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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 a

mechanical interpretation of the geometrical

p rin cipie s g uidin g th e art of s tere oto my fo r d esig ning

masonry arches. The treatises on the

oupe es

pierres -even those published after the birth of

m odern structural m echanics- deal w ith the design

o f v au lte d str uc tu re s fro m a n e ss en tia lly ge om etric al

point of view. For instance, the m ain issue of cutting

voussoirs, as con cerned the inclination of the joints,

was dealt with in geom etrical term s without taking

any statical consequences into account. W ith

reference to this problem , the

oupe es pierres

develops tw o geom etrical criteria: the first requires

that the joints converge at a single point e.g. Villard

de Honnecourt ; the second requires that the joints be

p erp end ic ular to th e intr ad os of the arc h e.g . F rz ier .

In order to determ ine the degree of stability

corresponding to these geometrical criteria, the

present paper analyses the problem of stonecutting in

statical terms by considering the equilibrium of

voussoirs in the absence of friction and cohesion. T he

works of Coulomb, de Nieuport and Venturoli are

exam ined and the statical form ulation of the problem

is 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 RE OT OM Y

From the M iddle A ges to the 18th century, stereotom y

was considered the most important construction

technique. By m eans of geom etrical principIes, in

Dan ila A ita

fact, it allow s one to visualize a tridim ensional object

by m eans of a bidim ensional reproduction and to give

an appropriate form to each of the voussoirs m aking

up a vault. Tn this way, it is possible to construct

vaults, domes and squinches and to perform an

in 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 the

design of com plex vaulted structures seem s to hark

back sim ply to the solution of geom etrical problem s.

I n a ntiqu ity , th e arc h w as co nsid ere d a s a pr e-e min en t

example of geometrical perfection, containing in

itself a principie of statical perfection: the com mon

conviction was that geometry, not statics, could

p ro vid e th e sa fe st p ro por tion s f or de sig nin g arc hes .

The ancient Egyptians, Greeks and Romans cut

stones into large blocks, so that they form ed sound

constructions and their weight took the place of

mortar.

W ith the passing of tim e, efforts were made to

reduce the dim ensions of the stones constructing the

structure, so as not to place excessive organisational

d em an ds o n th e b uildin g s ite. H en ce th e fir st o bjec tive

in perfecting techniques for cutting stone is finding

stability com parable to that w hich w ould be obtained

using m uch bigger stones, w hile using sm aller ones.

A second problem relating to stonecutting is linked

to the fact that stone is characterised by a high

resistance to compression and a low resistance to

traction and to bending. For this reason in ancient

tem ples the m axim um distance between the axes of

the colum ns did not exceed 4-5 m etres. H ence

t

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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6 2

s e c o n d objective in improvement of hewn stone

construction techniques is to solve the problem of

getting over bigg er inter-ax is sp aces an d co vering s.

T he s o- ca Jle d encorbellement method (Fig. la) was

the first solution, used starting from antiquity. The

construction principie is very sim ple: it consists in

using overhanging (i.e., corbeJled) stones, w ith the

b ed s a lwa ys ho ri zon ta l.

Though this technique may appear unrefined and

prim itiv e, it m ade it po ssible to realise som e w ork s o f

inestimable value. One of the most ancient and

celebrated w as the so-called room of The Treasury

of A treu s, a m asterpiece o f M ycen ean arch itecture

done in the 13th c entury Be. From the 7th to the 2 d

centu ry B C the E truscans frequ ently used co rb ellin g

to cover some funerary chambers (one thinks, for

example, of the tombs at Casale Marittimo and

M ontagn ola) or to m ak e arch es (S akaro vitch, 1 99 8).

While encorbellement is a techn ique that carne in to

b eing fo r constru ctin g hew n stone stru ctures, the arch

and the tunnel vault carne into being as brick

constructions. They appeared starting from the

beginning of the 3td millennium in regions where

there was a shortage of wood, like M esopotamia and

the vaJley of the Nile, but carne to be part of the

sto necu tting techn ique o nly w ith th e in tro duction of

the

voussoir,

a w edg e-shap ed sto ne w ith tw o obliqu e

faces by means of which it rests on the adjacent

v 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 ue

to 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 back

to earlier than the end of the

4th century or the

beginning of the 3' . W e are referring to the arches

that cover the gates of fortifications at Eraclea of

Latmos and at Velia, which were ancient Phocian

co lo nies in C entral Italy, or the on es un der the vau lted

room s of some Macedonian tombs (Langhada,

Leucadia) or again the underground cham bers of the

th 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 the

lateral dissipation of the thrusts exerted by the vault

or by the arch is sol ved, in that the vault belongs to a

s tru ctu re th at is in terred o r th e arch co vers an ap ertu re

belonging to a wall. Down to the 2nd century BC all

s tru ctu re s w ith a rc he s o r v au lts a re o f th is

type

This

is stiJl the case in the very beautiful vaults of the

staircase at the Pergam os Gymnasium. It was the

Roman builders that, starting from the end of the

2

D.Aita

century BC, first made the vault a free volume: with

them , the vault show ed itself openly, carne out of the

ground, and becam e a noble construction, no longer

confined to subterran ean constructio ns an d fun erary

a 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 c entury AD

the

Philippopolis

theatre was built on the Jebel ed-

Druz: it contains some rampant arches and a cross

vault. Theodoricus' mausoleum is the only ltalian

m onu ment co mparab le, for stereo to mic virtu osity, to

th 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 architect

o rig in ally carn e from S yria (A dam , 19 84, 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 g

at the confines of the Roman and later Byzantine

Empire, an area where, for defence against Persian

invasio ns, the m ost elaborate fortificatio n system s

w ere b uilt. T he encou nter in the sam e reg o n b etw een

a lon g trad ition of ston e con stru ctio n, the k no wled ge

of the best Roman architects and engineers and

sp ec ific d em an ds o f m ilita ry arch itectu re c an p erh ap s

explain the perfecting of local craftsmen in the

realisation 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) and

C hoisy (187 3; 188 3), sto necu tting m ethod s ap pear to

have been brought from the East to the West by

crusaders. The development of stereotomy in the

South of France in the 12th and 13th c enturies is one

arg um ent in favo ur o f this thesis (S akaro vitch, 1 998 ).

The first problem that faced medieval builders in

the realisation of vaults w as how to cut the voussoirs

constituting a structu re. T hey seem to have an sw ered

th is q uestion from an essen tiaJly geom etrical p oint o f

view, without taking statical or structural

considerations jnto account. Indeed, stereotom y

treatises illustrate the rules according to which

voussoirs are to be cut in order to solve the different

geometrical

p ro blem s th at m ay arise.

The various methods with which stones can be cut

can be grouped into two big families: on one hand,

archaic methods, and on the other hand cutting

pa r

quarrissement and pa r panneaux. A rc ha ic m eth od s

are those w hich require no preparatory trace. There

are essen tially th ree: cu tting par ravalement, a la

demande

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 g

the stones w hen they are in place in the vault.

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B etw ee n g eo me tr y a nd m ec ha nic s

16 3

~.

.= -

.

~

~

--=:=J

t

~

,

//

\

h

~ :==

-=:J c=

=::J c=

~ ~

F ig ur e 1

Th e encorbellement m etho d (a ) an d th e m eth od of c utting

par rava lemen t

(b )

B efo re the y are p ut in the ir d efin itiv e p ositio n, the y

are roughly hewn, and only when they are in their

definitive position are they given their exact shape.

For example, in the room of The Treasury of

Atreus, where the

encorbellement

technique w as

used, the intrados of the vault was cut after the stones

were put in place, the excess stone which formed a

sor of upside-down staircase being rem oved, w ith a

--

...-

.'

a

Figure 2

Cutting 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 ture

than to stereotomy, this technique presents two

disadvantages. O n the one hand, it m akes it necessary

to put in bigger stones than are necessary and to cut

them afterwards in difficult w orking conditions. O n

the other hand, the ravalement rem oves the m ortar

an d h en ce it ca n o nly b e us ed in co ns truc tion s joints

vij:~

(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 for

subsequent retouching, in relation to the claveaux

already put in place on w hich it is to rest. This type of

technique, used for example in Romanesque

architecture, is very slow. The advantage is a great

versatility of use, with relatively little m aterial and

work, since it is possible to choose for each case the

rough stone that best approxim ates to the claveau to

be m ade (Sakarovitch, 1998; C happuis, 1962).

Probably in order lo accelerate the speed of

construction on sites, cutting techniques were

perfected and better exploited the potentialities of

geometry.

Cutting p ar qu ar ris seme nt, also known as

de robement,

consists in cutting the stone w ithout the

help of

panneaux,

using the heights and depths

delim iting the voussoir to be m ade.

W ith the method

p ar p an nea ux ,

instead, the

b

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16 4

volum e of each voussoir is determ ined starting from

the surface of each of its faces. Efforts are made to

inscribe a voussoir in the smallest possible

paraJlelepiped rectangle. In order to do this, the

parallelepiped can be rotated at a certain angle w ith

resp ect to th e v ertical. A ll referen ces to it h av in g b een

lost, it is necessary to use panneaux, i.e . m od els

reproducing the shape of the faces of the voussoir

w ith th e tr ue d im en sio ns .

C UT TIN G V OU SS OIR S: A G EO ME TR IC AL P RO BL EM

OR A STA TIC O NE?

In order to highlight the peculiarities of stereotom y,

which lies somewhere between geometry and

stru ctu ral m ech an ics, it seem ed p articu larly u sefu l to

analyse some of the main treatises on coupe des

pierres, dwelling in particular on one problem: the

determ ination of th e in clination of the jo ints w hen the

arch in trad os an d ex trad os h a v e b een assig ned .

R eg ard in g th e

t ui ll eu rs d e p ie rr e,

1 have i de nt if ie d

tw o m ain sch oo ls o f th ou gh t .

A first theory maintains that the straight lines

rep resen tin g th e d irectio n o f th e jo in ts m ust co nv erg e

at a point, whatever the arch intrados and extrados

curves are like. This theory is found, for example, in

V illard de H onnecourt (l3th century) and in M illiet

Dechales (1674). It is based on the executi ve

simplicity of the use of a rope to mark out the traces

of the joints, but takes into account neither

constructive nor statical factors (only in the case of

the platband, as we shall see, does the theory

correspond to a correct statical solution to the

problem ). Perhaps it w as precisely because of the lack

of consideration for constru ction problem s that this

theory did not enjoy great favour. The fact is that it

contem plates the possibility of realising both acute

and obtuse angles in cutting the stone, and this

certainly constitutes an element of executive

d iff ic ulty a nd c on stru ctio n w ea kn es s.

In a sketch by ViJlard (Fig. 3), we find an

explanation of how to trace out the wedges of a pair

o f arch es w ith a su sp en ded in term ed iate cap ital, u sin g

a rape to m ark out the traces. In this case -exam ined

al so b y M iJliet D ech ales (F ig . 4 )- arch -cap ital-arch

is assim ilated to a sin gle v au lted stru ctu re.

A second theory, instead, maintains the

perpendicularity of the jo ints to the intrados line (l).

D . A it a

Figure 3

V ill ar d d e Ho nn ec ou rt 's

Carnet

(13 th cen tu ry ): tracing oU t

the voussoirs of a pair of arches w ith a suspended

i nt ermed ia te c ap it al

..

F igur e 4

M illiet D ech ales (1 67 4): De areu in alias figuras

degenerante

This theory is present, for example, in Frzier

(1737-1739). It is exceJlent from the construction

viewpoint, since the right angle is the easiest to

ex ecu te an d th e m ost u nifo rm ly resistan t.

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B etwe en g eo me tr y a nd m ec ha ni cs

16 5

In Frzier' s treatise, stereotom y is view ed as a set

of prevaIently geometrical rules. For Frzier the

expression 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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16 6

join ts d e lits. A etu ally th e so lu tion s p re se nted by

Frzier eorrespond to the statieaIly correet one

p ro po se d b y C ou lomb (F ig . 6 ).

..,

F igu re 6

Tracing o f the inclination o f the joints in a platband for

Frzie r (1737-1739)

T he g eo metrical ch aracter o f stereo to my , at least a s

it w as conceived dow n to the start of the eighteenth

century, culm inated in the treatise by Desargues

(1640), who applied his universal methods to the

tech nique of ston ecuttin g, endeav ouring to solve the

p articu lar problem s of stereotom y w ith a single rule.

Unlike what happened in all other treatises on

stereotomy, which until the 19th century were

presented as more or less complete collections of

c ases, D esarg ues stu dies a sin gle arc hitec to nic o bject,

the descente biaise dans un m ur en talus (F ig . 7 ). T he

term descente indicates a type of cylindrical vault

w hose axis is not horizontal; the term

hiuise

implies

that the angle between the axis of the vault and the

wall e n tu lu s ( no t v er tic al) is g en er ic .

After defining the technieal terms, Desargues

defines the planes and straight lines that will be

req uired fo r referen ce: th e plan de face, w hich is the

plane of the w all; the essieu, whieh is the axis of the

tunnel vault and gi ves the direction to the

g en er atr ic es ; th e

plan dro it a l'essieu,

which is the

plane perpendicular to (he

essieu,

which bears the

s ec ti on d ro it e

o f th e v au lt.

D.Aita

After setting up these prelim inary hypotheses,

D esargues seeks to solve the geom etrical problem of

obtaining the true dimensions of the faces (or of the

an gles) req uire d fo r cu ttin g th e sto ne.

F igu re 7

Th e descente biaise dans un mur en ta/us stu died b y

Desa rgues (1690)

As is well known, it was only in the eighteenth

century that the arch w as at last studied in a statical

key.

Philippe De La Hire, a versatile and illustrious

French scholar kn ow n fo r his

T ra it d e M e ca ni qu e,

is

com mon ly rem em bered as tbe first au th or to have dealt

w ith th e th em e o f arch es an d v au lts fro m a statical p oin t

of view. Indeed, later scientists in the 18th and 19th

centuries referred to him , considering his theo ries as

first m ore o r less su ccessfu l atte mp ts to u se m ech an ics

to acco un t fo r co nstru ctio n ru les, w hich u ntil th at tim e

h 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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etween

g eo me tr y a nd m ec ha ni cs

16 7

and engineering. He was an outstanding m em ber of

the Acadmie Royale des Sciences; taught

m athematics at the College de France and also gave

lectures at the A cadm ie d' A rchitecture. T here are no

printed versions of these lectures, but two

manuscripts:

A rc hite ctu re C iv i/e

e

Trait de la coupe

des p ie rre s; the latter w as a subject he taught for over

tw en ty y ea rs .

Th e

Trait de la coupe des pierres

(late 17th

century) has not been published; how ever, Frzier, in

his

T ra it d e S t r ot om ie ,

takes up som e topics from

it. In De La Hire's m anuscript we find the most

com mon argum ents reJating to stonecutting, but the

ge om etric al c on stru ctio ns a re v ery co mp lex . O f m ajo r

interest is the start of the treatise, where De La Hire

affirm s that L es ouvriers appellent la science du trait

dans la coupe des pierres, celle qui enseigne a tailler

et a form er sparm em plusieurs pierres, en telle sorte

qu' tant jointes toutes ensem ble dans l' o rdre qui leur

est convenable, elles ne composent qu'un massif

qu'on peut considrer comme une seu le pierre. In

this passage (for the first tim e in a treatise on

stereotom y) it is stated that a necessary condition for

the stability of a vaulted structure is the absence of

kin em atic m otio ns b etw een th e pa rts, i.e. e qu ilibriu m

be tw ee n th e pa rts.

A s regards the inclination of the jo ints de te te , from

som e drawings present in the

Trait de la coupe des

pierres it c an b e ob se rve d tha t it m ust be pe rpe nd icu lar

to the tangent to the intrados curve in the point of

d iv is ion o f the join t. T he hy po th esis o f orth ogo na lity

of the joints to the intrados was also to persist in the

two works on mechanics by De La Hire, i.e. his

Trait

d e Mecani qu e

(1695) and his subsequent m em oir of

1712 en ti tl ed Sur la construction des voates dans les

edijices

(1731). In these works reference is m ade to

tw o fundam ental problem s: one relating to the figure

of the arch and the other concerning the sizing of the

piers. In the Trait there is an intuition, though a

confused one, of the pathway that was soon to lead to

the solution of the first problem ; the 1712 memoir

o ffers the f irs t im pe rfec t bu t pro mis ing so lutio n w hich

through successive passages was to lead in future to

co ll ap se ca lcu la ti on .

Perhaps precisely because statical approaches to

the arch were inaugurated by a scholar com ing from

the world of stereotomy, the orthogonality of the

joints to the intrados appears like an implicit

h 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 down to

Coulom b, such as Charles Bossut, Claude Antoine

Couplet, Giordano Riccati, Mariano Fontana and

A nto n M aria L org na .

In the panorama of historical treatises on arches

a nd v au lted stru ctu res , it is in tere stin g to o bs erv e th at

the problem of the inclination of the joints in an arch

is only studied from a statical point of view by a few

authors, such as C oulom b, D e N ieuport e V enturoli.

In his Essai (1776), Coulom b sets out to solve the

problem of determ ining the direction of the joints in a

vaulted structure w hose im rados and extrados curves

have be en assigned, so that the structure will be in

equilibrium in the absence of friction and cohesion

b etw ee n th e jo in ts .

Le t

P

e Q(.J) be the components, horizontal and

vertical respectively, of the resultant of the forces

acting on the part

aGMq

of the vault (Fig. 8).

:

~

>

A ,

l :

/1>

~

./

...

.B

~r '-

D

i/

V

l-: w';

:

/~//~

--

l

;

~:... ,

,~,~.I

e

e

F ig ur e 8

T he p ro blem 0 1' t he in clin atio n 0 1' t he jo in ts a cc or din g to

Coulomb (1776 )

There are tw o conditions to respect for the vault to

be in equilibrium in the case of the absence of friction

an d c oh es ion b etw een th e jo ints:

- the resultant m ust be perpendicular to the joint

Mq, whose direction forms an angle

.

with the

vertical; i.e. it m ust be:

Q(

=

P ta n f

(1 )

- the resultant must always pass between the

points

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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16 8

In his

Elem enti di M eccan ica

( 18 06 ), V en tu ro li

g oes b ack to C oulom b' s treatm ent of the equilibrium

of arches in the absence of friction and cohesion

betw een the joints, w ith the intention of proposing a

re read in g o f th e p ro blem in d iffe re ntial term s.

Venturoli considers an arch E'aE, symmetrical

with respect to the vertical axis AR , made up of

in fin ite v ou ss oir s w eig hin g

MmnN

co ntig uo us, b ut

not connected to one anothef and resting on the

m otio nless pu lv in ars w ith ou t fric tio n an d co hesio n

E e, E 'e '

( Fig . 9 ).

F igu re 9

T he p ro blem o f th e in clin atio n o f th e jo in ts ac co rd in g to

Ven tu rol i (1806)

Let one project orthogonally the intrados curve

AM E on the vertical axis AR . The generic point M

will be identified by the coordinates

AP

= x

e

PM

=

z;

let one d eno te w ith h( o) the length of the generic bed

Mm,

to w hic h th ere c orre sp on ds th e a ng le

o

an d th e

coordinates (z,x).

C alculating by m eans o f an alytical trig on om etry

the area of the infinitesim al quadrilateral MmnN,

co mp ris ed b etw een th e jo in t

Mm,

id en tif ie d b y o, and

t 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 al

vo usso ir. Fo r (1), w e w ill have:

dQ

=

Pdf

cos'

f

H en ce w e o btain the eq uation :

D . A ita

~ h(f)2df

+

h(f)[dx sin f

+ dz co s f]

=

Pdf

2 cos2

f

(4 )

by means of which, knowing the intrados curve and

the law of the inclination of the joints, it is possible to

c alc ula te th e J en gth h(o) o f each jo in t Mm an d h en ce

the thickness of the arch; or, vice versa, if h(o) is

assigned, it is possible to find the direction of the

joints, in order to satisfy the first equilibrium

condition.

Another scholar that considered the influence of

vou ssoir cutting on the equ ilibrium of a m aso nry arch

was de Nieuport (1781). Starting from De La Hire's

theorem , he considered the fact that, in general, in an

a 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 intersection

points of the straight lines of the joints. D e N ieuport

studied not only cases in which the joints are

orthogonal to the intrados or converge at a point, but

also m ore gen eral cases. lt is necessary, th en, to m ake

reference lO the curvature radius and consider the

voussoirs as infinitely sm all but having thickness.

The three fundamental curves are connected to one

an other b y th e eq uilib riu m relation s. K now ing tw o of

them through the equilibrium conditions, one

determ ines the third curve (R adelet de G rave, 1995).

The memoir continues with the elaboration of

complex anaJytic developments, backed up by

g rap hic resu lts (F ig . 1 0).

lt is thus possible to determ ine the law governing

the inclination of the joints in vaulted structures

having the intrados and extrados assigned, so as to

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B etw ee n g eo me tr y a nd m ec ha nic s

16 9

ensure the equilibrium in sliding, even in the absence

of friction and cohesion betw een the voussoirs.

H ereafter, w ithout illustrating the m athem atics of

the problem , I give here some graphic results for

some structural typologies present in stereotom y

treatises. T he geom etrical constructions proposed by

the ta illeu rs de p ie rr e presuppose that the straight

lines representing the inclination of the joints will

converge at a single point, or w ill be orthogonal to the

intrados. However, the equilibrium solution in thc

absence of friction and cohesion betw een the joints

does not coincide w ith the stereotom ic solution (A ita,

2001). By way of example, in the case of a circular

arch w ithout friction or cohesion betw een the joints,

equilibrium in sliding is ensured if they are inclined

as in Fig. 14: hence they w ill not prove perpendicular

to the intrados (a hypothesis always implicitly

considered both in the stereotom y treatises and in the

statical 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 hip

between stereotomy and mechanics, it is perhaps

useful, at the end, to observe that the arch model

adopted by Coulom b, de Nieuport and Venturoli for

the inclination of the joints from a statical point of

view is that of a system of rigid heavy blocks,

perfectly sm oothed and devoid of friction, analogous

to the one first proposed by De La Hire. In effect, the

p re se nce o f fric tio n a nd c oh esio n e nsu re s the sta bility

Figure 11

Inclination of the joints in a platband w ith a horizontal

e xt ra do s a nd an nt ra do s e n chape : equil ibrium solut ion in

a bs e nc e o t r ic t io n a nd c o he sio n

Figure 12

Inclination of the joints in a platband w ith extrados and

intrados en chape: equilibrium solution in absence of

f ri ct io n a nd c oh es io n

Figure 13

Inclination of the joints in a circular arch: the hypothesis

a lw ay s im plic itly c on sid ere d b oth in th e ste re oto my tr ea tisc s

and in the sta tical one s before C oulom b

of vaults m ade in accordance with the principies of

stereotom y. A t al] events, it is interesting to observe

th at s ta tic al m o de llin g -a la De La Hire- did not

influence

la thorie et la pratique de la coupe des

pierres, which instead developed on the basis of

geometrical principies and empirical rules that

sedimented in the course of tim e, perm itting the

construction of architectures of inestim able value

a 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 his

he]p in preparing the graphic m aterial of this paper

a nd 1 'o r h is p ra ct ic al s ug ge st io ns .

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17 0

F ig ur e 1 4

In clin atio n o f th e jo in ts in a c ircu lar arch : eq uilib riu m

sol uti on i n a bs en ce o f f ric ti on a nd c ohes ion

NOTES

T he term

i ntr ad os li ne

refers to th e cu rv e d eterm in ed

b y th e intersection of th e intrad os su rface o f the v ault

and aplane used to draw up the

pure.

It is u su ally

orthogonal to the axis of the vault, but can also be a

vertical plan e if this is suited to m akin g the

trail

easy,

o r an oth er su itab le p la ne.

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