur 2.9)

Figure~~~~~~.2 esrdadCmue ogtdnlSrisa37. Inhsfo lsetIPaet

I Ioiv

I-43


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TEST 1, LOCATION A3L

a 10

I,r

I I

TIME (MSEC)AXIAL STRAIN

EXPERIMENTAL DATACLCULmET. uT ....(b) A3L Strain Gage Location (See Figure 2.9)

figure 3.12. (cont.) Measured and Computed LongitudinalStrains at 37.5 Inches from Closest End Plateto Zxplosive44


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deformation. However, the two strain gages, which were placedat location C and were separated 180 degrees in thecircumference, provided quite different longitudinal strains.This discrepancy in readings could have been caused due toobserved water insertion in the strain gage at location Cl.

Accordingly, the numerical result was not in goodagreement with the experimental data as shown in Figure 3.14.The sensitivity of the location were the strains were computedwas also evident for location C as shown in Figure 3.15. Theexperiment showed a compressed longitudinal strain while thenumerical solution showed an oscillation of the strain fromcompression to tension. A small distance away from location Cgave a different solution. However, the experimental data wasstill different from the numerical result. The longitudinalstrain at gage location B was compared in Figure 3.16. Theresult was similar to that at gage location C. It was notclear at this time what the major cause of the discrepancywas. However, this was the first test of a series ofunderwater experiments to be performed. The discrepancy willbe investigated from the following experiments with moreavailable experimental data and numerical simulations.

C. NUMERICAL RESULTS OF NEAR-FIELD EXPLOSIONThe study of the dynamic response of cylindrical shells

subjected to a close-in explosion was performed. The stand-offdistance was 12 inches from one end plate. The overall

45


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TEST l, OCATION AIL

0 'S'I I'%

I 8 I Ss I #Z I8 I s

0 a 4 a a 1

* /

TIME (MSEC)AXIAL STRA'NEXPERIMENThAL DATAT QI-P-

AIL Strain Gage Location (See Figure 2.9)

E0

Figure 3.13. Measured and computed longitudinal strains at38.5 inches from closest end plate toexplosive46


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TEST I, OCATI*J CIL

.v

0 2 16810TIME (MqSEC)AXIAL STRAINEXPERIMENTAL QATAC'LHUL~U QRTFM

(a) lL Strain Gage Location (See Figure 2.9)

Figure 3.14. Measured and Computed Longitudinal Strains at4.5 Inches from Closest End Plato toExplosive47


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TEST lo LOCATION C3L

1%

a%10

0- 2 6TIM (ME

Stan at 4. Inhe fro Clss lt

0t 2 1 6 8 10TIE48SC


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TEST 1, . LOCATION CIL

1z

TIME (MSEC)AXIAL STRAINEXPERIMENTAL DATAUHLUULH TLUUHOTH-

CiL Strain Gage Location (See Figure 2.9}

Figure 3.15. Measured and Computed Longitudinal Strains at3.5 Inches from Closest End Plate toExplosive49


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TEST 1, LOCATION B2L

I.J1,

,\'z

0 4 6 IOTIME (IM5EC)AXIAL STRAINEXPERIMENTAL DATA

B2L Strain Gage Location (See Figure 2.9)

Figure 3.16. Measured and Computed Longitudinal Strains at20.75 Inches from Closest End Plate toExplosive50


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geometry with finite element meshes is shown in Figure 2.1. Atwo pound explosive charge of pentolite was used in thenumerical study. The dimension of the cylinder was the same asfor the far-field explosion except for a length of 43 inchesinstead of 42 inches long and the addition of ring stiffeners.

1. Pressure Wave StudyThe explosive charge was refined in the numerical

model to avoid discontinuities of the sphere as it expandedwith time. The pressure wave propagated spherically throughthe finite element mesh as shown in Figure 3.17.

The wave propagation was studied especially at thefirst interaction between the pressure wave and one end plate.The cylinder was accelerated when the wave first interactedwith the end plate. At about 0.3 milliseconds the fluid andend plate velocities break away for a short period. The fluidvelocity was lower than that for the shell. Consequently,tension would be induced in the fluid but since fluids cannotexperience tension, the water particles break away creating avacuum. This effect, known as hull cavitation, can be observedin Figure 3.18.

2. Ring Stiffener StudyThe three cylinders studied were an unstiffened

cylinder, a one-ring stiffener cylinder, and a two-ringstiffened cylinder. The ring stiffeners were loca.ed at equalintervals between the end plates. In other words, one ring

51


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. .. . .. .. ... . . . . . . .. . . . . . . . . . . . .

-~ ~ a tim= .392E....s ..

. b tim .798E0......

-Cim - - 9-4--psfr ng evel----- -----. ... ... ...- ..I -*-59---- -5. ----7.48E-. .rssr unt (gcm.m..cr..ose...**..h2..)-9-- 7 -----figure~~~.- P-ssr -avrpgtoitr in N--ar--Field Explosion Model

52-- -- - - -


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frinI IevIAtimeT- II194E0. . I I-6I48. II I- II I rTT I I I II f-I

pressure~~~~~ ntcm* miIse 2 I.4I2IaI

Figi~~~~~~~~ I8aIato at En Plt Clss toeaI-FielExplosion

frng3evl


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stiffener was placed at the center of the cylinder as seenpreviously in Figure 2.2a and the two ring stiffeners wereplaced at one third distances from the end plates respectivelyas seen previously in Figure 2.2b. The stiffeners were oneinch high and 0.25 inches thick. The nominal material propertydata of a 6061-T6 Aluminum alloy was used for this study.

The ring stiffened cylinders were compared to theunstiffened cylinder to provide insight into the effectivenessof tnh ring stiffeners in affecting stress wave propagationand deformation of cylinders. The numerical model was labeledin specific locations where the dynamic response was ofinterest. These locations are indicated in Figure 3.19 andthey include the largely deformed area at the front end of thecylinder, F, the stiffener locations: Si , S2, and S3, and thelocation close to the remote end plate, R.

The initial shock pressure produced a severe localdeformation about three inches from the front end plate forall three cases of cylinders as shown in Figure 3.20. Thecylinder yielded identically in all cases because thestiffeners had no effect on the initial deformations near thefront end plate until the pressure wave passed through thestiffeners. Comparison of the circumferential strains atlocation S1 between the unstiffened and two-stiffener modelswas given in Figure 3.21. In the figure legends A and Bdenoted shell elements located just before and after thestiffener. The stiffener reduced the compressive peak strain

54


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Pc , , ,i

F: 2.6 inches from end-plate closest to chargeSl: First stiffener of two-stiffener modelS2: Stiffener of one-stiffener modelS3: Second stiffener of two-stiffener modelR: 40.4 inches from end-plate closest to charge

Figure 3.19. Near-Field Model Locations Chosen forAnalysis

55


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at the location of the stiffener as expected. However, theunstiffened cylinder had a larger relief of the compressivestrain than the two-stiffened cylinder. Comparing the same atlocation S3 indicated that the stiffener not only reduced thecompressive peak strain, but also relieved the strain more atlater times as shown in Figure 3.22. The circumferentialstrains at the two stiffeners were plotted in Figure 3.23. Thefront stiffener had a much larger strain than the backstiffener. The large plastic strain at the front stiffenerrestrained the recovery of the circumferential strain at theadjacent shell elements at later times compared to theunstiffened cylinder as shown in Figure 3.22.

Comparison of the unstiffened and one-stiffenercylinders revealed that the stiffener caused a large relief ofthe circumferential strain. The center stiffener also causeda big difference in the circumferential strains between theshell elements just before and after the stiffener. The strainwas greatly reduced after the stiffener as seen in Figure3.24. The longitudinal strains were compared in Figures 3.25and 3.26 between twc-stiffened and unstiffened. Figure 3.25was the comparison a+ location S1 and Figure 3.26 at locationS3. The ring stiffener at location S1 altered the longitudinalstrain before and after the stiffener as shown in Figure 3.25.The stiffener at location S1 induced a higher longitudinalstrain at the shell element located just before the stiffenercompared to that at the same location of the unstiffened

56


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(a) nstiffened model

(b) ne-stiffener model

(c) wo-stiffener modeltime = .35999E+03 gi s

Figure 3.20. Local DtformatioflS for Near-Field Models57


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CA:, 13.3 ;.n. !r=n c~cse endC0

- C;.j w 0

M."M.43

in uI -O

t iffmr(a) nstiffened modelA .3 n. frzm zlise endC j. 5: .. -.:-se end

CL

SK0

4 t(b n-sifne oe

Figre3.1. CicufeenialStais t ot Sde oStfee lss oApoieCag

~ C58


68/84

, Xll-e, : ": z- , :-:Se e."dc I wA43

C - Oil.

0 _n f4L :_:se en.d0

I !m f I ioff Un.ffnd oe

so* mm* IM 03

0I SK 03

0. I I I I 1 t

(a) Onstiffener model

.1 699 03

(a) nestiffene model

Figure 3.22. Circumferential Strains at Both Sides ofStiffener Farthest from Explosive Charge

59


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I O t

C

c $ .

( S enlo exlsv

IV

t

(b) Stiffener 28.5 in. from end close to explosive

oL"*

.: I l

figure 3,23. Circumferential Strains at Both Stiffeners ofTwo-Stiffiner Model

60


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cylinder. In addition, the stiffener relieved the strain nearto zero at the shell element located just after the stiffeneras time elapsed. The effect of the stiffener at location S3 onthe longitudinal strain was much smaller compared to that atlocation S1. No accordion mode was observed for all threecases of cylinders.

61


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, sK s.A: 2X. in fr:m ::Ise endaSm . n. : =~ .zse end

M 1

L

SK 6,. 0 * e .

fr . :1050 e.

t me(a)Onstiffener model

Figre .2. Crcmfeenial Strin Comarson foUntfee n neSifnrMdl

Ii IU~e62


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0 41VL 11

6~ Il 233!-ri:'s na IA: in. ::n :.zse end

t ;I.(a) Unstiffened model

o /

r - sI

S iffnd USifnrLcainCoett

Charge

3II '3 "4 - .n r..:r :-:$e er.d

t I I *i A. I S I I

(b) Two-stiffener modelFigure 3.25. Longitudinal Strains (Unstiffened and Two-Stiffened) at Stiffener Location Closest toCharge

63


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IK #F I~0

-I I

3 8K 3 A: Z'.3 ;n . fr:m zlose endI ow3 3: 28. in. from close end

t i l

(a) Unstiffened modelIWn

0U

I off29I W03

.01I '-lIII hu W uI W E

3e 4 A. i"3 ft=c --.- se endIV ej V 4'

(b) Two-stiffener model

figure 3.26. Longitudinal Strains (Unstiffened and Two-Stiffened) at Stiffener Location far fromCharge64


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IV. CONCLUSIONSThe numerical and experimental study was performed to

investigate the nonlinear dynamic response of cylindricalshells to end-on explosions. Unstiffened and ring-stiffenedcylinders capped at both ends were subjected to near-field andfar-field explosions. Both finite element and boundary elementmethods were used for the numerical study. The experiment wasfulfilled using a 6061-T6 aluminum alloy cylinder and a sixtypound HBX-icharge with 28 feet stand-off distance.

The far-field explosion resulted in a nearly symmetricdeformed shape of the cylinder about the center plane betweenthe two end plates, while the near-field explosion caused veryunsymmetric deformation about the center plane. Both far-fieldand near-field explosions induced localized deformations nearthe end plates. The localized deformations were located closeto both end plates for the far-field explosion and a verysevere, localized deformation occurred close to the closestend plate to the explosive charge for the near-field explosionregardless of the existence of ring stiffeners. The accordionmode was observed for the cylinder subjected to the far-fieldexplosion but not for the cylinders subjected to near-fieldexplosion.

A parametric study was undertaken to examine the effect ofend plates on the deformation of the cylinder subject to the

65


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far-field explosion. A variation of the density and stiffnessof the end plates caused a significant change in the stresswave propagation and the deformation of the cylinder. However,the change of stiffness had more significant effects than thechange of density.

The circumferential strains under the far-field explosionhad large initial compressive peaks and returned to the steadystate values quickly. The circumferential strains were largernear the end plates than around the center due to the localdeformation. The localized deformations were caused by thecompressive shock pressure applied on the end plates. Thelongitudinal strains under the far-field explosion werecompressive at the shell near the end plate closer to thecharge but tensile at the opposite location.

The failure of pressure and several strain gages preventeda quantitative comparison between the numerical andexperimental results. A qualitative comparison was possiblebetween the two solutions only at a few locations. Thecomparison was better near the remote end plate than the nearend plate to the charge. A good explanation for thisdiscrepancy was not possible due to the lack of information inthe experiment. However, this was the first test among aseries of experiments to be performed. More detailedinformation from the next tests may provide a betterunderstanding of the discrepancy.

66


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The effect of ring stiffeners was larger on thecircumferential strain than on the longitudinal strain. Thering stiffener located close to the severe local deformationhad a more pronounced effect on the strains. The ringstiffeners in general reduced the circumferential straincompared to the unstiffened case and stiffeners also gave morerecovery of the strain. However, when the stiffeners had alarge plastic deformation, they allowed less relief of thecircumferential strain.

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APPENDIX: UNIAXIAL TENSION TEST DATA FOR 6061-T6 ALUMINUM

A uniaxial tension test was performed on a test specimenof 6061-T6 Aluminum to find its material properties. Thespecimen was cut to ASTM E8-69 specifications for rectangulartension test specimens.

A Material Test System (MTS) model 810 was used for thetest and the gage length was set at two inches for the tensiletest. The tensile test was performed at a rate of 400 sec/inchand a lad versus displacement curve was plotted as shown inFigure A.I. The MTS model did not have data recordingcapabilities and therefore, some points were taken from thegraph after the test to plot the appropriate stress-straincurve. Some sample data points are listed in Table A.I.

From the stress-strain curve in Figure A.2, the yieldstress was found to be close to 43 ksi using a 0.2 percentoffset and the modulus of elasticity was 10800 ksi.

68


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L. -:: -- -------

-4f :4.

Load SO .(LB) ZLZzj:

Displacement (inches)Figure Al. Load-Displacement curve for 6061-T6 Aluminum

69


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50000'

40000-

30000 /

u, 20000 /

10000-

0/0.000 0.002 0.004 0.00O6 0.008 0.010

STRAIN

Figure A. 2. Stress-Strain Curve fo r 6061-T6 Aluminum70


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TABLE A.1LOAD (LBF) STRESS (psi) STRAIN0.0 0.0 0.0500 10526 0.0010625 13158 0.001251000 21053 0.00201250 26316 0.00251500 31579 0.0031251625 34211 0.0033751750 36842 0.0036251875 39474 0.0038752000 42105 0.0046252075 43684 0.0066252100 44210 0.008125

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LIST OF RZFEREZNCZS

1. Budweg, H.L. and Shin, Y.S., "Experimental Studies on theTripping Behavior of Narrow T-Stiffened Flat Plates toHydrostatic Pressure and Underwater Shock," Proceedings ofthe 58th Shock and Vibration Symposium, Vol. 1, NASAPublication 2488, pp. 61-95, October 1987.

2. Jones, R.A., and Shin, Y.S., "The Response and FailureMechanisms of Circular Metal and Composite PlatesSubjected to Underwater Shock Loading," Proceedings of the61stt Shock and Vibration Symposium, Vol. 3, Pasadena,California, pp. 163-178, October 16-18, 1990.3. Kwon, Y.W., Fox, P.K., and Shin, Y.S., "Response of a

Cylindrical Shell Subject to a Near Field Side-OnExplosior," Proceedings of the 62nd Shock and VibrationSymposium, v.2, pp. 483-492, 29-31 October 1991.4. Banerjee, P.K., and Butterfield, R., Boundary ElementMethods in Engineering Science, McGraw Hill Book Company(UK) Limited, London, 1981.5. Stillman, D.W., and Hallquist, J.O., VEC/DYNA3D User'sManual (Nonlinear Dynamic Analysis of Structures in ThreeDimensions), Software Technology Corporation, 1990.

6. DeRuntz, J.A., and Rankin, C.C, Applications of the USAand USA-STAGS Codes to Underwater Shock Problems, v.1,v.2, Palo Alto, California, 1990.7. Geers, T.L., "Computational Methods for TransientAnalysis," Boundary-Element Methods for TransientResponse, Elsevier Science Publishers, p. 231, 1983.8. Hallquist, J.O., LS-TAURUS An Interactive Post-Processorfor the Analysis Codes LS-NIKE3D, LS-DYNA3D, and TOPAZ3D,Livermore Software Technology Corporation, 1990.9. Stillman, D.W., and Hallquist, J.O., LS-INGRID A Pre-Processor And Three Dimensional Mesh Generator for thePrograms LS-DYNA3D, LS-NIKE3D and TOPAZ3D, LivermoreSoftware Technology Corporation, 1991.

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10. Dobratz, B.M., LLNL Explosives Handbook, LawrenceLivermore Laboratory, University of California, p. 8-21,1981.11. Askeland, D.R., The Science and Engineering of Materials,Second Edition, PWS-KENT Publishing Company, p. 421, 1989.12. King, F., Aluminum and its Alloys, Halsted Press, p. 226,

1987.13. Timoshenko, S.P., and Gere, J.M., Theory of ElasticStability, McGraw-Hill Book Company, Inc., p. 459, 1961.

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INITIAL DISTRIBUTION LIST

No. of Copies1. Defense Technical Information Center 2Cameron StationAlexandria, Virginia 22304-61452. Library, Code 52 2Naval Postgraduate SchoolMonterey, California 93943-50023. Professor Y.W. Kwon, Code ME/Kw 2Department of Mechanical EngineeringNaval Postgraduate School

Monterey, California 939404. Professor Y.S. Shin, Code ME/Sg 1Department of Mechanical EngineeringNaval Postgraduate SchoolMonterey, California 939435. Department Chairman, Code ME 1Department of Mechanical EngineeringNaval Postgraduate SchoolMonterey, California 939406. Dr. Thomas T. Tsai 1Defense Nuclear Agency6801 Telegraph RoadAlexandria, Virginia 223107. Dr. Kent Goering 1Defense Nuclear Agency6801 Telegraph RoadAlexandria, Virginia -23108. Mr. Frederick A. Costanzo 1David Taylor Research Center

Underwater Explosion Research DivisionNorfolk Naval Shipyard, Bldg 369Portsmouth, Virginia 23709

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9. Dr. B. Whang, Code 1750.2Hull Group Head, Submarine Protection DivisionDavid Taylor Naval Ship Research and DevelopmentCenterBethesda, Maryland 20084-500010. Professor Thomas L. GeersDepartment of Mechanical EngineeringCampus Box 427University of ColoradoBoulder, Colorado 8030911. Dr. John A. DeRuntz Jr.Computational Mechanics SectionLockheed Palo Alto Research Laboratories3251 Hanover StreetOrganization 93-30

Palo Altu, California 94304

luis a. boticario- dynamic response of cylindrical shells to underwater end-on explosion

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