4 . b . 3

THE MOMENT -MAGNIFIER METHOD APPLlED TO BRICK WALLS

CARL TUR KS T RA

JOSE OJINAG A

Department of Ci,)il Engineering and Applied Mechanics , UcGill University , !1ontreal , Canada

TllE MOMENT- MAGNIFIER METHOD APPLIED TO BRICK rvALLS

This paper summarises the results of a study of the

ultimate capacity of vertically loaded brick walls .

The study is based on the moment -magnifier method

which is used in steel and concrete column design in

North America . Extension to brick masonry design

would provide a consistent approach in limit states

design methods presently under development in Canada .

In the method, bending moment-axial force interaction

relationships for shor t walls are used . However , the

bending moments found from conventional structural

analysis are multiplied by a magnification factor de ­

pending on wall height, stiffness and end constraints .

In the paper the basic assumptions of the method ar e

presented briefly and relevant inf ormation on stress ­

strain characteristics and br ick wall behaviour are

r eviewed . The method is applied to a large collection

of published test data from Europe and North America

which has been set up in a computerised data bank .

Test data includes a variety of brick- mor tar combi ­

nations, wall slenderness and end conditions, and load

eccentr1:ci ties .

The limits of applicabil i t y of the method ar e di scussed

and conclusions drawn as to the advantages and di s­

advantages of the approach in wall design o

DIE MOMENTEN-VERGROESSERUNGSMETllODE

IN DER ANi.'ENDUNG AUF ZIEGEV1AUERWERK

Diese Untersuchung fasst die Ergebnisse einer Reihe

von Versuchen zusammen, die die Bruchlast von verti­

kal belastetem Mauerwerk bestüwnen sollte? Der Unter­

suchung liegt die Momenten- Vergrosserungsmethode zu­

grunde , wie sie auch für den Entwur f von Stahl - und

Betonsáulen in den USA angewandt wird .

Eine Uebertragung dieser Methade auf den Entwur f von

Ziegelmauerwer k WÜY'de zu einer konsequenten Annâne­

rung an die derzeitig in Kanada entwickelten Entwurf9-

methoden fúnr en . In dieser '4ethode wird das Zusammen ­

wirken von Biegemoment und Axialkraft bei kurzen

Mauerl<)erksteilen zugrunde gelegt . Jedoch we1'den die

aus üblichen Untersuchungen gefunderten und bekannten

Biegemomente mit einem Vergrosserungsfaktop multi­

pliziert , der von der 'Iandhohe , der Steifheit und

der Einspanmmg abhá·ngt .

In der vorliegenden Untersuchung werden die der

Methode zugrunde gelegten Annahmen erortert. Es

f olgt eine kurze Uebersicht úôer Charakterika der

Spannungsbeansp~Achung und úôer das Verhalten von

Mauerwerkskorpern . Die Methode bez ieht sich au f eine

grosse Anzahl ver offentlichter Versuche au s EUr opa

und den USA , die in e1:ner l<omputeY'gesteueY' tc<n Daten­

bank zusammengef asst wurden .

Die Versuchsdaten schliessen verschiedenartige Mortel ­

kombinationen , Schlankheiten, Einspannungen und aus­

mittige Lasten ein .

Die Gr enze der Anwendbarkeit dieser Methode wird dis ­

kutiert und schliesslich werden Folgerungen füp den

Entwurf von Mauerwepkskó'rpern gezogen .

4 . b . 3-0

LA METHODE DU " MOMENT - MULTIPLICATEUR "

APPLIQUEE A DES MURS EN BRIQUES

Cette communication résume les résultats d 'une re­

cherche de la capacité limite des murs en briques

chargés verticalement . L ' étude est basée sur la

méthode du " moment - multiplicateur " largement

utilisée en Amérique du Nord pour le calcul des co­

lonnes d 'acier et de béton . Une extension de cette

méthode à la maçonnerie serait une étape importante

dans la direction du calcul des valeurs limites qui

sont actuellement développées au Canada .

Dans cette méthode on part du rapport moment de

flexion - force axiale pour des murs courts . Les

moments de flexion trouvés par les méthodes de cal­

cul conventionnelles sont toutefois affectés d 'un

multiplicateur, basé sur la hauteur de paroi , la ri­

gidité et les tensions limites .

Dans cette communication les hypotheses de base de

la méthode sont brievement présentées ainsi que d ' im­

portantes constatations en liaison avec le rapport

tension - transformation et le comportement des murs

en briques .

La méthode est appliquée à un grand ensemble de ré­

sultats qui ont été traités par ordinateur et publiés

en Amérique et dans le Nord de l 'Europe . Les données

d 'essai comprennent une grande variété de combinaisons

de mortiers - briques, d 'élancements de murs et de

conditions limites , ainsi que d 'excentricités .

Les limites du champ d 'application de la méthode sont

discutées , de même que les avantages et inconvénients

pour le projet de maçonnerie .

DE METHODE VAN DE "MOMENT- VERMENI GVULDIGING "

TOEGEPAST OP BAKSTEENMUREN

Deze mededeling vat de resultaten samen van een

onderzoek naar de grenskapaciteit van vertikaal

belaste baksteenmuren . De studie is gebaseerd

op de 'moment-magnifici " methode welke in Noord­

Amerika veel gebruikt wordt voor de berekening

van staal en betonkolommen . Uitbreiding van deze

methode tot metselwerk ware een belangrijke stap

in de richting van de grenswaardenberekening die

thans ontwikkeld wordt in Ranada .

In deze methode wordt uitgegaan van de verhouding

buigmoment - axiale kracht bij korte muren. De

buigmomenten gevonden door konventionele reken­

methoden worden echter vermenigvuldigd met een

multiplikator, gebaseerd op wandhoogte , stijf­

heid en randspanningen .

In de mededeling worden de basishypothesen van de

methode kort voorgesteld, samen met de belangrijk­

ste vaststellingen in verband met de verhouding

spanning-vervorming en het gedrag van baksteenmu-

ren .

De methode wordt toegepast op een grote verzameling

van resultaten gepubliceerd in Amerika en Noord­

Europa die in een computer behandeld werden . De

proefgegevens omvatten een grote variatie van bak­

steenmortelkombinaties , muurslankheden en randvoor­

waarden alsmede excentriciteiten .

De grenzen van het toepassingsgebied van de methode

worden besproken alsmede de voor- en nadelen voor

het ontwerpen van metselwerk .

I[HROOUCTIO~J

To de velop m8thods of limit states design procedures for preoictlng average ultin~te load capacity. a know ­ladge of ths va riability of capacity about these aV3ratSS mUSL ~8 estaol ished . The objective Df thls paper is to sxamlne the LHe Df the momant magnifier a~proach i n tne special case of singl o wythe unre ­illforc8d brid, \·la 11s subjected to 8ccentric vertical end loads .

The design prob lem involved is the conventional one of relatin~ saction capac ity to loa ding conditions . end constraints, l'lull geomet r y. and the properties Df st andard reference specimens . However. the problem ia rela~iv9 1y difficult in orick wall design bar:ause of the ;,xis tence of two components with different ms chanical properties leading to complex strGss distributions and failure criteria [1,2) . For til e case considered . a numb5r Df simplified theories (3. 4 .5. 6 . 2) and emp iri ca l metilods (7) have been de­veL::ped .

PraGtical design methods should be relatively simple and consiste nt ~ ith princip18s of strL~tural mechanics so that s ituations beyond test conditions can be tro~cod. Df cünsiderabls interest is the possibility Df using tne morr.ent -magnifier approach suggested by Yoke l. Ma they and Dikkers [8 . 91 for concrete b lock rr. ~ 3'Jnry and b rick mesonry (10). Such an app roa ch would provi de cons istency with steel and reinforced concrete desii~ matho ds . Mo reover, it would provide a r3tion al basis fo r consideration of general wall end ~nnditions. l~ading . and pcssibly the effects Df reinfCl rcing .

This paper examines application Df the simplest posslble form Df tlle momen t-rrlagni fi el' appT'oach to­g2ther with re15ted wall properties. Attention is restr i cted to variations in wall capacity . The question Df safa ty factors to be used in design is not considered.

REFERENCE PROPERTIES

In the absence Df reliable models rela ting masonry prcperties to the properties nf bricKs and mDrtars . s mall piers or prisms hava be8n used as a standard measure Df masonry behaviour . The choice Df a stan­dard rsfer8nce lnvo l v8s considp.ration Df the effects Df test conditions together with t he easo of fabri­cation and is somewhat arbitrary . I n this st udy . the single wythe prism with a haight to thickness ratio Df 5 tested between flat rigid plates (11) was adopted as a practical reference . Al I wa ll capacities P per unit length are referred to the capacity P per unit length of s lJch prisms . o

Numerous tests (12 .13 . 14 . 15) have shown that the stress-strain chBracteri stics Df brick masonry is generally non - linear . Some tests sugges t that seg­ments Df a parabola may be used out the behaviour in the region Df high compressive stresses does not saem to have heen extensive ly investigated . Tensile stress capacity has not been well defined . In exarnination Df short section behaviour the three idei:llize d stress - stra i n diagrams Df Fig . (1) were used.

The initial tangent modul us E has a significant random variat ion . Exarr.inat ion Df 142 experimental results (16) innira~ed a very good correlation with brick com­prassive strengt h fb . The equation

E 145 . 000 + 220 .6 fb (psi) (1 )

had a correl ation Goefficient Df 0.941 and a standard error Df e sti ma~ e of 360 .000 psi . Waak lima mortars

4 . b . 3-1

have bean excluded .

SHORT SECTION CAPACITY

Fundamen tal to any study Df wall behaviour is the capacity Df a short sRctl on subj ec tod to an axial f orce F' and a bending moment M about the centroidal axis . For brick wal ls. several cases must be con­sidered dape nding on whether the section is in com­prassion through its thickness or axperiences ten­sile st r ain . In the latter case . bahaviour depends on ~hether or not the tensile capacity has been ex­ceeded .

For an assumed linear strain variation through the wall . resultant axial forces and bs nd ing moments can readily be calculated for any assumed stress­strajn charactgristics . Shovin in Fig . (2) are the axial fo rc e - banding moment relationships for the three ideal1zations Df Fig . (í) for cases corres­ponding to a ttainment Df the u ltimate rupture strain E. on the compression face . Results have been non­dimensionalized using the reference ca~acity Po an d the conventional linear kern bending moment Po t/12 . Also sh:Jwn are wall test results fro rn Ref. (7) wi t :, a slanderness ratio kl/r less tnan 40 for walls tested axially . witn one end flat . or in double curv­ature, and lass than 30 f or single curvatura son­ditions . Such walls can be expact~d to havB somewhat less than the short section capacity .

Comparison Df the analytical results show that non­linearity reduces the kern Bccentricity and inCI'eilSeS axial load capacity fo r a gil/en eccentri city . The experiment a l resul t s in dicate t ha r cólpaclty decreases less rapidly with eccentricities up to t/3 tnan even a linear-rectangul ar stress-strain diagra!l1 without tensile stI'ength would predict . A simi l ar "strain­gradient effect " was noted by Yakel et aI (8) fo r con­crete block masonry . Such effects may be oue to changes in transverse stress patterns under ncn-uni­form strain and suggest thet the a xially 10i:lded prism is an inadequate measure Df mechanical properties.

THE MOMENT - MAGNIFIER METHOD

As wa ll slenderness increases . late ral deflect j.ons be­come s i gn-Lfican t as axial forces act throü"h the de­flectians to modify the distribution Df bendirog rno­ments along the \~all heigM . As a r"s ul t . the l ocation and magnitude Df tha maximum moment is va1' iable . For exarnple. an eccentric load within the [,erro at the ends cm lead to eccontrici ties beyond tile I,ern at mid ­height . Details Df behaviour depend on the wall height 1. radius Df gyration r. end condHions. and the shape of the moment diagram found fro m elementary anal ysis.

The moment-magnifier method is a device for converting the bending moments alo"g the lsngth Df a beam-column to equi.val,mt short section bendir,g mO!l1ents . The section is then designe d using the shor~ 3ection in­teraction relationsh ips. the applied axial forces , and the larger Df the applied end moments or aqui valent "magnified " moment o Tha methoo was davelope d for steel sections and has been ad~pted to concret~ column de­sign (171.

For the load case under study . the equi valent moment M can be I-Iri tten in ter-ms Df the axial force and m~~~mum applied end eccentricity e in the form

max

M mag LC Pe max

(2)

tl . b . 3-2

TE.

(a)

15

10

5

-5

- 10

- 15

-2G

St ress Stre 5S

+-. E. Str ain I

I

St r ain

Li near ( b ) Paraboli c

FIG. 1 r DEALI ZED STRESS

E r ror

... ...

... ... X

O X

X

(% po )

... ...

... ...

X ...

... 50

...

O

...

X

X

X

1.0

0 .5

Stress

E. E. 2

Strain

(c) Linear Rectangle

STRAIN DIAGRAMS

Axia I Force

1.0 2 .0

FIG~ 2 SHORT SECTION INTERACTION DIAGRAMS

Effect ive Length Factor

X O

""

K = I K 0.7 K = 05

150 200 X Slenderness

Ratio = _k_1 r

FIG. 3 ERRORS FOR AX IALLY LOADED WALLS

3 .0

where

L 1

----- p-- (31 1 - p-

c,

c ( 41

p cr

( 51

ll18 fectur L ll1r: l uooc; the effect. s of r'lateri éll stiff­nass tllrough ths fIlodulus Df e lasticity t: , section goome try through Lhe moment Df i nertjoa I , the length x" emd en,J s'Jpoo,-t condi1: io n th l'ougn the effective lsngth filet!),- :'.. -;-he shape Df the e lfomentary í11Dme nt oiagralil l5 introouced b", the faetcr C. To dea l with nominally axiéllly loaded Cilses, a mllumum a r acci ­d",ltaI anel dccentricity G1l1s t be i ,-,troducB-J .

lhe n~thud was deve lope d fGr lineilr sl a stic sections ,;iLh c:onstant c r uss-s e C1:i OrI3 an d melst be approached ~fth caution in mason r v as in concrete design o Non ­:Lj r,eari ty Df mechanica l pl'operties can increase l~Leral deflections relativs cO an alysis based on the j ~ itial tangent modulus. le nsi 18 cracking before failure Jeads to variatLuna in c:ross-sectional geo ­metry ':ln d can lead t,) stability failures not con ­,, 1dered 1.;'1 the method . In con c rete design these pf+ects hciVO ~een i nclujed by the ~SB Df modified fl~xurAl riRidities t I and J i rrdtations on the range 'o r app ] il'6t LJiI l i: p recIedo ~;t clbi Iit:, f aj lures .

1\3 cn ir.:!.tiiJl stap i. n , ~\/Jluc t i'l n CJ f 'thc app l ication DF t~, e approach t u prEH.dc·;:. :i..l.lrl of \Jariat.ions j.n wall r:'lr,Bc it y , the sjompl est possibln fo r ;]', was a pplied to plJblished wall data in ,,J hich uricK 3trength , pris:n stcength, a nd test condit1ons wa~8 elearly stated . oI ,;a ini tial modu l us frClm Eq o (1; ,483 uscd together cá th a linear st r 2ss -s traJn ,j ~agra rr" zero te nsi l e ~apacity and ~r:e uncracked s8c~io n r ad i us af gy ­,'at i on .

In p;enera l, íour caSE,S rêust toe sxamj.ned lef1dir,g to the fo llowing fo~r eq~etjO"5

Case 1 Uncracked S8ct i on . Fai lure Rt ~n d

[ ~ 01= c

1 + 6 e /t I!lax

0 . 5 < ~- < 1 o

Ca'-ip 2 Uncrilcked ::J8cti.Cln , Failure Al c n g I-ieight

5 <' ~ < n. __ p _ o

Case 3 Crilcked Saction , Fai l ure a t En d

~ [ 1 - 28/t l 4 max

L

0 < ~ <[J 5 - p - . u

Célse 4 Crack~rl SRct i on , Fai l ure Along Length

[J<~ <0 . 5 o

( 61

(7)

( 81

(9)

4 . b . 3 - 3

Fcr the give n values Df th e anti ecco n t~i c1ti 8s , the t heorstir:al capac i ty reduction fa e i,c r (P/P ) TH is t he least of tM8 fou r Bol utio ns t o these equat ~~n~ .

To cC I~pact the re~IJ I te for ô var'Lety uf brick dnd mor t ar strengtl's the deviatüm Df the theo r etical capaci t y from the exp erime ntal capacity hós oean calcu15ted as a pe rc e ntage cf tha prism c~pacit~ to obtai~ tl18 relativ8 8rrc r

A negati v8 valu8 cf t hi s Rrror in djcates ~ conser­vative tneor8 tic~1 predi utio n .

( 101

As mentioned pre viously , ml. nl mUm eccentrtcities must be assumed to predict the hehaviour uf nomi nal]y axially loaded walls . Since the form J& such eccen ­tricities a long t he height of ô wa ll must aIs o be assumed , a va riety Df ch oices are possiole . In analysis it was conservat.ivel y assume d that the rúni ­mum eccentricity was constant A1Dng the height 18arl ing to single curvature .

Shown in Fig . (3) are the errors obcair led for t he da ta cf Ref . l71 wi t h a min i mum eccentricity 8f ~ per c8nt Df the wal l thickness . lt can be seen that lhe methcd generally predicts the variati on Df capacit y wi th heigh t end end conditi o ns with an abso lut e error thdt mi ght docrease with slenderness . tiy su i ;:able choice of minimum e ccentri c iti es , t hB grror i n p~ediction can be adjusted towards more cons8 rv~t t lie r2su l t s ~nd the variat i cn with slerderne~ s ca n be 0'':lr:ged .

ECCENTR IC.A,LL'( L.O,\,Jéll !t/.A, ~ LS

In the case Df eccentrically 1030",rj 'NC" ls, t!,fe. bd~ Le variables are the effective Rlende~nes s r dti n kt/r , t he r e lati ve end eccentricities eJe~ ,md the maxin:um 8nc1 eccentricity e . Shown ü , F i ~ . (dl ""re U-,e ab­solute predicti~n ~~~ors obt~inB ~ fo r the Ja ta oi' Ref . ( 7 , 181 . These data indicate t:hilt 'lhR rnethod is gensrally conse rv ê:ti V2 wi t I"! A i") q rl' ~ r 7 t:bt Ch::c:'Ra:::y,;

with sle ndernBss ,

DISCUSSIClN

In evaluation Df the ability of th'l morr.enr. - m,~gni fi8 1 '

meth od to pre~ict ~ariations in wall capaclty, th9 assumptio lls adopted rnust be considered . {l,s shovm iq Fig. (2) , use Df a linea r stress -st rai n diagram sV's ­temati cally underestimates section capacity for 8ccentric loads . USB Df the i ni tial modulus Df p Jes ­ticity to c:a l c ulate criti caI l oa ds le a ds to unds; ­estimation Df t he effe c ts Df lateral deflection s wlth decreasing erro r as t he slenderness increases, As ~

res ult , the method can be e xpected te oe conse rl:et~ 'ie

for relatively s hort ec csntrically 10adeG wa l ls , As s l endernes s inc r easE:s I.htJ noncor,5el'võtisr.1 Df tnG effects Df the laterõ o" [J8Fl8ctiom; comUones w~t,-, the conservatis!l! of sh ort e e C'tin ll ~ r:)p",; ·ti es l Cilding to a reouced total el"rr:('o

The r e sult s Df 'lhis i> t l!dy SLlg6est l.r,at USE c,f Ju,oar elastic mode l for h ri.cK masonry 1s no t war('anteo . ,4

more realistic defini tion ~f t he st,es s-strai n charac ­teristics i s required . S"C'.cessfL!l app l i cõtion Df' t.he moment-magnifi e r app roacr wi ll ; 'equire coth lmprol/C'c shol ,t section prc;p8rtios and ~ cetal l ed ~tudy (1f l at­eral deflections TO ystabllsh ~hp val~e5 cf f rlnd r to De use d in designo

4.b.él-4

Fl nallv ~t shQ~lrl be noted that some variations ob­tsined resu lt fr nm the fact that prism capacity is normally found from amall samples . As a result the ratio (P/r lEXP may indicate a greater vari atio n in wall capac~ty P than ac t ual l y exists. Related studies (19 ) and repeoted wall testa suggest t ha t wall capaci ty may not be as vari able as uni t and priam props rt ies .

REFERE I~CES

1. Hilsdorf, H.F., " Investigation i nto the Failure Mechanism of Brick Masonry Loaded in Axial Com­pression°, Oesigning, Engineering and Constructing with Masonry Products. F . B. Johnson. ed., Gu lf Publishing Company. Hou5ton , Texas , 1969 , pp . 34-41.

2. Sahlin , Se ven, Structural f'lasonry , Englewood Cliffs , N.J.: Prentic Hall, 1971 .

3. Chapman, J . C. and Slatford. J .• "The Elastic Buck­ling of Brittle Co lumns", Pr'oceedings of t he In­stitution of Civil Engi neers. January. 1957 , Vol. 6. Paper No. 6147, 107-125.

4. Poulsen, E. and Risager. S " "Tl;e Bearing Capacity Df Linear Elastic Brittle Colurnns". Bygnings­statiske Medde leser . Vol. 36, No . 3 , Copenhagen . Oenmark , 1365.

5. Hallar, P .• "Load Capacity of Brick Masonry" , Oe­signing Enginee ring and Constructing wit h Masonry Products . F.B . Johnson, ed ., Gu lf Publishing Com­pany, Houston. Te xas, 1969.

6. Monk, Clarence B., "Column AGtion of Clay Masonry Wal ls", Oesigning, Engineering and Construct i ng with Masonry Products , F. S . Johnsoll, ed. , Gulf Publistüng Cmnpany, Houston , Texas, 1969 .

7. Struct ural [1 ,,'1 Produots Insti tute . Recommended Practice for Engineered Elri~ f', f1asonry . McLean, Virginia : SCPI , 1969 .

8. Vokel, Fe l ix V., and Mathey , Rebert G., and Oikkers, Robert O .• "Compressi'Je Strength of Slender Concrete Mas onry Walls ", National Bureau of Standards , Building Sc~encB Series 33. Oec. 1970.

9: Vokel. F.V. and Oikkers , R. O .• "Strength of Mason-1''1 Walls under Compressive and Transverse Loads ", National Bureau of Standards (U.S.). Bldg . Sci . Ser. 34, March 1971.

10. Vokel. F . V. and Oikkers , R. O. , "Strength of Load Beari rlg Mas on ry Walls ", Structul'al Di vision, Proceedi ngs Df the A. S . C. E .• May 1971 . Vol.97, 1593-16Ll9 .

11. Canadian Standard s Association , "CSA A23.2.13 , '1973. Te s t for Compressive Strength cf Moulded Concreto Cyllnders ", Rexd~le, Ontari o , 1973 .

12. Glõ,',vi lle. W. H . • an d Ba rnett. P , W., "Mechanical PrOpeI'ti85 of Bricks and Brick\<lOr k Masonry" , Oepartme nt of Scientifi c and I ndust r ial Research. Building Research. Specia l Report No. 22 . Building Research St atiorl. Ga rston. Watford. Herts, His Majesty's Stôtiorlery IJffice. London, 1934 .

13 . SeR , "Compressiv8 and Transve rse Tests of Five Inch Brick Wa lls" Structural Clay Products Rosearch Foundation. Research Repurt No . B. Gene va , 11lino1s, May 196 5 .

14 . SCR , "Comp ressive,lrans vérse, and Racking Strength Tests of Fou r-Incl; Bri ck Walls " , Structural Clay Products Research Feundat ion . Research Re port No. g, Gbn'l 'Ja. Ill1n015 , Aug . 196 5 .

15. SCR . "Compressive and Transverse StrenEth Tests of Eight-Inch Brick Walls" . Structural Clay Products Research Foundation. Research Repurt No. 10, Geneva , Illinois , october 1966 .

16 . Eskenazi , A., and Ojinaga, J . and Turkstra , C. J .• "Some Mechanica l Properties of Brick and Block Masonry ; Interim Report" , St ructura l Masonry Series 75-2, McGi11 University , 1975.

17 . MacGregor , Jarnes G .• Breen, Joh n E .• and Pfrang. Edward O .• "Oesign o f Slender Concrete Columns". American Conc reta Instituto Journal, VaI. 67, No. 1. January 1970, pp 6-28 .

18 . Watsein, O., and Al l en, M.H., "Compressiv8 Strength of Brick Walls with Large Eccentricities", A. S.C.E . Nation al Structural Engineering Meeting , Meeting Preprint 1400, Baltirnore, f'la ryland. April 19 -23. 1971.

19 . Fisher , K . • "The Effect of Low Strength Bricks i n High Strength Brickwor~," . Proceedings Df the British Ceramic Society. Load-Bearing Brickwork (4), No. 21 . April 1973.

E rror (% Po )

10

~--------~-----------r---------'~---~---~ 50 100 i50

X 200

- 10

kl

-20 e1/e2 = LO Single Curvature

X,e Max=t/6

-3 O X ... e Mox = t/3

-40

-50

E rror (% pc)

10

50 8 xlOoA 150'" 200 kl

... X X X ,--

O X O X~ -10 O X

X e 1 /e 2 = O ... \ e " t /6 -20 X X Max-... eMax = t/3 ...

-30 X ... O eMax =tl2,4 ... "-

Error (% pol

10 ... ... ... ... ,& kl

X --r--x-... 50 100 150 200

-10 X e I le2 = - 1.0 Double Curvatut'e ... ... X e Max= t /6 -20 X ... eMax = t /3

-30 X X

FrG, 4 ERRORS FOR ECCENTRICALLY LOADED ~IAU .S