u-rBvst Availab~e Co '

NLEVETECH4NICAL REPORT SL.79-15PL

0 PROJECTILE PENETRATION IN SOIL AND ROCK:ANALYSIS FOR NON-NORMAL IMPACT

Ro ~ ~ ebert S.BrarDnilCCriho

Final Report *LDL-ýU L

Prs~are for Defense Nuclear AgencyJ Washington, D. C- 20305

Under DNA Subtask Y99QAXSB21 1, Work Unit 20

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SECURITY CL.ASSIFICATION OF THIS PAGE (Who et D nfe.4)

4 &q.~LH PAGEBFR OPEIGFR1. 0ER NUMIIUE ý WF5/71 2. GOVT ACCESSION No. 3. ACSLPENT'S CATALOG H4UMBER

Technical Repor L-7915 r.02!.4. TITLE (sail S..bile S l O 44OI aPEF0acoe

rdTPROJECTILE gENETRATION IN SOIL AND ROCK: ANALYSIS Final re .#TbR NON-NORYAL MC- A J A

Daniel C./Creighton

S. PERFORMING4 ORGANIZATION NAME ANO ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASKC

U. S. Army Engineer Waterways Experiment StationStruactures Laboratory SeBok1P. 0. Box 631, Vicksburg, Miss. 39180_____________

11. CONTROLLING OFFVICE NAME ANO ADONIS$Defense Nuclear Agency // Decm ~r 1979Washington, D. C. 20305 -_

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17. OISTRIVIUTION STATEMENT (of the .hetroat ontm lot Block ". It Mff..w~t from. Rspo*i)

I*- SUPPLEMENTARY NOTES

..'This earch was sponsored by the Defense Nuclear Agency under Subtask/,Y99Q.AX!23 Work2 1 .Ui 20jŽ"P etrat ion Support."

It. KCEY WORDS (C-..~U .Idi nec...r OW IdontifVP by block 0,webw)

Ballistics PenetrationComputer analysis ProjectilesImpact Rock penetrationNumerical analysis Soil penetration

-An empirical rigid-projectile anal~ysis has been developed for non-normalimpact and penetration into soil and rock. The soil penetration analysis usesYoung's S-number as the penetrability index. The rock penetration analysis

calclats rsisanc-topentraionusing density, strength, and Rock QualityDesignation. Expressions for compressive normal stress are integrated numneri-cally over the projectile surface using the differential area force law in the

(Continued)

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S1CUISItY CLAWFIPCATIWU OF h PAGS(wo" Does ~an-

20. ABSTRACT (Continued).

PENCO2D computer code. The analysis includes free-surface and vake separation-reattachment effects.

The PENCO"D code is used to make comparison calculations for Sandia/DNA

reverse ballistic tests and several full-scale tests conducted by Sandia

Laboratories. A parameter study shows effects of soil penetrability, impact

velocity. initial attack and obliquity angeles, and projectile geometry onunderground trajectories.

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PREFACE

This investigation was sponsored by the Defense Nuclear Agency

under Subtask Y99QAXSB211, "Penetration," Work Unit 20, "Penetration

Support." This study was conducted by personnel of the Structures

Laboratory (SL), U. S. Army Engineer Waterways Experiment Station (WES),

during October 1978 through May 1979, under the general supervision of

Mr. Bryant Mather, acting Chief, SL, and Dr. J. G. Jackson, Jr., Chief,

Geomechanics Division (GD), SL. Mr. R. S. Bernard formulated the

theory, and Mr. D. C. Creighton developed and implemented the computer

analysis, both under the technical guidance and direction of Dr. B. Rohani,

GD. Messrs. Bernard and Creighton prepared this report.

COL John L. Cannon, CE, and COL Nelson P. Conover, CE, were Command-

ers and Directors of WES during the period of research and report prepara-

tion. Mr. F. R. Brown was Technical Director.

KciessionFo

---- S±•• For

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iii

CONTENTS

PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I

CONVERSION FACTORS, U. S. CUSTOMARY TO METRIC (SI)UNITS OF MFASUR1F4EMNT . 6

CHAPTER I INTRODUCTION ..................... 7

1.1 Background ............ . . . . ... 71.2 Purpose and Scope ............... 7

CHAPTER 2 PENETRATION ANALYSIS FOR MOTION IN TWO DIMENSIONS • • 9

2.1 Projectile Motion in Two Dimensions .... ........... 92.2 Stress Distribution for Rock and Concrete ........ . 112.3 Stress Distribution for Soil ..... ............... .... 122.4 Wake Separation and Reattachment .... ............. . .. 132.5 Free-Surface Effect ........... ................... ... 16

CHAPTER 3 CALCULATIONS AND EXPERIMENTAL DATA ...... ......... 23

3.1 Background ..... ........................ 233.2 Reverse Ballistic Tests in Sandstone. ........... 233.3 Full-Scale Penetration Tests in Soil .... ........... ... 26

CHAPTER 4 PARAMETER STUDY FOR SOIL PENETRATION ............... 46

4.1 Background .............. ........... ................ 464.2 Baseline Conditions ................. .................. 464.3 Variation of Target Penetrability and

Impact Conditions ................ .. . .4.4 Independent Variation or Projectile

Parameters for Straight Aftbody ..... ............. .... 484.5 Variation of Aftbody Flare ........ ................ .4.6 Variation of Coupled Projectile Parameters ........... . .. 4a

CHAPTER 5 CONCLUSIONS ..................... qo

REFERENCFS ................. .............................. 61

APPENDIX A PENETRATION THEORY FOR ROCK AND CONCRETE ...... 65

A.1 Empirical Theory for Normal Impact ..... ............ 6rA.2 Extrapolation to Non-Normal Impact ....... ............ 07

APPENDIX B PENETRATION THEORY FOR SOTI ....... .............. 73

B.1 Fnpirical Theory for Normal Impact ........... ............ 7(B.2 Extrapolation to Non-Normal Impact ....... ............ 77

APPENVIX C NOTATION ................ ........................ .II

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LIST OF TABLES

Table Page

3.1 Soil penetration data for non-normal impact .... ....... 28

3.2 S-number profile for TTR Antelope Dry Lake site. . . . .. .. 9A.1 Penetration data for rock and concrete ......... 6)A.2 Projectile and target parameters .... ............ 70B.1 Typical S-numbers for natural earth materials ........ 70

LIST OF FIGURES

Figure

2.1 Projectile motion in two dimensions ... ........... ... 192.2 Three-dimensional view of projectile ... .......... . 10

2.3 Projectile surface geometry and velocity components . . . 202.h Wake separation for purely axial motion ........... .. 202.5 Separation and reattachnent on a rotating projectile . 12.6 Rear axial view of projectile cross scetion .... ....... 212.7 Projectile orientation with respect to free surface . . 223.1 Scale drawing of RBT penetrator ......... ............. 303.2 Comparison of calculated and measured lateral accelera-

tions for Sandia/DNA RBT, 3-degree attack angle . .. 313.3 Comparison of calculated and measured axial decelera-

tions for Sandia/DNA RBT, 3-degree attack angle ..... .. 313.4 Comparison of calculated and measured strains for

Sandia/DNA RBT, 3-degree attack angle ..... .......... 323.5 Comparison of calculated and measured lateral accelera-

tions for Sandia/DNA RBT, 20-degree obliquity .... ...... 323.6 Comparison of calculated and measured axial decelera-

tions for Sandia/DNA RBT, 20-degree obliquity ........ 333.7 Comparison of calculated and measured strains for

Sandia/DNA RBT, 20-degree obliquity ..... ........... 31N3.8 Calculated lateral force for different values of free-

surface parameter, Sandia/DNA RBT, 20-degreeobliquity ..... ........................

3.9 Calculated pitch moment for different values of If-ee-surface parameter, Sandia/MNA HBT, 20-degreeobliquity ................... ........................ ., ,*

3.10 Calculated lateral force for different values of wakeseparation angle, Sandia/DNA RBT, 20-dejret"obliquity ................................ 3

3.13 Calculated pitch moment for different values of wakeseparation angle, Sandia/DNA RBT, 20-degreeobliquity ..................... ........................ 35

3.12 Calculated lateral force for different values of wakeseparation angle for Sandia/DNA RBT, 3-degreeattack angle . . . . . . . . . . . . . . . . . . . . . .

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Figure Page

3.13 Calculated pitch moment for different values of wakeseparation angle for Sandia/DNA RBT, 3-degreeattack angle ..... .... ...................... 36

3.14 Calculated lateral force for different values ofobliquity, Sandia/DNA RBT .......... ................. 37

3.15 Calculated lateral force for different values ofattack angle, Sandia/DNA RBT ......... ................ 3T

3.16 Calculated pitch moment for different values ofobliquity, Sandia/DNA RBT ............ ................. 38

3.17 Calculated pitch moment for different values ofattack angle, Sandia/DNA RBT ....... ............... ... 38

3.18 Variation of maximum calculated lateral load withattack angle and obliquity, Sandia/DNA RBT . ........ 39

3.19 Variation of maximum calculated pitch moment withattack angle and obliquity, Sandia/DNA RBT ..... ........ 39

3.20 Calculated lateral force for different values of targetstrength, Sandia/DNA RBT, 3-degree attack angle ......... 40

3.21 Calculated pitch moment for different values of targetstrength, Sandia/DNA RBT, 3-degree attack 4angle. ....... 20

3.22 Calculated lateral force distribution for differentvalues of target strength, Sandia/DNA RBT, 3-degreeattack angle ............. ....................... .... 41

3.23 Effect of wake separation angle 0min on calculatedtrajectories, Sandia Test R800915 .... ............. ....41

3.24 Effect of wake separation angle *min on calculatedtrajectories, Sandia Test R800916 ...... ............. 214

3.25 Effect of free-surface parameter k on calculatedtrajectories, Sandia Test R454025-23 ....... ........... 42

3.26 Effect of wake separation angle Omin on calculated trn--jectories, Sandia Test R454025-23 ................. 43

3.27 Effect of wake separation angle on calculated tra-jectories, Sandia Test R454025-22. ......... ........... 43

3.28 Comparison of calculated trajectories for Sandia pene-trators, ýmin = 1 degree ......... ................. .... 44

3.29 Comparison of calculated trajectories for Sandia pene-trators, *min = 2 degrees .......... ................. .44

3.30 Comparison of calculated trajectories for Sandia pene-trators, Omjn - 4 degrees .......... ................. 45

3.31 Comparison of calculated trajectories for Sandia pene-trators, $min = 8 degrees ......... ................. ..... c,

4.1 Calculated trajectory for baseline conditions .......... 514.2 Calculated external forces for baseline conditions . . . .4.3 Calculated pitch moment for baseline conditions ........ 24.4 Effect of soil penetrability on calculated trajectories . . 24.5 Effect of impact velocity on calculated trajectories . 5 534.6 Effect of attack angle on calculated trajectories ..... 534.7 Effect of obliquity on calculated trajectories ........ ... 944.8 Effect of projectile weight on calculated trajectories . •;44.9 Effect of projectile diameter on calculated

trajectories . . . . . . . . . . . . . . . . . . . . . . . 55

4

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Figure Page

4.10 Effect of projectile nose shape on calculatedtrajectories ... .................... 55

4.11 Effect of projectile mass moment of inertia on cal-culated trajectories .......... ... .............. ... 56

4.12 Effect of projectile length (as an independentparameter) on calculated trajectories .......... 56

4.13 Effect of CG location on calculated trajectories ... ..... 574.14 Effect of aftbody flare on calculated trajectories .... 574.15 Scale drawing of projectile with 3-degree aftbody

flare ....... ......... .......................... ... 584.16 Combined effects of weight, length, CG location, and

mass moment of inertia on calculated trajectories .... 58A.1 Nondimensionalized penetration data for rock and

concrete ... ......................... 71B.1 Comparison of calculated penetration depths with data

from TTR Main Lake area ......... ..... ................ 80

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5

CONVERSION FACTORS, U. S. CUSTOMARY TO METRIC (SI)UNITS OF MEASUREMENT

U. S. customary units cf measurement used in this report can be con-

verted to metric (SI) units as follows:

Multiply By To Obtain

feet 0.3048 metres

inches 0.0254 metres

feet per second 0.3048 metres per second

pounds (force) per foot 14.5939 newtons per metre

pounds (force) per square inch 6894.757 pascals

pounds (mass) 0.4535924 kilograms

pounds (mass) per cubic foot 16.01846 kilograms per cubic metre

pounds (mass) per inch 17.85797 kilograms per metre

pounds (mass)-square inches 0.0002926 kilograms-square metres

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6

PROJECTILE PENETRATION IN SOIL AND ROCK:

ANALYSIS FOR NON-NORMAL IMPACT

CHAPTER 1

INTRODUCTION

1.1 BACKGROUNID

In recent years, due to growth of the experimental data base,

development of empirical formulas, and implementation of new theoretical

and computational techniques, it has become possible to predict the

approximate motion of stable earth penetrators after normal impact in

many types of geological targets (Reference I-L). Results of some non-

normal (yawed/oblique) impact tests have also been predicted success-

fully, although the data base is small compared to that for normal impact

(References 5 through 7).

Previous analyses by the U. S. Army Engineer Waterways Experiment

Station (WES) of non-normal impact have used the Cavity Expansion Theory

(CET) for rock, and an empirical formu-lation similar to Young's equation

for soil (References 5 and 8). These analyses were considered prelimi-

nary when they were first developed, and it was hoped that more credible

analyses would arise in the future.

1.2 PURPOSE AND SCOPE

The present investigation concerns the development of a generalized

znalysis for oblique/yawed impact and penetration in soil and rock (or

rock-like) targets, based on the modification of previous analyses and

the interpretation of recent test data. The objectives are (a) to

obtain external load-time histories sufficiently accurate for structural

response calculations and (b) to calculate postimpact trajectories well

enough to predict the terrndynamic performance and stability of a given

projectile.

The two-dimensional (2-D) penetration theory, including equation of

motion, stress distribution, and free-surface tnd wake separation-

reattachment effects, was developed. The penetr ation theory was computer

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Icoded into PENCO2D to make comparison calculations with experimental

data from both reverse ballistic and conventional penetration tests.

Using PENCO2D, the effects of initial impact conditions, projectile

geometry, and soil penetrability were examined in a brief parameter

study.

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CHAPTER 2

PENETRATION ANALYSIS FOR MOTION IN TWO DIMENSIONS

2.1 PROJECTILE MOTION IN TWO DIMENSIONS

Consider a rigid, deeply buried projectile (no free-surface effects),1

whose motion lies in the XZ-plane, as shown in Figure 2.1. The rota-

tion is described by the angular velocity 0 , and the translation by

S.. Z , the X- and Z-velocity components of the center of gravity

(CG). 2 ' 3 For analysis, it is convenient to express the CC velocity in

terms of its lateral and axial (x- and z-) components V and V ,4 x z

respectively:

V X cose -e sin e (2.1)

V z sin 0 + cos 8 (2.2)

For purely axial motion (Vx 0 , 6 = 0), the stress distribution on

the projectile is symmetric, and the net force is in the axial direc-

tion, producing no pitch (turning) moment. The introduction of rotation

(6 0 0) or lateral motion (Vx 0 0) destroys this symmetry, and it is

necessary to specify the asymmetric stress distribution in order to

calculate the resulting force and moment.

Suppose that the compressive resisting stress a is always normal

to the projectile surface (no tangential stresses). On any surface area

element dA , the x- and z-components of a are, respectively,

1 The XZ-coordinate system is fixed in the target (Figure 2.1).2 A dot above any quantity indicates differentiation with respect to

time.For convenience, symbols and abbreviations used in this report are

4 listed and defined in the Notation, Appendix C.The xz-coordinate system is fixed on the projectile axis, with itsorigin at the CC (FiFure 2.1). Equations 2.1 and 2.2 representthe projections of the absolute velocity vector along the x- andz-axes.

9

o a a cos ,n co0 8 (2.3)x

a a o sinrn (2.li)a

where

* * azimuthal angle (Figure 2.2)

tan n s ulope of projectile surface at a given point, with respectto axis of symmetry (Figures 2.1 and 2.3)

The lateral and axial, force components (Figure I.') acting on surface

element dA are, res!,ectively,

dF - -o dA (2.5)x x

dF -o dA (2.6)

where dA in the differential element of surface area. The lateral

aid axial net force comivnentts acting on the entire projectile are then

F - -f o dA (W.7)x x

F - -f o. lA (2.8)

where the integral is evaluatied over the entire projectile ,urfaece.

The translational (i2G) equations of motion are

M X - F eos 0 + F nin 0 (2W.)

M - -F ,in 0 + F cos 0 (W. 10IX

where M in t he proje1e t. iI mnn rmind X iuih ntrc the omponeInit.S of'

at-oeleration In the X- nnd '.7-M rect. l.r *, restlpe-,t ively. The projlect, ile

is nhitmiied to be rigid, zio the roeatn Iton]i erint. ion of' mot.ion is

1;;- f , - f x ,IF 12.11)

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where I In the trmaivernt' maps nitimoet of' Inert ia aboitt the C0, 0 Is

the nngular nceeleration, andi x mid r, mire the lateral And atxial

dlantacea, respeot ivelvy, from the CO (Figitre 0.01. 'Me right-hand s~ide

of FEquation Xli1 in the t.otAl moment exertet tin the pro~jeotilie ikbouit.

ItS CGt, Andi thIe integraln are ev~unt~ed over the etit ire nren or the

iprojeect le ourfnoe.

2~. " STMI OI)TTHPIVION FtPROCK( AND CONCRFTF

For 1'enetrat ion into conorete mtid IntAot rook (Appendix A), the

net force on the projleeti Ic cn lip oalcUlAttd appIrx~xItmAtely from the

following norMal st~rens dist ribut. tn:

where

Y - unconfined oompirnsin vi sttrength or lutacnt tntrget maiterint

11 target. mnts delirlity

v nb'oo i t e I oal vet colt y (i tA a ~ (n poinut. ton projcoft le miiwt'nov(Figrp 2. 3)

v oiftitintd ormnl e.lmpotiont or' v t, I1ur' 2.

ci' i ned in tvermrn of' V *V , 0 ,Anti 7 , the iliitAt.it.Ies v Andx

v kre

v j~~ (V, 4,

I f tierorsnry , mitllnc 00t-'et I ,'is Y onii l'~' ffidietforC I h' .pim I It ycr the rook or for t he rinimi mInnn~~e~te~ II I the con..I'Mret(Append i A'). Any tirt. or imits otnn ber iinc! III F'qunt ion 212Hitfhoy mint. I'e I mrns lonnlI ly vompnt. 11, 1r (n. #. s' Iign , feert

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vn V. sin n + (Vx + zi) cos n cos * (2.14)

It is assumed that a given surface element must have a net motion into

the target material to produce a resisting stress; hence, a vanishes

when vn < 0 in Equation 2.12.

2.3 STRESS DISTRIBUTION FOR SOIL

For penetration into soil (Appendix B), the net force on the

projectile can be calculated approximately from the following normal6stress distribution:

V V V+5 r v >0

S v S p v $2 v n

a. (2.15)

0, v <0

where

1.96 x 105 lb/ft, constant 7

8 6.25 x 106 Ib2-sec 2 /ftT, constant

y U 1.56 x 105 lb/ftt3 , constant

S a Young's S-number (Appendix B), soil penetrability index

rp - local cylindrical radius at a given point on the projectilesurface

Z - vertical depth of a given point on the projectile surface,measured from the target surface

The expressions for v and vn are still given by Equations 2.13 and

2.lh, respectively; and the requirement that vn 0 for o 0 applies

in soil as well as in rock or concrete.

The quantities u , p , and y are constants, independent of soil

6 Any set of dimensionally compatible units can be used in Equation 2.15,

7 as long as all quantities are converted to those units.A table of factors for converting U. S. customary units of measurementto metric (ST) units is presented on page 6.

12

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type and projectile characteristics. Thus, the only parameter charac-

terizing the soil is Young's empirical S-number, which can be obtained

for a given target from previous penetration data or from correlation

with geological descriptions (Table B.1).

2.4 WAKE SEPARATION AND REATTACTUENT

Flash X-ray photographs and two-dimensional (2-D) finite-difference

calculations have shown that loss of contact between target and projec-

tile occurs somewhere on the nose, while reattachment may or may not

occur on the aftbody. The current analysis is not sophisticated enough

to predict the actual onset of separation. Nevertheless, it is neces-

sary to include a simple model for the kinematics of wake separation,

especially when 2-D projectile motion in soil is investigated.

It is assumed that there is a minimum angle of approach, or wake

separation angle, *min required between the target and the projectile

contact surface (Figure 2.4) in order for contact to be maintained.

For a given surface element, the local angle of approach 0 is deter-

mined by the instantaneous orientation8 of the projectile such that

Vsin =n (2.16)

where V is the CG velocity and V is its outward component normalnto the projectile surface, given by

V n V sin n + V cos n cos i (2.17)n z x

Thus, separation occurs whenever < <min ' i.e., whenever the local

angle of approach is less than the wake separation angle. 9

Separation is only part of the problem; reattachment may or may

The rotational velocity 6 is not considered in formulating thecriterion for wake separation. This quantity does, however, playa strong role in wake reattachment.For the case of no yaw (no angle of attack), Equation 2.16 reducesto sin * sin ni

13

-.. r - S !

not occur, depending on the relative motion of the wake and the aftbody

downstream of the separation point. It is expected that the wake separa-

tion angle +rmin will generally be small (<10 degrees), in which case

the angle of attack a required to maintain contact will also be small.

In any case, reattachment will be analyzed based on three assumptions:

1. Constant axial velocity (V )2. Constant angular velocity (6)

3. Constant angle of attack (a)

Obviously, these three quantities change as the projectile penetrates

into the target. Nevertheless, if the quantities do not vary rapidly,

the quasistatic approximation is adequate.

The cavity formed by the wake is a gently curved cylindrical tube

of increasing radius (with time), and the projectile rotates inside

this tube as it travels forward. The curved axis of the cavity is

fixed in the target, and aftbody reattachment occurs wherever the

projectile rotates into the cavity wall (Figure 2.5).

Disregarding any stresses that may oppose the radial expansion of

the cavity, conservation of mass and incompressibility in the target

requires that

r constant (2.18)

where r is the local cylindrical wake-cavity radius. EvaluatingcEquation 2.18 at the separation point, it then follows that

rc c = r V tan (2.19)

where r is the projectile radius at the separation point. For smallo

values of ,min ' r is approximately the radius corresponding to

n = @rmin (Figure 2.4). Furthermore, for straight- and tapered-aftbody

projectiles, r is approximately the radius at the base of the nose,o

Denoting axial distance aft of the nose tip by C , time deriva-

tives can be related to V and C by the transformationz

14

d= v d (2.20)dt zdý

and Equation 2.19 reduces to

drr c cr tan (2.21)

for which the solution is

r c 4r2 + 2r (- )tan in ->O (2.22)

where o is the value of C at the separation point. For < o

the cavity radius is the same as the projectile radius, since no loss

of contact has occurred.

Using Equation 2.20 again, the differential equation for the

lateral displacement 6 of a point on the projectile axis, relative

to the cavity centerline (Figure 2.5), is

dt dS (2.23)dt z dr

Treating V and e as constants, the solution for 6 isz

f 2 6 (2.24)z

The initial displacement 8 = . sin a is the displacement due to theo10

angle of attack a (treated here as a constant), so Equation 2.24 is

replaced by

10 It is assumed that small attack angles do not significantly change

the cavity geometry. Thus, if 6 0 , the cavity centerline willbe straight; but if in addition a $ 0 , the projectile axis willbe skewed relative to the cavity, producing displacements 6Ssin c between the projectile axis and the cavity centerline.

15

I

C. .. ( 2.25

2 1 - )2 + • sin a( )

where

Q -tanI (2.26)

At any point on the projectile surface aft of the separation point, the

criterion for no contact is

2oa r 2 2 <S4 ") * sin a -rp PCos r< r sin2 '0, <_ _

z 2

(2.27)

r Cose 2 sia< r 2 r 2 si

Hence, if m min and Equation 2.27 are satisfied, there is no con-

tact, and 0 0.

The foregoing analysis is only a rough method of accounting for

wake separation and reattachment using a single input parameter, the

wake separation angle jmin * Nevertheless, this analysis does make

it possible to assess the relative effects of separation and reattach-

ment on projectile stability, as will be shown in Chapter 3.

2.5 FREE-SURFACE EFFECT

In brittle materials, such as rock and concrete, a crater is always

formed during (or just after) impact. Postimpact inspections in rock

(Reference 9) indicate that these craters may be as much as 10 projec-

tile diameters across at the target surface. The crater width decreases

Figure 2.6 presents a rear axial view of the projectile and cavitycross sections at an arbitrary location aft of the separation point.The figure is valid only for small displacements, i.e. 6 -<<

16

rapidly with depth, however, approaching a value equal to (or slightly

less than) the projectile diameter after a few calibers of penetration.

The same phenomenon occurs in soil, though to a lesser extent than in

rock, since soils are more ductile than brittle in behavior.

For normal impact or for near-normal impact with slight yaw, it is

not necessary to account for the cratering phenomenon (i.e., free-

surface effect) in calculating the loads on the projectile with the

current analysis. On the other hand, for oblique impact with no initial

yaw, the free-surface effect is probably one of the dominant mechanisms

creating an unbalanced lateral force on the projectile, especially in

rock and concrete. Thus, for analyzing non-normal impact in general,

the presence of the free surface should be acknowledged, even if only

by a crude representation of its effect.

The influence of the free surface is modeled herein with an on-off

stress criterion, governed by the location of a given projectile surface

element dA with respect to the free surface (Figure 2.7). Whenever

the radial distance from the projectile axis (through dA) to the free12

surface is less than some prescribed value r , then the stress onsdA is set equal to zero. Quantitatively, this condition is expressed

by setting a = 0 whenever

ZCG + z cos e < rs sin e cos j (2.28)

where ZCG is the vertical distance from the free surface to the CG.

Assuming r to be directly proportional to the local projectile radiuss

r , Equation 2.28 is replaced by the stipulation that a = 0 wheneverp

ZCG + z cos 8 < krp sin 8 cos k (2.29)

where the free-surface parameter is

12 rs is the maximum perpendicular distance from projectile axis,

through dA , to target surface for which stress relief due to thefree surface can occur.

17

r

k • a constant (2.30)p

This model for the free-surface effect produces a stress distribu-

tion that may be partially or completely turned off on the top side

(-w/2 < 0 < w/2) of the projectile, depending on the geometry, depth,

and orientation with respect to the target surface. The deeper the

penetration and the more vertical the orientation, the smaller the

influence of the free surface on the lateral loads.

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I

9f

WoT9 I z C0o.C'Att SVSItU,,%to No T%( YAKUOT

mCO0ROOMATI SVITO C

A1140 010 P*MJCF.tqW?[ A

Figure 2.1 Projectile motion in two dimensions.

Figure 2.2 Three-dimensional view of projectile.

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"1b-

fI

Figure 2.3 Projectile surface geometry andvelocity components.

V..i.20OINT

V,Vz

Figture 2.4 Wake separation for purely axial motion.

20

\, \ \.-,..

OFF.•

Poo. I WAfL

( '

me ~ I atV4~d

"%. S%

Figure 2.5 Separation and reattachment ona rotating projectile.

WAK1-CAVITY WALL

PR JCTL .v~

NOTE. THIS DRAWING VALID ONLN FOR

DISPLAý'EMENT RELATIVE TO WANkE-CAVIT'tCROSS SECTION AFT OF SEPARATION POINT

Figure R. ear axial view otf pro'jec.til]e

cross section.

21

Figure 2.7 Projectile orientation with respectto f~ree surface.

22

/

CHAMMI2 3

CALCULATIONS AND FXPERIMENTAL DATA

3.1 BACKGROUND

Detailed information concerning non-normal impact effects is scarce,

mainly because additional parameters that go beyond those important for

normal impact are Involved. In the latter case, the important quantities

are the weight, size, and geometry of the projectile; the Impact veloc-

ity; the final depth; and the axial deceleration record. For non-normal

tests, however, the additional important parameters are the moment of

inertia and CG of the projectile, the lateral compor.ents of the impact

velocity, the 2- or 3-D trajectory (including projectile orientation),

and the lateral components in the deceleration record.

Usually, the only way to obtain all this information is by con-

ducting a reverse ballistic test (RBT). Still, if the projectile

parameters are known, much Insight can be gained from knowing only the

impact conditions and the final position, particularly with regard to

stability.-In this chapter, comparisons will be made between calculationsI and

test results for RBT's and conventional penetration tests. The main pur-

pose will be to investigate the general credibility of the analysis and

the parameters used therein. Although the data are too few to achieve

the degree of verification that can be obtained for normal impact, there

are still enough benchmarks available to check for unreesonable predic-

tions. With this done, some credence can be given to the parameter

study in Chapter 4.

3.2 REVERSE BALLISTIC TESTS IN SANPSTONE

Tn 1Q77, Sand iL Liborattoris cornductd l'our DNA-t:-.potorced H1VV'":i In

sandstone. A completc description of the tte-;tto Is given In Reference 10,

The PENCO2D computer code, which solves the penetration problem In two

dimensions, uring the annlysis presented In Chapter 2, was used forall calculations reported herein.

23 • n n

and the processed data are presented in References 11 and 12. I'o of

the tests (Nos. 2 and 3) were done at a 3-degree attack angle, and one

(No. 4) at 20 degrees obliquity (no angle of attack). The impact veloc-

ity was approximately 1500 Wt/e In each case. Nominal target properties2

are

p * 130 lb/ft3

Y - 3400 psi

Rock quality designation (R(D) w 100

A scale drawing of the projectile Is shown In Fig•tre 3.1; the projectile

parameters are'

Length (L) - 18•.19 Ia

Weight (W) - (•. 48 l1b

Diameter (D) - 1.9 In

Caliber radlun head (CRH) w 6.0O (beveled tip)

r " .X1.Q Ib-in

For each caae, the load dt atribut, iont on the projIect le, calculated

using the P NCO0;1 computer code (with output moditfioation..), have been

used an Input to a daynmilc ,truottural-response code (W1AMI1) by

T. Ilelytschko.3 lT'redIottoun were generated for acceleromter Rtid .train-

gage outputs, aamplen of whtich are compared with tesit resu lts in

Figures 3..' through 4.7. In the WHAKS ai.culat, ionu, the nceierometprs

indicated In Ft lgre, 3.; and .. 3 were located on the ax Ir, 16.3', 16. In

from the nose tip. The .itrainn gng was on the out.itIde

"The units naowaa were cahosaena for convente n'e of expw',e.ijoin. I oal -culations they must be oonvert.ed to a iompntlble .ayst m (0'.g.1, r ugs, *feet, seconds,).Pernonal oonmaun I latt ton, r'ceivedl •anrch IaQT, tfrom T. Velyt..ehko, Inc.,Chicago, IllT., to R. Blernard , Geomeohannitos Dvi .ota, Utl. alte trrlaboratory. a

2. I•

I

bottom (w - w), 9.6 inches from the nose tip.

For the 3-degree attack angle calculation, there is clearly a

phase difference between the calculated and measured lateral accelera-

tions (Figure 3.2), the cause of which is not yet understood. Also, the

initial positive peaks in the calculated results are somewhat low, but

the later negative peaks are in fair agreement with the test data.

Better agreement exists for the axial decelerations (Figure 3.3). Fair

agreement is obtained for the strain gage output (Figure 3.4), although

the initial negative strains are somewhat underpredicted.

For the 20-degree obliquity calculation, the calculated results are

in much better agreement with the test data (Figures 3.5 through 3.7),

although the phase-difference problem is still apparent in the lateral

acceleration (Figure 3.5) data and the strain gage output (Figure 3.7).

In using the rENCO2D computer code to generate the input for

Belytschko's WIAMS calculations, the free-surface parameter was set at

k - 8 , and the wake separation angle was set at *min a 0 . Thlese

values were chosen after examining the effect of varying k and 0min

in WES predictions for total lateral force and pitch moment (Figures

3.8 through 3.13). The value k = 8 is reasonable in light of pene-

tration-crater measurements in sandstone (Reference 9), and the n 0

value is reasonable, since the wake separation angle should be small in

hard targets.

In order to examine the effects of attack angle and obliquity on

lateral load and pitch moment in the WLE3 theory, a series of calculations

has been made using the 3- and 20-degree RST's as baseline cases. The

results are shown in detail in Figures 3.14 throut-,h 3.17 antd summarired

in Figures 3.18 and 3.19. A similar series of' calculatlon, was made,

varying the target strength for the 3-degree Ri1T. T'he results, shown

in Figures 3.20 and 3.21, indicate that the tNi tinI peaks in the lateral

force (at 0.12 ms) are much more dependent on strr'i-tfh than arc the

later peaks (at 0.7 to 0.8 ms). The reason for the similarity in the

Variation of k has no discernible effect on the 3-degree rPTcalculation.

25

later peaks can be seen by examination of the lateral force distribution

(per unit length) at 0.7 me (Figure 3.22). The amplitude of the force

distribution is a strong function of the strength, but the positive and

negative contributions are such that there is little difference in the

net lateral force at 0.7 me.

3.3 FULL-SCALE PENETRATION TESTS IN SOIL

Sandia Laboratories has conducted at least four full-scale soil

penetration tests that are suitable as benchmarks for the WES 2-D pene-

tration theory. Two of these tests, Nos. R800915 and R800916 (Refer-

ence 13), were conducted in the Pedro dry lake site on Tonopah Test

Range (TTR). The other tests, Nos. R454025-22 and R454025-23 (Refer-

ence 14), took place in the Antelope dry lake site on TTR. Pertinent

information for all four teats is summarized in Table 3.1, and Young's

S-number profile (Reference l4) for the Antelope site is given in

Table 3.2.

Using the information given in Tables 3.1 and 3.2 as input,

calculations have been made for the penetrator trajectories. The re-

stilts are compared with the Sandia test data in Figures 3.23 through

3.27.

Calculations for tests R800915 and R800916, presented respectively

in FIgures 3.23 and 3.211, show the effects of varying the wake separa-

tion angle. Here the expetimental results are inconsistent. The tests

were identical except for a 20 percent difference In impact velocity,

yet there was a 500 percent difference in lateral (horizontal) displace-

ment.5 One of the reasons for the discrepancy may have been a differ-

ence In the amount of wake separation In the two tests. Separation

alone, however, Is not enough to account for the difference, since even

the calculatlon for #min * 0 predicts 8 feet of lateral displacement

for test R800015. Tn these caculations the free-surface parameter was

5 The average vertical penetration resistance apparently was the same,because the depths can be calculated with same S-number.

26

set at k = 0 , due to its lack of influence on near-vertical impact

problems.

Test R454025-23 is a good example for assessing the effects of the

free-surface parameter independent of the wake separation angle effects.

Here the initial obliquity is large, and k and 4min both influence

the Qalculated trajectory (Figures 3.25 and 3.26). For this test the

best simulation of the observed trajectory is obtained for k = 4 and

imin = 0.6 Nevertheless, it is clear from the calculations that thewake separation angle has a much stronger influence than the free-

surface parameter, even for large obliquity.

Test R454025-22 was conducted with a shorter projectile (L/D - 8)

than the other three tests (L/D - 10). In this case, the penetrator

impacted at 30 degrees and rotated 80 degrees before coming to rest 13.5

feet below the target surface. With the free-surface parameter set at

k a 4 , calculations made for different values of the wake separation

angle (Figure 3.27) indicate that a value of *min = 4 degrees best

reproduces the observed final position of the penetrator.

In other (normal impact) tests7 in the Antelope site, Sandia has

found L/D = 10 to be significantly more stable than L/D - 8 (Refer-

ence 14). For comparison of the two projectiles in this study, calcula-

tions were made for the shorter projectile (from test R454025-22) using

the S-number and impact conditions of test R800916, setting k = 0 , and

varying *min * The results (Figures 3.28-3.31) indicate that the

stability of either projectile is strongly dependent on the value

selected for the wake separation angle.

6 To simulate the test conditions (Table 3.1), the calculation was

started with the nose tip at Z = 8 inches0 = 2 degrees, a < 0.5 degrees.

27

4 '-

I -

I -'

pI j� � .4v'� *�. �

..4

I�'j ..4 -*

4" "4 .4

"4 44'

0.4444,, .-.144' 1I�.jo� -

44' 44'

.4 � i*

p..

4., 4., 44'

3*. .44, 3.44*4

�** .444., 4.' -4

4., .40

a4- �"I 4-1 344 2.44 4-.

.. .. .4)

S � 4 I� �. � a- ...-. iha * *� - 1.4.4 41 .41.... n £3.4 4; � * �'44a �L. '� �

44�4�JJ . 4

- . '� 'S .40I�'4 -4 -4

* WI'I 0 ' 444-4

-4

. --- '.4 n'� .4.4 4.4 .� )

1I'4 44.4 1044.

'4) 4" 0.4

44' 0.44H 40 .4, .4

'£3 -4 '£3

'£3 '£3 '£3

£3 .0

4:4 42--4 '-4

�444:

"4 '4, I..

44' *.�4) 4'J.4' .4�4 '4

I.. 44)-4 4 �

.4. 44. dM0

28 4.

Table 3.2. S-number profile for TTRAntelope Dry Lake site.a

*

Depth, ftFrom To S-number

0 27 5

27 35 2.5

35 50 5.5

50 80 7

80 85 3

85 109 8

109 170 11.5

S170CM 13.5

a From Reference 14.

29 r-i~r- rcy4~ ioly

I

AXIALDISTANCE

FROMNOSE TIP

STATION INCHES SIGNIFICANCE

B 3.80 NOSE BASE

CG 8.11 CENTER OF GRAVITY

F 8.20 FORWARD ACCELEROMETER

R 17.30 REAR ACCELEROMETER

S 9.60 STRAIN GAGE

NOT[E FLARE UEGINS AT .64IN AMNDENDS Al 13.04 INTOTAL LENGTH L 2 I18.I5IN

CG 8

R 3 F

I I I .... I

20 IS 1O 5 0

DISTANCE FROM NOSE TIP, INCHES

Figure 3.1 Scale drawing of RBT penetrator.

4

30'

i ,

........... T4 A &L DATA

TIST IO DAT------ lYv)O.T4

4v

I .-IS

) ,,o~ .t*,•......ctta,,,-r"%\

-444

Figure 3.2 Comparison of calculated andmeasured lateral accelera-tions for Sandia/DNA RBT,3-degree attack angle.

4*

- • a ALCUL At~cO, re *41 . v&fS ........... 111t I co•r.STEST *OA0*

-4.

ai

S14071: *IlAA , tCg l bitT ¢11 8K1IS•I

Q. a a 4 a's .O ,,ýv,•l %

Figure 3.3 Comparison of calculated andmeasured axial decelerationsfor Sandia/DNA RBT, 3-degreeattack angle.

31

mesure stais o Snda

DN - BT 3I-tS5AA atac

anle

Figure 3.54 Comparison of calculated and

mesiedstans for Sandia/DART20DNA r obli3qegreittac

32le

....... *TIST 4 GAT&

* aA

J

*Aai A 6 9

4I I

Fiur 3. oprsoefolua dadmn

0ue strin for GA tlilM IN* .

obliquit oliqity

33

-r,,, TCAL..uaAt'lY C'OVU?)

91016I1SNPWACC PARAMaTER

J0

0 0.2 04 06 @0610 0uIS s

Figure 3.8 Calculated lateral force for different valuesof free-surface parameter,* Sandia/DNA RBT,20-degree obliquity.

PRIER-34,111ACE PARAMETER i

-- 4

-12

0 0.8 0.4 0.0 0.6 1.0 1.2

Time, MS0

Figure 3.9 Calculated pitch moment for different values offree-surface parameter, Sandia/DNA RET, 20-degree obliquity.

334

SWAKE SIEPARATION ANGLE t. QEG

S0". ..... .

-A -- 4

I--0 0

$ U.

o.9 04, o.6 0 10, 1

-rime, M 3

Figure 3.10 Calculated lateral force for different valuesof wake separation angle, Sandia/DNA RBT,20-degree obliquity. ,

10,- WAKE siEPARATiON ANGLE 01"• . 01EG

0

.... o..o.o... 244

8

j3:

U

'--

&

- ,o . .. . .. I, , I

0 0.2 04 0. O'a ,.aTIME. MSFigure 3.11 Calculated pitch moment for different values of

wake separation angle, Sandia/DNA RBT, 20-degree obliquity.

35

*Sow S O L

- 0 .M~

............ 2

a4

ItI

WAKE SEPARATION ANGLE •m O EO

J _- 4

1-S

10

-4 0 L I I . . .. , . . . . .

a a 6 as -. 0

'Time. MS

4 I,

Figure 3.12 Calculated lateral force for different valuesof wake separation angle for Sandia/DNA RBT,3-degree attack angle.

Io WAKE SEPARATION ANGLE DEG

0............... 2

Iz /P " .,

"0 / " ........

I.:

0

-0

i -,oi .. . i

o 0.2 0.4 0.0 0.0 1.0 1.

TIME, MS

Figure 3.13 Calculated pitch moment for different values ofwake separation angle for Sandia/DNA RBT, 3-degree attack angle.

36

c~k

aJ

INITIAL OBLIQUITY ANGLE O,, OEG

' 210

-, I - 4 0Il

04

U

U.

4 0 ......

NO0

0 Oa 04 Os O I0 I2

TIME, MS

Figure 3.14 Calculated lateral force for different valuesof obliquity, Sandia/DNA RBT.

INITIAL, ATACIK ANGLE do . DIG

0

. ... ......

- IQ-oI I1I

T-, II NI S

Figure 3.15 Calculated lateral force for differ-ent values of attack angle, Sandia/DNA RBT.

37

*0 - INITIAL oSLIQUITY ANGLE 9Q .

-......--......

J A

I-

S..D

S~NOTE: •G 0

-go I I II , I

0 0.2 0.4 0.6 0.6 1.o 1.2

TIME, MS

Figure 3.16 Calculated pitch moment for different values ofobliquity, Sandia/DNA RBT.

ooI hINITIAL ATTACK ANGLE a%. OEG

....... I3

7

...... ... ..........i-,

20

U

IL

NOTE: @o0 ao

-io I I I

0 0.2 0.14 0.0 0.3 5.0 1.2

TIME, MS

Figure 3.17 Calculated pitch moment for different values ofattack angle, Sandia/DNA RBT.

38

OMN -on

I Figure 3.18 Variation of maximum

calculated lateral I"rA ATAC AfU Co agload with attack N,-,* AT,*. ANGLO • , 56

angle and obliquity, •Sandia/DNA RBT.

0 I , , I , I

so 50 30 40 so

INITIAL OULIQU.IY ANGLO 0, 0140

; ' INITIAL ArAI 'oL c.N Figure 3.19 Variation of maximum

01

Icalculatedpic

pitch

moment with attackr angle and obliquity,Sandia/DNA RBT.

SI

INITIAL •0 A

39

,,,,I T• ,L .N ig r .9 V r a l n o a i u

TARGET STRENGTH, PSI

...... - 6000

- 6,000

0IL

0

* ~ %.........

a0* 04 0.e ofV1.Time. MS

Figure 3.20 Calculated lateral force for different valuesof target strength, Sandia/DNA RBT, 3-degreeattack angle.

t0

TARGIET STRENGTH, PSI

.................. 2.000

-I $ -¶ .0w0

30

L

-o00 1.2 0 14 0 16 06 ¶a0I 1i

Figure 3.21 Calculated pitch moment for different values of,target strength, Sandia/DNA RBT, 3-degreeattack angle.

4o~hOJ

6s .

ill........ .... L.ow

Figure 3.22 Calculated lateralforce distribution I

for different valuesof target strength, \ IISandia/DNA RBT, 3-degree attack angle.

Mork: IYP 'tD CTLAO .5O

AISTANCEM•.NOO TA Most TY I

ACTUAL TPAJECTOW1

.CALCU..ATtO TRAJItCTO0M&II 6

I Figure 3.23 Effect of wakeseparation"angle *minon calculated-I

S Itrajectories,I e-."... Sandia Test

SI 4 R800915.SI

4 0

14S. , I I ..

NOSYRION*IlL. P•OSITlON, Ir'

k4l

Pm 01"I/lt oTl-V, "WI¢TN 0WI

!-.

ACTUJAL T"JgCT*M- CA.OJLAIUO TRAJECTOMICS. 0.

00 " I "T ' *, I

*me a. ae a a."6O416ftYAL POSITION. IT

Figure 3.24 Effect of wake separationangle Omin on calculatedtrajectories, Sandia Test

o800916.

LEGEND

ACTUAL TRAJECTORYCALCULATED TRAJECTORIES, 0m 0

O 0 =8 II

IL.

go I I I I

-20 0 20 40 60 60 100 Izo 140

HORIZON'TAL POSITION, Fr

Figure 3.25 Effect of free-sutrface parameter k on calculatedtrajectories, Sandia Test R454025-23. • £

4.2

oN,

:ta

CALCULATED OgTRAETORN, k 0

'4-0 * 0 too ISO SO 94 00

HONSIOI.IAL POSITIoN, r?

~~Figure 3.26 Effect of vake separationanl

~~~trajectorieCs, Sandia Test 402W OnCluae

CAL5 LA025 *Q4r022.

143

.a .-.. 4 J

* Cam

048406-M 6 So I

so v._o n

S a

As SO SO

- - 1p I•

""O" '0P6YAU POSIT,*", .?

Figure 3.28 Comparison of calculated tra-jectories for Sandia pene-trators, *mi 1 degree.

.e SO SO

-... .. ,-,,' . ..o•

a

* as

U . , P.,LL, .-

Figure 3.29 Comparison of calculated tra-

jectories for Sandia pene-trators, i n 2 degrees.

.434

f

"" ,. v e .,,T

41

Figue 3 mar of c-O L V, .V , ¶S

Figure 3.30 Comiparison of calculated tra-jectories for Sandia pene-trators, n * 4 degrees.

min

-. , .. /o,'

- s/

* ~I

*0

Figure 3.31 Comparison of" calculated tra-[ jectories for Sandia pene-,, ~trators, *mi a S degrees.

•. I m5

.ICHAPTER 4

PARAMETER STUDY FOR SOIL PENETRATION

4.1 BACKGROUND

The PENCO2D computer code predicts trajectories that seem reasonable

in light of existing non-normal soil penetration data (Chapter 3),

although inconsistencies in the test results leave unresolved the ques-

tion of wake separation and its quantitative effect on stability. Un-

certainties notwithstanding, enough benchmarks do exist so that some

credence can be given to a study of the effects of varying the projectile

parameters, the impact conditions, and the target penetrability.

A baseline projectile, target, and set of impact conditions are

specified in this chapter, and calculations are presented in which indi-

vidual parameters have been varied one at a time. In some cases the

variation of a given quantity independent of other quantities may be un-

realistic in a practical sense (e.g., changing the projectile length

without affecting the moment of inertia). However, the objective in

such instances is to show the influence of a particular parameter in the

calculation itself. In a practical design-parameter study (Section 4.6),

coupled parameters (e.g., weight, length, and mass moment of inertia)

have to be varied together.

24.2 BASELINE CONDITIONS

The projectile chosen for this study is similar to the one used in

Sandia Test R454025-23 (L/D a 8) and represents a marginally stable

design, according to Sandia's experience. The pertinent projectile

quantities are

W a 300 lb

D 6 in

L a 48 in

CRH - 6.0

CG location CCG a 26.4 in from nose tip

246

14 2I u 14 X 10 lb-in

Aftbody taper angle *aft = 0

The baseline impact conditions are

V a 1500 ft/sec0

a 0 -4 degreeso

0e a degrees

The baseline target properties are

S 8

ýmin = 2 degrees

All of the above parameters will be varied except k and min whose

effects have already been demonstrated in Chapter 3. Aside from the

individual quantity or quantities being varied, the input will be the

same as the baseline conditions. 1

Figures 4.1, 4.2, and 4.3 show the trajectory, external forces, and

pitch moment, respectively, calculated for the baseline conditions.

Since the baseline calculation uses a constant wake separation angle

(fmin - 2 degrees), the penetration path is curved, rather than straight,

and there is some residual oscillation in the pitch moment as aftbody

contact alternately increases and decreases.

U. 3 VARIATION OF TARGET PENETRABILITYAND IMPACT CONDITIONS

Figure 4.4 indicates the degree to which the target S-number in-

fluences the trajectory. The main effect seems to be an increase in

the total path length with increasing S . Although the horizontal dis-

placement increases with S , there seems to be little variation in the

The baseline calculation will appear in all comparisons of trajectories

except Figure 4.7, and will be represented by a solid curve (e.g.,Figure 4.1).

47

°.t

overall path shape. The same thing can be said for the impact velocity

(Figure 4.5).

The initial attack angle has a mild effect on the trajectory (Fig-

ure 4.6), aside from a marked increase in path curvature and horizontal

displacement between 0 and 2 degrees. The 0-degree case goes straight

in because there is no angle of attack (or obliquity) to initiate rota-

tion. Large obliquities (a° U 0, e0 z 60 degrees) can produce ricochet

(Figure 4.7), although lesser obliquities seem to have a gentle effect

on path curvature.2

4 .4 INDEPENDENT VARIATION OF PROJECTILEPARAMETERS FOR STRAIGHT AFTBODY

Figures 4.8, 4.9, and 4.10 show the effects of varying the weight,

diameter, and nose shape, respectively. Treated as independent param-

eters, none of these variables has much effect other than changing the

path length. The horizontal displacement increases with path length,

but the shape of the trajectory changes only slightly.

It might be argued that the mass moment of inertia alone should

have a big influence on the trajectory. This is not the case, however,

as can be seen in Figure 4.11. Although the mass moment of inertia has

a moderately stabilizing effect, it is overshadowed by aftbody separation,

which allows the projectile to rotate with little opposition during most

of the event (see Figure 4.3).

Figure 4.12 shows the consequences of treating the projectile length

as an independent parameter, keeping the geometric position of the CG

constant (CCG/L - 0.55). A reduction in projectile length has a stabi-

lizing effect, due to reductions in the moment arm and the total (lateral)

loaded area on either side of the CG. This is the least realistic of

all the foregoing trajectory comparisons because a physical change in

length changes all other parameters (Section 4.6).

The most important single parameter is the geometric CG location

The initial obliquity required for ricochet would be smaller if a

larger value has been chosen for ýmin

48

I J~eI

CCG/L , as is indicated in Figure 4.13. If the CG is far enough forward

(CcG/L = 0.45), the negative moment (after CG entry into the target)

outweighs the initial positive moment, and a rotation reversal occurs.

If the CG is far enough aft ({cG/L = 0.65), the initial positive moment

is so great that the projectile ends up going sideways.

4.5 VARIATION OF AFTBODY FLARE

Figure 4.14 shows trajectories calculated for different values of

the aftbody flare angle *aft * In each calculation the aftbody merges

with the ogive nose section at the point of tangency, where n =aft

An illustration for *aft = 3 degrees is shown in Figure 4.15.

If *aft = *min , there is no loss of contact with the target, andthe penetrator goes almost straight in. On the other hand, if

0 < aft < ,min ' the trajectory is straighter than the baseline case

(straight aftbody, *aft= 0) but it is still noticeably curved. Flare

angles greater than Omin reduce the final depth, due to increased base

diameter (decreased W/A).

From this comparison, the optimum flare angle appears to be a value

less than the wake separation angle, producing a slightly curved trajec-

tory but leaving the depth essentially unaffected. The flare must be

installed without shifting the CG toward the tail.

4.6 VARIATION OF COUPLED PROJECTILE PARAMETERS

The projectile parameters that were varied independently in Sec-

tion 4.4 are coupled to one another in practice. As far as stability is

concerned, the most important coupling occurs among weight, length, CG

location, and mass moment of inertia. A serious parameter study should

consider the simultaneous variation of these quantities.

A simple but fairly realistic way of accounting for the change in

W , CG ' and I with L is to add or subtract cylindrical aftbody

sections of constant linear density3 from the baseline projectile, while

Constant weight/unit length = 300 lb/48 in = 6.25 lb/in.

49

asauming I W ? Figure 4.16 shows trajectories for projectiles

whose properties were varied in this way.

According to the calculations, if the projectile is short enough

(L/D < 5), the CG wll lie in an unstable position, ultimately producing

sideways motion. Otherwise, the effect of the changing parameters is

little more than an increase in path length with projectile length.

141

See also Figure 4.13.

50

S~ '~ "

/

-ai.

I0

a.0

V, - Iw o"11S

541E

to..

IS L.O •

|00

140 I I I I-so 0 0 to 44 40 IGO in0

4II00IZONTAL POSITION, PT

Figure 4.1 Calculated trajectory forbace ine conditions.

I I iS0 40 t0 IO SoO 140 I0O

TieI, Me

Figure 4.2 Calculated external forces for baselineconditions.

51

i

I-

-00 fOI I % -. q I-I

-- !

Figure 4.4 Effect of sol pntnbli

P:L~~~uo calCi culated p~c trnjecYt ot* tien. t

5P.. . A A

to

J " \

14 . .. L ..... J . . L I .. l I"40 0 00 4S4 44 00 00 00

Figure },Ji Effect of soil penetrabilityon calculated traJectorien.

5;,

____ ____ __ /

/k

II

MIAC? 'lLOCIl'Y V' P11S

-- I000SO-

40

J

too N,

10%

I* *o

-40 .I 1L.

-10 0 *6 dO 00 *0 loG lao

MOU•OA014AL POSITION, FT

Figure 4.5 Effect of impact velocity oncalculated trajectories.

INITIAL A'GL1 OF AtA< 1 O.EO

-4

o~~~~~~o ... a• t •.---

000

-10 0 *0 40 0 s too IS *OIAOPZONTAL. 002I'ON. PT

Figure 4.6 Effect of attack angle oncalculated trajectories.

53

INITIAL. AN .4[ 09 O .1LIQIJTv 1, 010

"o i \ ...\ . .. ...... ..-(0

z'o

a so

14

wN

ROTLE %* O

-to

•40 I I I,

Figure 24.7 Effect of obliquity on calcu-lated trajectories.

PUOJE[CTILIr OfIN IItl

0"

som0

I-*

"'-S

.50 • io 40 a0 so ice 81o

MMI01ZONtTAL. "OSIIONI, Vf

Figure 4.8 Effect of projectile weighton calculated trajectories.

*5)4

~!

,~5 , - .. . .. .. .. t,\e . '..

* tor• ~ ~PNOJV[CILE 0ODME1UR[ 0. I N

*10 • 6.0

40100

SI O

U4,SV

i

S•~40 I I I I I10 0 so 40 80 s0 too IS0

""ONRIZONTAL POSITION, PT

Figure 4,.9 Effect of projectile diameteron calculated trajectories.

IOJrECTILI NOSI SI4APZ. CM4

...... .. ... 1.2s

--- 2.20

6.0

00

040

o-

ISO

140-10 00 o0 00 I&0

MORPIZONTAL PO•ITION, PT

Figure 4.10 Effect of projectile noseshape on calculatedtrajectories.

55

" 8 0 • 4 ' to '

40

Figure 4.21 Effect of projectile massmoment Of inertia on cal- . ......culated trajectories.

GL l dr = 149 III • 141. , O. C "C . "[ M ra *'40

It

9..

~\ /

'40

- '3.I, * , A . 0 O * g l e, 0 'Fig-are .12 Effetect o f P roject ile ngth

eter) oWen ccuiatea octe.trajectories.

56

~~O~ g ?I L * 0'It I. C E W U~ M k ~ /l

I II III I III lllll lll 14 I 'L a i 0,i iiiilli1 II

01$YANCE[ FRM0, NOSETIP TO CO Ce., I,'

21.A 0.a524.0 0.30

-a. .8 0DA031.2 0.5*

FO PROJECTILE GOING 3SOEWAYS

i i "-,£40IL

a tES.j

100

Itol

-Ito 0 LO O so so too IIO,45ORIZONTAL POSITION, FT

Figure 4.13 Effect of CG location on"calculated trajectories.

APTSOOVY FLAtE ANGLE #,t . DIE

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

-o -;

<&

IL

0.5J

4,

* 100

Izo NOTE: * 2 DEG

"140 I I I I I-- o o 20 Ao so So 00 fo

HORIZONTAL POsITION, F'T1

Figure 4.14 Effect of aftbody flare on

calculated trajectories.

0-

57

-~ IF-

0 I FT0 -A

Figure 4.15 Scale drawing of projectile with 3-

degree aftbody flare

!. +1 I 'D •e I , LlU.S41

• i 4ll l 0.71 0.i O .10

23 • ,9 9 0 1.6. * 104

-in • 0 0.56 4.0 104

- 37 0 % ,I

-Il

40

.009

I I I I -J

144-o0 0 a t 66 4 0 .0 0

PIUI ONL0S,?I"OV. P,

Figure 4.16 Combined effects of weight,

length, CG location, and

mass moment of inertia on

calculated trajectories.

58

.>I

CHAPTER 5

CONCLUSIONS

An empirical theory has been developed for analyzing impact and

penetration in soil and rock. Calculations with this theory reproduce

penetration data for normal impact at least as well as other empirical

analyses (e.g., Young's equation). The extension from one- to two-

dimensional motion was obtained by extrapolation, with simple models

postulated for the free-surface and wake separation effects. The entire

non-normal analysis was then incorporated into the PENCO2D computer code.

Benchmarks for the theory consist of strain and acceleration data from

RBT's in sandstone, and trajectory data from conventional penetration

tests into soil.1Projectile acceleration and strain predictions, generated from WES-

calculated external loads, agree fairly well with the RBT data. There is,

however, a tendency to underpredict the lateral loads for yawed impact

somewhat. There is also an unexplained phase difference between the cal-

culated and observed lateral accelerations for both yawed and non-yawed

oblique impact. On the other hand, the calculated variations of maximum

lateral force and pitch moment with initial obliquity and attack angle

are quite reasonable (essentially linear). In any case, the recommended

values for the wake separation angle and free-surface parameter, respec-

tively, are #rin = 0 and k - 8 for rock and rocklike materials. 2

Comparisons of calculated and observed penetrator trajectories in

soil are inconclusive, since the test data are not consistent. Concern-

ing terradynamic stability, however, the most important target parameter

is the wake separation angle,3 whereas the most important projectile

WHAMS code (dynamic structural response) calculations made by2 T. Belytschko, with PENC02D calculations as input.

These values gave reasonable external load predictions for the RBT's.At this point, it is impossible to specify the most likely values ofthe free-surface parameter and the wake separation angle. A reasonableguess for the respective ranges of values, however, is 0 < k < 3and 0 < min <8 degrees

59

parameter is the CG location. Aftbody flare also has a beneficial

effect, if the CG is not shifted toward the tail and the base diameter

is not increased enough to significantly reduce the final depth.

One phenomenon not predicted by the present theory is the marked

improvement in projectile stability achieved by going from LID a 8 to

L/D a 10 , which was observed in tests conducted by Sandia Laboratories

(Reference 14). WES calculations indicate a gradual improvement in

stability with projectile length (see Figure 4.16). If LID a 8 does

represent a critical value for projectile stability, this would indicate

that some basic mechanism is missing in the theory, either in the wake

separation analysis or in the stress equations.

One mechanism conspicuously absent in the present wake separation

analysis is the resistance of the target to the (cylindrical) expansion

of the wake cavity, which would lead ultimately to the rebound/collapse

of the cavity on the aftbody. The probable variation of min with

velocity, depth, nose shape, and target penetrability (S-number) is also

neglected. Finally, the stress equations may be underpredicting the

effects of lateral motion. Future work should consider these problems

and any others that might bear on the stability question.

The latter conclusion was also drawn from the preliminary analysis(Reference 8).

60

l I ~i l

//

REFERENCES

1. C. W. Young; "The Development of Empirical Equations for Pre-dicting Depth of an Earth-Penetrating Projectile"; Development ReportNo. SC-DR-67-60, May 1967; Sandia Laboratories, Albuquerque, N. M.

2. C. W. Young; "Depth Prediction for Earth-Penetrating Projec-tiles"; Journal, Soil Mechanics and Foundations Division, AmericanSociety of Civil Engineers, Vol 95, No. SM3, May 1969; pp 803-817.

3. C. W. Young; "Empirical Equations for Predicting PenetrationPerformance in Layered Earth Materials for Complex Penetrator Configura-tions"; Development Report No. SC-DR-72-0523, Dec 1972; Sandia Labora-tories, Albuquerque, N. M.

Z4. R. S. Bernard; "Depth and Motion Prediction for Earth Penetra-tors"; Technical Report S-78-4, Jun 1978; U. S. Army Engineer WaterwaysExperiment Station, CE, Vicksburg, Miss.

5. R. S. Bernard and D. C. Creighton; "Non-Normal Impact andPenetration: Analysis for Hard Targets and Small Angles of Attack";Technical Report S-78-14, Sep 1978; U. S. Army Engineer WaterwaysExperiment Station, CE, Vicksburg, Miss.

6. Y. M. Ito and others; "Analysis of 0.284-Scale RBT for 30Yawed Impact"; Informal Report, Contract DNA 001-76-C-0383, Oct 1978;California Research and Technology, Inc., Woodland Hills, Calif.

7. D. Henderson and E. J. Giara, Jr.; "Earth' Penetrator TechnologyProgram"; Draft Report DNA 4571, Jan 1978; Avco Systems Division,Wilmington, Mass.

8. R. S. Bernard; "Earth-Penetrator Trajectories in Soil: Prelimi-nary Analysis and Parameter Study"; Letter Report to Defense NuclearAgency, Oct 1978; U. S. Army Engineer Waterways Experiment Station, CE,Vicksburg, Miss.

9. C. W. Livingston and F. L. Smith; "Bomb Penetration Project";Apr 1951; Colorado School of Mines and Research Foundation, Inc.,Golden, Colo.

10. W. R. Kampfe; "DNA 1/3-Scale Pershing Penetrator Tests";Letter Report to Defense Nuclear Agency dated Oct 1977; Sandia Labora-tories, Albuquerque, N. M.

11. D. C. Creighton; "Correlation of Reverse Ballistic Test Data";Letter Report to Defense Nuclear Agency, Dec 1977; U. S. Army EngineerWaterways Experiment Station, CE, Vicksburg, Miss.

12. D. C. Creighton; "Correlation of Additional Reverse BallisticTest Data"; Letter Report to Defense Nuclear Agency, Aug 1978; U. S.Army Engineer Waterways Experiment Station, CE, Vicksburg, Miss.

13. C. W. Young; "Results of December 1977 Davis Gun Tests at TTR";Memorandum dated 13 Jan 1978; Sandia Laboratories, Albuquerque, N. M.

61

.714. C. W. Young; "Status Report on High Velocity Penetration

Program"; Report No. SAND 76-0291, Sep 1976; Sandia Laboratories,Albuquerque, N. M.

15. R. S. Bernard and S. V. Hanagud; "Development of a ProjectilePenetration Theory, Report 1"; Technical Report S-75-9, Jun 1975; U. S.Army Engineer Waterways Experiment Station, CE, Vicksburg, Miss.

16. R. S. Bernard; "Development of a Projectile Penetration Theory,Report 2"; Technical Report S-75-9, Feb 1976; U. S. Army Engineer Water-ways Experiment Station, CE, Vicksburg, Miss.

17. R. S. Bernard and D. C. Creighton; "Projectile Penetration inEarth Materials: Theory and Computer Analysis"; Technical ReportS-76-13, Nov 1976; U. S. Army Engineer Waterways Experiment Station, CE,Vicksburg, Miss.

18. J. A. Canfield and I. G. Clator; "Development of a Scaling Lawand Techniques to Investigate Penetration in Concrete"; Technical Report2057, Aug 1966; U. S. Naval Weapons Laboratory, Dahlgren, Va.

19. W. J. Patterson, Sandia Laboratories, Albuquerque, N. M.,Untitled Letter Report to: R. S. Bernard, Soil Dynamics Division, U. S.Army Engineer Waterways Experiment Station, CE, Vicksburg, Miss.;26 Aug 1976.

20. D. U. Deere; "Technical Description of Rock Cores for Engineer-ing Purposes"; Rock Mechanics and Engineering Geology, 1964, Vol. 1,No. 1, pp 16-22; International Society of Rock Mechanics, Springer-Verlag, N. Y.

21. D. J. Dunn; "Bomb Penetration of Earth-Covered ConcreteTargets"; Proceedings of the Seminar on the Attack of Earth, Stone, andConcrete Barriers by HE Projectiles (15-16 May 1974), Part 1, ReportBRL-R-1872, Apr 1976; U. S. Ballistic Research Laboratories, AberdeenProving Ground, Md.

22. R. S. Bernard; "'Empirical Analysis of Projectile Penetrationin Rock"; Miscellaneous Paper S-77-16, Nov 1977; U. S. Army EngineerWaterways Experiment Station, CE, Vicksburg, Miss.

23. W. J. Patterson; "Projectile Penetration of In Situ Pick";Report No. SLA-73-9831, Nov 1973; Sandia Laboratories, Albuquerque, N. M.

24. S. W. Butters and others; "Field, Laboratory and ModelingStudies on Mount Helen Welded Tuff for Earth Penetrator Test Evaluation";Report No. TR 75-9, Aug 1976; Terra Tek, Inc., Salt Lake City, Utah.

25. D. K. Butler and others; "Constitutive Property Investigationsin Support of Full-Scale Penetration Tests in Dakota Sandstone, SanYsidro, New Mexico"; Technical Report S-77-3, Apr 1977; U. S. ArmyEngineer Waterways Experiment Station, CE, Vicksburg, Miss.

26. W. J. Patterson; "Physical Properties and Classification ofSeven Types of Rock Targets"; Report No. SC-TM-68-621, Sep 1968; SandiaLaboratories, Albuquerque, N. M.

62

S..... ............ . . .. G O

27. C. W. Young and G. M. Ozanne; "Compilation of Low VelocityPenetration Data"; Report No. SC-RR-306A, Jun 1966; Sandia Laboratories,Albuquerque, N. M.

63

k'

APPENDIX A

PENETRATION THEORY FOR ROCK AND CONCRETE

A.1 EMPIRICAL THEORY FOR NORMAL IMPACT

Early penetration analyses at WES (References 15 through 17) relied

heavily on the Cavity Expansion Theory (CET), whereby the coefficients

a and b and the function f(V) in the resisting stress

a = a + b f(V) (A.1)

could be obtained explicitly in terms of the density, strength, and

elastic moduli of the target. Examination of penetration data from

several sources, however, has failed to show a consistent correlation

between penetrability and standard engineering properties, especially

for soil. Although the most consistently accurate predictions have

been obtained for concrete, additional factors such as aggregate size

may overshadow elastic effects.

In the long run, a "fundamental" theory like the CET requires

empirical modification in order to be of general use as a predictive

tool. An empirical approach from the outset seems better, thus keeping

the functional relations among the parameters as simple as possible.

For concrete and rock, a stress equation of the form

a = a + bV (A.2)

does seem adequate for correlating penetration data obtained with a

given projectile/target combination. In general, however, the coeffi-

cients a and b will vary with the characteristics of the projectile

and the target. Based on penetration tests in concrete (Reference 18)

and intact rock (Reference 19), the following empirical formula is pro-

posed for the stress distribution on the projectile1

All dimensional quantities must be expressed in compatible units;

e.g., slugs, feet, and seconds.

65_

Z4OT

OY + 3VV-p) wrain(.3

Relating Equation A,3 to A.2, it in clear that a a 6.328 Y ioiITand that b a 14.7i46 /pY -in n . When Equation A.3 in subntituted into

Equationto 02.4, and '.8, the equation of motion (Equation P.10, 0 - 0)

for a fully embedded projectile become-i

N 4Y +~ W4v7) (A.4~)re"

where D In the maximum diameter and N IIl the none performance

coefficient In rock and concrete, given for ogives by

r, i il . ]I/i,

N o.863 LRil1 (A.5)

#%nt| for coneri by

N -re ,0.-- (A. ()

with ni repreiet.titig the cone hl f-nnmgle nnd ('Ilii reprenent. ig the

rmdiua of curvitture of the ta'ngent ogive in enlib ers (multiplen of P).

1isrepgnrding the changing croton seetion during thi nolte-emhetment

proeen, the, "i al-depth soluth Ion to EqwIit ion A.) i In

iiN Mi Iv ~ ri- Nmax * ."� " ,1 - In 1 + V (A.')

whrre V ino iio Im1, t'. velovity.0

Tt aiddIt, It e to the parame•f•ern iven In II ' Fu'.i ,n A.7, thrert I1" nt.

Scan:t, on" mere, par lte,.,,r t~hat. lf~l izn't '�e prti trabi ]i ty. Iin "'ok t hi :

(l•ut.it.y In the Nlook uin lit.y P•en I 'tioh (03,11)), t.ro th',d by POvert,

(leferelnc e 20). "hi, 8Ii , an I :iderx for t.he Ierrrv- or in stltm fr'itot.m-itin

in a given site, is obtained using a modified core-logging procedure:

All solid pieces of core that are 4 inches long orlonger are added up, and this length is called the mod-ified core recovery. The modified core recovery is di-vided by the total length of core run, and the quotientmultiplied by 100 percent is the value of the RQD.

In concrete, the maximum size of coarse aggregate D apparently has aagmild effect on penetrability (Reference 21), via the ratio Dag/D .

Letting Y represent the intact unconfined strength for rock and

concrete, the effective strengths for penetration can be calculated from

. Y" O (A.8)

( ) Y " 2 (A.9)

Thus, to calculate penetration in less-than-intact rock (RQD < 100), the

quantity Yr should replace Y in Equations A.3, A.4, A.7, and 2.12.

Likewise, when the maximum aggregate size is specified for a given

concrete, Y should replace Y in the same equations. 2

cFigure A.1 shows a comparison of Equation A.7 with nondimension-

alized penetration data for rock and concrete. The concrete data were

obtained for a case in which D/Dag = 3 ; i.e., Yc = Y . Primarily,

the figure shows the degree to which Equation A.7 is able to collapse

the data to a single curve. There is some residual scatter, but this

correlation represents an improvement over previous attempts (Referencesand 22).3' TabTle A.1 presents the data in dimensional form, whereas

Table A.2 gives the target properties and projectile parameters.

A.2 EXTRAPOLATION TO NON-NORMAL IMPACT

Equation A.3 was obtained by starting with Equation A.2 and

In structural concrete, Dag usually varies from 0.75 to 1.5 inches.

3 The RQD is always assumed to be 100 for concrete.The improvement comes from the addition of a nose-shape effect and thereduction of the RQD effect initially used in References 4 and 22.

67

, ...i

/,I

adjusting coefficients until a good data fit was achieved for combined

rock and concrete penetration data (Figure A.1) taken from References

18, 19, and 23. Previous experience with these materials had shown

that a and b should be proportional to Y and 4P? , respectively;

the Tan-n-proportionality (nose-shape effect) was drawn from Young's

observations of soil penetration and was presumed to hold true for

harder targets. With the addition of the RQD effect on the apparent

strength, the correlation shown in Figure A.1 was achieved.

The form of Equation A.3 is such that it can readily be extrapo-

lated to the case of nonax~al motion. Recognizing that for axial

motion

Vsin n - (Aa.1)

V

where Vn = component of V normal to projectile surface, it is reason-

able to replace V and V , respectively, by the local velocity vfl5and its normal component vn . Thus, for 2- or 3-dimensional motion,

the expression for the compressive normal stress on the projectile

surface is

o = 1.582 (4Y + 3v÷ /) 1 J1 (A.11)

where

v a absolute local velocity relative to fixed target

vm = compopent of. v normal- te projectIll' surfac "

In order for a stress to act on a given projectile surface element,

there must be contact with the target (Section 2.4) and net motion into

the target (v > 0). Otherwise, it is presumed that a 0n

Additional material property information is given in References 24-26.For purely axial motion, V and Vn are identical with v and vnrespectively. In extrapolating to a local definition of stress fornonaxial motion, the most direct route is to use the local velocity.

68

I /

Table A.1. Penetration data for rock and concrete.

Impact PenetrationVelocity Depth

Test Source Target Material fps in

Canfield and Clator Concrete 1005 8.0Canfield and Clator Concrete 1025 9.0Canfield and Clator Concrete 1250 10.0Canfield and Clator Concrete 1485 14.5Canfield and Clator Concrete 1775 16.5Canfield and Clator Concrete 1975 23.5Canfield and Clator Concrete 2020 19.5Canfield and Clator Concrete 2325 26.0Canfield and Clator Concrete 2350 24.oCanfield and Clator Concrete 2430 27.5Canfield and Clator Concrete 2535 29.0Canfield and Clator Concrete 2655 29.5

Sandia/DNA Welded Tuff 1220 87Sandia/DNA Welded Tuff 1350 102Sandia/DNA Welded Tuff 1560 142Sandia/DNA Welded Tuff 16hO 132Sandia/DNA .. . .... Welded Tuff . 1650 132

Sandia/DNA Sandstone 1455 140

Sandia/DNA Sandstone 1505 146

Sandia No. 120-77 Welded Agglomerate 1065 156

Sandia No. 120-112 Sandstone 824 122

Sandia No. 120-103 Sandstone 880 1 2 0 a

Sandia No. 120-106 Granite 860 150

a In this test the sandstone was covered by 30 inches of soil, so the

total depth of penetration is 150 inches. However, the penetrationresistance of the soil is negligible compared with that of the rock.

69

-- ".. e

/ 3 S ~ GP

C)~Q : CCJ CU 0 -

022

.0

UN ~ U-\ H 4

ft00O0 43 04

t t-- O H 0 ý4 0\ t- ~4-02 4.0

34-413

qu4 4.) Hi

4) -) 4- 02 43 0.)0C.) 4.) *t1 +)

U.4 W m tC-

44~ u iuH~ 434)4

~~~~~i UU*0 cj c~ ~ ') H0

434* 4 0 0~t 0 '\J 0 0 Hl\ 3*

4-.) 43 9 . . . .0

m EH H 00\0 0 c C

04.1 4 )H

000

+- 43

HH 3

0 4 41) 4413 ý

A oCO 00 0 c4.Q4)

HM Hq 1 02(1

00

04- 4) ))

"Ol So

07

ri)

I

LEGEND

SYMBOL TARGET SOURCEOF DATA

0 CONCRETE CANFIELD & CLATORA WELDED TUFF SANDIA/DNA TESTS

o1 SANDSTONE SANOIA/ONA TESTSA WELDED AGGLOMERATE SANDIA TEST NO. 120-77* SANDSTONE SANDIA TEST NO. 120-112* SANDSTONE SANDIA TEST NO. 120- 103V GRANITE SANDIA TEST NO. 120-106

z0o.8 0

0 I

0 2 4. 6 6

V(p/Y)0 5s (100/RQD)

0 `

Figure A.1 Nondimensionalized penetration data for rockand concrete.

71

low

APPENDIX B

PENETRATION THEORY FOR SOIL

B.1 EMPIRICAL THEORY FOR NORMAL IMPACT

To date, Young's equation (References 1-3) has been the most con-

sistently successful formula for calculating penetration depths in soil.

Rather than using standard engineering properties to describe the

target, Young lumps the penetrability into a single empirical variable,

represented by S . In U. S. customary units,1 Young's equation is

0.0031 S N VVA (V - OO) , Vo > 200 ft/s

o.53 3 Ns V_ In (l + 100,000)' V0 < 200 ft/s

where

Z = final depth, ftmax

W = projectile weight, lb

A = 1 = maximum cross-sectional area, sq in

V = impact velocity, ft/s

N = nose performance coefficient in soil, dimensionlesss

S = soil penetrability index, dimensionless

The S-number can be obtained directly from previous penetration

data for a given target, or it can be estimated from data for geologi-

cally bimllar targets (Table B.I). "Whatever the case, Young's equation

represents a scaling relation between the projectile characteristics

(size, weight, geometry), the impact velocity, and the maximum depth of

penetration.

Since Young's equation gives only the final depth, it cannot be

used to calculate the instantaneous resisting force, only the average

resisting force. This is not a drawback for normal impact, since the

1 Young's equation is not dimensionally homogeneous. All quantities

must be specified in the units given.

73

*n uC=llM piACM M.&zK NOT FflMsDi

Swg

I

axial resisting force is often a step-pulse (i.e., uniform, except for

some oscillation) in a given layer. On the other hand, for non-normal

impact, both the axial and lateral forces must be known if the 2- and

3-D motion is to be calculated. Since Young's equation gives no indica-

tion of the lateral forces, a more detailed analysis is needed.

In Reference 4 a projectile equation of motion was developed for

normal impact and penetration in soil, based on data originally used

by Young. Postulating that the axial force should (a) have the form of

a step-pulse and (b) exhibit linear dependence on velocity, it was

"found that an equation of the form

dV (-M- a + bV + cZ (B.1)

would satisfy both these criteria, provided that

b (B.2)

The explicit appearance of the projectile mass in the velocity coeffi-

cient b was puzzling but absolutely necessary to generate a step-

pulse resisting force. Moreover, it brought the mass dependence of the

maximum depth into agreement with Young. Physically, however, the mass

should occur only on the left side of Equation B.1, with parameters such

as diameter, nose shape, and S-number appearing in the coefficients a

b , and c .

The mass dependence of the coefficient b is now thought to be a

size effect in disguise. This is reasonable since many of the larger

projectiles used by Young had masses roughly proportional to D3 2

Con3equently, an alternative to Equation B.2 is

2 The deep earth penetrators, built and tested by Sandia Laboratories,

had L/D ratios of about 10 to maintain stability. For many of theseprojectiles the weight (in pounds) can be related to the maximumdiameter (in inches) by

3A

W = 1.3 D + 30%

7 4

6esZ7 ,~' FO\ ~P /

b IýD' a(B.3)

In either case, the approximate solution for the final depth Z ismax

a [-(a + bvy + bV v)2 + McV 2 (B.4)

and the dependence of a and c upon S and D is the same as in

Reference 4:

Da- (B.5)

D2c-D2 (B.6)

s 2

which insures that for low-velocity, shallow penetration,

z S (Rp7)max D

in accordance with Young's equation. When the nose performance coeffi-

cient N is added, the coefficients a , b , and c can be expressed5

in terms of three "universal" constants y , B y

a D -(B.8)S .N . • . I . . ..

b 'G(B.9)

SN

2c S 2 Ns2 (B.10)

With these expressions for a , b and c , it follows from Equa-

tion B.4 that

75

AIp

I

Zma -- (B.u1)D"

where

1 1n < 2.5 (B.12)

and Young's nose performance coefficient in soil is approximated by

N o0.632 L (CRx 'lJ ogives (B.13)4 CRH - 11

"N , cones (B.114)f sin nc

The values of the constants u , 8 , and y have been obtained.

The ratio uf O/y was adjusted until step-pulse deceleration curves

were achieved. The values of p and y were then determined by

fitting Equation B.4 to shallow penetration data (Reference 27) for

projectiles with the same weight and diameter3 in the Main Lake area of

Tonopah Test Range (TTR). The values for the three constants are

a 1.96 x lo5 lb/ft

B a 6.25 x 106 lb 2 s 2 /ft7

y 1.56 x 105 lb/ft 3

Figure B.1 compares Equation B.4 with normalized 5 deep- and

4 W 224 pounds, D = 4.4 inchesEquations B.1, B.4, B.8-B.l0, B.15, and B.16 are dimensionallyhomogeneous. Any set of dimensionally compatible units may be usedtherein, as long as all quantities are converted to the same systemof units.Normalized using Equation B.4. Amount of scatter in normalized dataindicates degree of validity as scaling relation.

76 GOO

'A -

shallow-penetration data for TTR Main Lake. The same data are given in

original form in Reference 2, where Young gives the S-number profile as

S a 5.2 from 0 to 8 feet and S = 2.5 from 8 to 25 feet. Equation B.4

* agrees best with the shallow data using S = 4.3 and with the deep

data using S - 3

Equation B.4 appears to do about as well as Young's equation in

extrapolating and interpolating soil penetration data, but as a formula

for depth prediction, it offers no advantage over the latter. On the

other hand, the underlying equation of motion (Equation B.1) contains

* some additional information concerning the explicit relation between

the axial resisting force and the velocity, depth, diameter, and

geometry of the projectile.

The analysis of nonaxial motion requires that the stress distribu-

tion be specified on the projectile. In the present context, any stress

distribution is potentially correct if it generates Equation B.4 with

the values of a , b , and c given by Equations B.8-B.IO. The

simplest distribution that accomplishes this, giving approximately the

right dependence of Ns on projectile geometry, is a compressive normal

stress defined as follows:

0.625 + 5 sin + 5 sin n (B.15)S r p S2

where

rp = local cylindrical radius of projectile

, local angle belweeJ projectile surface and axis of symmetry

Z = vertical depth from target surface to point on projectile

surface (Figure 2.7)

B.2 EXTRAPOLATION TO MON-NORMAL IMPACT

Nov that the stress distribution is defined by Equation B.15, the

extrapolation procedure for soil is the same as for rock and concrete

(Appendix A). Thus, for 2- or 3-D motion, the function sin n is re-

placed by v/v, and V by v ,so thatn

77

I

" rp + 59 (B.16)

As in Appendix A, it is assumed that for a stress to act on a given

projectile surface element there must be contact with the target (Sec-

tion 2.4) and net motion into the target (vn > 0), otherwise a - 0

78 •,..,.-,•a 0 9.1

__L .. i ....

Table B.1. Typical S-numbers for natural earth materials.

S Materials

1-2 Frozen silt or clay, saturated, very hard. Rock, weathered,low strength, fractured. Sea or freshwater ice more than10 feet thick.

2-3 Massive 'gypsite deposits.b Well-cemented coarse sand andgravel. Caliche, dry. Frozen, moist silt or clay.

4-6 Sea or freshwater ice from 1 to 3 feet thick. Medium dense,medium to coarse sand, no cementation, wet or dry. Hard,dry, dense silt or clay.c Desert alluvium.

8-12 Very loose, fine sand, excluding topsoil. Moist, stiff clayor silt, medium dense, less than about 50 percent sand.

10-15 Moist topsoil, loose, with some clay or silt. Moist, mediumstiff clay, medium dense, with some sand.

20-30 Loose, moist topsoil with humus material, mostly sand andsilt. Moist to wet clay, soft, low shear strength.

40-50 Very loose, dry, sandy topsoil (Eglin AFB). Saturated, verysoft clay and silts, with very low shear strengths andhigh plasticity.d Wet laterltic clays.

a Taken from Reference 3.b White Sands Missile Range, New Mexico.c Tonopah Test Range, Nevada, drn lake playss.

Great Salt Lake Desert and bay mud at Skaggs Island.

79

, ,

3.1

3* /

//30 ~

It /

/ •/

a a /

• mm .,mm r•C~ ON .4.02d l

//1 0/. ,

• a" ,00 000/

NI _PA' CT' 'f/

FIr&gur B.1 Cc:lupwrion of CILtlevited penetration depths vith data fO

M Main L~ke area.

80 GOO~OALZET

-jc

nnn• • "6

i * ar ce ~l Il ac 100 zoo''ll[Ir'"'

APPENDIX C

NOTATION

a, b, c Empirical force coefficients

A Maximum cross-sectional area of projectile

CRH Tangent-ogive radius of curvature, in calibers ("caliberradius head")

dA Differential element of surface area on projectile

dFx, d z Differential force components in the body-fixed lateral andaxial (x- and z-) directions, respectively

D Maximum projectile diameter

D Maximum coarse aggregate sizeag

f(V) Function of velocity

F', F z Net force components in the body-fixed lateral and axial(x- and z-) directions, respectively

g Gravitational acceleration (32.2 ft/s )

I Transverse mass moment of inertia about CG

k Free-surface parameter, rs/rp

L Projectile length

M Projectile mass, W/g

n Exponent

N Nose performance coefficient in rock and concreterc

N Nose performance coefficient in soil

r c Cylindrical radius of wake cavity

r° 0 Cylindrical radius of projectile at separation point

r Local cylindrical radius on projectile surface at a givenP point

r s Maximum perpendicular distance from projectile axis,through dA , to target surface, for which stress reliefcan occur due to free-surface effect

81

RQD Rock quality designation

S Soil penetrability index

t Time1

v Absolute local velocity of any point on projectile surface

v Outward normal component of local velocityn

Vx' vz Local velocity components in body-fixed lateral and axial(x- and z-) directions, respectively

V Projectile CG velocity

Vn Outward normal component of CG velocity

V Impact velocity

V V Components of CG velocity in body-fixed lateral and axialx, Z (x- and z-) directions, respectively

W Projectile weight

x Lateral (body-fixed) position in projectile

X Horizontal position in target

X, Z Components of acceleration in the X- and Z-directions,respectively

Y Unconfined compressive strength of intact rock or concrete

Y Concrete strength, corrected for aggregate sizeC

Y Rock strength, corrected for RQDr

z Axial (body-fixed) position in projectile, measured forwardfrom CG

Z Vertical position (depth) in target, measured downwardfrom target surface

Z Vertical position (depth) of CG, measured downward fromcc target surface

Z Maximum (final) depth of penetration of nose tip

max

A dot over any quantity indicates differentiation with respect to

time.

82 NJ\a'GOO'

a Angle of attack

mo Initial angle of attack

0 Constant, 6.25 x 106 lb 2 -s2 /ftT

Y Constant, 1.56 x lo5 lb/ft 3

a Lateral displacement of point on projectile axis, relativeto wake cavity

a Initial displacement of point on projectile axis0

C Axial distance aft of the nose tip

Co Axial distance from nose tip to separation point

CG Axial distance from nose tip to CG

Local angle between projectile axis and line tangent toprojeci.ile surface

"c Cone half-angle

o Obliquity angle, measured counterclockwise from target-fixedZ-axis to projectile-fixed z-axis

e Angular velocity

o Angular acceleration

0 Initial obliquity angleo

P Constant, 1.96 x 105 lb/ft

p Mass density of rock or concrete

a Local compressive normal stress on projectile surface

ax a z Components of local compressive normal stress in body-fixedlateral and axial (x- and z-) directions, respectively

Local angle of approach between projectile surface elementand target, based on CG velocity components

Oaft Aftbody flare angle

#min Angle of approach at which separation occurs (wake separa-tion angle)

Azimuthal angle on projectile

83

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In accordance with letter from DAEN-RDC, DAEN-ASI dated22 July 1977, Subject: Facsimile Catalog Cards forLaboratory Technical Publications, a facsimile catalogcard in Library of Congress MARC format is reproducedbelow.

Bernard, Robert SProjectile penetration in soil and rock: analysis for non-

normal impact / by Robert S. Bernard, Daniel C. Creighton.Vicksburg, Miss. : U. S. Waterways Experiment Station ; Spring-field, Va. : available from National Technical InformationService, 1979.

88 p. : ill. ; 27 cm. (Technical report - U. S. Army Eagi-near Waterways Experiment Station ; SL-79-15)

Prepared for Defense Nuclear Agency, Washington, D. C.,under DNA Subtask Y9OQAXSB211, Work Unit 20.

References: p. 61-63.

1. Ballistics. 2. Computer analysis. 3. Impact. 4. Numericalanalysis. 5. Penetration. 6. Projectiles. 7. Rock penetra-tion. 8. Soil penetration. I. Creighton, Daniel C., jointauthor. II. United States. Defense Nuclear Agency. III. Series:United States. Waterways Experiment Station, Vicksburg, Miss.Technical report , SL-79-15.TA7.W34 no.SL-79-15

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