AMRA TR 66-07

ANALYSIS OF EDGE NOTCHESIN A SEMI-INFINITE REGION

TECHNICAL REPORT

by

OSCAR L. BOWIE

CU L E II 0H USFOR FEDERAL SCIENTIFIC A.ND

N. TECHNICAL INFORMATION ( 1fi ara'dop Hiorof Laho ep

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MATERIALS ENGINEERING DIVISION

U. S. ARMY MATERIALS RESEARCH AGENCYWATERTOWN, MASSACHUSETTS 02172

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ANALYSIS OF EDGE NOTCHES IN A SEMI-INFINITE REGION

f

Technical Report AMRA TR 66-07

by

Oscar L. Bowie

June 1966

D/A Project 1P014501B33AAMCMS Code 5011.11.858

Research in Mechanics

Subtask 36912

Distribution of this document is unlimited

MATERIALS ENGINEERING DIVISIONU. S. ARMY MATERIALS RESEARCH AGENCY

WATERTOWN, MASSACHUSETTS 02172

U. S. ARMY MATERIALS RESEARCH AGENCY

ANALYSIS OF EDGE NOTC•ES IN A SEMI-INFINITE REGION

ABSTRACT

A procedure for solving the plane elastic problem for edge notches in asemi-infinite rcgion is presented. A modification of the conventionalMuskhelishvili technique is made by utilizing a mapping function whichdescribes both the physical region and its reflection with respect to thestraight edges. An effective power se~rles development in the parameter plane

is then described and illustrated by the numerical solution for semi-ellipticalnotches in a semi-infinite sheet in tension at infinity parallel to the edge.

•-q 1•,•"•-'• m ,•. Wz.•'K ••.•.•-"•-- -.. " --. - - - .. . .. .

CONTENTS

Page

ABSTRACT

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . I

INITIAL FORMULATION OF THE PROBLEM ................. 1

A POWER SERIES FORMULATION OF THE PROBLEM .......... ............. 3

DETERMINATION OF THE SERIES COEFFICIENTS.......... . . . . . . 4

SEMICIRCJLAR EDGE NOTCH ............ . . . . . . . . . . . .. 6

SEMI-ELLIPTIC NOTC• IN A SEMI-INFINITE SHEET . . . . . . . . . . .. 9

ACKNOWLEDGENT ............ .......................... ... 12

LITERAIURE CITED . . . . . . . . ................... 13

- •mm.----'- - - -

INTRODUCTION

Edge notches in a semi-infinite region identify a familiar problem classin plane elasticity. The difficulty of finding a fepresentation of the solu-tion which allows a systematic consideration of boundary conditions on boththe straight line and notch sections of the geometry is a challenging matter.Several particular geometries have been successfully analyzed by a variety ofrather ingenious methods. Perhaps the most familiar is the original Maunsell 1

solution for the problem of a semicircular notch in a semi-infinite plate intension parallel to the edge. Later, Ling 2 discovered an integral represen-tation for this problem as well as a class of problems corresponding tocircular notches or mounds and straight boundaries. Edge cracks have been suc-cessfully studied by several investigators, e.g., References 3 and 4. Estimatesof the maximum stress for edge notches have been made by approximating argumentsby Neuber.

5

For general notch shapes a more uniform approach would be desirable. Onecould, for example, consider the mapping technique of Muskhelishvili 6 as suchan approach. However, several practical difficulties arise in the applicationof this technique to this problem class. Frequently the exact mapping functioncontains branch point singularities necessary for the description of the junc-tions of the notch and the straight line boundaries. Effective use of theexact mapping function in such cases is limited, as the problem ceases to heone of linear relationship. The alternative of polynomial approximation of theexact mapping function can be used, but frequently a suitably accurate poly-nomial representation is awkward to find. A good example of this difficultyis shown in the recent work of Mitchell 7 in his solution for a semicircularnotch.

In this paper a power series technique is presented which appears to pro-vide an effective and fairly uniform approach to single edge notches in a semi-infinite region. Initially, Muskhelishvili's reflection argument is used totranslate the problem into one of finding a suitable analytic stress functiondefined exterior to the closed region bounded by the notch and its reflectionwith respect to the real axis. A departure from the conventional mappingapproach is then made by introducing a mapping function defined on the unitcircle and its exterior to describe both the given physical region and its re-flection with respect to the real axis. A power series representation in theparameter plane is a natural consequence of this formulation. Furthermore, inmany practical problems, the explicit occurrence of the branch point singular-ities in the mapping function referred to above is eliminated. Finally, thestructure of the solution can be predicted in a fairly systematic manner byutilizing most of the consequences of the unmodified Muskhelishvili theory.

INITIAL FORMULATION OF THE PROBLEM

It will be assumed that the material occupies the notched lower half-planeS- defined in the z-plane, Figure 1. The boundary of the notch C- will beassumed to be an arbitrary continuous curve, and the straight edges are assumedto lie on the real axis, y-0.

-,ll~l•t,,ifl. .. . . ..... t,,• ~ ll lli li ll ~ i, . -. -- t. - -.,-

/ /

Figure I. SINGLE EDGE NOTCH IN A SEMI-INFINITE REGION

In the initial stage of the formulation, the reflection argument ofMuskhelishvili 6 will be utilized. The essential details of this argumentwill be briefly summarized in this paragraph. The stress components in carte-sian coordinates can be expressed in terms of two analytic functions 0(z) andY(z) in the following manner:

0 + c0 - 2 [0(z) + 0(z)) (1)y x

0- Y + 2ir-'y . 2[10'(z) + O(z)], (2)

where primes denote differentiation and bars denote complex conjugates.

In the present problem, the region of definition of O(z) and Y(z) is S-.On the other hand, using the extension principle of Muskhelisihvili, the defi-nition of O(z) can be extended into S÷ by defining

0(z) - 0(z) - z'(z) . •(z) for ,,s÷, (3)

where

Z(z) f ' ). (4)

The stress function T(z) can now be expressed in terms of the extended defini-tion of 0(z) as

z(z) O(z) -(z) - zO'(z) for zeS'. (5)

2

For simplicity of presentation, self-equilibrating load systems acting onthe boundary of the notch will be considered at this time. Actually, this isa minor restriction since loading at infinity or along the straight edges canbe handled directly for the unnotched half-plane in terms.of elementary func-tions and Cauchy integrals, e.g., Reference 6. By considering the superpositionof solutions, it is clear that the problem difficulty rests in solving thecases of self-equilibrating load systems acting on the notch boundary. Forself-equilibrating load systems,

V(z) and '/(z) - 0(1/z 2 ) for large Izi. (6)

Thus, from (3) it is evident that -(z) is analytic in both S- and S+.

The character of 0(z) on the straight edges of the boundary is clear from

01 - irxy = 0(z) - 0(Z) + (z-i--)0'(z). (7)

If the straight sections along the real axis are considered as load-free, then,since z-- 0 on the real axis,

'(x) - 0+(x) = 0, (on the straight boundary). (8)

The notation 0-(x) and 0+(x) denotes the value of 0(z) as z-.x through S- andS', respectively. From (8) and the previous discussion, O(z) can now beconsidered as an analytic function in the region of the plane exterior to thecontour C=C-+ C+ where C+ is the reflection of C- with respect to the realaxis, Figure 1.

Finally, the boundary conditions on C- must be considered. For the pres-ent purpose, we assume that the normal and tangential stress components, Nand T, are prescribed on C-. Then, if a denotes the angle between the x-axisand the outward normal,

N - iT - 0(z) + 0(z) - e 2 ia[(C-z)V'(z) - C(z) - 0(z)] for zeC-. (9)

A POWER SERIES FORMULATION OF THE PROBLEM

For a. power series representation, it is obviously desirable to deal witha parametric region consisting of a circle and its exterior (or interior).Therefore, consider an auxiliary complex plane, the c-plane, such that theunit circle ý =o-=eie and its exterior are mapped into C and its exterior,respectively, by the analytic function

z - cc(•. (10)

A departure from the conventional mapping approach is made here since both thephysical region and its reflected image are described by the auxiliary plane.The stress functions 0(z) and I(z) can be considered as analytic functions ofthe parameter 7 since z and ý are related by the analytic function (10). Thenecessity of introducing considerable new notation can be avoided by

3

designating 0(z)=4[w( )] as o( , etc., which leads to such relationships as0'( z) =0')/'( ) , etc. Due to the symmetry of C with respect to the x-axis,-(z)j--.) in the notation described.

The boundary condition (9) can now be expressed as a function of Cr. Dueto the symmetry of C, the range 0_8<77 will be considered as corresponding toC+ and the range 7r.0<.27Tr, corresponding to C-. The applied load can be con-sidered as a function of oa, i.e.,

N - iT g(o,), ._<••2i. (11)

Thus,

0(0o) + YTM - e2ia{[_COFOT - 60(0-)]t (oicv() -(1 =(o - *(oi

7<0< 277 (12)

where

e 2ia= a r2a()/j(T). (13)

On the unit circle itself, it is possible to encounter singularities in0(ý) either by load distribution or as a result of geometric irregularitiesin the notch shape. For severe types of singularities, e.g., poles occurringat crack roots, logarithmic singularities corresponding to point loads, etc.,it is generally possible to anticipate and introduce the proper structure asa separate part of the representation. The remaining arguments will, there-fore, be confined to the cases of nonsevere boundary singularities, i.e.,

O0

ct A C ", > (14)=2 n ~~

with at least conditional convergence assumed on the unit circle r.

DETERMINATION OF THE SERIES COEFFICIENTS

The analyticity of 0(ý) permits the expression of the coefficients interms of the well-known integrals

2niA~ = f$1i'•)d•/•n, n - 2, 3, ... (15)

or

27r

2TAn .r 0'- n (o-)d6, n = 2, 3, .(16)

0

in addition,277

o 1 0'no()de, n - 1, 0, -1, -2, ... , (17)0

4

but if the solution exists in the hypothesized form, the conditions (17) can

be considered as redundant.

If the function F(o') is defined as

F(o) = O(U) -0(a) = 0, O5<7<

= OM e 2 [77g- . CO-) - OW] - CO-)

+[w(o-) . ca4o-)] $'(o-/w(o-) ,<e<2,,, (18)

then the boundary conditions of the present problem are equivalent to

F(o-) = 0, 0<0<2Y. (19)

The Fourier criteria for the vanishing of F(or) is thus

Zwr 21

f O1'F(O-)dO f o-nF(o-)do 0, n = 0, ±1, ±2, ... (20)0 IT

If we denote by Fj,

27Fj= f o-JF(No)d, j = 0, ±1, ±2, ... (21)

IT

it is evident that computationally (16) and (17) are equivalent to

F. 0, j - 2, 3, 4, ... (22)

F1 =0, j = 1, 0, -1, -2, ... (23)

respectively. For an infinite system, the redundancy of (17) implies theredundancy of (23).

A reasonable plan of solution conceptually is to introduce the series

(14) into the set of conditions (22). This leads to an infinite set of linear

equations in the unknowns An, which can be solved by successive truncations

of the system. Early numerical solutions indicated, however, that the rate

of convergence of this truncation procedure was too slow to be practical. The

difficulty shows up numerically in a lack of dominance of the principal diag-

onal of the coefficient matrix.

Conceptually, the difficulty can be traced to the series truncation.

Let OA(C) be defined as a truncation of (14),

M+1

h = W A C'n. (24)n=2

In general, the corresponding function FA(cr) is a polynomial (or a rational

function) of o, involving both positive and negative powers of cr and (19)

cannot be identically satisfied by a choice of the M available coefficientsAn. Furthermore, the Fourier conditions (23) which cannot be considered asredundant for a truncated stress function are ignored in the plan describedabove.

A combination of two procedures was found to be the most effective com-putational plan. For the leading coefficients, the system

F. = 0, j = 0, ±1, ±2, ... , ±P (25)

was solved for several values of P with M = 2P + 1. A rather dramatic improve-ment in convergence of the leading coefficients was -found by this plan. Inthe illustrative problems, the first twenty coefficients were found with re-liable accuracy with values of P<20. On the other hand, a critical systemsize is reached where the higher ordered coefficients begin a rapid growth ininaccuracy and the determinant of the coefficient matrix becomes very small.This can be anticipated due to the redundancy in the limit of P + 2 of theequations in (25).

The procedure used to calculate the higher order coefficients consistedof solving the set of conditions

-f {F(o)} 2 d9 - 0, i - 2, 3, ... , M + 1. (26)• 0

This set of conditions corresponds to the requirement that (19) be satisfiedfor a finite set of coefficients in the least square sense. The system (26)eliminates the previous stability difficulties for larger systems and can becompactly expressed as a suitable linear combination of the elements F..Although (26) could be used to determine the entire set of coefficients, theleading coefficients converge somewhat more slowly than in (25), and the max-imum system size is reduced by using a combination of the two procedures.

SEMICIRCULAR EDGE NOTCH

The simplest illustration of the preceding analysis is provided byMaunsell's problem, i.e., a semicircular notch in a semi-infinite plate intension, T, parallel to the edge, Figure 2. The radius of the semicircle willbe chosen as the unit length, and polar coordinates (r,e) are introduced in themanner shown. Fcr illustrative purposes, the mapping notation will be retained,thus,

z = cv(•) = c. (27)

The stress functions will be denoted by 0i(1) and 'V(C) where

o(•) = T/4 + t(C) =T/4 + T 7- A '-nn= 2 n

=I(C) -T/2 + v(•). (28)

6

S+

/\

Figure 2. SEMICIRCULAR EDGE NOTCH

Since the particular solution in (28) corresponds to or, T everywhere, it isclear that the determination of 0(t) and W(•) is a problem type with theproperties assumed in the previous analysis. Furthermore, since a - 77 +

g(o) = -T/2 (1+o- 2 ). (29)

From symmetry considerations, the series coefficients in (28) can be writtenin the form

A2k = B2k, k = 1, 2, 3,

A2 k.l= iC 2 k-1, k = 2, 3, 4, (30)

where B2 k and C2 k.1 are real.

Corresponding to (18),

F(0) = 0,7

= + C(O--) _+ a2[p(o) +

+ (T/2) (l+a2), -<0<(2-u. (31)

7

Thus,

OD

Fo 77B2 -8 Y C2 k.l/(2k-3)(2k+l) + /2k=2

F, 0

02

F2 = 8 k22 C2 k. /(2k-3)(2k-1) + 7/2

F2 j = 277(1-j)B 2 j + (2j-1)7r B2 j. 2

+8kZ 2 8 2k (2k-1)(2j-l)/[(2k-2j) 2 _1][2k-2j-3] j - 2, 3, (32)

00-iF2 j I =-( 2j-l)C2j - 1-27(j-l)C2 .3-16 2 B2k( 2J- 3 )(j-l)/[(2j.-3) 2 -4k 2] [2k-2j+l]k= 1

4(j-l)/(2j-1)(2j-3), j = 2, 3,

F_2j-= B2j+2-8 1 C2k.l( 2 j+l)(2k-l)/[(2k+2j) -1][2k-2j-3], j = 1, 2,k=2

"-iF'2j+l= -C2j+1 +16 2 B2kj(2j+l)/[(2k+2j) 2 "1] [2k-2j-1] -4j/(4j2-1),

k=1I = 1, 2,..

The conditions corresponding to (26) can be conveniently expressed interms of Fn. Since,

S•f{F()} 2 de = 2 " F(o-) ýF(-) do = 0, (33).1 7?r 7T

we find from the even and odd coefficients, respectively,

(2-2j)F.2j + (2j+l) F. 2 j. 2 + F 2 j. 2 = 0, j = 1, 2, ... , M

(2j-l)F. 21 - 21 F 1 21 + F 2 .3 0, j 2, 3, ... , M + 1. (34)

The first twenty coefficients were found to five decimal place accuracyby using (25) for several values of P < 20. The higher order coefficientswere then calculated by solving the system (34) for several values of M.Forty of the higher order coefficients were obtained to five decimal placeaccuracy with th6 use of systems in which M<50. The comparatively rapid decayof the coefficients is shown in Table I.

Due to some recent controversy as to the accuracy of Ling's2 numericalresults, e.g., Reference 7, the stress at the notch root, 0 = -7T/2, wascalculated. In terms of the present formulation,

8

Table 1. STRESS FUNCTION COEFFICIENIS FORSEMICIRCULAR EDGE NOTCH

k B2k C2k 41 k B2k C2k _ 1

1 .0.79248 -0.66283 11 -,.00 147 -0.00074

2 40.65206 +0.33908 12 -0.99110 -0.009170

3 -0.04406 +0.05755 13 -0.00084 -0.0006.5

4 -0.02325 +0.01765 14 -0.00065 -0. oo0.595 .0.01358 +0.00616 15 -0.00 51 .0.000536 -0.00850 +0.09 199 16 -0.00040 .0. (90047,

7 -0.00561 +0.A0032 17 .0.00032 -0.000428 .0.00385 -0.00138 18 -0.0C025 -0.00"389 -0.00272 -0.00066 19 -0.00020 -0.00034

[1 -0.00198 -0.00074 20 -0.00016 -0.00930

0 (0 -7-/2) = T (1-4 [ (C 2 k+l-B 2 (35)k= I2k)(-1

Using the coefficients in Table I, •X/'T = 3.0656 at the notch root. Allowingfor round-off in the present data, it would be reasonable to estimate thatx/T = 3.065 1 0.001 which can be compared with Ling's result, -X/T=3.065.

Estimates of the accuracy of the solution at the free corner, e =0, weremade from the data of Table I and additional higher order Bk coefficientsnot recorded here. Excellent agreement was found which further indicates thatreliable peak stresses at notch roots can be obtained by this procedure, par-ticularly since series convergence would be expected to be more rapid at suchpoints.

SEMI-ELLIPTIC NOTCH IN A SEMI-INFINITE SHEET

A final illustration of the analysis is provided by a semi-elliptic notchin a semi-infinite sheet in tension at infinity parallel to the edge, Figure3. Only an approximation of this problem has been previously available.Again, the straight edges of the sheet are assumed to lie on the real axis,y a 0.

An appropriate fcrm of the mapping function is

z = w( ) = c-B" 1, 0<VB<1. (36)

the ratio of the axes of the ellipse, X u alb, is clcarl.,

,X ( l-B)/( 1.+B). (37)

For illustrative purpo.fes• the analysis was carried out in terms of thestress function O(c) and ot,(.4. where

9(38)

S÷

S \C+

t

T

/ /

Figure 3. SEMI-ILLIPTIC EDGE NOTCH

The preceding arguments carry through in a straightforward manner and theboundary condition can be written as

-cx) €(o- = '(ai •(o-) + W(o-)'- -F •

-aw'(o) Ico(o) gCoi)dc, rt<0<2w. (39)-1

If €1(0) denotes the required stress function, then

-= (T/4)(•-B") + *(•) (40)

where

11+1= T2 an'+. (41)n=2 n

The pievious argument was applied to the function wo'(Oi€(O). Again fromsymmetry,

a2k = 4k, k = 1, 2,.

a 2k-1 = ic2k-I, k = 2, 3, (42)

10

Since

g(crý -(T/2)[D+ o-2 '()/w(1] , (43)

it follows that in the boundary condition (39),

f 1o'((o)g-) d T 7 -(T/2)( I +B)(o- -'o . (44)-1

Thu s,

2~7r---f. = If {oW'(0-) -wa)(0-) + €(0"o [a(o1) -

7 T

-w (o-) (T/2)(I +B)(o-o')}do, j 0, ±I, ±2, ... (45)

The least square conditions similar to (26) can then be written as

[2 +B-2(1+ B)j] fl-2j [2j(l +B) - 1 f-2j-1

" f2j- - Bf 2 j- 3 - j, = 1, 2,

[2(l+B)j - (l+2B)] f 2 2 j - [2(1+B)j - (2+3B)] f-2j

"- f2j-2 -Bf 2 j- 4 = 0, j = 2, 3, ... (46)

The numerical analysis was then carried out for several values of B inthe manner previously outlined. For brevity, only the peak notch stresses forvarious values of X =a/b wil! be presented. In terms of the present formulation,

0rmax = Orx( =i)= T{I+ 4 Y [(1- 2 k)b 2k+2kc 2 k+ I] (_1)k/(I.B)}. (47)k=1

Comparison with the conventional approximation to this problem was made bycalculating

Q = 0-max/O-*max, (48)

where ar* is the corresponding peak stress for an elliptical hole in an infi-nite sheet with a corresponding uniaxial tension at infinity. The lattersolution is well known, e.g., Beference 8, and

o* = T(1 +2X-1) (49)

The results are shown in Table II where the limiting value of Q for K = 0 isbased on Koiter's result. 4 The results listed can be considered accurate towithin 0.1 of one percent. In the calculation of the coefficients on adigital computer, it was found advisable to use double precision.

11

S. . . . : , _---

Table II. MAXIMUM STRESSES FOR SEMI-ELLIPTIC NOTCHES

B \ max /T max/T Q

0.0 1.0000 3.065 3.000 1.0220.1 0.8182 3.540 3.444 1.0280.2 0.6667 4.136 4.000 1.0340.3 0.5385 4.910 4.714 1.0420.4 0.4286 5.948 5.667 1.0500.5 0.3333 7.412 7.000 1.059

0.6 0.2500 9.625 9.000 1.0690.7 0. 165 13.320 12.333 1.0800.8 0.1111 By interpolation - 1.092

0.9 0.0526 By interpolation - 1.106

1.0 0.0000 Koiter's result - 1.1215

ACKNOWLEDGMENT

The author is indebted to D. M. Neal who programmed the several phasesof the analysis.

12

LITERATURE CITED

I. MAUNSELL, F. G. Philosophical Magazine, v. 21, 1936, p. 765.

2. LING, C. B. Journal of Mathematics and Physics, v. 26, 1947.

3. BUECKNER, H. F. Some Stress Singularities and Their Computation by Meansof Integral Equations. Boundary Problems in Differential Equations.University of Wisconsin Press, Madison, Wisconsin, 1960.

4. KOITEB, W. T. On the Flexural Rigidity of a Beam Weakened by TransverseSaw-Cuts. Proc. Kon. Ned. Ak. Wet. Amsterdam, v. B, 59, 1956; also,Journal of Applied Mechanics, March 1965.

5. NEUBER, H. Kerbspannungslehre. J. Springer, Berlin, Germany, 1937.

6. MUSKHELISHVILI, N. I. Some Basic Problems of the Mathematical Theo-cyof Elasticity. P. Noordhoff, Limited, Groningen, Holland, 1953.

7. MITCHELL, L. H. Stress Concentration at Semicircular Notch. Journal ofApplied Mechanics, v. 32, Series E, December 1965, p. 938-939.

8. TIMOSHENKO, S., and WOODIER, J. N. Theory of Elasticity. McGraw-Hill,New York, 1951.

13

I INT.A.S T FT FD

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3. REPORT TITLE

ANALYSIS OF EDGE NOTCHES IN A SE'4I-INFTNITE REGION

4. DESCRIPTIVE NOTES (Type ol report and inclu.ive dates)

5 AUTHOR(S) (last name. first name, initial)

Bowie, Oscar L.

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13 ABSTRACT

A procedure for solving the plane elastic problem for edge notches in asemi-infinite region is presented. A modification of the conventionalMluskhelishvili technique is made by utilizing a mapping function which describesboth the physical region and its reflection with respect to the straight edges.An effective power series development in the parameter plane is then describedand illustrated by the numerical solution for semi-elliptical notches in a semi-infinite sheet in tension at infinity parallel to the edge. (Author)

DD JAN 15, J473 UNCLASSIFIED

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3. REPORT TITLE: Enter the complete report title in all (4) "U. S. military agencies may obtain copies of thiscapital letters. Titles in all cases should be unclassified, report directly from DDC. Other qualified usersIf a meaningful title cannot be selected without classifica- shall request throughtion, show title classification in all capitals in parenthesis hrimmediately following the title. •"

4. DESCRIPTIVE NOTES: If appropriate, enter the type of (5) "All distribution of this report is controlled. Qual-report, e.g., interim, progress, summary, annual, or final. ified DDC users shall request throughGive the inclusive dates when a specific reporting period iscovered.

If the report has been furnished to the Office of Technical5. AUTHOR(S): Enter ihe name(s) of author(s) as shown on Services, Department of Commerce, for sale to the public, indi-or in the report. Enter last name, first name, middle initial. cate this fact and enter the price, if known,If military, show rank and branch of service. The name ofthe principal author is an absolute minimum requirement. 11. SUPPLEMENTARY NOTES: Use for additional explana-

6. REPORT DATE Enter the date of the report as day, tory notes.

month, year; or month, year. If more than one date appears 12. SPONSORING MILITARY ACTIVITY: Enter the name ofon the report, use date of publication, the departmental project office or laboratory sponsoring (pay-

7a. TOTAL NUMBER OF PAGES: The total page count ing for) the research and development. Include address.

should follow normal pagination procedures, i.e., enter the 13. ABSTRACT: Enter an abstract giving a brief and factualnumber of pages containing information, summary of the document indicative of the rc.port, even though

it may also appear elsewhere in the body of the technical re-7). NUMBER OF REFERENCE&: Enter the total number of port. If additional space is required, a continuation sheetreferences cited in the report. shall be attached.

8a. CONTRACT OR GRANT NUMBER: If appropriate, enter It is highly desirable that the abstract of classified re-the applicable number of the contract or grant under which ports be unclassified. Each paragraph of the abstract shallthe report was written. end with an indication of the military security classification

8'b, 8c, & 8d. PROJECT NUMBER: Enter the appropriate of the information in the paragraph, represented as (TS): (S),military department identification, such as project number, (C), or (U).subproject number, system numbers, task number, etc. There is no limitation on the length of the abstract. How-

9a. ORIGINATORWS REPOR{T NUMBER(S): Enter the offi- ever, the suggested lenpth is from 150 to 225 words.cial report number by which the document will be identified 14. KEY WORDS: Key words are tech:ically meaningful termsand controlled by the originating activity. This number must or short phrases that characterize a report and may be used asbe unique to this report. index entries for cataloging the report. Keý words must be

9b. OTHER REPORT NUMBER(S): If the report has been selected so that no security classification is required. Iden-

assigned any other report numbers (either by the originator fiera, such as equipment model designation, trade name, -nili-

or by the sponsor), also enter .his number(s). tary project code name, geographic location. may be used askov words but will b.- followed bv an indication of technical

SonteXt. The as.igo•mnt of links, rules, and weights is

Soptional.

UNCLASSIFIED•-urtyClsifca tio n