STRULiURAL ANA LYSIS OF HISTO RJCAL CONSTRUCf IONS n P. Roca, l .L González, E. Onate y P.B. Lourenço (Eds.)

© ClMNE, Barcelona 1998

MECHANICA L BEHAVIOUR OF ARCHES AND VAULTS

Jaeques Heyman Depanment of Engilleeril!g U"i~'ersit)" 0/ Cambridge CUlIlbridgt! CB2 I PZ

SUMMARY

Masonry. as a unilateral materiaL is incapable of carrying lensi le force~. and will craek in response to displaeements imposed by the ex terna I e nvironment ( for example. seulemenl o f foundations). The resulting deformations of a masonry structurc are ve ry much larger lhan any "elastic " strai ns: llloreover. si ncc lhe imposed exte rnai d isplacemen!s may be ul1known. a convel1lional analys is will y ield littl e relevan! information. However, an examination of poss ible Slares of eqlJilibrium eonfinns lha! lhe stability of masonry resides in its geomclry - lhe shape of lhe structure tnust be Correel.

L'ITRODUCfION

Figure I, from a 1934 drawing by Po l Abraham. shows typica l cracks in a quadripanite masonry vault; an ai s le vault is illustrated. but similar defec ls may be seen in thê central high vaults of a major church. The displaeements necessary 10 produce sueh craeking are large - lhe spal1 of lhe vault in figo I may have increased by 100 or 200 Olm. Despile such movemenls, whieh are oLJl side lhe experience of a modern engineer used to lhe design of steel or conc rete fcames, there may we ll be general agreement Ihat lhe vault of figo I is not in a dangerous state. On lhe contrar)', a masonry structu re is thought of as robusl. ab le to withstand s ignificanl movements of foundal iol1s, 10 survive eanhquakes and indeed bombardment in lime of war.

Howevcr, a vague belief Ihal lhe vault of figo I is sa fe must be supporled by a more exael assessment of its state. How does Ihal Figure I. Typica l craeks in GOlhic vau lt s

2 STRUCTURAL ANA LYSIS OF HISTORI CAL CONSmUCTIONS 11

vault C~\rry its loads? (The loads are in general duc only to ils own weighl. ) And why has lhe vault cracked in precisely lhe way skctchcd? (Thc crading is typical o I' 1110:-[

vaults.) Some sort of slruclural analys is is in facl needed to c:-.tablish lhe mcchan ics 01' lhe vauh. Without such a basic undl.!rstanding repair work may bc direclc el to wrong cnds. or indeed may bc hannful rather Ihan benefic iaI.

The tools available to lhe structural cngineer are wel! kn owll . On ly thrcc Iypes 01' equation lTlay be wrille n , and fro m these lhe s tructura[ quantitics may be c <.I Je uhucd . First, equilibrium must bc sa(Ísficd - lhe externai forces ac ting 0 11 a strlll.:lUrc mu~t hc balanced by lhe internai slress resultants. Cenaill assumption ... and approxirnation ... lllu~1 be madc in formul aling these equations of equilibrium. bul in fael lhe equations :U.I.:

unambiguous , and may be Irusted . Secondly. material propcnics are introcluccd inlo the analysis , and here there may be largc areas of uncerlainty . The Cl;1311 C and plasli c properlies of sleel may bc well knowll - Ihcy are Ic ...... certain for rcinforced concrctc. For masonry. which is all asscmbJage of stonc (in itself inhomogencous) anti monar (lf variable or unknown propcrlies. lhe material par:l111CICrS in tcnsioll and comprc ... si on can bc very differenl - an assemblagc of masonry elemenls ma)' hc unablc to acc,,: pl t..:n ... ik Slress. for example . and this musl bc allowcd for sornehow in lhe ana lys i ....

Finally, lhe geomclrical equalions. lhe so-callecl equations of eomp;.lIibili ty . may hl.: almosl impossible 10 forrnulale. These equalions relate lhe illlernal clcfonnalions oI' a struelu re lO lhe externai displacemenls, and are exc/llplified by "houndary condilion ... '· . Knowledge of Ihese displaeemcnl boundary eondilion~ is essential for lhe elasli c (Navi n ) So lulion of a hyperslalic SlrUClure. bUI il is a facI Ihal ve ry ... mall chang~s in Ihe ... c geomelrical eonstraints - lhe precise span of an areh, for example - \ViII cause \'cry largc changes in lhe internai stale 01' lhe slruclure. Disrcgarding Ihis fo r lhe rnOlllent. an e!aslil.: so lution for a typ icalmasonry Slructure will indiealc first. Ihallhe slres~c ~ are very Jo \V compared wilh lhe basie slrenglh of lhe male riaL and second. thal lhe dellcxion ... ma)' well be markcdly less than say 10 mm. compareci wilh lhe mOVC lllcnl of 100 o r 200 tllm lha! may be imposed on a masonry vault by di splacc tllcllt of its SUppOrls.

Fortunale ly, significanl Slalemcnls abou! lhe behaviour of masonry Sll"llc turcs may bc made on lhe basis solely of equilibrium. lhe m03\ rcliablc of lhe Ihree type~ of equalioll Ihal ean be used in struetural analysis.

2 THEMATERIAL

Masonry madc from Slones, or brieks . or indeed oI' sun-dl"icd mud (adobc. wilh OI"

without a rei nforcemenl of slraw) has litlle tens ilc slrenglh. 011 lhe OlheI" hand . a~ has becn noted. compressive Slresses are low. and lhe usual "engineering" assumplion ... for masonry are Ihal

(a) it has no tensile strenglh. (b) il has virtually infinite compressive slrenglh. and (c) slip does no! OCCUI" IJClweelll:omponcnts ofthe slructurc.

Wi th these assumplions, a structural analysis of musonry may be made by using only equalions of equilibr ium, and disregarding elaslie distortions and conditions of compalibilily. The analysis faUs within lhe framc\Vork of plastic ity thcory. and powerful theorems may be invoked.

3 THE SIMPLE ARC H

Thc bchaviour of the simplest possible masonry slruelure, lhe voussoi r arch. sho\Vs how an ana lysis can be made. Figure 2(a) shows lhe arch: during construction lhe masonry must be su pponed on temporary falsework. Once Ihi s cenlcring is rcmoved. the-

J. HEYMA N I Mechanical behaviour of arches and vaul ls 3

eompleted areh wi ll require support from its abutmenls - the thrust of the areh will force lhe abu tmenls 10 give way slight ly. and the span of the areh wil! increase.

Figure 2(b) shows how the areh aeeommodates itself to the inereased span - the figure is mueh exaggeraled . The areh will eraek aI joints between voussoirs: there is no strength in these joints. and three hingcs have fonned (which in praetiee may be revealed by cracking of any mortar Ihm may be presenl). The Ihree-hinge areh is a well known and pe rfectly slable strueture: there is no suggeslion Ihal the arch of figo 2 is in a state of distress. On the contrary. il has mercly responded in a scnsi ble way to attack from a hostile environrnent. A first fundame ntal statement relating to lhe asseSSl11enl of arehcs may bc made on lhe basis of this simple example: cracks in masonry are nOI necessarily signs of d islress. Craeks renect the natural stale of a stone slruelure: the secre l of the behaviour of masonry is thal il can adjusI to randomly imposed externa i deform­ations by such harmless crackin g.

" <b,

Figure 2. Stable slate 01' a cracked voussoir areh

The dotled linc in figo 2(b) indicales lhe way lhe forces are transmilled wÍlhin lhe arch. Each of lhe voussoirs pushes againsl its neighbours: at lhe crown of the arch thcre is only a single poinl of contac!. and lhe linc of thrusl must certainly pass through Ihat point. as il muSI through lhe hingcs aI the abutments. The shape of lhe line of Ihru:-t in figo 2(b) was identified by Robert Hookc in 1675 - il is Ihat of lhe corresponding inverted weighl less cord. carrying lhe same loads in tcnsion (ralher Ihan lhe comprc:-:-ion of the arch).

The dotted line in figo 2(b) is located precisely. sinec il must pass through the th ree hinge points. Howevcr, if lhe arch is nOI subjccled 10 lhe particular 1110vements 01' lhe abutmenls impl icd by fig o 2. then lhe line of Ihrust - lhe inverted hanging cord - c:ln nOI be drawn immediately. Figure 3 is based on a skelch illustraling a ninclcenth-ccntury examinal ion of lhe equilibrium of arches: Ihree lincs of thrust are shown. of which the stcepesl corresponds lO lhe dOlled curve of figo 2(b). The shallowe~( curve in figo 3 records lhe stale of lhe arc h ShOllld lhe abultncnts approach ralher Ihan spread. as thcy mighl if the arch were an internai span of a multi-span bridge. Belween the!)e IwO extremes is shown a curve follow ing roughly the centre line of lhe arch. and in facl an infinite number of sueh curves can be included within the thickness of lhe masonry.

Figure 3. Altemative lines of thrust

4 STR UCTURAL ANALYSIS OF HISTOR ICA L CONSTRUCTIONS 11

Very smal!. almosl infinitesimal. movements of lhe abUlmcnts will cause lhe line af thruSI to adopl one rathc r Ihan another of lhe possiblc locations. Howevcr, Ihese abutment movements, defining lhe geometrical boundary conditions for lhe arch. are nOI only unknown. bul unknowable. depending as Ihey do 011 changcs in soil conditions. lhe passage of a heavy load. even wind load ing from a gale. and so 011. A second fundamental statcment aboll! lhe behaviour of mllsonry may be rnadc on lhe basis of Ihis example of lhe simple arch - [here is no way 01' determining lhe "actual" equilibrium statc af a hyperstatic struClurc.

However. lhe powerful "safe thcorem" of plasticity theory may bc used. For lhe masonry arch, if one state of equ il ibrium can be fou nd for which the structure is slablc. then ir will be stable absolutely: it is not necessary to try 10 dete rm ine lhe "actual" state . Thus. in figo 3. any one o f lhe three inverted hanging cords represents a line o f thrust lying wholly wilhin the masonry, and is hence a demonstratian that the arch is safe . Very small unknowable movemenls of the abutments may cause lhe line of Ihrusl to move vialent ly within the masanry, bUI it can never escape.

Thc arch must. of course, have lhe righ l shape. Thus a sem icircu lar arch. for example , which is able to carry its own weighl. must have a minimum thickness (whic h tums out to be just over 10 per cenl of the radius). A Ihinner arch will not stand: a Ihicker arch will always be able 10 conta in a Ihrust line corresponding to lhe inverled hanging cord.

A Ihi rd general fundamental statement may be made on lhe basis o f these argumenls. Stabi lity af a masonry slruclUre is assured by ils shape - lhe geomelry must be correcI. In modem times mathemalics may bc used la verify lhe geomelry. cithcr by direct calculatian, ar on the drawi ng baard, by the use of graphic statics . Sefare the use of mathemalics, lhe structure itself confirmcd stab il ity - lhe facl lhat a masonry structure is seen to exist is a demanslratian. wilha ul calculatian. Ihal a set of forces exists within lhe fabric which equilibrales lhe weight a f the structurc. Thercaftcr smaJl movcmenls of lhe cnviranment may cause cansiderablc changes in the equilibrium state. and they may cause cracking, bUl they can never of themselvcs cause collapse.

4 THEMASONRYVAULT

The defecIs in lhe quadripartite vault in fi go 1 may be interprcted by these s implc ideas about lhe masonry structure - aI the same time , as will be seen. a dear view is oblained of lhe way lhe forces are carried in lhe vault.

A firs t slep Iies in lhe examination af a simple barrei vault, figo 4, which is na thing more Ihan a Ih ree· di mens ional trans lat ian a f the arch of figo 2. The cross·section in figo 4 is drawn rough ly to scale (say a vault with thickness 300 mm over a span of 12 m). The vault is maintained by ex ternai buuressing. The arch af the vau lt is toa Ihin to carry ils own weight right down to lhe springings. and rubble fill is shawn backing the haul1ches so that lhe thrust (lhe inverted hang ing cord) can escape from the vault proper. In figo 4(b) the buttressing system has given way slightly, and lhe craek paltem of

'"

'" Figure 4. Hinging crack in barrei vau lt

J. HEYMAN I Mcchanical behaviour af arçhes and vaults 5

Fig. 2(b) has developed. The hinges in the extrados near the fiU \ViII not be seeo from below, but the more or less central hinge will be visible.

The cross-section of lhe vault in figo 4 is the same down lhe le ngth of the barrel. Figure 5 shows a s ingle bay of a quadripartite vaul! formed by the intersection of two slightly pointed barreis. If now the buttressing of lhe vault gives way, lhe portion running eas!­west will crack as before. and the single hinge line at or near lhe crown will be seeo from within the church. The ehange in geomelry eaused by the increase in span wiU result in a drop of lhe erown of the vault.

There is a severe geomet rical mismatch in the intersecting barrei whieh runs north-south - lhere is nOI enough masonry to fiH the inereased north-south dimensiono A eraek pauern WJII

whieh allow$ the vaull 10 crack

E

" "i,,~c line II --.' " ,

w

la)

(b,

'"

defonn in virtually strain-free monoJithic pieees is skelched in figs 5(b) and Cc) . Sabouret cracks (narned after lhe arehi­!eet who made an analysis

Figure 5. Crack patterns in a quadripartite vault

of vault5 in 1928) have opened in the vault a1 a distanee of a metre or 50 from lhe north and south wall s (usually eonlaining windows). In addition, eraeks may have opened adjaeent to the walls. These çracks in lhe north-south barre i represent complete separation of the masonry, and both Iypes of crack - hinging and separation - may be seen in the sketch of figo 1.

Once again. such crack patlerns do no! indicate lha! lhe vaul! is on lhe point of collapse. Ralher. lhey give more or 1ess prec ise indications of responsc to externa!ly imposed movemen1S. Moreover, il is clear that the edges of thul the edges of the vau!t are acting as simple arches, as is indiealcd schematically in figo 6.

Figure 6. The edges of a vault aet as simp!e arches

6 STRUcrURAL ANALYSIS OF HISTORICAL CONSTRUcrlONS 11

5 CONCLUS ION

Masonry is a material which, in ilse lf, is virtually rig id and infinite ly slrong in comprcssion. lt canna!. howevcr. accept tcnsion, and will crack in response to attcmpls to impose tensile forces. Any distortions which may be vis ible in a masonry st ruclure are lhe rcsuhs of lhe development of such cracks:

The cracked s tate is lhe natural statc of masonry.

Cracks indicate lha! lhe environment has imposed ex ternai defonnations on lhe st ructure - movements of lhe foundalions, for example. Very smalJ movements oI' tbis sort can lcad to vcry large changcs in lhe equil ibrium statc of lhe structure - lhe way in wh ich lhe loads are carried. Sincc such mQvements are unpredictable. lhe caJcu lat ion af lhe "actual" state of a structure cannot be made:

There is no unique calculable equilibrium state for masonry.

Thus an elasti c analysis of a masonry struclure. pcrhaps by using finitc c1ements. will certainly gencrate an equilibrium solution. However. the elaslic analysis relies on known boundary condit ions, and since these conditions are essentially unknowable. the solution is one which will not be observable in practice. Further. claslic displaccll1cnls fo r masonry are very small. and negligible in comparison with distortions imposeel by lhe environmenl: no significant information will result from an eJastic detlexion anaJysis.

Instead, reliance may be placed 00 the "safe" theorem of plasticity theory. If any one equilibrium solution can be showo to be satisfactory - lhe forces lie wilhin the surfaccs of lhe masonry - lhen lhi s guaranlees the safely of lhe structurc. This is a slatcment aboul geometry:

The correct shape of a masonry structure ensures its safety.

6 BLBLIOGRAPHY

l . Heyman. El esqueleto de "iedra. Mecânica de la arqllileul/rll de fâbrica. Madrid (Instituto luan de Herrera). 1998. [The S/one Skele/oll. Cambridge Univcrsily Press. 1995 .]

J. Heyman. Teoría, hisroria y resrallración de eslnu.:/llras de fábrica. Madrid (Instituto Juan de Herrera), 1995.

J. Heyman. Arches, vat//rs muI bll/(resse.~ . Aldershol (Variorum Press), 1996.