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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
=
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]
=
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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