between geornetry and mechanics - a re-exarnination of the principies of stereo to my from astatical...

8/8/2019 Between Geornetry and Mechanics - A Re-exarnination of the Principies of Stereo to My From Astatical Point of View

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Between g eo rn etry and rn ech an ics:A re-exarnination of the principies of stereotorny

frorn a statical point of view

The main objective of this paper is to give amechanical interpretation of the geometricalp rin cipie s g uid in g th e a rt of s tere oto my fo r d esig ningmasonry arches. The treatises on the coupe despierres -even those published after the birth ofmodern structural mechanics- deal w ith the designo f v au lte d str uc tu re s fro m a n e ss en tia lly ge om etric alpoint of view. For instance, the m ain issue of cuttingvoussoirs, as con cerned the inclination of the joints,was dealt with in geometrical terms without takingany statical consequences into account. W ithreference to this problem , the coupe des pierresdevelops tw o geom etrical criteria: the first requiresthat the joints converge at a single point (e.g. V illardde Honnecourt); the second requires that the joints bepe rp end ic ular to th e intr ad os of the ar ch (e .g . F rz ier ).

In order to determine the degree of stabilitycorresponding to these geometrical criteria, thepresent paper analyses the problem of stonecutting instatical terms by considering the equilibrium ofvoussoirs in the absence of friction and cohesion. T heworks of Coulomb, de Nieuport and Venturoli areexam ined and the statical form ulation of the problemis e xte nd ed to so me s ter eo to mic c on stru ction s.

T HE O RIG IN S O F S TE REOTOMY

From the M iddle A ges to the 18th century, stereotom ywas considered the most important constructiontechnique. By means of geometrical principIes, in

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fact, it allow s one to visualize a tridim ensional objectby m eans of a bidim ensional reproduction and to givean appropriate form to each of the voussoirs makingup a vault. Tn this way, it is possible to constructvaults, domes and squinches and to perform anin fin ite v ar ie ty o f b old te ch nic al o pe ra tio ns .

In this context, it is interesting to observe that thedesign of complex vaulted structures seems to harkback sim ply to the solution of geom etrical problem s.I n a ntiqu ity , th e arc h w as co nsid ered a s a pr e-e min en texample of geometrical perfection, containing initself a principie of statical perfection: the com monconviction was that geometry, not statics, couldp ro vid e th e sa fe st p ro por tio ns f or de sig nin g ar ch es .

The ancient Egyptians, Greeks and Romans cutstones into large blocks, so that they formed soundconstructions and their weight took the place ofmortar.

W ith the passing of time, efforts were made toreduce the dimensions of the stones constructing thestructure, so as not to place excessive organisationald em an ds o n th e b uilding s ite. H en ce th e fir st o bjec tiv ein perfecting techniques for cutting stone is findingstability com parable to that w hich w ould be obtainedusing m uch bigger stones, w hile using sm aller ones.

A second problem relating to stonecutting is linkedto the fact that stone is characterised by a highresistance to compression and a low resistance totraction and to bending. For this reason in ancienttemples the maximum distance between the axes ofthe co lum ns d id no t exceed 4 -5 m etres. H ence th e

Proceedings of the First International Congress on Construction History, Madrid, 20th-24th January 2003,ed. S. Huerta, Madrid: I. Juan de Herrera, SEdHC, ETSAM, A. E. Benvenuto, COAM, F. Dragados, 2003.


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s e c o n d objective in improvement of hewn stoneconstruction techniques is to solve the problem ofgetting over bigg er inter-ax is spaces an d co vering s.T he s o- ca Jle d encorbellement method (Fig. la) wasthe first solution, used starting from antiquity. Theconstruction principie is very sim ple: it consists inusing overhanging (i.e., corbeJled) stones, w ith theb ed s a lwa ys ho ri zon ta l.

Though this technique may appear unrefined andprim itiv e, it m ade it po ssible to realise som e w ork s o finestimable value. One of the most ancient andcelebrated w as the so-called room of The Treasuryof A treu s, a m asterpiece o f M ycen ean arch itecturedone in the 13th c entury Be. From the 7th to the 2"dcentu ry B C the E truscans frequ ently used co rb ellin gto cover some funerary chambers (one thinks, forexample, of the tombs at Casale Marittimo andM ontagn ola) or to m ak e arch es (S akaro vitch, 1 99 8).

While encorbellement is a techn ique that carne in tob eing fo r constru ctin g hew n ston e stru ctures, the archand the tunnel vault carne into being as brickconstructions. They appeared starting from thebeginning of the 3td millennium in regions wherethere was a shortage of wood, like M esopotamia andthe vaJley of the Nile, but carne to be part of thesto necu tting techn ique o nly w ith th e in tro duction ofthe voussoir, a w edg e-shap ed sto ne w ith tw o o bliqu efaces by means of which it rests on the adjacentv ou ss oir s, la te ra lly tr an sf er rin g th e v er tic al f or ce s d ueto its ow n w eight and any other loads.

Th e first exam ples o f arch structu res in the G reek-R om an w orld, w hose dating is certain, do not go backto earlier than the end of the 4th century or thebeginning of the 3'". W e are referring to the archesthat cover the gates of fortifications at Eraclea ofLatmos and at Velia, which were ancient Phocianco lo nies in C entral Italy, or the on es un der the vau ltedroom s of some Macedonian tombs (Langhada,Leucadia) or again the underground cham bers of theth eatre at A lin da in C aria . In th ese d ifferen t ex am ples,as in the Egyptian vaults, the problem posed by thelateral dissipation of the thrusts exerted by the vaultor by the arch is sol ved, in that the vault belongs to as tru ctu re th at is in terred o r th e arch co vers an ap ertu rebelonging to a wall. Down to the 2nd century BC alls tru ctu re s w ith a rc he s o r v au lts a re o f th is type. Thisis stiJl the case in the very beautiful vaults of thestaircase at the Pergam os Gymnasium. It was theRoman builders that, starting from the end of the 2" "

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century BC, first made the vault a free volume: withthem , the vault show ed itself openly, carne out of theground, and becam e a noble construction, no longerconfined to subterran ean constructio ns an d fun erarya rc hite ctu re ( Sa ka ro vitc h, 1 99 8) .

A t all ev ents, the crad le of stereo to my w as palaeo-Christian Syria. In the m iddle of the 3t" century ADthe Philippopolis theatre was built on the Jebel ed-Druz: it contains some rampant arches and a crossvault. Theodoricus' mausoleum is the only ltalianm onu ment co mparab le, for stereo to mic virtu osity, toth e c on str uc tio ns o f p ala eo -C h ris tia n S yr ia m en tio ne d-indeed, it is even supposed that the architecto rig in ally carne fro m S yria (A dam , 198 4, 20 7).

H en ce s kilf ul a rc hi te ct ur e c la v e ca rn e in to b ein gat the confines of the Roman and later ByzantineEmpire, an area where, for defence against Persianinvasio ns, the m ost elaborate fortificatio n system sw ere b uilt. T he enco unter in the sam e reg o n b etw eena lon g trad ition of ston e con stru ctio n, the k no wled geof the best Roman architects and engineers andsp ec ific d em an ds o f m ilita ry arch itectu re c an p erh ap sexplain the perfecting of local craftsmen in therealisation of arch or vault stru ctures (M ang o, 19 93).

According to a hypothesis based on nineteenth-century studies by V iollet-le-D uc (1854-1868) andC hoisy (187 3; 188 3), sto necu tting m ethod s ap pear tohave been brought from the East to the West bycrusaders. The development of stereotomy in theSouth of France in the 12th and 13th c enturies is onearg um ent in favo ur o f this thesis (S akaro vitch, 1 998 ).

The first problem that faced medieval builders inthe realisation of vaults w as how to cut the voussoirsconstituting a structu re. T hey seem to have an sw eredth is q uestion from an essen tiaJly geom etrical p oint o fview , without taking statical or structuralconsiderations jnto account. Indeed, stereotom ytreatises illustrate the rules according to whichvoussoirs are to be cut in order to solve the differentgeometrical p ro blem s th at m ay arise.

The various methods with which stones can be cutcan be grouped into two big families: on one hand,archaic methods, and on the other hand cutting pa rquarrissement and pa r panneaux. A rc ha ic m eth od sare those w hich require no preparatory trace. Thereare essen tially th ree: cu tting par ravalement, a lademande an d a la p erch e ( Sa ka ro vitc h, 1 99 8).

Cutting pa r r av al eme nt (F ig . lb ) co nsists in cu ttin gthe stones w hen they are in place in the vault.


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~. .= - .~~--=:=J t ~, // \ h~ :==-=:J c==::J c=~"""""""""""""""~""F ig ur e 1Th e encorbellement m etho d (a ) an d th e m eth od of c uttingpar rava lemen t (b )

B efo re the y are p ut in the ir d efin itiv e p ositio n, the yare roughly hewn, and only when they are in theirdefinitive position are they given their exact shape.For example, in the room of The Treasury ofAtreus, where the encorbellement technique w asused, the intrados of the vault was cut after the stoneswere put in place, the excess stone which formed asor! of upside-down staircase being removed, w ith a

-- ...-.'

"" aFigure 2Cutting par quarri s semen t ( a) a nd par p anneaux (b )

c uttin g m eth od par raval ement . C lo se r to sc ulp turethan to stereotomy, this technique presents twodisadvantages. O n the one hand, it m akes it necessaryto put in bigger stones than are necessary and to cutthem afterwards in difficult w orking conditions. Onthe other hand, the ravalement rem oves the m ortaran d h en ce it ca n o nly b e us ed in co ns truc tion s jointsvij:~ (S ak aro vitc h, 19 98; C ho isy , 18 99 ).

I n c ut tin g la dem ande, each stone is hewn forsubsequent retouching, in relation to the claveauxalready put in place on w hich it is to rest. This type oftechnique, used for example in Romanesquearchitecture, is very slow. The advantage is a greatversatility of use, with relatively little m aterial andwork, since it is possible to choose for each case therough stone that best approxim ates to the claveau tobe m ade (Sakarovitch, 1998; C happuis, 1962).

Probably in order lo accelerate the speed ofconstruction on sites, cutting techniques wereperfected and better exploited the potentialities ofgeometry.

Cutting p ar qu ar ris seme nt, also known asde robement, consists in cutting the stone w ithout thehelp of panneaux, using the heights and depthsdelim iting the voussoir to be m ade.

W ith the method p ar p an nea ux , instead, the

b


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volum e of each voussoir is determ ined starting fromthe surface of each of its faces. Efforts are made toinscribe a voussoir in the smallest possibleparaJlelepiped rectangle. In order to do this, theparallelepiped can be rotated at a certain angle w ithresp ect to th e v ertical. A ll referen ces to it h av in g b eenlost, it is necessary to use panneaux, i.e . m od elsreproducing the shape of the faces of the voussoirw ith th e tr ue d im en sio ns .

C UT TIN G VOU SS OIR S: A G EOMET RIC AL P RO BL EMOR A STA TIC O NE?In order to highlight the peculiarities of stereotom y,which lies somewhere between geometry andstru ctu ral m ech an ics, it seem ed p articu larly u sefu l toanalyse some of the main treatises on coupe despierres, dwelling in particular on one problem: thedeterm ination of th e in clination of the jo ints w hen thearch in trad os an d ex trad os h a v e b een assig ned .

R eg ard in g th e t uil le ur s d e p ie rr e, 1 have i de nt if ie dtw o m ain sch oo ls o f th ou gh t .

A first theory maintains that the straight linesrep resen tin g th e d irectio n o f th e jo in ts m ust co nv erg eat a point, whatever the arch intrados and extradoscurves are like. This theory is found, for example, inV illard de H onnecourt (l3th century) and in M illietDechales (1674). It is based on the executi vesimplicity of the use of a rope to mark out the tracesof the joints, but takes into account neitherconstructive nor statical factors (only in the case ofthe platband, as we shall see, does the theorycorrespond to a correct statical solution to theproblem ). Perhaps it w as precisely b ecause of the lackof consideration for constru ction problem s that thistheory did not enjoy great favour. The fact is that itcontem plates the possibility of realising both acuteand obtuse angles in cutting the stone, and thiscertainly constitutes an element of executived iff ic ulty a nd c on stru ctio n w ea kn es s.

In a sketch by ViJlard (Fig. 3), we find anexplanation of how to trace out the wedges of a pairo f arch es w ith a su sp en ded in term ed iate cap ital, u sin ga rape to m ark out the traces. In this case -exam inedal so b y M iJliet D ech ales (F ig . 4 )- arch -cap ital-archis assim ilated to a sin gle v au lted stru ctu re.

A second theory, instead, maintains theperpendicularity of the jo ints to the intrados line (l).

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Figure 3V ill ar d d e Ho nn ec ou rt 's Carnet (13 th cen tu ry): tracing oU tthe voussoirs of a pair of arches w ith a suspendedi nt ermed ia te c ap it al

..F igur e 4M illiet D ech ales (1 67 4): De areu in alias figurasdegenerante

This theory is present, for example, in Frzier(1737-1739). It is exceJlent from the constructionviewpoint, since the right angle is the easiest toex ecu te an d th e m ost u nifo rm ly resistan t.


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In Frzier' s treatise, stereotom y is view ed as a setof prevaIently geometrical rules. For Frzier theexpression eo up e d es p ierre s does not so much mean" . . . l' o uvrage de l' a rtisan qui taille la pierre, as


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join ts d e lits. A etu ally th e so lutio ns p re se nted byFrzier eorrespond to the statieaIly correet onep ro po se d b y C ou lomb (F ig . 6 ).

..,

F igu re 6Tracing o f the inclination o f the joints in a platband forFrzie r (1737-1739)

T he g eometrical ch aracter o f stereo tomy , at least a sit w as conceived dow n to the start of the eighteenthcentury, culminated in the treatise by Desargues(1640), who applied his universal methods to thetech nique of ston ecuttin g, endeav ouring to solve thep articu lar problem s of stereotom y w ith a single rule.Unlike what happened in all other treatises onstereotomy, which until the 19th century werepresented as more or less complete collections ofc ases, D esarg ues stu dies a sin gle arch itec to nic o bjec t,the descente biaise dans un m ur en talus (F ig . 7 ). T heterm descente indicates a type of cylindrical vaultw hose axis is not horizontal; the term hiuise impliesthat the angle between the axis of the vault and thewall e n tu lu s ( no t v er tic al) is g en er ic .

After defining the technieal terms, Desarguesdefines the planes and straight lines that w ill bereq uired fo r referen ce: th e plan de face, w hich is theplane of the w all; the essieu, whieh is the axis of thetunnel vault and gi ves the direction to theg en er atr ic es ; th e plan dro it a l'essieu, which is theplane perpendicular to (he essieu, which bears thes ec tio n d ro it e o f th e v au lt.

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After setting up these prelim inary hypotheses,D esargues seeks to solve the geom etrical problem ofobtaining the true dimensions of the faces (or of thean gles) req uire d fo r cu ttin g th e sto ne.

F igu re 7Th e descente biaise dans un mur en ta/us stu died b yDesa rgues (1690)

As is well known, it was only in the eighteenthcentury that the arch w as at last studied in a staticalkey.

Philippe De La Hire, a versatile and illustriousFrench scholar kn ow n fo r his T ra it d e Me cani qu e, iscom mon ly rem em bered as tbe first au th or to have dealtw ith th e th em e o f arch es an d v au lts from a statical p oin tof view . Indeed, later scientists in the 18th and 19thcenturies referred to him , considering his theo ries asfirst m ore o r less su ccessfu l atte mp ts to u se m ech an icsto acco un t fo r co nstru ctio n ru les, w hich u ntil th at tim eh ad b ee n e ntr us te d to p ra ctic e a nd in tu itio n.

Philippe D e La H ire w as a disciple of D esargues,an d dealt w ith m echanics, astron om y, m athem atics


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and engineering. He was an outstanding member ofthe Acadmie Royale des Sciences; taughtmathematics at the College de France and also gavelectures at the A cadm ie d' A rchitecture. T here are noprinted versions of these lectures, but twomanuscripts: A rc hite ctu re C iv i/e e Trait de la coupedes p ie rre s; the latter w as a subject he taught for overtw en ty y ea rs .

Th e Trait de la coupe des pierres (late 17thcentury) has not been published; how ever, Frzier, inhis T ra it d e S t r ot om ie , takes up som e topics fromit. In De La Hire's manuscript we find the mostcomm on argum ents reJating to stonecutting, but thege om etric al c on stru ctio ns a re v ery comp lex . O f m ajo rinterest is the start of the treatise, where De La Hireaffirm s that L es ouvriers appellent la science du traitdans la coupe des pierres, celle qui enseigne a tailleret a form er sparm em plusieurs pierres, en telle sortequ' tant jointes toutes ensem ble dans l' o rdre qui leurest convenable, elles ne composent qu'un massifqu'on peut considrer comme une seu l e pierre. Inthis passage (for the first time in a treatise onstereotom y) it is stated that a necessary condition forthe stability of a vaulted structure is the absence ofkin em atic m otio ns b etw een th e pa rts, i.e. e qu ilibriu mbe tw ee n th e pa rts.

A s regards the inclination of the jo ints de te te , fromsom e drawings present in the Trait de la coupe despierres it c an b e ob se rve d tha t it m ust be pe rpe nd icu larto the tangent to the intrados curve in the point ofd iv is ion o f the join t. T he hy po the sis o f orth ogo na lityof the joints to the intrados was also to persist in thetwo works on mechanics by De La Hire, i.e. his Traitd e Mecani qu e (1695) and his subsequent memoir of1712 en ti tl ed Sur la construction des voates dans lesedijices (1731). In these works reference is made totw o fundam ental problem s: one relating to the figureof the arch and the other concerning the sizing of thepiers. In the Trait there is an intuition, though aconfused one, of the pathway that was soon to lead tothe solution of the first problem; the 1712 memoiro ffers the f irs t im pe rfec t bu t pro mis ing so lutio n w hichthrough successive passages was to lead in future toco ll ap se ca lcu la ti on .

Perhaps precisely because statical approaches tothe arch were inaugurated by a scholar coming fromthe world of stereotomy, the orthogonality of thejoints to the intrados appears like an implicith y p o t h e s i s i n t he con si de ra ti on s o f a lmost a ll a ut ho rs

that deal w ith vaulted joints, from that time dow n toCoulomb, such as Charles Bossut, Claude AntoineCouplet, Giordano Riccati, Mariano Fontana andA nto n M aria L org na .In the panorama of historical treatises on archesa nd v au lted stru ctu res , it is in tere stin g to o bs erv e th atthe problem of the inclination of the joints in an archis only studied from a statical point of view by a fewauthors, such as C oulom b, D e N ieuport e V enturoli.

In his Essai (1776), Coulomb sets out to solve theproblem of determ ining the direction of the joints in avaulted structure w hose im rados and extrados curveshave be en assigned, so that the structure w ill be inequilibrium in the absence of friction and cohesionb etw ee n th e jo in ts .

Le t P e Q(.J) be the components, horizontal andvertical respectively, of the resultant of the forcesacting on the part aGMq of the vault (Fig. 8).

:~>"A ",l" : /1>~ './... .B~r '-" Di/V l-:!w';: /~//~ -- l! ;~:... , ,~,~.Ie eF ig ur e 8T he p ro blem 0 1' t he in clin atio n 0 1' t he jo in ts a cc or din g toCoulomb (1776 )

There are tw o conditions to respect for the vault tobe in equilibrium in the case of the absence of frictionan d c oh es ion b etw een th e jo ints:

- the resultant must be perpendicular to the jointMq, whose direction forms an angle . with theve rtic al; i.e. it m ust be :

Q( f ) = P ta n f (1 )- the resultant must always pass between thepoints M an d q.As anticipated, Coulomb shows that in a platband the

straight lines of the joints have to converge at a point.


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In his E lem en ti d i M ec ca nic a ( 18 06 ), V en tu ro lig oes b ack to C oulom b' s treatm ent of the equilibriumof arches in the absence of friction and cohesionbetw een the joints, w ith the intention of proposing are read in g o f th e p ro blem in d iffe re ntial term s.

Venturoli considers an arch E'aE, symmetricalw ith respect to the vertical axis AR , made up ofin fin ite v ou ss oir s w eig hin g MmnN co ntig uo us, b utnot connected to one anothef and resting on them otio nless pu lv in ars w ith ou t frictio n an d co hesio nE e, E 'e ' ( Fig . 9 ).

F igu re 9T he p ro blem o f th e in clin atio n o f th e jo in ts ac co rd in g toVen tu rol i (1806)

Let one project orthogonally the intrados curveAM E on the vertical axis AR . The generic point Mwill be identified by the coordinates AP = x e PM = z;let one d eno te w ith h( o) the length of the generic bedMm, to w hic h th ere c orre sp on ds th e a ng le o an d th ecoordinates (z,x).

C alculating by m eans o f an alytical trig on om etrythe area of the infinitesim al quadrilateral MmnN,comp ris ed b etw een th e jo in t Mm, id en tif ie d b y o, andt he jo in t Nn, id entified by o +do, we obt ai n:

Area MmnN = + h ( f )2d f + h ( f ) [dx sin f + dz co s f](2)The are a of MmnN, calculated in (2), is

proportional to the w eight dQ o f th e in fin ite sim alvo usso ir. Fo r (1), w e w ill have:

dQ = Pdfcos' f H en ce w e o btain the eq uation :

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~ h(f)2df+ h(f)[dx sin f+ dz co s f]= Pdf2 cos2 f (4 )by means of which, knowing the intrados curve andthe law of the inclination of the joints, it is possible toc alc ula te th e J en gth h(o) o f each jo in t Mm an d h en cethe thickness of the arch; or, vice versa, if h(o) isassigned, it is possible to find the direction of thejoints, in order to satisfy the first equilibriumcondition.

Another scholar that considered the influence ofvou ssoir cutting o n the equ ilibrium of a m aso nry archwas de Nieuport (1781). Starting from De La Hire'stheorem , he considered the fact that, in general, in ana rch th ere are th ree fu nd am en ta l cu rv es: th e in trad os,th e ex trad os and th e cu rve form ed by th e intersectionpoints of the straight lines of the joints. D e N ieuportstudied not only cases in which the joints areorthogonal to the intrados or converge at a point, butalso m ore gen eral cases. lt is necessary, th en, to m akereference lO the curvature radius and consider thevoussoirs as infinitely sm all but having thickness.The three fundamental curves are connected to onean other b y th e eq uilib riu m relation s. K now ing tw o ofthem through the equilibrium conditions, onedeterm ines the third curve (R adelet de G rave, 1995).The memoir continues with the elaboration ofcomplex anaJytic developments, backed up byg rap hic resu lts (F ig . 1 0).

lt is thus possible to determ ine the law governingthe inclination of the joints in vaulted structureshaving the intrados and extrados assigned, so as to

_A~.." Y e 7/

$:JS":.'


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ensure the equilibrium in sliding, even in the absenceof friction and cohesion betw een the voussoirs.

H ereafter, w ithout illustrating the m athem atics ofthe problem , I give here some graphic results forsome structural typologies present in stereotomytreatises. T he geom etrical constructions proposed bythe ta illeu rs d e p ie rr e presuppose that the straightlines representing the inclination of the joints willconverge at a single point, or w ill be orthogonal to theintrados. However, the equilibrium solution in thcabsence of friction and cohesion betw een the jointsdoes not coincide w ith the stereotom ic solution (A ita,2001). By way of example, in the case of a circulararch w ithout friction or cohesion betw een the joints,equilibrium in sliding is ensured if they are inclinedas in Fig. 14: hence they w ill not prove perpendicularto the intrados (a hypothesis always implicitlyconsidered both in the stereotom y treatises and in thestatical ones before C oulom b, Figure 13).

CONCLUSIONS

In o rd er b ette r to u nd er sta nd th e d elic ate r ela tio ns hipbetween stereotomy and mechanics, it is perhapsuseful, at the end, to observe that the arch modeladopted by Coulomb, de Nieuport and Venturoli forthe inclination of the joints from a statical point ofview is that of a system of rigid heavy blocks,perfectly sm oothed and devoid of friction, analogousto the one first proposed by De La Hire. In effect, thep re se nce o f fric tio n a nd c oh esio n en su re s the sta bility

Figure 11Inclination of the joints in a platband w ith a horizontale xt ra do s a nd an nt ra do s e n chape : equil ibrium solut ion ina bs e nc e o " t " r ic t io n a nd c o he sio n

Figure 12Inclination of the joints in a platband w ith extrados andintrados en chape: equilibrium solution in absence off ri ct io n a nd c oh es io n

Figure 13Inclination of the joints in a circular arch: the hypothesisa lw ay s im plic itly c on sid ere d b oth in th e ste re oto my tr ea tisc sand in the sta tical one s before C oulom b

of vaults made in accordance with the principies ofstereotom y. A t al] events, it is interesting to observet ha t s ta ti ca l mode ll in g -a la De La Hire- did notinfluence la thorie et la pratique de la coupe despierres, which instead developed on the basis ofgeometrical principies and empirical rules thatsedimented in the course of time, perm itting theconstruction of architectures of inestim able valuea nd , p ar ad ox ic ally , o f g re at s tr uc tu ra l in te re st.

AKNOWLEDGEMENTS

A special thank to Geom. Gabriele Mazzei for hishe]p in preparing the graphic material of this papera nd 1 'o r h is p ra ct ic al s ug ge st io ns .


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F ig ur e 1 4In clin atio n o f th e jo in ts in a c ircu lar arch : eq uilib riu msol uti on i n a bs en ce o f f ric ti on a nd c ohes ion

NOTES

T he term in tr ad os lin e refers to th e cu rv e d eterm in edb y th e intersection of th e intrad os su rface o f the v aultand aplane used to draw up the pure. It is u su allyorthogonal to the axis of the vault, but can also be avertical plan e if this is suited to m akin g the trail easy,o r a no th er su itab le p lan e.

REFERENCE L IST

Adam, J.-P. 1984. La construction romaine, matriaux ette ch niq ue s. P ic ard . P aris: 20 7.

A ita , D . 2 00 1. U na p oss ib ile r ile ttu ra d el p ro ble ma d ell'a rc otra geometria e meccanica. Atti del XV ConvegnoA/M ETA di M eccanica Teorica e Applicala (Taorm ina,26-29 S ep lemb er 2 0( 1) .

Benvenuto, E. 1981. La scienza delle costruzioni e il suos vi lu pp o s to ri co . S an so ni . F lo re nc e.

Bossut C. 1778. Recherches sur r quilibre des votes.M moires de l'Acadmie Royale des Sciences, amze1 77 4. P ar is : 5 34 -5 66 .

C happuis, R . 1962. G om etrie et structure des coupoles surpendentifs dans les gliscs romanes entre Loire etPyrnes. Bu ll el in monumenl al , 1 20 : 7 -3 2.

C ho isy , A . 1 87 3. L 'art d e b atir chez l es Romai ns . Ducher .Paris.C hoisy, A . 1883. L 'ar! de batir chez les B yzantins. L ibrairied e la S oc i t a no ny me d e p ub blic atio ns p r iod iq ue s. P aris.

Choisy, A. 1899. Histoire de l'architecture. Gauthier-V i ll ar s. P a ri s.

Coulomb, C. 1776. Essai sur une application de regles dem axim is et minim is a quelques problem es de Statique,r el at if s a r A rch it cc tu re . M m oires de M alhm alique el dePhysique prsenls a l'Acadm ie R oyale des Sciences,

D . A ita

par divers Savans, el Ias dans les Assembles, anne1 77 3. P ar is : 1 7.3 43 -3 82

Couplet, C. A . 173]. D e la pousse des votes. M m oires d el'Acadmie Royale des Sciences, anne /729. Paris:79-117.

C ouplet, C . A . 1732. Seconde partie de l' e xam en des votes,Mmoires de l'Acadmie Royale des Sciences, anne1730. Par is : 1 17 -1 41 .

D e La H ire, P. 1695. Trait de m chanique, ou ron expliquetout ce qui est ncessaire dans la pratique des A rts, et lespropriets des corps pesa nts lesquelles ont eu plus grandusage dan s la P hysique. Im prim erie R oyale. P aris.

De La Hire, P. End of 17th c entury. Trait de la coupe despierres. M anoscritto n. 1596 B ibliotheque de l' i nstitut deF ra nc e. P ar is .

D e La H ire, P. 1731. Sur la construction des votes dans lesdifices. M m oires de l'A cadmie Royale des Sciences,anne 1 7/ 2. P ar is .

De La Rue, J. B. 1728. Trait de la coupe des pierres. P.-A.Ma rt in . P a ri s.

D e N ieuport, C . F . 1781. E ssai analytique sur la m chaniqued es v o te , 1 8/5 /1 77 8. M m oire s d e I 'A ca d mie im p ria lee l ro ya le d es sc ie nc es e l b elle -I ettre s d e B ru xe lle s: V ol. 1 1.

Desargues, G. 1640. Brouillon project d'exemple d'unemaniere universelle du S. G . D . L . Touchant la pratiquedu trait a preuves pour le coupe des pierres enarchitecture; et de I'clarissement d'une maniere derduire a u petit pied en perspective com me en gom tral,et de tracer tous quadrants plats d'heures gales au soleil.Paris.

D e H on ne co urt, V . 1 3th ce ntU ry . C ar ne t.Frzier, A. F. 1737-1739. La thorie et la pratique de la

coupe des pierres et des bois pour la construction desvotes et autres parties des batim ens civils et m ilitaires outrait de strotomie a l'usage de l'architecture.Strasburg-Paris.

M illiet Dechales, C. F. 1674. Cursus seu mundusmathematicus r . . . l. Lyon.

M an go , C . 19 93 . A rc hite ctu re b yz an tin e. G allim ar d/E le cta .Milan.

Radelet de Grave, P. 1995. Le De curvatura fornicis deJacob Bernoulli ou I'introduction des infiniment petitsdan s le calcul des votes. In: Radelet de Grave, P;Benvenuto, E. (eds). Between mechanics anda rc hi te ctu re . B ir kh au se r. B as le .

S akarovitch, J. 1998. E pures d'architecture. D e la coupe despierres it la gom etrie descriptive, X Vle-X IX e siecles.B ir kh au se r. B as e! .

V io lle t- Ie D uc , E . 1 85 4-1 86 8. D ic tio nn air e d e I 'a rc hite ctur efran~ aise du X I au X VI siecle. P aris: item T rait.

V enturoli, G . 1806. Elementi di M eccanica. Per i FratelliM asi e C om pa gno. B ologna: 66-76.

V ettone, R . 1996. B atir, m anuel de la construction. PressesP oly te ch niq ue s e t u niv er sita ir es R om and es . L au sa nn e.

between geornetry and mechanics - a re-exarnination of the principies of stereo to my from astatical...

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