STRUCTURAL ANALYSIS OF HISTORICAL CONSTRUCTI
ONS
11
P.
Roca
, J.
L.
Gonúlez,
E.
Ona e y B.
Lo
u enço (Eds.)
© CIMNE.
Ba
celona 1998
ANAL YSIS OF THE STRUCTURE OF GOTHIC CATHEDRALS
APPLlCATION TO BARCELONA CATHEDRAL
Pe e
Roca,
LaTa
Pe
ll
eg
ini,
Eu
genio Onn e nnd Alex Hanganu*
U
nil
'e
s
ila/
Po
lil
ec
lli
C
{j
de Cow/!
lIl
ya
·Inle na io lol Ceme
o
Nwne ical Me hods
in
Ell
ginu ing (CIMNE)
G a
ll
Capi à
S/
li
08034 Ba
c
IO
1(J.
Spai
ll
SU
MMARY
The s lldy
a
ancien s uc u
e
s composed
a
Slane a ches and aul s equi es
h
e use
o
speci ic echniques
a
analysis endowed wilh powe ul p ocedu cs a lhe
madelling
o
geomel
y,
he simula ion
o
lh
e mechanical espanse
a
lh
e a iaus
ma e iais (including ashla blacks and
ma
sa
n
y.
backings and ills) and
h
e s
im
ula ion
a
lhe possible agenlS (g a i y,
Vind
, ea hquake, se lemems) w
hi
ch may a ec lhe
s uc u e
o
may ha e a ec ed
i
h augh
hi
sla
y.
Ma e speci ically,
Ih
ose
e
chniques
mu
sl be able 10 epToduce lhe aclu
al
condi ions
a
h usl equilib ium be ween lhe la ge
numbe o cu ed. linea o wo-dimensional membe s in ol ed.
The alidi y
o
se e al modelling echniques wi
h
a e
y
di e enl le ei
a
sa
phis ica ion
is
sludied Ih ough Ihei use
in
lhe s udy
a
a pa icula Galhic
cans uc ion: Ba celona Ca hed al. The a ious lechniques used we e de eloped
a
lh
e
School
a
Ci il Enginee ing a lhe Tech
ni
cal
Un
i e si y
o
Ca alonia and ha e al eady
been used
in
he analysis
a
a se! a hislo ical cons uc ions, including lhe Basi
li
ca
o
San Ma co (Onale e
ai
.,
19
97) and lhe C yp
o
lhe Colõnia Güell (Roca, 1997
).
The p ese
n
s udy
o
Ba celona Ca hed al, now unde de elop nen ,
is
Ihus no only
ai
med a gaining a
beue
unde s anding o a ious
as
pec s
o
lhe Tesis ance
o
Go hic
cons uc ions (and he sou hem a
nd
Ca alan Go
hi
c, in pa icula ) bu also a
app aising lhe possibili ies and limi a ions
o
a ious analy ical echniques.
232 STRUCTURAL
ANALYSIS
OF
HI
STO
RI
CAL CONSTRUCTI
ONS
11
1
SOME
CONSIDERAT
IONS
REGARDING
GOTHIC
STRUCTURE
1.1
O e all s uc u al dcsign
The esis ance
c
Go hic cons uc i
on
ga e ise
o
a cc ain amoun
c
con o e sy
a e Viollcl-le-Duc's
i
s se ious a emp
lo
p opose a alienal ision
c
i s
mechanical p incipies and pa icula s uc u aI o de ing.
Hi~
cnlhusias ic
in e p e a i
on
o
GO
lhic s uc u e as a ully alianal objec was la e con adic ed by olhc
esca che s (see
B.
Bassegoda, 1974). Ra he lhan de ending Violle 's poin s
a
iew.
hey ound ins ances
o
s uc u al a bi a iness and unce ain ies which we e only o bc
unde s ood as composi ional mo i s, cons uc ion de ices,
C
pu ely a consequence
c
he limi ed knowl
edge
c
in uilion
c
lhe his o ical dcsigne s.
Violle 's heo y p esen ed Go hic s uc u al o de ing as a sys em sl iclly based on
equilib ium and jus i ied by
i .
Fu he mo e,
ali
de ices
o
Go hic S
l u
clu e we e laid
Oul
so
as
lo
co
n ibu e o he achie emen o
an
o e all duc ili y.
In
hi
s con e
x
,
du
clili y musI be unde slOod as he eapacily o lhe sl uclu e
o
accepl signi icanl
de o ma ion wi hou sus
ai
ning damage o ailing. I
is
his duc ili y which allowed he
Go
hi
c s uc u al componen s (pie s, a ches,
au
l
s)
o al ain hci imp essi e
d
im
ensions and hei slende ness.
Mo e ecenlly, olhe esea che s ha e ound nolable s uc u al weaknesses
in
Go
hi
c
eons uc ions. Ap opos
o
lhe sludy o he Ca hed al
o
Sla. Ma ía,
in
Vi o ia, Spain,
C oci el a
I.
(1
995) obse ed weak poin s such as (I) a e y Iimi ed in e connec ion
be
lween lhe s uclu al elemenls, p oducing a lype
o
isos a ic consl uc ion; (2) lhe
excessi
e
slende ness
o
pie s and ela ed possible inOuence
o
second·o de e ec s
in
lhe equiJib ium, (3) lhe dc o mabili y o bu esses a
nd
ly
in
g a ches, (4) he
eccen ici y c ea ed o e he pie by bu esses a ached lo pie ex ensions which
usualJy do nol sland di ec 1y o e lhe pie , (5) lhe weakcning e eel causcd on suppo
elemenls by lhe i o ium and possible open
in
gs, wi h special ega d lo hei shea
e
sis ance, and (6) lhe acl Ihal ib sp ingings
in
lhe aisle and a cade a ches may
sl ongly educe lhe local esis an! sec ions
o
lhe uppe pie exlens
ion
s.
The
combina ion
o
!hese aspec s a e lhe cause
o
majo de onna ions and dangc ous oul·
o
-plumbness. Aeco ding o lhe esea che s sma
l!
design o cons uc ion impc eclions
may lead o high de o mabili y and c acking.
Fu he mo e, equen damagc pa c ns ha e been idenli ied which
in
some cases a e
conside ed "ch onic", such as
lh
e so·called Sabou eL c acks lypica
ll
y obse ed
in
la ge
g oined
auh,
(Heyman, 1983, Ba hel, 1993).
The s udy
o
se e a1 majo GOlhic cons uc ions, including lhe case
o
lhe Basilica o
Sl. F ancis
o
Assisi a e he Seplembe 1997 ea hquake (C oci, 1998
),
al50 showed
he weakness
o
Go hic sl uc u es o special agen s such
as
ea hquake, inlense wind
and la ge ounda i
on
sellle nenls.
1.2 The ole
o
lhe s uc u al elemen s
The clea and uni ocal ole ha Viollel·de-Duc assigned
lO
lhe a ious sL uclu al
elemenls (diagonal, l ans e se and c1e es o y a ches
in
aul ing, bu esses and lying
P. ROCA e aI. I Analysis
o
Ba celona Ca hed al 233
bu esses, e c.) and
lh
e e y assumplion lhal a
li
hese elemenls a e esse
n i
al
in
lh
e
Go hic g amma , a e also oday dis ega ded by some specialisls. Some new
unde s anding ca ne om lhe obse a ion
o
lhe s ales
o
esidual equilib ium a ained
by
Go hic cons uc ions a e pa ial desl
u
clion caused by
i
e, bomb
exp
losions
(du
in
g Wo ld Wa
11
) o simp
ly
lack
o
main enance. Rema kably, some aul s
main ained a pe ec equi
li
b
i
um a e los
in
g
h
e ans e se o diagonal ib
s.
O he
cons uc ions los some bu esses o lying bu l esses wi hou immedia e damage
be
in
g obse ed
in
lhe sys em
o
aul s (
8.
Bassegoda,
1974
).
Wi h ega d o lhe ole
o
lhe eleme
nl
s
in
a c oss aul , some s udies e ealed ha he
diagonal, longi udinal and ans e se elemen s con ibu e li le, i any hin
g,
10
es
is ance. The s abili y
o
lhe aul is due mainly o he capaci y
o
h
e webbing
o
he au
l
s o wo k as a s
heH
wi h double cu a u e (Ma k, 1982, 1993).
I
is
in e es ing
o
no e ha his con adic s some classical opinions
wh
ich assigned
lO
lh
e diagonal
a ches he main s uc u al ole and elegaled he webbing lO li lle mo e
h
an
a simple
cJosu
e.
The ole
o
he
iH
placed o e he
a
ul
s
is
also subjecl
o
opinion a
nd
p obably has
e
y di e en unc ions depending on i s o ig
in
and ma e ial composilion. The
backing -placed
aI
lhe haunches
o
lhe aul s -
is
o
u mOs
l impo ance o Slabi
li
se lhe
au
l
s subjec
o
g a ily load. Addi ional
iU
placed o c lhe aul migh also p oduce
a ma ginal s ab
ili
si
ng e ee when lhe aul
is
only subjee
o
g a i y l
oa
d
s.
Howc e ,
such a
iU
may ha e e y di e en e ec s
in
lhe case
o
agen s o he Ihan g a i y. The
s udies ca
i
ed ou on he Basilica o
SI.
F ancis
o
Assisi a e lhe ea hquakc showed
he non-cohesi
e
illing exis ing o e lhe au
l
s, which was nol o iginal1y placed he e
on
pu pose bu me ely accumula ed o e lhe decades and had a e y un a ou able
e ec du ing he occu ence
o
lh
e ea hquake leading o
he
collapse
o
some
o
he
au
l
s (C oei,
198
8).
The
iU
w
hi
ch exis s
in
some sou hem Go hic eons uc ions (such
as
in
Sl
a.
Ma ia
deI
Ma
in
Ba
ce
lona and Ba celona Ca hed al i sel ) consis s
o
a cohesi e mass
o
lime
mo a and adi ional ce amic essels and
is
hus cha acle ised by conside able
s eng h
co
mbined wi h e y
li
mi
ed weigh
(1.
Bassegoda,
19
83). Supposedl
y,
I
hi
s
kind
o
co
nc e e
iH
makes a signi ican con ibu i
on
o
esis ance
in
lhe cases
men ioned, whe e a aul webbing much hinne han
in
mosl Go hic eons ue ions has
also been obse ed.
2 TECHNIQUES APPLlED TO THE ANALYSIS
OF
GOTHIC STRUCTURE
2.1
Di ieul ies
o
he analysis
Because
o
lh
e many di icul ies which a e encounle ed
in
lhe sludy
o
GOlhic
cons uc ions, ew seien i ic s udies ha e been pe onned o da e wi h a iew o
gaining a deepe unde s anding
o
hei mechanical and esis ance p openies. The
main di ieul ies a e posed by
h
e
o
llowing aspec s:
Because
o
he agile na u e
o
s one mason y, analy ical s udies based on he
hypo hesis
o
linea elas ie
i
y a e o e y Iimi ed
a
li
di y. In ael, wi hou a g ea deal
234 STRUCTURAL ANALYSIS
OF
HI
STORICAL CONSTRUCTIONS
11
a
(unhe
in e p e a ian by lhe analys . hey may only in o m abou lhe se icc
beha iou p o ided lha! s esses keep wí hin mode a e alues.
The
knowledge
a
bo h lhe in e nai composi ion
a
lhe s uc u al membe s and lhe
mechanical p opc ies
a
lhe exis ing ma e iais a e no mally e y limi ed, o se e al
eascns. Fu he mo e,
bOlh
lhe composi ion and lhe ma e ial p ope ies may show
conside able
sca
e ing h oughou
lh
e en i e cons uc ion.
The
mcchanical beha iou
is
deeply dependen on lhe his o ical cons uclion p oces
s,
which
is
mos ly unce ain, Addi ional unce ain y is p o ided by conlingencies 5uch
as
acciden s and pa ial ebuilding, du ing cons uc ion
a
immedia e cmedial mea
su
e
s
o cons ain p eeocious de o ma ion.
Fina
ll
y,
as
we
a e eminded by Cassinello (1998), hose
co
ns uc ions whieh ha e
eached lhe p esen day ha e usually undc gone majo ans onna ions as a
consequence
o
p og essi e
O
pennanen de onna ion, e acking, physieal
o
ehemieal
deg ada ion
o
ma e iais,
o
possible eons uc ion al e a ions o eplacemen s. Again
acco ding
o
Cassinello, he elabo a ion
o
an accu a e model shou
ld
hus ake in o
accoun he h ee main
aspeClS
de ennining he s uc u al beha iou , namely he
(de onned) geome y, in e nai eomposi ion. and ma e ial p ope ies.
The
in eg a ion
o
lhe his o ieal ans o ma ions should be ex ended o hese h ee aspecls.
2.2 Signi ican s udies and de elopmen s
A widesp ead se
o
echniques has been used
in
he s udy
o
GOlhic cons uc ion,
including pho oelas ic modelling, classical heo ies bascd on plas ic heo em
s,
con en ional ma ix calcula ion and lhe ini e elemen melhod.
Pho oelas ic modellillg
The
esea ch ca ied ou by Ma k (1982) by means
o
pho oelaslic models, eXlended
o a la ge numbe
o
majo cases such
as
he calhed als
o
Amiens, Beau ais. Bou ges,
Cha es and Palma de Mallo ca, cons i u es one
o
lhe i s and mo e success ul
a emp s o quan i a i ely analyse lhe sl uc u e
o
hese cons uc ions lo gain a deep
unde s anding
o
lh
ei pe onnance when subjecl
o
a ious agen s (wind
in
pa icula ).
Thc
s udy enables us o en isage he essen ial cha ac e is ics
o
lhe
buildings, compa e lhei di e en design, and be e unde s and lhe ole
o
hei
di e en s uc u al elemenls. l also p o ides
an
unde s anding
o
he o igin
o
some
o
lhe exisling damage, such as ce ain c acks and pennanen de o malions.
Plas ic analysis
The
possibili ies
o
classical plas ic analysis
in
lhe s udy
o
Go hic ca hed als, based
on he plas ic limi heo ems, ha e been shown by Heyman (1998). Due o lhe alidi y
o
lhe hypo hesis assumed by classic heo y. e y ealis ic collapsing mechanisms may
be de ised h ough i s ca e ul applica ion. Howe e , classical analysis is
o
li le use i
applied o condi ions o he han
o
a emo ed em ailu e. Thus, classical heo ies
P.
RO
CA
cl
alo
I A
nal
ysis
o
Ba celona Calhcd al 235
a e
no e y use ul
o
in e p e lhe cause and ex en
o
c acks, de o ma ion o o he
damage no di ec ly ela ed o he gene a ion
a
a
co
llapse. Fu he no e, hei p ac ical
bu igo ous use beco
nc
s ex emely di icul
in
he case
o
complex s uc u es wi h
mui iple elemen s (as encoun e ed by se e al au ho s, incJuding Cau in and S agni o,
1998
).
Acco ding
lO
Chassagnou e
aI.
(1998), lhe e alua ions ob ained using classical
heo ies end o
be
oo pessimis and may lead
o
epai s hal a e mo e a - eaching
han ac ually needed. This is so because
o
lhe ype
o
addi ional hypo hesis ha mus
be assumed o make he use
o
hese Iheo ies easible.
CO l e l ional ma ix
calc
ula io l
Using con en ional ma ix o mula ion, Leon a
nd
Cassa i (1997. 1998) dc eloped a
delailed pa amel ical s udy
o
he main geome
i
cal and ma e ial a iables in luencing
lhe
s uc u al esponse
o
Le
6n Ca hed al
in
Spain. Thc s udy was based on a ca e ul
modclling
o
lhe main ans e se sec ion
a
lhe building and he na
e
au
l
s
by
means
o
an
equi alen sys em
a
one-dimensianal elcmen s.
Fini e elemen me h
od
The main limi a ion
o
mo
s
ini e elemen onnula ions, whcn applied o mason y
elemcn s, lies
in
he h
YPo
lhesis
o
he con iouum, which makes
il
di icul o c cn
impossible o simulale
h
e pa icula kinema ics o he mason y modes o ailu c.
Elaslic analyses based 00 he ini e elcmenl me hod a e only sui ab
le
o
cha acle ising wo king condilions subjecl
o
s a es o e y modc ale s esses
in
which
comp essi e s esses a e la gely p edominan . Howe e . he adoplion o
su
i able
cons i uli e equa ions which accoun o he main phenomena eJa ed
o
he ailu c
o
he ma e iais (c acking unde ension, c
u
shing unde comp ession, e c.) makes
il
po
ssible
o
ep oduce mo e ad anced slages
o
lhe esponse and e en 10 simula e
ailu e mechanisms simila 10 hose p edic ed by lhe classical
Ih
co ies.
C oci e
aI.
(1995) ca ied oul a ini e elemenl analysis
o
lhe Ca hcd al o S a. Ma í
a,
in
Vi o ia, Spain. The analysis, applied
o
he main ans e se sec ions
o
lh
e building
and o lhe na e aul s, ollowed an inc emen aI s a egy o accoun o c acking
du
e
o
ension o shea s esses,
as
wcll as he equilib ium second-o de e cc s. This analysis
showed some
o
he main weaknesses
o
lhe building, as has al eady bcen men ioncd
in
Sec ion 2.1.
Simila analyses we e also used o lhe s
lUd
y
o
h
e co
ll
apse o Bcau ais Ca
h
ed
al
(C oci e al., 1998b) and he e ecIs
o
h
e ea hquake
o
Sep embe 1997 on lhe
Basilica
o
Assisi (C oci, 1998
).
The ini e elemen me hod, also combined wi h sui able cons i u i e equa ioos,
ha
s
been success ully used o s udy Go hic c oss aul s (Ba hcl, 1993
).
Fo his pu posc,
e y de ailed inile elemen models we e elabo a cd and used
in
combinalion wi h a
ma e ial ea men enabling lhe simula ion o c acking as well as sliding be ween a ch
ing join s.

236 STRUCTURAL ANALYSIS OF HISTORICAL CONSTRUCT
IO
NS
11
Ca
u in and S agn
i
o (1993,
19
95) ca ied oul e y
in
e es ing s
lU
dies using bo h
c1assical plas ic analysis and nan-linea analysis
by
lhe ini c ele nc
n
melhad. Thei
me hod was success ully applied o analyse lhe cen al na e
a
Reims Ca hed al.
The
in
i
o cleme
n
me hod has beco also u i
li
sed o s udy se e al Spanish Go hic
ca hed als (Bu go
s,
Se ille) by Izquie do (1997; see al50 Cassinell
o,
1998).
Cha
ssagnou e l
aI.
(1998) ha e de eloped a echnique o lhe gene ic modelling
a
Go hic c oss aul s, al50 based on
lhe
ini e elemen melhad. wi h
lh
e a
im
a
making
a ailable a
1001
o lhe p ac
i
cal e alua ion
a
h
ei s uc u al esponsc.
Ona e cl ai (
199
7) ha e de eloped a ini e elemen damage model w
hi
ch was
success ully used o he anal
ys
is
o
SL
Ma ks Basilica
in
Venice. This model, b ie
l
y
discussed in a la e sec
i
on, is one
o
lhe echniqucs chosen o he nume ical s udies
o
lhe
Ca
hed al
o
Ba celona p esen ed
in
his pape .
3
BARCELONA
CAT
H
EDRAL
3.1
ln oduc ion o lhe
s
udy
The
choice
o
Ba celona Ca hed al (Figs. 1-6) o lhe pu pose o
h
e p esen s udy is
pani
a
ll
y due lo he acl ha abundan in o ma ion exisls on i s geome
y and olhe
cons uc ion aspcc s, made a a
il
able lo lhe au ho
by
lhe Diocesan A c
hi
e
and lhe
Chap e
o
Ba celona Ca hed al. Funhe mo e,
h
e e a e some publica ions wi h
in e es ing his o ical and cons
u
c ion in o ma ion (Elías, 1926,
1.
Bassegoda, 1968,
19
81, 1983). Ba celona Ca hed al also shows ce ain s uc u
al
singula ilies which
ga e
i
added in e es
as
a subjec
o
s udy.
• Fi s , i
is
a cons uc ion in lhe so
-c
alled Ca alan Go hic s yle, wh
ic
h exhibi s
no able spec
i i
c ea u e
s.
Ca alan Go hic cons uc ions we e ypically designed as
e y diaphanous spaces wi h la ge-spanned na es.
•
The
dimensions
o
lhe aul s a e ema kable due o lhe la ge gap be ween
lh
e pie s
nOI only
in
Ihei ans e sal di ec ion, bu also 10ngi udinal1
y.
• The lay-ou
o
lhe l ans e
se
seclion
o
lhe building
is
cha
ac
le
i
sed by
h
e
ai
sle
na es being a
lm
os (buI
nOI
quile) as high
as
he cen al na e. Because
o
lhis
sl uclu a
ll
ay-ou , lhe aisle aul s h;we an impo an! "I nclu al unc ion in elain
in
g
lhe Ih uSl
o
lhe cenl al aul
s.
In
a celeb aled leclu e on Mallo ca Ca hed al, Rubió i Bell c (1912) dis
in
guished
belween lh ee di e en ypes
o
Go hic calhed als wi h ega d lo lhe s uc u al
o ganisa ion
o
lhei ans e se sec ion.
In
lhe i s ype, lhe na e and lhe
ai
sles a e
buil o lhe same heighl (as in Sa agossa, Munich, Pe ugia and o he s). The second
ype, in w
hi
ch lhe aisles a e almos bu nol qui e as high as
h
e na e, is by a lhe leas
widesp ead since il only inc1udes lhe cases
o
Ba celona Ca hed aJ and
S a
. Ma ía dei
Ma .
The
hi d ypc, in which lhe aisles a e signi ican ly lowe lhan
h
e na e,
is
by a
lhe mos equen (being lhe case
in
Amiens, Reims. Beau ais, Cologne, Milan,
Toledo, Ma
ll
o ca and many o he s).
P.
ROCA el
a!.
I Anal
ys
is o( Ba celona Ca
lh
ed al 237
~'''
IDlln''U:UU~iIIl l
1!U
n.
:~
II
I!i:
'jI(U'UVl
;
SllIl I:U1I1UJ 11U!
Fig.
I Longi udinal and ans e se sec ions
o
Ba celona Calhed al ( acsimile
ep oduclion
o
plans d awn
in
1864
as pa
o
lhe sludies p e ious o lhe cons uc ion
o
lhe new cimbo io).
238 STRUCTURAL ANALYSIS
OF
HISTORICAL CONSTRUCTlONS
11
.
,
Fig. 2 Plan
o
he building (1864 plans).
Ui'l1i8DR1Hií
I~R8 h ni'l
PI."I.
'",.,-.!
'"
I •
.yp."AI.
:
~~"-::'
"'
.'_
n.
.
.........................
.
~
,
""""J
.
..,
,,,,..
: !:'''
,....
....
,.
'
.....
;
:::.
~
:
Z':,;
..2
;
'Y({E-'A~
"
~
..
. n
....
::
~::!!:..
' n
....
~
......
..
-
......
"
......
.......
.....
" J .
...
..,...,
..
...
.,,,"'''''--
......
,,,,,,
J'"
U.
J •• 1
.......
J
..
.,-"
... ~
a...i
~ :
:':'hL:."·-
....
~
..
">:,,
..
-
" .
1"
....
__
..
"
".
'''--
..
J
..
"...,....
.....
",,"",--
" "
...
_
....
..
.
,.
...
-
:~~7",
..
"'''''
............
...................
,
...
"J
J
""
....
" J
-'.
J
""'-,
'-'::
:"T,C"
.
~.~.
P.
ROCA el
alo
I Anal
ys
is
o
Ba celona Ca hed al 239
Following Rubió. in
lh
e case
o
consl uclions belonging
o
he second ype lhe h us
o
he
na e aul s is comple ely e ained by lhe
ai
sle aul s so hal
no
lying a ches a e
needed
ai
ali. Whe e hey exisl. as
in
Ba celo
na
Ca
lh
ed al, Ihey a e non-s uc u al and
Ih
e
i
only unc ion is p obably d ainage.
Th
e p ominenl ole
o
lhe
ai
s
le
aul s as s abi
li
s
in
g eleme
n
s may be elaled
10
Iw
o
cons uclion de ices w
hi
ch a e consis en wi h hem and which we e p obably
in
co
p
o a ed
in
o de
10
sl englhen Ihem.
In
pa icula ,
lh
e aislc aul s a e su cha ged
w
i h
an
amou
n
o co
nc ClC
ill
g ea e
Ih
an ha placed o e lhe na e, as i
o
inc ease
hei coun e ac ing h us
l.
~
I
O,5m
Fi
g.
3 T ans e se sec ion
o
lh
e na
e
pie s a
nd
lh
e pie s susl
ai
ning
Lh
e c
im
bo io
as ep esenled
in
lhe 1864 plans (E 1:40
).
3.2 B ic desc ip ion
o
he building
Cons
u
c
i
on o lhe na es
o
Ba celona Ca hed al was begun
in
129
8 and las ed o
mo e lhan a
ce
n u
y.
As usual. lhe apse was cons uc ed i sl, being inished
in
1327,
while lhe cons
u
c
li
on
o
lhe en i e na e conlinued un il
141
7.
In 1422 wo
k
s opped,
lea ing lhe cimbo
io
un inished and a p o
i
sio
nal
wall
cJo
su e as a açade. Mosl o
lhe o iginal ea u es
o
he building a e assumed
10
be all ibulable o he mas e
builde Jaume Fab e, who wo ked
on
h
e Calhed
al
om
1317
.
The plan
o
lhe emple shows a
e
y uncon en ional lay-o
ul
in
which he cimbo io is
nOI
e ec ed o e lhe c ossing bu o e
lh
e i s b
ay
o
lhe na e elose o lhe açad
e.
while he c ossing
is
delimi ed by lhe wo majes
i
c elock owe s, mo e commo
nl
y pa
o he açade. A simila dis ibulion can be obse ed
in
e y ew case
s,
such as lhe
Ge man ca hed als
o
Ulm and F ibu g.
The building
in
cludes h ee na es (
lh
e na e and wo aisles) al hough, as a
consequence
a
i s pa icula design, i appea s
o
enclose wo addilional aisles. This
246 STRUCTURAL ANALYS
IS
OF
HI
STORICAL CONSTRUCTIONS
11
Fig 10. Dc o mcd shape
c
a aul! quad an mode!!ed as a amed sys em, subjecI
o
dead loading.
Fig. 11 Fo ce ajcc o ies
in
a
GO
lhic aul subjecl o dead loading (Ma k. 1982).
Allalysis
o
lhe aul usi /g
3D
solid /inile elemen s
Thc FEM modeJling a lhe au
l
accoun s o lhe ib sys em, lhe webbing, lhe nlbble
backing a1 lhe aul haunches, he conc
e e
iH
and lhe la e al walls, ali
de
ined
by
hei
co
esponding mechanical p ope ies. Gi en lhe
li
mi ed in o
ma
ion a ailable
on he ma e ial p ope ies, some assump ions had o be made
in
a de
o assign
mechanical pa ame e s o he a ious ma e
i
aIs (Table
1).
The
g ea es sou ce
a
unce ain y was he cone e c ill, composed
a
ce amic po e y and
li
me mo a , he
p ope ies
o
which had o be oughly es ima ed. The ashla mason y and ubble we e
es ima ed, on he basis
o
empi i
ca
l c i e ia, om he a e age comp essi e s eng h
(74.2 MPa)
o
he Mon
jui"
c sands one used o he cons uc ion
o
he building.

P.
ROCA
e
aI.
I Analysis o Ba celona Ca hed al 247
ig.
12
FEM model and dis ibu ion
o
s esses caused
by
o al dead load
00
lhe
na e aul (maio comp essi e s esses
in
Pa).
Fig.
12
shows he dis ibu ion
o
s esses which
is
ob aioed o he aul subjecI o
o al dead load. Some
o
he ohse a ions allowed
by
lhe simpli ied ame model can
be
con i med, sue h as lhe signi ican ole
o
lhe webbing
in
ans e ing he o ces
10
he pie s, and lhe majo con ibu ion
o
lhe ans e se a ches bu small o null
con ibu ion
o
he cle es o y ibs and diagonal in e sec ions.
248
STRUCTURAL ANALYSIS
OF
HI
STOR
ICAL
CONS
TRUCTI
ONS
11
Table I. Mechani
ca
l p ope ies de ined o lhe a ious ma e iais
Ma e i
al
D
e o
ma ion Poisson Comp esi e Tensile Dens
i
y
modulus coe icien s eng h sl engh
(kN/m3)
(
Mp'
)
(kP.)
(kP
a)
Ashla
8000
0
.2
8000
400 2,7
mll
so
n y
(*)
Rubble
(**)
2500 0.2 2500 100
2,5
Conc c c
i
11
500 0.2 500 50
1,0
(*) in
pi
e
s,
ibs and aul wcbbing
(**) a lhe co e o pie s and
aI
aul haunches
The
ana
lysis p ediclS
an
un calis ica
l1
y high ul ima e capacily (o
e
lh ee imes lhe
o al
dead
load
in
g) b
eca
use lhe la e al displace ncn s
c
lhe sp ingings a e de ined as
u
ll
y cons ain
ed
in lhe indi idual ca men!
o
lhe aul
.
Thus. lhe simula ed ul ima e
m
ec
hani
sm
does nOI aceoun
o
lhe lexibili y and
li
mi
ed
capaci y c lhe e aining
sys em
p
o ided
by lhe aisle auhing and bu esses.
Mo
e ealis
i
c esu
l
s a e
ob ai
ned
when, as expla
in
ed in Sec ion
4.3.
lhe aul is
in
se
ed
inlO
a mo e comple e
model
also
including ali s uc u al elemen s
in
he I ans e se sec ion.
4.3 S udy
o
a na e bay
S udy using Gene alised Ma ix Fo mula ion
The
pu pose
o
Ihis sludy was
10
analyse lhe possibili y
o
s udying
lh
e equilib ium
o
Go hic s uc u e up o lhe ul ima e condi ion by means
o
a simple model. In spi e
o
his simplici y, he model was elabo a ed o oblain an accu a e desc iplion
o
lhe
geome y
o
lhe na e sec ion, lhe dis ibu ion
o
lhe dead load and lhe esponse
o
lhe
ma e i
ai
s.
Using
he
acul ies
o
lhe me hod, he a ious s uc u al elemenls we e modelled as
s aigh
o
cu ed lin
ea
elemen s wi h a iable and
co
mpo
si e c oss-sec ion (Fig
s.
13-
14
).
l
sho
uld
be
no
ed
Ihal,
in
a e y
simpli
i
ed
way, he aul s a e also ea ed as a
cu ed
compos
ile beam whe e each
ma e
ial (maso
n
y. ubble, conc e e
i
l!) is de ined
by i s pa icula mechanical p opc ies.
The
analysis a
cco
un s o lhe e ccls
o
mas
on y
c acking
unde ension o c ushing und
e
com
p
ession as desc ibcd in
Sec ion 4.1.
By
g adual1y applying he g a i y load up o and beyond
h
e dead load alue we we e
able o isualise lhe wo king condi ion
o
he sl uc u e, lhe p og essi e d
e
elopmen
o
c a
c
king
and
lhe inal appea ance
o
a numbe
o
c ushed zones leading o ailu e
(Figs. 14-16).
Th
e applicd load was kep p opo ional o lhe dead load h oughou lhe
p oce
sso
Failu e
wa
s eached o a o al applied g a i y load
equa
l o wi
ce
he
(O a
l
dead
load.
As
ca
n be
se
cn
in lhe igu
es
men ioned abo e, lhe sec ions a ec ed by deep c acking
o
c ushing
de e
lop a highly p onounced cu a u e and hus may be iden i ied wi h
P.
ROCA
c
aI.
I Analysis o Ba celona Ca hed al 249
hinges, which would lead
o
an
ul ima e mechanism acco ding
o
lhe plas ic heo y.
The se
o
hinges leading o he ul ima e mechanism appea ed
in
he na e aul .
lhe
uppe egion
o
he pie sha s and he aisle au
l
s.
In
bo h lhe na e and aisle
a
ul s
he hinges de eloped a hei c own. Howe e
,
he ob ained mechanism
is
nOI
iden ical lo he heo e ical one Ihal would be ob ained by applying lhe plas ic Iheo y.
because
o
lhe conl ibu ion
o
lhe gene al de o ma ion o lhe elemenls
lO
he
ins abili y. Owing o lhis con ibu ion, lhe ailu e mechanism becomes poss
ibl
e e en
i some hinges a e no ully de eloped, as a lhe na e aull c own, o no de eloped
aI
ali,
as
occu s in he base
o
lhe pie s whe e lhe mobili y needed
10
de elop
lh
e
mechanism is gi en simply by he g ea nexibili y o he sha
s.
A ull lheo c ic
mechanism would equi e a se
o
se en hinges.
Fig.
13
Equi alenl ame model.
Du ing lhe p ocess, lhe a e age comp essi
e
s ess
a!
he base
o
lhe pic s a ies om
a alue
o
3.5 MPa, o dead loading applied, o a alue
dose
o he comp essi e
s eng h (8 MPa).
The esponse is almos linea up o ailu e, wi h only e y sligh losses
o
s
i ness
Ih oughou lhe de elopmen o he hinges and a inal, agile ailu c. I should be
menlioned ha his agile esponse may esul
as
a limi a ion
o
lhe nume ical
p ocedu e
o
con inue lhe analysis beyond some local se e e damage.
The pa ame ic s udies showed lha lhe ul ima e load
is
nOI
dependen upon lhe
comp essi e s eng h when alues
hi
ghe lhan 8 MPa a e conside ed, which means
lha lhe ailu e
is
cons an ly caused by lhe gene al ul imale mechanism. This is wha
should be expec ed om lhe dassical plas ic analysis.
In
con as!, lhe ul ima e load
dec eases
in
p opo ion
o
he comp essi e s eng h when alues
o
less Ihan 8 MPa
a e de ined because he ailu e is hen dele mined by he c ushing
o
lhe mason y
ai
lhe base
o
he pie s.
250
STRUc URA
L ANALYSIS OF
HI
STORICAL CONSTRUCTI
ONS
"
As
demon
s alcd
by
lhe pa amcl ic s udies, lhe
gene a
l
ca
pacily wus
nOI
signi ican ly
dependem on lhe
s
i ness
o
he
conc ele i!!
( o
a
lu
cs
a
lhe modulus
o
de o ma
ion a yi
ng
be
lwecn
50
o 1000 MPa).
Simi
la ly. a
modc
a c a ia ion
o
i s
a e age densi y h
ad
almosl no c eel upon lhe ul ima c load.
,
""
Fig.
14
Dis ibu i
on
o axia
l s esscs and
c acking
o lhe s nlc u e subjccl o
dead
load.
Fig.
15
Dis ibu ion
c
axial s esses and c acking
aI
ailu c occu ing o a dcad
load ac o ised
by
2.
P.
ROCA
Cl
aI. I Analysis
o
Ba celona
Ca hed ai 251
Fig.
16
De o med shape
close o ailu c.
This mode! was also used o simulale lhe e ecl a
an
ea hquake by applying a se
o
ho izon al equi alen s a ic loads consis enl wi h he disl ibu ion 011 mass. The 11011-
linea analysis p edicled lhe ailu e o a seismic coe icien alg eqllal o 0.12, which
can bc conside ed sa e enough wi h cga d o lhe seismic illlensÍly
o
lhe egion
o
Ba celona.
The applica ion
o
lhe Spanish seismic code would lead
lO
lhe conside a ioll
o
a
coe icien equal o 0.086 o a mason y building
in
a simila condi ion and o a
i
sk
pe iod
o
one
hund ed yea s. As showll
in
Fig.
17,
he dis ibu ion
o
s esses
is
nOI
signi ican ly al e cd wi h espec o lhe dead load condi ion when a mo e mode ale
ea hquake is simula ed.
The
co esponding de o med shape
is
shown
in
Fig.
18.
; .
...
Fig.
17
Dis ibu ion
o
s esses caused by a se
o
s a ic equi alen o ces
simula ing
an
ea hquake cha ac e ised by a coe icien alg equal o 0.086.

252
STRUc URAL
ANALYSIS DF HISTDRICAL CDNSTRUCTlDNS
11
Fig.
18
Geome y de o med
byea hquake.
Ob iously, lhe li ni ed a
li
di y
D
lhe l ca men
o
lhe ea hquake as an equi alen! se
D
s a ic o ces should be
a
ken in o conside a ion and lhe ob ained alue only
ega ded
ao;
a coa se app oxima ion o lhe possible seis mie esponse D lhe building.
Fini e elemell( damage modelling
In o de o keep lhe o al numbe
o
equa ions wi hin easonable limi s, only a qua e
o lhe ans e se sec i
on
was disc e ised in o e ahed al solid elemen s o ca y
Ou!
lhe
damagc analysis.
Damage i s appea s a lhe c
QWIl
o
lhe ans e se a ches
D
lhe na e. As add
i
ional
load
is
p og essi ely app
li
ed, damage sp eads co e
i
ng
la ge egions
o
lhe s uc u e
(Figs. 19, 20). Addi ional damage ocuses a e obse ed
aI
lhe c own
o
lhe aisle
aulls.
ai
lhe haunches
o
lhe aisle l ans e sc a ches and a lhe bases
o
lhe pie s.
Failu e occu s o a o al load applied
equa
l lo Iwice lhe dead load. caused by lhe
gene a ion
o
a collapsing mechanism simila o lh
al
ep oduced by he
GMF
mode .
As can be seen
in
Figs. 19-21, bo h damage and cu alu e a e concen a ed in he
egions whe e ei he se e e c acking
o
in ense comp essi e s esses appea ed in he
GMF
mode!.
The
main di e en
ce
be ween lhe wo models lies
in
lhe loca ion
o
he
maximum
damage in he aisle aul , which
in
lhe FEM model eaches i s highes!
in ensi y nol aI he c own bul in he sec ion eloses o he aul haunch nea lhe pie .
This
is
also he p obable localion
o
lhe heo e ieal hinges leading o
lh
e ul ima e
mechanism.
As
in
lhe
GMF
analysis,
an
ahnos linea ela ionship is ob ained belween he amoun
o
load and lhe displacemenls
o
lhe sl uclu e wi h only a e y sligh , g adual loss
o
s i ness obse ed wi h he ex ension
o
damage. This almos linea esponse
is
ob ained up o he uhima e load, wi h no u he so ening b anch being p edic ed.
AIso as in he
GMF
analysis, lhe agile esponse migh be a pu ely nume ic
al
e ec!
caused by insu icien mesh e inemen o by he
li
mila ions
o
lhe me hod used
in
he
solu ion s a egy.
P. ROCA c
aI.
I Analysis
a
Ba celona Ca hcd al
253
Fig.
19
Dis ibu ion
o
damage
a
loading le eis co esponding
o
dead
load
ac,o s
o
0.5, 1.0,
1.5
and 2.0 (see damage seale
in
Fig. 20)
254
STRUCTURAL ANALYSIS
OF
HISTORICAL CONSTRUCTIONS
11
.
11 2
.
110
.
019
· "',
,
:)11
.
UI
·
l.:)
.2"
·
172
·
8621:-1
Fig. 20 Dis ibu ion
o
damage a loading le eIs co csponding o dcad load
ac o s
o
0.5.
1.0
and 2.0.
P. ROCA el aI. J Analysis
o
Ba celona Cn hed al 255
Fig.
21
De
a med shape o g a ily load
5 CONCLUSIONS
Two al e na e numc ical app oaches cha ac c ised by Ihci c y di e en deg ee
o
sophis ica ion we e sa is aclo ily used o
ca
y
ou a nan-linca analysis
o
he
sl uc u c
o
a Go hic cons uc ian -namely. Ba celona Ca hed al - aking in o accoun!
he mo e p ominen ea u es
o
he mechanical esponsc
o
lhe s uc u al ma e ial.
The wo app oachcs - he Gene alised Ma ix Fo mula ion and he 3D ini e clemen
nodelling wi h a damage model -p oduced e y simila p ediclions o lhe espanse
a
lhe s uc u e h oughou lhe loading p ocess, lhe collapsing mechanism and lhe
ul ima e capaci y
in
he analysis
o
a na e bay.
Bo h app oaches. bul
in
pa icula he much simple
GMF
p ocedu e. awc / ci
success
in
analysing mason y consl uc ions
o
!hei abilily o accu a ely simula e
h
e
essen ial ac o s dCle mining Ihei s uclu al esponse. Fi sl. / ey sha e he capaci y 10
de_sc ibe com pie x geome y and
10
accu alely com
pUle
and disl ibu e dead loads on
lhe
s uC
lu c. Second, hey include sui able cons i u i e equa ions lo mode! lhe
essenlial cha acle is ics
o
lhe beha iou
a
he ma e ial
(l
he inabili y
o
mason y
o
ca y ensioll,
in
pa icula ).
The main limi a ion
o
he
GMF
s ems om i s inadequacy
o
ea! Iwo-dimensional
elemen s such
as
Go hic aul s.
I
an
accu a e s udy
o
a aul
is
o be pe o med. he
use
o
a de ailed ini e elemen mode ling becomes una oidablc. Fu he mo c,
il
equi es e y ealis ic ma e ial modelling
o
lhe a ious sl uc u
al
elemen s, ma e iais.
su
cha gc and suppo condi ions in ol ed (including ibs. webbing. mason y
haunches. and possiblc conc e e
iH).