4 . b . 7

L1MIT-DESIGN PROBLEMS FOR MASONRY ARCH- FRAMEWORKS

WITH ELASTIC PERFECTLY PLASTIC BEHAVIOUR

Ao BOVE

S o BR ICCOLI BATI

!3 o LE GG~R I

I s tituto di Cos t r uzioni , Uni versiti degli Studi di Firenze , Firenze , Italia

LIMIT- DESIGN PROBLEMS FOR MASONRY ARCH- FRAMEWORK.5

YI TH ELASTIC- PERFEC2'LY PLASTIC BEHAVIOUR

The prDblem o f arch frame',Jor ks is treated s earching

p7,a8t?:e hinge po Z:n ts wll'ich ar e necsssary o.nd suffi ­

c-ient cond::tion to dete r mine s t rw:tur-c collapse .

A dir 3ct computat-ional p ,oocedur'e i3 s et up 100der the

hypotl-zes i s o f eZ':ls t i c -pel'.fect l y p l astic behavi our

Oj" 7;'"S 0 1U'lj . SU:Jh a me tlzod is de velaped f or' ar'ch

me;r!beí'G but Ct h8l" m1sonr'y pm blems can be treated by

means of th-is generalizad analysis .

A short r e ference is given for technical and compu­

tati,mat sem'ches and resu l ts related to the above ­

mentioned approach .

LE CALClfL SUIVANT LA METHODE DES ETATS- LIMITES

POUR LES AHCS E'N MACONNERIE A COHT'ORTEMENT

EL4STIQUE-PLASTIQUE IDEAL

La communication traite le calcul des arcs en

maçcr;nerie pour l ' analyse des eharnieres p las ti­

ques qui sont déterminantes pour la ruine de

l ' ouvrage .

Une méthode de caleul qui est adaptée aux besoins

du caleul par ordinateur, est présentée . Cette

méthode est établie sous l 'hypcthes e d ' une maçon­

nerie à comportement élastique - plastique idéal.

La mithode a été développée pOUl' des arcs, mais

pe ut étre app liquée à d ' autres f orme s de maçonne -

n e.

Quelques :r>éférences sont données ainsi que des

:r>ésultats obtenus par la méthode approX1:mative

décrite plus haut .

PROBLEME DER BRUCHMETHODENBERECHNUNG VON

BOEGEN IN MAUERWERK MIT ELASTISCH­

PLASTISCH IDEALEM VERHALTEN

Behan de lt wird die Bereehnung von Bogen aus Mauer­

werk dureh die Analyse von p lastischen Scha1'Yli er ­

punk t en die den Br uch de r Konstruktion bestimrren

k8nnen .

Eine Rechenmethode, die direkt für Corrputeroere ch ­

nung geeignet ist wird aufgestellt, wlter d.er An­

nahrrte , dass das Mauerwerk sich ideal e lastisch­

plastisch verhiilt .

Die Me thode ist rÚY' Bogen entwickelt worden, aber

kann auch auf andere Mauerwe r ksformen angewandt

wer den .

Am Schluss werden Quellen angegeben und über Ergeb­

nisse berichtet, die mittels de r obe"! beschr iebenen

Niihe rungsverfahren erreicht wur den .

PROBLEMEN IN VERBAND MET DE BEREKENING

T/OL(;ENS DE BREUKTHEORIE VIlN GEMETSELDE

BOGEN MET IDEAAL ELASTISCH- PLAS'!'ISC!l GSDRAG .

Behandeld wordt de ber ekening van bogen uit metse&werk

door middel van de analyse van p lastische schar-nier­

punten die de br euk van de konstruktie kunnen bepalen .

Een rekenmethode die direkt voor computerberekeningen

geschikt is , en waar bij uitgegaan wordt van de ver­

onderstelling dat het metselwerk z ich elastisch­

plastisch gedraagt, wordt voor gesteld .

De methode is ontwikkeld voor bogen, doch kan ook op

ander e vormen van metselwerk worden toegepast .

Op het einde wor den de br onnen aangegeven, alsmede

resultaten die door middel van de bovenvermelde me­

thode bekomen werden .

I~nRODUCTIO~1

The main purpose cF the pmsent paper may be to reduce th8 E~~ b9lh~en mesollry str"ucturo computational tech­niques ar,c t!'032 ',!hich are com~unly used to compute soll~ions Df elestic problema with many unknowns . A p~rlicular sim is t~8 extension of t he limit stata de­si;n procedu~6s - viz . e1dstic-plastic theory - to a broad clas3 Df rrasonry structures ; so , the mason ry aI cl, is examini"ld in detail as a sigr,ificant example of sllcn an ap;J roacn . Therefore , the method descri bed in what fol lowE is intends d to be a generalizati on of prav10us doproaches Igraphic or analyti c onesl . Other interesti~g tjpes Df structures can bs treated under thE5S ü3SU1T'j":-ltions : for instanc8 . lhe brood class of frame structures , i.e . Elastic (aid , as u pa rticular case , ri gidl reol.;f,dé,nt Trames . I') tne fullo'."iClg we shall ooint Ou~ two ~iff8renc ssts Df nypothesüs: t he first i= rslated t8 the classicr:i l treat:flE:!I"1t Df masonry strush_d~25 J tlie 38C'lr,:j allows haildl in g th2 sO ~ fOclls~Jed

probleni in a compute~-ori.;nted ôpp r oach.

CLASSICAL HYPOTHESES DF MASDNRY BEHAVIOUR

The basie assLll.lptions usually made aiJnlJt masonry struc­tIJl~r.S ('"')nr'crn pilyco i.cõl la;,.Js in lin8l31' lr2haviour of the wllo::'o frclm'?; tr,ey iHB the following , rou['hly su",mar­iZ2d :

il no tsnsi18 strength i5 supposed throughout the stn .. cture

iiJ vE11I~S c'f rorrpr2ssion stress are 1m>! in comparison \,\Iith L 11Jú:-;!-'3 'véllLi'::_s

iiil strai" le'mls am ,wite neglectable

ivl th~ 2~:)")\jO n:,;ntiO:l2c1 '1y psl,;he3eS lSôd to the rigid ar ri~iCi-ol-?stic 6n,lysjs . P-s a mntter of fact .. in t~? cas~ Df ~~snnly arch .. s~~8ti~e5 the t h r ee hingRs :1rch rr·Qcel ih!3 C:8en u5ed ~ éS an EXôr:lple of rig~d an~ly?is (+)

: rlS m2éir-.s tr.2 r.3;;Ju,-:-tir.g of alI str8ss-str"in data 2\/é.i.lé:l:ll8 :'r t~G BC tU3: elasLic =:-r8'10'.2na . r'Jr t her-~,r3 ~ ~uch e~ ~p~r03cn forhi~; rhe d8termi,c~j.:Jn of th? l.úLi::jU9 :;,.tu.Jl 2"=1:Ji1 i.Drj~url \1.slu8s : f-J::;ui lic.,riurn B~d :3ts ir t .!.:J'3st on3 t:;cluJli.iJriur ,~, c:Jn fi:jL!Ié...tiGl1 is c'Jtri..1R.:.le in the r-S:jull-~cnt rl[~L:; rn,Jdel which ~as bnen a3SU:T.9r] . A.n':"lther c3~ISLli l:Jtion :,-Ji-1ich lJl.'gllt to be dis-cussad is thac L:e;Jally r;,~jGe ~b;Jut hn~lci:;on2.ity ()f

Tat :J:. ... i:::ll . T~3 ... lo]l-Knu~-J:l '/cL;~solr ~rch i: assD'ltia ll y é) ::'c3r:.i.':L,Jer fre.rr.3'Norr, mer:9 üy rigio elcrT,ents . however J

t~e :,,:[,,11 n"r.d'Jiollr callC10t inc1 ucJe effEcLs Df non­h:tr·.IZ",r,;lty , fcr alI tl">3 nLl~""r·ts e r s ident i.ctl l. The pl·éi-'t':"rs :f sirr:Jlifvj.l~ ectudl st ructur~b in Dl dor to O.Jt~i.1 t!"3':- .2:J1e ic18dli::'Rd r12ch3nlcal n,ocals f-Jéi= great lr-'Jurt.=m::. i'l ~,.'~~t r21c:t[:~ to t-h::'! 8e.Jr:18tr~1 Df BTlB l ysed fC'~~"'?.J:JT~ . In f3:t J ~h~ i'.:;j n :j.n~ Df l:í19 .-3l~ch is a]I·'·:Y'S 1'3~:::rc8j a3 t;~8 Q,lly sti1~-f::. r..i..ng elorr,8,l1: in the (J\'t~ra.-!.] -r:1SClnr~' fr.=:r:,E:! j it i3 8êSy to re:1l.C f? the oroblem i"l t:li; ~JaJ. to a [.v!c - nirr,'=?nsicna l case . FLJrth2 r r.10r2 ~

tt-,i: -3:--,ro.=sh S'~8rTS to t::=: -~t!it2 uc csptaole to vsri.fv -und8:- ~omE: :.;)nser\)é7:i\'~ tl._SI..<T~,tiúil S - st:'::I;Jj lity of allc:isilt ma::o;:r:l strlicturSG (SE8 . for insl:arice 3 Clare Colleg2 ~ ri d~e ,:md Telfúl"'G I 3 El~id ge at r;~'J? r ) . On the ot'ldr 'tlr"lci , 1:hR iTlOd5 r n l:Jols Df st rL:ctLJral me chanics a1 d rsc:Jnt l1e\JelC;llllsnt in ;r,e.5u~ry [Juildi llg óllo'\I'J to hnp8 fuI' a d~~igf1 - (1ri9n tecJ CurTJ;:Jut éticJn of str uctur ·es, using rr8t~~d3 which are av~ilaole ' also ' for veri fy in g stabi lity Df pa rticular existing structures ,

Col See, for instanc8 , the collapse confi guration éSS'.Jndd by /I .J. 5 , PiflPi'll'd DroO J . flukE:r [see raf. I Wh~l'l' p::;sition Df fe;ur ;Jlastlc hinges is asslI'lled to üe y,nO.· .. l .

DE FINITION DF A GENERALIZED METHOO FOR ANALYZI NG MASO NR Y FRAMES

4 , b . 7- 1

Near alI above mentio ned r estricti ve hypotheses can be remo ved in orde r to search desi gn-oriented solutions within c1~ssical linear DI' li mit state behaviour , The basic ideBs of t his approach are in the consideration Df general a l astic prob l ems and th eir treatment in li near form , A mechanical discretionization model has been e~~ l oyed which is essentially a finite ele ­ment model , it is st!'ictly si milar to the common voussoir scheme , this one may be deduced as a par­ticular case of a finite element model o Figure 1 . , shows three adjacent finite elements which are supposedly each made ,Df homogelleoua elastic lelastic­plastic , ri gid , etc . I material . 50 , elEstic contin ­uum is cut in vcus soir-finite elements : their nwnber directly influences Lhe dimens ions of the resulting system of linear algebraic equations . Therefore , numcer and pcs iticlJ1 s of cuts must be determingd ttlking in to 3ccount the actual dsgree Qf nGn-homogene1ty of structure and the capabili ty Df computatioilôl tools .

Let {xJ ed {xJ be the vectors Df internaI atress and displacerr.ents r'espectively , F and Dl t he loading vectors in tsrm3 of forces and dis~la(;emgnts , and let K relat e to t he squ ar8 matrtx DF s l ast~c constants sollJtion ir1 tenT,s DF di s placements only leads tu :

whence:

{x J

In the examples whtch have been discussed a standard Cho les ki. algo rithrl has been used to comput e solutions Df li near systems wl1e re c08fficien t matrices arE< al ­wajs symmetrtc B:ld positive-definite . Such an a1go­rithm allows ren~ rk able storage sDvtng taklng i nto accGunt a I s o tho band struct ure cf coe f fici Rnt ma ­tr1ces . lrrpo rtaClt adVa:ltôg8s Df this n',a thuC: are in the snarching of thp un1quB elast1 c snlution, 9nd neD rly alI clas si cal flrc c3d~r9s Igraphic search Df ' possiole ' thrusê 11n8 , fur instencel can bü deduced as parti cula r cases of this generalised ünalysi3 .

LINEAR I ZEO COL LAP SE ANALYSIS

Collaose analysis under the assllmptions Df elastic perfaL:tly Dl~sti c ~JfJr,al!ioIJl' of met eri,Jl is oscúm­;Jli3hed in fou r l insdr staps . Four plasttc hi nge s are determined together with the collap se vülu8 Df load paramelcr . O:.]ljicus 1y , the oV8rall beh aviouT' of equi ­li8rium path is slôckening in a lI diff8T'e~ t l03ding con::Jttionq . l:-:e plastic ,"ome" t in mDsun ry li . 8 .

mGm9Gt value leading to plasticityl has been deter­mined experimen tally .

REFERE~ICES

1 . Baker , J .F., HDrne , M, R. , and Heyman , J , : Plas tic 8shavi our and De sign , Cambridge , 1956 .

2 . Baldacci , R. , Ce radini, G., Giangreco , E, : La Plasticit~ , Milano , 1971 .

3, De Josselin , De Jong : St atics and Kinematics in the failable zone Df a granular material , Delft , 1959 .

4 . [lrucKe::, , O.C .: Coul omtJ Friction , Pl asticity and Limit Loaejs , J. ,~;J:J 1. Mech , 21, 71 - 4 11 9541 .

4 . b.7-2

5. Drucker , D. C., Prager , W. : Soil Mechanics and Pl astic Analysis or Limit Design , Quart . App l . Math . , 10, (1952) .

6 . Heyman , J . : The stone skeleton , Int . J . Solids & Structures , 2 , (19 66) .

7 . Heyman , J . : On She l l Solutions for Maso nry Domes , In t. J . S . S . , 3 , ( 1967) .

8 . Heyman , J .,: The Safety Df Masonry Arches , Int . J . 5 . 5 . 2 , (1969) .

9 . Heyman , J . : Plastic Oesign Df Frames , vol . 2 , Cambridge , (197 1 ) .

10 . Kooharian , A.: Limit Analysis Df Vo ussoirs (s8gmenta) and Concrete Arches , Proc . Am . Concr . Inst. 88 , (1953) .

11 . Massone t - Save : Calco l o a rottura delle strutture VoI. 1, Zanichelli , Bologna , 1967 .

!t

Fig . 1

12 . Pippard , A. J . S ., Trauter , E. , and Chitty , L. : The Mechanics Df the Voussoir Arch , J . Inst . Civ . Eng . , 4 , (1936) .

13 . Pipp ard , A. J . S . , Ashby , R. J .: An Experimental Study Df Voussoir Arch , J . Inst. Civ . Eng . , 10 , (1938 ) .

14 . Prager , ""., Hodge, P . G. : Theory Df Perfectly Plast i c Solids, New York, (1951) .

15 . Prage r, W. : Einfuhrung in die Kontinuous Mechanik , Basel , (1961) .

16 . Prager , W. : Introduzi one alIa t eo ri a della Mi l ano (1969 ) .

17 . Ware , S . : A treatise Df the Properties of Arches and their Abutment Piers , (1809) .

18 . Casapieri , A. , De Leonibus A., Indagine teorico sperimenta le sulla possibilità di applicazione de I calcolo a rottura a l Ie s trutture murarie con partico l a r e riferimento ag I i archi, Re I . Prof . S . di Pasquale , a . a . 1972-73 .

I

Arch Ring and Discretioni z ation Mede l

4 . b . 7-3

---- -eJ---------------------------------

t

À3~---------~~ À2~-----À 1 -+-__ -}c::

--r-----r-----~

Fig . 2 Fig . 3 Slackening Lineari-zed 81Ui Ziôriwn Path for

Central &;lading Plastic H-~nges for different Loading Condi tions

ColZaose rrndtiDliers :

À1

= 2. 00797541 8 +6

À2 = 6 . 64367227 8 +5

À3 = 4. 13173315 8 +5

À1 + À2

+ À3 = 3. 08551596

i 8 +6 1\

\ \

-......... ~ j. --+-I I

/(1 _~ /(3

Loading position s

Fig . 4

r-....

4) I

CoLZaose Multil?liers versus Loading Positions

À (]) = 1. 745959 8 +6

À(2) = 1 . 726459 8 +6

À (3) = 1 . 811492 8 +6

À (4) = 2. 007975 E+6

1 I 1

V --I. ~/ T

I I