STRUCTURAL ANALY5lS OF HISTORlCAL CONSTRUCTIONS II P. Roca, j.L González, E. Ofiate y P.B. Lourenço (Eds.)

© ClMNE, Barcelona 1998

THE SURVEILLANCE OF BRUNELLESCHI DOME IN FLORENCE

A. Chiarugi, G. Bartoli and S. G. Morano Civil Engineerj"8 Department U"h'usity fi! Florena Via di S. Marta. 3 - 50139 Florena lta/y

SUMMARY The Dome of Santa Maria deI Fiore Calhedral in Florence is affected by a system

of cracks which appeared as soon as the monument was completed; the crack panern have been expanding during time, so that many instruments have been placed during the centuries 10 conlrol il. In 1987 a large digilalJy controlled moniloring syslem has been installed to observe and cOnlrol tne propagation of cracks. Acquired data allow both an understanding of some of the mechanical and structural properties of lhe behavior of the Dome and a suggestion for some consideral ions aboul the use of monitoring systems in the field of lhe structural preservation of ancient monumenlS.

In the paper, lhe main characlerislics of the syslem are reported as well as some of lhe main results obtained from lhe data analysis. A final paragraph reports the studies conceming the mechanical identificarion by means of numerical F. E. models.

I. THE BRUNELLESCHI DOME ANO, THE CR'ACKS LA YOUT Conslruction..qf lhe S. Maria dei Fiore Carhedral began in 1295 and lasled for ali

of lhe nexl cenlury. "fhe Dome was designeo by Filippo BrunelJeschi and was finished in 1434, while the cJerestory was set in 1472. The oclagonal structure incJudes an internai thick dome and an ex ternai t~in one. These two domes are structurally linked by joining elemems, which are constituted by masonry walls.

The. first historical infonnation about structura l damage was reporled in early seventeemh century, bUI they were mentioned even before. One of lhe most complete description went back to 1757 when a complete survey of the çracks was done and 13 different crack seis were desc ri bed. Two main cracks were noticed, localed in web no. 4 and no. 6(1). They slarted from lhe tambour and continued as far as lhe higher part of lhe Dome, passing through bOlh the two slructural internaI and externai domes.

(11 The eight webs of lhe dome will be numbered f TOm one [O eig/II. I being lhe web facing lhe nave and following lhe counter-clockwise direclion. From a geographical poinl of view. web no. 7 is lhe northern one. while no. 3 is lhe southern one. Even webs are located Qver lhe pillars. while lhe uneven numbered webs are localed Qver lhe arches.

338 STRUCTURAL ANALYSIS OF HISTORICAL CONSTRUCTIONS 11

Nevertheless, other (WQ maio cracks which are nowadays presenl in web no. 2 and no. 8 were nQt mentioned. These cracks formed arter 1757. and are af lhe sume lype cf (hase in webs no. 4 and no. 6 (i ndicated as Type A cracks). Overall, lhe main crack pattern in lhe dome is, ai lhis presen l time, qui te symmetric about ils cenlcr. Olher Ihan lhe cracks already mentioned lhere are three additiona l types af cracks which are present 00 lhe dome (see Figure 1). These include:

several vertical cracks oear lhe circular windows (cyes), JUS! abovc the kcyslOncs cf lhe arches, in lhe uneven webs (type B); some minor vertical cracks, even ir with Icss ampliwde with respec! to type A, .ind no! passing th rough lhe widlh Df lhe tWQ domes, around lhe eight internai edges of lhe dome (type C); fom eracks in the upper internai part of the unevcn webs whieh do nol pass through the dome (Iype D).

•

comer uock.s te)

'W~b croclcs (A - O)

Figure I - Plan view of lhe main craeks on lhe Brunellesehi p ome

The presence of this complcx crack pattem has modified lhe slruelural behavior of lhe Dome. Inslead of a circular shell , lhe Dome now behaves like four dri fling half­arc hes, linked jusI below lhe uppe r clerestory which backs are constituled by lhe pillars, lhe chape ls and lhe nave of the church. Thc crack panem has evolved Ihrough the centuries, and cracking can be ascribed lo lhe dome's geometry and the combined effeel of its sel f-weighl and insufficient resistance o f lhe tambour [IJ. Di ffe rences in cracking between even and uneven webs, is due lo lhe varialion of lhe struClure below lhe dome. Four heavy pillars support lhe lambour in correspondence of lhe even webs, while lhe uneven webs are localed over four arches. lt can be argued Ihal lhe tensile strcsses in lhe parallcJ di rect ion, ai lambour levei and due to lhe shell slructura l behavior, are parti ally ba lanced by the compress ive Slresses due to lhe presence of lhe arches in the uneven webs, so limiling lhe cmcks amplitude (see Figure 2). The presence of lhe arches themselves is also responsible for lhe inerement of lhe tensilc strcsses (and eonsequenlly af lhe cracks amplitude) j uS! above lhe pi ll ars.

A. CH IARUGI et aI. f Brunclleschis Dome in Florence 339

The cracks have atways been of a concern, 50 several conlrol devices have been installed in order to measure their evolution. At the beginning of this century, some mechanical control systems were installed. At the present time, two of these are still working: lhe one installed by lhe Opera dei Duomo (0.0.) in 1955 and the one placed by lhe Soprintendenza ai 8eni Ambientali e Architettonici (S.B.A.A.) in 1987. These systems have helped to illustrate the long-term evolution of the cracks, but no! to describe reliably the struclural behavior and the link belween environrnental loads and the variations in width of the cracks.

C!erestory

Tensile Slress Zone

Figure 2 - Static ske tch of the Bnmelleschi Dome

In arder 10 better understand the structural behavior and 10 describe lhe link betwcen environrnental loads and lhe variarions in width of lhe cracks, a large digital manitoring system was instaJled in 1987. This system incJuded 166 instruments, and ir has been working since 1988.

2. MONJTORlNG SYSTEM DESCRlPTION The goals of lhe monilOring syslem were mainly three: (a) 10 provide descriptions

of the dome movemenls and the time varia!ions of the width of lhe cracks; (b) lhe study of lhe correlation between temperatures and the time evolulion of lhe cracks; (c) 10 provide early warning of potential structura! problems, through lhe designation of threshold values of cerlain paramelers being measured.

The monitoring system needed to allow lhe gauging of time variation of the crack widths as weU as the movements of the struClure. These two quantities \Vere believed 10 be lhe mos! imporlant in lhe description of the present condition af lhe Dome. Moreover, due to former sludies, lhe monitoring system needed to allow for the logging of temperature values. These \Vere thought to be lhe ma in cause cf the crack \Vidth variations. In order to address this probtem, bath air and masonry temperatures needed [O be measured.

340 STRUCTU RAL ANALYSIS OF HISTORICAL CONSTRUCTIONS 11

Disp lacemenl transducers have beco placed mai nly across the fuU depth cracks and near lhe wedges of lhe dome, aI five different heights. In addition other instruments have beco pos it ioned near lhe lower edge of lhe circular openings, in a rder 10 evaluale possible relali ve sinking betwee n lhe Dome basement and lhe superstructure. Ali lhe instruments are markcd wÍth lhe in it ials DFn-mm, where DF ind icates lhe instrument Iype, fi is lhe number c f lhe web in which lhe instrument is localed and mm indicates lhe pos it ion. T he 72 insta ll ed displacement transducers are induc live types. They are precise to about ±O.02 mm. The logged data was recorded relatively [O lhe day in which lhe inslrumenls were ph\ced, 50 lha! the data are ab le to represent o nly variatio ns o f the cracks width. Because o f lhe position of the dev ices, they show oniy cracks' width variation in lhe "tangenlial" Dome direction (i.e. in the plane tangent to lhe Do me surface).

Re lative horizontal displacements between. the pill ars and the lambour o f lhe Do me are measured by means o f eight plumb- li nes placed near to the wedge o f the dome whose in-plane positi on is recorded at three different heighls by photoelectric cell s (in lhe sequei indicated as telecoord inometers). Th is system allows lhe eval uation o f horizontal di splacements. The data recorded by lhe different instnllnen lS are ind icated by lhe ini tials TLn- mmd, where TL indicates lhe inslrument Iype, !l the web in which Ihey are placed. !!l!lllhe he ight aI which the data refer lo and º lhe direct ion o f the datum (i.e. X or Y with respect 10 a given reference co-ordinate system).

Vert ical displacements of lhe horizontal plane jusI below the edges o f lhe c ircu lar opcn ings are measured by means of a system o f eight levei instruments (markcd with LlVOOm, fi being the number o f the instrument) linked by a hyd raulic oil c ircuil. The acquircd data permit the evaluution of both rigid movemenlS of the whole dome (i.e. fotations o f lhe plane) as well as relative movement among d ifferenl parts o f lhe tambour. T he instalJed instrument s alJow measurement o f d isplacemenls up 10 0.8 mm.

Thermometers were insta lled on each web at the second corridor levei, in such a way so as to meaSure the internai dome temperature in lhree d ifferent points and the ex ternai dome one in two. Webs 2 (facing north) and 7 (southwest facing) have been provided wi lh additional thermometers, since these structures are under maximum and minimum solar exposure . The 60 installed thermometers provide data fo r evaluation of temperature grad ients. The instrumenls (marked as TM n-mm (or TAn-mm), indicating thermometer in the masonry (TM) or in air (IA) in web number n and pOSiliol1 mm) are resistive ones , allowi ng a prec ision of abou! ± 0.05 oCo

To evaluate variation in lhe levei of underground water, a piezomete r has been placed externally of lhe monument , in the vicinit y o f web no . 4.

Four devices linked to a central control un it com pose thc acquisirion system. The devices are placed pe ripherall y with respect to lhe central unit, and provide for data logging, analogic-to-digital conversioll, and the temporafy storagt: uf uüt<l. The L:entral cOnl rol uni t co nsists o f a personal computer which functions lO manage and control the acqui sition system, as well as lhe pcrmanenl data storage.

Data can be logged cither at certain pre-fixed times 01' over uni formly d istributed inlervals during lhe day. The operator also has the option o f collecting add itional data points. The syslem checking is performed automntically by the computeI', indicating every malfunc tions (i.e. temporary off-line of the instruments or out- scule measurernents). Recorded data are then converted into physical units and co ntroJled.

A peripheral contro l dev ice has been installed aI lhe Engineering Faculty of the Univers ity of Firenze. This enables remote cont ro l and monitoring o f lhe whole system. The system is programmed lo automaticall y log data every six hours fo r ali instruments, starting at 6 :00 a. m.

A. CHIARUGI et aI. I 8runelleschis Dome in Florence 341

3. ANALYSIS OF THE EXPERiMENTAL DATA The acquired data has beeo analyzed in arder to:

investigate the structural behavior of the monument, with a particular regard to the correlation between environmental actions and structural response; study the long-term behavior by means of trend analyses, based on the meao-rate estimation of crack amplitude variations; assess a contro l procedure capable to signal unusual readings.

3.1 Structural behavior understanding The recorded data is first corrected for data errors . These errors can be caused by

electrical problems, bad protection against lightning, and other causes. Missing data is replaced by a linear interpolarion between the last available "good" data points.

Next. the recorded time-histories of ali the different instruments are evaluated for correJation. Two correlation functions have been studied: correlation between signals recorded by instruments of the same kind (i.e. temperatures vs. temperatures. displace­ments vs. displacements) and between differenr types (i.e. displacements vs. temperatures).

Based on this evaluation, the follow ing observations are made.

3.2 Seasonal behavior From visualization of the time-histories of displacement and temperature data (see

Figure 3 as an example), Iwo periodic phenomena are observed. The first (and main) phenomenon involves long-term variation of lhe data, showing a periodic annual behavior (Fig. 3a). The second one is for between day and night (Fig. 3b).

1.5 ,----r-----r---,---,.----,--7.(.:;"") '.0

/ -1 .0 . -/. . . ......... -

-1 .5 c[_m_m,,1 _,,,97,,,,1',---,::,,19,,,',"9,,1 __ ,-,,,,9,,°,:[ __ '"',,91,,l"":-""-",,99,,'c'[_---';· :,:19"',,,,3 o /' 500 1000 1500 ........ 2000

/' ........ ........(dayS]

05 f----~--~---~----....,---~--'::~':l

00

690 700 720 730 740 750 [days)

Figure 3 - Time-history of recorded data by one displacement transducer in web no. 4: a) long-period evolution; b) short period evolutioo.

342 STRUCTURAL ANALYSIS OF HISTORICAL CONSTRUCn ONS 11

3.3 Temperature distribution Cross-correlation curves between Icmperalures measu red by differenl instruments

placed in lhe same web show a lime-shifl compared 10 lhe auto-corrclation function (Figure 4a). This is related 10 lhe thermal diffusion betwcen diffcrent layers af lhe dome and to lhe (hermal inert ia cf lhe masonry. Because cf lhe fact Ihal lhe curves look ve ry simi lar, temperatu res on different layers can be modeled by lhe samc harmonic funcl ion. In (his way, lhe small time-shifl is neglected while lhe funclions simply possess diffcrent amplitude due to diffusion phenomcnon.

Cross-correlalion curves between data recordcd ai lhe samc leveI in different webs show good agreement (see Fig. 4b). This fac l implies thal lhe Icmperalurc dislribution can be assumed as conslant nlong lhe parallel di rection of lhe Sl!'ucturc.

Similarly, correlal ion curves belween inslrumCnlS placed aI differenl leve is in lhe same web and at the same dislance from the cente!' of lhe dome cxhibit ve ry small varüuion (see Figure 4c). The lemperature can Ihen be Ihoughl as uniform along lhe meridian direct ion 100.

On lhe whole, because of lhe previous considerar ion, lhe tempen.llure disl ribut ion can be assumed constant along both meridian and para llel d ircclion, leaving only a gradienl along lhe rad ial direcl ion, in view of a si mulation of lhe slruclural response by finite e lement techniques.

1 .0 .-----;-~--,----,.-'-A-"-'O-_oo<"T, T,-.-"-~-"-T-M-2-04-.-T-M-206--'

2 Cross.torrelallOn TM204-TM206 05 . J Cross orrelation T~204.1M208

0.0

·0.5

1_-= __ JL~=--L ____ ~~ ____ ~~ __ -L __ ~~la~1 ·1.° Õ 500 1000 1500 2000

·0.5

[days]

1 Auto-correlabon TM201·1M201 2 Cross-correlation nJi201·TM204 J - Cross~rrela!ion T~20'.lM209

lei . , ,O Lo-~-.L~5-00---1.--~I.,.oooL----1~15"'OO::--L-~2"'OOO':CJ

[days)

Figure 4 • Correlation funClions belween data acqui red from Ihermometers in several webs: a) mdial di rection; b) parallel direct ion; c) meridian direction.

A. CHIARUGI ct aI. I Brunellesc his Dome in Florence 343

3.4 Cracks behavior Along the main cracks, a different behavior is seen in lhe lower part with respecl

to lhe higher part of lhe crack. While the inferior part is opening, the superior on e is closing, and vice versa. This can be seen from the correlation graph (Figure 5a) which shows a phase of delay of sl ightly more Ihan 900 among curve no. 1 (auto-correlat ion of lhe instrument placed in the higher pan of lhe internai dome in web no. 4) and nos. 4 and 5 (cross-correlalion belween the prev ious and those placed in the lower pari ). It is to be nOled Ihat greater opening of the cracks in those areas of the Dome that are freer to expand follows incrcases in temperature. The opposite behavior is found in those areas where effective reslraints are present (especially due 10 nave and chupeis), and 50 thermal dilalation can only take place by a diminution of cracks. The phenomenon of variations in crack width therefore cannOI be explained simply by cons idering lhe crack as a "Ihermal joint" (i.e. uniform beh;l\'ior along its extension) aS was previously thought [2 ]. Instead, the behavior is characle ri zed by a considerable complex ity that is related to the inherent complexity of the structure (i nteraction belween the dome and other structu ral elements. such as lhe nave. lhe chapels and 50

on). In the rad ial direction , every crack sho\Vs a behavior similar to lhe previously

described one. Crack opening and closing are nOI in phase between the internai pan of the Dome and the externai one (see Figure 5b).

3.5 Correlation between temperature and cracks di splacements The correlation bctween temperature values and crack amplitude variat ions

(Figure 6) confi rms what was previously believed. For example, an increase in temperature (warm months) induces a different behavior in lhe cracks near the tambour (curves no. 3 and 4) and in Ihose situated in the upper part of the Dome (curve no. 2). This fact implies a very strong correlation betwccn thermal inpul and cracks movemcnts. as long as il is concerned with the seasonal behavior of the st ructure.

3.6 ESlimation ofthe trend Wilh a belter understanding of the seasonal behavior of the dome, the second parI

of the analysis focused on the stud y of cracking trends, lhe understanding of lhe long­term behavior of the crack layout.

Long. lerm cracking can be belter understood by considering the drifting behavior of the four half-arches nowadays constitu ti ng the dome. It is important to determine if cracking is increasing with time. This can be evaluated by removing seasonal effects from the recorded data and analyzing the residuais.

A procedure has been established (as described in (3)), based on a recursive approximation of the recorded signal. A first trend removal is performed, by fitting the experimental data by a linear plus harmonic function with a lime-period of one year. The period of one year was chosen because of the particular shape of the au lo­correlarion funcl ion, clearly showing this periodicity. Due 10 the fact Ihat the au lO­conelation of the residuaIs oblained in this way sho\Vs another strong periodicity of abou t six l11onths, a new fit with a harmonic function having the same period is performed. The resulting auto-correlation function of lhe residuais does nOI show any parlicular periodicity. So, the residuais can be considered as almost being delta­corrclated (white noise) with a dislribution very similar to a Gaussian one. The obta ined approx imation seems lo be sufficiently reliable, and lhe derivative of the linear componen! gives Irend estimalion.

344 STRUCTURAL ANALYSIS OF HISTORICAL CONSTRUCTIONS 11

1.0 r---'---~--:--......,.--:-"-'--.,.,,-'=::-:::=--' I I I A~IQ'COrrelalion DF101-0f,401

08

0.6

o.

o.,

0.0

-0.2

-0.4

-0.6

I 0.8 · 1·

0.6

o .•

o., I

0.0 I

-0.2 I.

I -0.4

I -0.6 . I

-0.6 I

. , .. "I" I 2- CIQSI'cp~(elalion OF10l.D~.402 . 3 Crcss-cbrr.lation DF 01-01;"406

·1 . 4 Cross-eOrrelation OF~OH)F4tO'

. _. 1 I ·

(ai

500 '000 ' 500 2000 [days)

I I , Auto-Corr.el8 tion OFt05-0E'406

I I 2 CrO" 'IUele~on DFt06-0~407 . 3 Crosa- rrelalion DF 06-01;'408

I I l- I· I. .I. J.

I:

·1· I I I ·

T I . .. ·1· I I

Ibl 500 '000 ' 500 2000

[days)

Figure 5 - Correlation runct ions between data acquired from di splacement transducers in web no. 4: a) meridian direction; b) radial direction.

Figure 7 repon s lhe origina l signal and lhe superimposition af both recorded data and lhe approximating funclion. Table 1 shows some results obtained by this analysis on data coming from displacement transducers placed in web no. 4. Pararncters rcported in Table I refer to the following expression:

{ 2 · " } { 4 · " ) x(I) =A+B·I+A,· si 36S.2S I+<j>, A2 ·si 36S.2SI+<j>, (I )

where x(t) represents the least squares approximation of a generic recordcd time· history x(t).

This procedurc allows for the estimation of trend values. The reliabi li ty of ' obtained values is confirmed by the (acr tha! Ihermometers show, over lhe same time· period, negligible Irends. A slab ilily analysis of the trend estimarion with respect to the acquisition period (Ihat is, lhe innuence of lhe available data on lhe eva lualcd quantities) has been performed. This analysis has been done on tcmpcrature data, looking for the samplc Icngth that gives a stable estimarion of lhe trend. It is

A. CHIARUGI et ai. I Brunelleschis Dome in Florence 345

reasonable to think that temperature data must not show any linear trend over a long enough period of rime. When temperarure data no longer show this trend, we have assumed that, over the same period of time, the trend obtained from the analysis of displacement transducer data was reliable.

LO , O.,

O.

O.,

O.,

0.0 I

-0.2 I· I

-0..4 I

-0..6 . r -0..6 ·1·

-1.0.0 500

I Auto-cQrretation TMfo1-TM401

. 2. CroSi-Cp~(ete.lio'" T.Mlo.1-0f'402 . 3 cfOss-cbrrelation TM 0.1-01;406

4" CfOss-cprrelation TMf.01-0F41o. -

•• _2_ (

··1·······1·····-1; ·1· I'

'000 '000 [days]

Figure 6 - Correlation functions between data acquired from thermometers and displacement transducers in web no. 4.

1.5

1.0

05

0.0

-0.5 .1.

-1.0

-1.5 0 500

1.0

-1 .0 .... [ . ,.

o 500

'.000

r '000

1.50.0

r' . 1500

2000 da s]

1 . Ib)

2000 [days]

Figure 7 - Trend removal analysis (referred.to data recorded by DF406, a disp lacement transducer in web no. 4): a) recorded signal ; b) recorded signal and linear plus double

harmonic approximation

346 STR UCTURAL ANALYSIS OF HISTORICAL CONSTRUCTIONS 11

A B AI A2 lns lr. (mm/ '1'1 '1'2

(mm) century)

(mm) (rad) (mm) (rad)

DF401 0.258 -7.730 0.41 3 328.065 0.19 1 24 1.3 14 DF402 0.029 -2.625 0.35 1 306.588 0.21 6 232.984 DF403 0.246 -0.721 0.193 2 14.570 0. 149 191.590 DF404 0.001 -2.205 0.043 359.414 0.023 223 .352 DF405 0.095 0.340 0.123 154.891 0.045 81.097 DF406 0.441 -6.905 0.464 230.277 0. 172 215.618 DF407 0.584 -2.576 0.604 195.623 0.150 169.055 DF408 0.442 -2.302 0.502 180.321 0.130 144.166 DF409 -0. 104 6.053 0. 11 5 48.810 0.01 2 287.3 11 DF410 0.705 -4.827 0.841 209.898 0.129 177.91 6 DF4 11 0.587 -1.570 0.704 201.209 0.107 134.429 DF412 0.275 -0.971 0.390 210.468 0.035 167.740 DF4 13 0.352 0.204 0.377 180.728 0.152 168.651

Table 1 - Trend analysis results 0 0 displacement transducers in web no. 4

T he obtained results show that, Qvcr time-periods longer than 5 years, lhe tetn­perature data sceITI to be slable. Temperature data show trend values nOI greater Ihan

0.1220 °C per ycar, which is negligible. Tabie 2 sho\Vs, for lhe some displace menl transducers rcported in Table I. lhe trend eSlimalion perforrncd over differcnt periods af time. As it cao be secn, lhe results seem to slabi lize using a period cf 5-6 years for almost ali lhe instruments. Nevcnheless, since lhe monitoring syslem is still acquiring data, furt her analyses will be performed over longer periods af lime, increasing lhe reliabilily oflhe results.

Instrument years

(mm! 3 years (mm!

4 years (mm!

5 years (mml

6 years (mm/

Table 2 - lnOuence of time duralion on lrend ana lysis results for displacement transducers placed in web no. 4

A. CHIARUGl CI aI. I Brunelleschis Dome in Florence 347

4. ANALlSYS BY FINITE ELEMENT MODELS The behavior of the Dome under lhe effecI of thermal loads has been investigated

by using finile element models, comparing the deformations obtained by numerical analysis with the measured ones. Due to the complexity of the problems it could be difficult to Slart the study with a complete model and in a firsl time partial and simplified models has been prcpared 10 analyzc specific aspects. Finite element analysis has been performed using three kinds of models described below. slarting from lhe simplest plalle models, Ihen axial symmelric models and conduding with lhe mOSI complete tridimensional modelo

The analysis of the daily thcrmal variations effects have been carried out by means of p/alie mode/s. This can be done considering a strip of Dome a long the parallel direction, placed at a general levei and having a unit Ihickness. These models are necessary in order to understand the crack variation amplitude in lhe horizontal plane.

Concerning over the opening and lhe dosing of the main cracks which is not in phase between the lower part and lhe higher one Cyearly thermal variations), it has seemed basic to make reference, as a firsl approach, to simplified axial syl1/l1Iefric models, using in thi s part of stlldy lhe general thcory for she ll s fonning su rface of revolution. With these simple models it has been possible 10 undersland the general charactcristics of lhe physical phenomenon, in order to correetly reproduce the experimental measurements.

After these IWO steps afina/tridimensional model, which repraduces, as dose as possible, lhe Dome geometry and the part below il (tambour, pillars, chapels), has been elaborated, in order to undersland the constraint effecI due 10 lhe chapels.

In ali Ihese models lhe masonry has lhe mechanical properties of a linear, elastic, homogcneous. isotropic so lid (Young modulus E=500000 N!crnl; Poisson ratio v=O.!: thermal dilatation coefficient ~=O.8xIO·5 0C'I), ([I], [4]). The used structural analysis code was FEMAS90, made at Inslitut fOr Statik und Dynamik. Ruhr Universital • Bochum, Germany.

4.1 Analysis bv plane models The recorded data dcarly shows that lhe internai dome is isothermal with respecl

to lhe daily lemperature variations but a daily variation in widlh of the main cracks in bOlh lhe domes takes place. Comparing temperatures and cracks width (Figures 8 and 9) it can be obse rved thal an increase of temperatu re in the eXlernal dome in warrn hours produces the opening of the cracks along the whole thickness of the Dome and vice versa.

These obse rvation allow to affirm thal lhe cracks movements are not due to lhe dilation ofthe internaI dome. which is isothennal ali thraugh lhe day, bul to lhal of the externai one. This means that lhe dilation of lhe externai parts is able to induce a coercive nexural effecl on the parallel slrips, whose resuIt is the relative rOlation bClwecn the faces 01' the main cracks. The externaI dome produces a "drag effect" on the internai one because of the connection between lhe two shells (realized by comer ribs and by two equally spaced ribs for every panels), Ihus proving that , by means of the adopted building technology, a perfeet monoJithic conneetion of the IWO domes has been realized.

Curves in Fig. 8 and 9 have been obtained by averaging the approximating funclions of lhe recorded signals Cin a leas! squares sense), conceming the instruments hélving lhe same position in the. thickness of the Dome. These data have becn logged in IwO specific days. when measurements were carried out every 1.5 hours [5).

348 STRUCTURAL ANALYSIS OF HISTORICAL CONSTRUCTIONS 11

- - --- - I - - - --- - I -

I- -

- l­

I - - --I

I I

.... r

Figure 8 - A verage cf lhe dai ly tcmperulure in lhe th ickness cf lhe Dome (l: intrados internai shelJ; 2: ex lrados internaI shell; 3: middle externai shell: values in °C) .

... t~ . ( .. )

Figure 9 - A verage cf lhe daily cracks amplitude variations (second internai galle ry levei): (1: intrades internai shell; 2: extrados internai shell; 3: inlrados ex ternai shell;

values in mm [pos ilive variations = cracks closing]).

The behavior of a sl rip cf Dome in lhe parallel di reetian. Iying ar Ibe second internai gallery levei can bc considered. Because af lhe symmetry of lhe slructure and lhe thermaI loads, it is enough to cons ider a quartel' cf strip cut by tWQ ve rtical planes containing lhe Dome axis and onhogonal to lhe panels not crossed by lhe main cracks (Figures 10 and 11 ). 11 has been assumed Ihal the slrip can gel deformed onl y in its plane, and in lhe two sect ions previously described symmetry boundary conditions for lhe uncracked stl1lctu re were considered: lhese points musI move only in the radial d irect ion and their displacernent has 10 be establ ished in advance, otherwise lhe formulated stalic problem is indelerminate.

A un iform increase of temperalu re in the externai dome produces the deformed shape shown in Figure 10 (in this phase lhe radial movement of lhe symmelry secl ions is nol allowed). lt is possible 10 observe lhe clos ing of lhe crack along ali lhe thickllcss of lhe Dome. nol in accordancc with the experimenta l data recorded by lhe deformeters. In fact, curves in Figures 8 and 9 show Ihat an increase of lemperalure in' lhe externai shell produces lhe opening of lhe main cracks. Figure II report s the deforrned shape obtained seu ing lhe radial rigid body 1ll0vemCnl af lhe IwO symmetry secl ions in such a way as lo have the opening of lhe cracks along ali lhe thickness; in this way a cracks ampl itude variations aI lhe Dome intrados greater Ihan lhe one aI lhe exlrados can be observed, in accordance with the rcco rded data (Figu re 9 ).

A. CHIARUGI el aI. I Brunelleschi s Dome in Florence 349

Figure 10 - Plane model: defonnation under uniform increase of temperatu re in ext. Dome. No radial movemenlS along

symmetry sect ions.

Figure 1I - Plane model: deformation under uniform increase of lemperatu rc in ex l. Dome. Rigid body radial movements

of lhe !wo symmctry sections.

As a maner of fact, daily cracks ampli lude variations in the horizontal plane can be explained by lhe superpos ition of (wo effects: a) lhe relalive rotation between the two faces of lhe crack, due to the temperature grad ient ; b) the radia l displacemenl UR of lhe symmetry sections geoerated by lhe inleract ion belween "parallel st rips" aod "merid ian strips".

In .order 10 beller understand the previously described phenomenon the temperalure distribulion shown in Figure 8 can be assumed as thermal loads for the model. Obtained rcsu lts are shown in Figure 12, If the model \Vere able to exac lly reproduce the defonnalive behavior of the strip, we should have obtained, in every instant, Ihal lhe difference belween oblained di splacemenls and experimental ooe, should have been COnSlan! , represenling lhe described radial rigid body movement. ·This difference has been evaluated usiog data reported in Figure 12; lhe graph showing lhe radial displaceme.nt of lhe symmetry sections in lhe plane model is represenled in Figure 13.

Concerning the comparison belween experimental values of the main crack amplitude varialions, aod lhe theorelic one, some differences are necessari Jy prescnt because the plane model di sregards lhe flexural rigidilY of lhe "meridian sl rips". This implies that the relalive theoretic rotation between the t\\lO faces of lhe cracks is a1ways greater Ihan lhe experimental one, This proves thal, even if lhe Dome is actually divided in foue "segment" connected in lhe higher part (because of lhe main cracks), there is ali lhe same a certain tridimensional behavior: thi s effeel, typieal for lhe shell forming surface of revolution, shows Ihal meridian st rips sl ill g ive an "elaslic" constrain 10 lhe defonnation ofthe para11eJ slrips in their plane.

350 STRUCTURAL ANALYSIS OF HISTOR ICAL CONSTRUCTIONS 11

Figure 12 - Analysis wi lh lhe plane model: cracks variation amplitude undcr lhe thermal variations of Figure 8 ( 1, 2. 3 see Figure 9).

Figure 13 - Radial displacemen t CUR(I) of lhe symmctry seclions in lhe plane model (strip aI lhe second internai gallery levei; [positive values induce cracks opcning)).

4.2 Analysis by axial-symmetric mode ls In order to better undersland lhe physical phenomenon cf lhe phase diffcrences in

the crack amplitude variations (yearly movements), some simplified models have becn elaborated. The struclure composed by lhe "Dome plus lambeur" was considercd (Figure 14). This cne has becn modeled by means cf shells forming surface of rcvolution. Under an un iform increase of temperaturc (ãT>O) two ext reme cases for boundary conditions in the lower edge have been examined: a) withoLlt any restraints for the mdial displacemenls: b) rad ial displaccmcnls complctely fixed.

In lhe first case, there is no variat ion in crack widlh. and the dcformed shape would h'lve been lhe same even if lhe cracks in the model were removed. Thc radial displacemenl for cach poinl on lhe middle surface of the shell is proporlional 10 lhe parallel radius R ar thal levei (ã~ = ai ã T R), moreover any tensile stresses is present [6}. 011 lhe contrary in the second case, the rad ial resl raints produce a defonncd shape of lhe meridian st rips typical for prismatic bars supported by a conl inuous eluSlic foundation: Ihis is due lO lhe induced forces on lhe lower edge of lhe shell. This can explain how (because of lhe flexural deformation in the meridian direction ) un increase of temperature produces lhe c10sing of the main cracks near the tambollr and their opening in lhe higher p'1I1 of lhe Dome (Figure 14).

A. C HI ARUG I et aI. I Brunellesc hi s Dome in Florence

fE UNI~5I TA ' OBôLI $lIJOl OI FIR90CZa 2fKS

rAcOt.T,,· OI IHC~tEJlIA

OIPAR T" IMtf'lTO OI lI'(;. CIVILEl

Figure 14 - Axial symmelric model: deformation under Ihennalloads

35 1

Moreover, on the Dome a temperature gradient in the normal di rect ion C<tll be also observed , producing a relalive rOlalion belween lhe faces of lhe main cracks. The further temperalure dislribulion to be added 10 lhe previolls one is Ihen a non uniform one. caus ing lhe opening of lhe main cracks aI the intrados and Iheir cJosing ai lhe eXlrados. This effecI has 10 be superposed wilh Ihat previously described . due lo a un iform increase of temperalure. The final results of this superposi tion is: aI lhe int rados, a smaller c10sing of lhe cracks in the lambour and a grealer opening in lhe Dome; aI lhe exlrados. the opposite behavior according to lhe recorded data (Figure 15). The five differenl curves in Figure 15 are refe rred to the leveis where deformelers are placed along lhe four main cracks (J: extrados o I' lhe pillars; 2: lower par! of the ci rcular windows; 3, 4, 5 respeclively lhe firsl, lhe second and lhe Ihird internai gaJlery levei of the Dome ).

4.3 Analvsis by a tridimensional model A finile element model of a BnmelJeschi-type dome, accord ing to an overall

~ hape rcproduci ng lhe monument one, was defined introducing some necessary simplificalions and regularizalion. [n spite of lhe introduced simplificalions, lhe model mude possiblc to llllderstand the global interaction between the "Dome plus lambour" sl ructure and lhe chapels. As desc ribed before, lhe real boundary conditions assumed at lhe basis edge of lhe lambour are fundament al, in order to reproduce the main crilcks IllQvements. The geomelry of an ideal octagonal, regular, double shell. ~ymmelric dome. wilh c ircular vertical profiles a1 the comers, was cons idered . The two shell s are connec ted by comer ribs and by two equall y spaced·ribs for every panel

352 STRUCfURAL ANALYSIS OF HISTORICAL CONSTRUCfIONS 11

([IJ, [4J, and [7]). The F. E. mesh was built up for a quarter of Dome (induding lhe pillars and lhe chapels), CUI by tWQ orthogonal vertical symmetry planes (Figure 16). The mesh takes in 2248 isoparametric plane elements (with 3 ar 4 nodes) and 1722 nodes, for a lOtai c f 8538 degrees af frcedam.

la]

<> ~ ••

"--===:;;:~~ /.

- _/

Figure 15 - A verage af lhe best fi! for signals recorded by deformeters placed on lhe main cracks on inlrados internai dome; (values in rum [pos itive = cracks c1osing])

• í\"

~

/7 IEII)N'V!RS'T~' OEJ;LI S'r'I.()! OI F IREJ<ZEI 2re

F ~covrA' III IIC~lA

- D1PARTl M8 NTO OI I NG. CIV1 LEI

Figure 16 - Tridimensional mode!: deformatian under thcrmaJ loads. Maximum yearly variations a f lhe mean tcmperature in lhe externai dome

A. CHIARUGI el aI. I Brunelleschis Dome in Florence 353

The main crack posilion was defined by schemalizing lhe aClual cracking pattern, following two vertical planes symmetrically intersecting four oul of the eight dome panels (those at ±45° from the church main axes). This struclure was completely conslrained ai lhe nodes of lhe lower surface, whilsl on lhe side vertical planes obvious symmetry conditions were applied for lhe uncracked st ructure . In lhe following, lhe main resu lts oblained by analyz ing lhe struclure under yearly periodic temperalure varialions are shown.

The stochastic analysis of lhe recorded data allows defining lhe time-dependent temperature in lhe Ihickness of the slructure (Figure 17), as well as lhe lime-dependent crack amplitude variations (Figure 15).

This model correctly reproduces the deformalive behavior of lhe main cracks. The maximum yearly variations of the mean temperalure in lhe externai dome were considered as thermal loads (Figure 17). The deformed configuration due 10 Ihis temperature dislribut ion is shown in Figure 16. Figure 18 reports lhe comparison belween lhe experimental crack amplitude variations and Ihose obtained by means of lhe mede!: a curvilinear coordinale along lhe meridian line corresponding to lhe main crack is reported as abscissa in lhe graph. A fairl y good agreement between experimental and obtained results is evident.

:I 1 !

Figure 17 - A verage of the yearly temperature in 1988-1992 (values in °C fi: intrados int. dome; 2: extrados int. dome; 3: middle ext. dome]) .

. . ~-: .... \ .

.f ..... /.

.! I.

.I

. .. .

Figure 18 - Cracks variations amplitude: comparison belween experimental results (Figure 15) and Ihose oblained by means oflhe F.E. model (Figure 16)

(values in mm [posi live variations = cracks clos ing]).

354 STRUCTURAL ANALYSIS OF HISTORICAL CONSTRUCTIONS li

5. CONCLUDlNO REMARKS The digital monitoring system has becn a fundamental tool to achieve a betler

know ledge af lhe structural bchavior af 8runelleschi Dome. Following aspects have becn po inted out

temperature distribution is quite constant along both lhe mcridian and lhe parallcl direclion, only exhib iling a gradient in lhe radial direction, despi te lhe o ricntation w it h respect to lhe sun; yearl y and daily width vuriat ions are vcry wcll corrclated to lemperalure data; lhe externai do me behaves as a "Ihermul shie ld" for lhe internai Qne, while lhe two domes aCI like a uni! from a stati c poinl af view bccHuse af lhe presence af lhe structural linking betwecn (hem; meridian linc cracking does no! behave as a !herrnal joint, s ince there are some ou t-of-phase differences between data reeorded in lhe upper and lhe lower pari of lhe dome; Trend analyses have becn pcrformed on data recorded from the monitoring system

showing an average annual incrcase of width eraeks of about 0.06 mm. The struetural analyses under thermal loads earried out using finite c lements

models of the Brunelleschi Dome have mude possiblc 10 correetly reproducc lhe

behavior of the monument under these loads.

REFERENCES [IJ C HIAR UOI, A .. FANELLI. M. and OIUSEPPETII , O. - 'Analysis of a

Brunelleschi-type Dome includ ing thermal loads', IABSE Symposium 1983: Strengthening of Building Struetures, Veniee, It aly. 1983.

[2J OPERA DI SANTA MARIA DEL FlORE - ' Rilicvi e studi sulla Cupola deI BrunelJesehi eseguit i da ll a Commissione nominata il 12 Gennaio 1934', (i n It alian), Internai Report, Firenze, 1939.

[3J BARTOLl, O., CHIAR UO I. A. and OUSELLA, Y. - Monitoring Systems on Historie Buildings: lhe Brunelleschi Dome, l ouma! Df S/rucrura! Engineering . Yol. 122. No. 6, 663-673 (1 996).

[4J CH IARUO I, A., FANELLI, M. and OIUSEPPETII, O. - 'Diagnosis and Strengthening af lhe BruneJleschi Domc', (ASSE Sympasium 1993, Rome, Italy. 1993.

[51 BA VETT A, F. - "Prob!emi di cupole di grosso spessore C01/ pal'/icolare riferimelllO afta cllpola di Santa Maria dei Fiore a Fireflze", (in Ital ian) Final disscI1ation Thesis, University af Florence, Faculty af Engineering, 1994.

[6] V.A, - "Elaborazione ed inlerpretazionc di dati sperimentali da sistcmi di Illoniloraggio: la Cattedra le di Santa Maria dei Fia re', (i n Italian) Civil Engineering Department, University of Florence, Internai Repon, 1990.

[7J C HIARUor, A. and QU1LOH[Nl, D. - 'La cupola dei Bruncllcschi. La geometria'. (in Italian), in " La cupola dei Bnmclleschi: il convegno di Ravenna", Ravenna, Ilaly, 1984.