STRUCTURAL ANALYSIS OF HISTORICAL CONSTRUCTIONS P. Roca, J.L. González, A.R. Mari and E. Onate (Eds.)

C CIM NE, Barcelona 1996

ANCIENT DOMES ON SPAIN

J .M. Izquierdo 8emaldo de Quirós Instituto Técnico de Materiales y COIIsfrucciones ONTEMAO

Monte Esquina 30

28010 Madrid, Spaill

1. INTRODUCTION

It 's not for strange that one of the first p athology treaties known in history is related to t he study of a shell on masonry. The moment whell, after the second half of the 17th. century, charaderistics of matherials and their behaviour in a strudure, related to their capacity to whistand loads as well as to their surrounding enviroment was known, the path to begin a serious study of the causes why, on certain moments , structures stop behaving properly and begin showing damages , was set. And, most important , the risk those damages meant for the structure was beginning to be possible to evaluate.

Up to this point structural collapsing causes were, esentially:

Arcrutect's unskill, or Contractor's, as reflected even in Hammurabi's Code.

Rot of organic materiaIs, and naturaI alteration of inorganic ODes. (House leprosy, as shown in Numbers 34.45) .

• Natural catastrophes, such as earthquakes, fires or lightning bolts .

• Foundation settlements.

Gods ' will and sin attonement (cvcn aftcr grcck ralionalism, !lee Lk. 13.4,

Silohe's tower).

But Dome behaviour resisted to fit into common behaviour patterns. Architect 's

skill was demonstrated when, shortly after putting off scaffolding, ruin didn't show up. Employed matherials didn 't easily degradate. Not in alI cases was it

possible to apply to fundatíon settlement or natural catastrophes. And God's will was utterly díscarded when the collapsed building was one just devoted to

His culto

J. M. IZQUIERDO BERNALDO DE QUIRÓS I Andent domes on Spain 247

Figltn }- The Clergy's Dome, in SaJamanca

Figun 2- The lantern

248 STRUCTURAL ANALYSIS OF HISTORICAL CONSTR UCTIONS

Pigu~ 3- Cracking on Dome's intrados

photograph ta,ken frorn the inter:section af the Dome's axis with the floar (this inclination is common in the lanterns af Salamanca's buildings dated prior Lisbon's quake, all af thero bending in the saroe direction). In the base of the lantern, the derangement of Dome's ashlar could be seeo.

The Dome's history was not rallying at all to the technic who was to board its

study.

In 1650 the bitumen for the Dome was being made, what brings the suspicioD

that the balf orange was in ao advanced construction paiot. Later, in 1667, the current workmaster, Pedro Mato, S.J., charged six iron flanges for binding the maio Dome, as some cracks had shown 00 it. Some time befare, ou year 1661,

Marto Jimnez went to Madrid looking for a solution to cover the Dome, due to exudation shown beside previous bitumen disposal.

Prof. G. de Ceballos, on his studies about Salaroanca's Barroque, concludes that the cracking happelled when the Dome recived the huge lantern's load.

Even lacking of the original docuroents, it seems odd to me that binding was a result of repairing decission caused by cracking immediatly produced after lantern erection, for that would mean that seventeeo years were spent on this craft, wruch seems too much time specially tak.ing ioto account that only nine

J. M. IZQUIERDO BERNALDO DE QUIRÓS I Andem domes on Spain 249

And, even then, failures happened. And failure modes, in rnany cases, were incomprehensible. There was a generalized idea of the existance of an analogy between Domes' and archs' behaviour, wruch were , from Etruscans, far better known, but third rumension 's influence on shell resistance couldn 't even be concieved. With the partial breaking of a Dome, with the fall of part of the nervature (and even of the keystone), it was difticult to understand what has happening without reffering to supernatural adion.

We present in this conference a summary of some of the interventions in Domes on historical buildings we have performed across 20 years of activity in the Rehabilitation and Pathology department of INTEMAC, during the study of which non-traditional fadors wruch have proved to be determinant to the monument 's condition have appeared.

2. A SINGULAR DOME W1TH SINGULAR PROBLEMS

It 's not exagerating to say that the Clergy's Dome at Salamanca is a quite singular construction. And not so because of its dimensions, wruch are relatively modest, with 13,40 m of internal diameter and a base thickness of 1,40 rn, as much as for the abnormal proportions of its lantern, whose internal diameter is 4,90 m , and his height 10 m , calling our attention when we approach the town on road , surpassing other constructions.

When spanish Culture Ministry charged to our Institute the study of damages, the situation was alarming. Few months before, a moulding had fallen off, st riking one of the church's benches and not causing personal damage by sheer luck, and all symptoms seerned to point to a possible collapse in middle terrn , with the irreparable lose tbis event would cause.

Stone's degradation on the outside was too noticeable for a building counting only some 200 years old. Vast zones in wruch ashlar surface was compIetely disagregated could be seen , as well as furrows caused by water's erosion. Particularly, the superior coroice in the lantern had fallen off in certain zones , taking the balustrade with it.

The Dome soffit was affected by important cracking in the meridian sense. The cracks , upon reacrung thole's height, tended to go through the axis of the windows, moving then to the coIurnns with operungs of 10-15 mm.

The Jantern shows an indination that becomes perfectly noticeable from the outside, translated in the inside as an excentricity of the vertical of the cirdes of the thole's base and the lantern 's Dome, which can be appreciated in a

250 STRUCTURAL AN ALY$IS OF I IISTORICAL CONSTRUCTIONS

F igun ~- Lantern 's inclination

F igun 5- Iron fl anges binding t he Dome

1. rvl. IZQUIERDO BERNALDO DE QUIRÓS I Andell( domes 011 Spain 251

years were needed to build traosverse archs, pendentives, basis, thole and maio Dome (we know that principal nane was installed on 1641). Aod much more strange are the worries for some leaks on the main Dome six years before, when according this hypothesis Dome was still in open air on rus crOWD.

50 we think t hat cracking and specialy leaking (this ODe due to joints operung as we will see later) both startt":d ;L<; I<.oon as the Dome was built, and they progressed in sucb a way that in spite of covering the Dome with lead plates C'of four pounds of lead every tierce on square", what , assuming that those measures were related to Castilian pounds and sticks, brings a medium thickness of about two millimeters), 00 1661, six years later, t he Dome's aspect was disturbing. And Marto Jimenez exclusive worrying about leaking proves, in our opinion, that on that year of 1661 cracking were nothing to be worried about, ar the appropiates measures would had been taken , specially taking into account that four yers before, the Augustines Recoletas church's Dome had collapsed .

Figure 6- Thole's cracking

Twelve years arter binding, a fur ther inspection was made, due to doubts about Dome's security, and 00 August 1685 Juan de 5etin was consulted. He then performed the first favourable report we have noti ced, warranting the

unexistance of any collapsing risk. Anyway, some other workmasters (among

252 STRUCTURAL ANALYS IS OF HISTORI CAL CONSTRUCTIONS

Pigure 7- Pendentives' cracking

those was Blas de Tamayo), disagreed with that opinion, and lead plates were removed, considering excessive their weight, (with an astonishing criterion , becallse the load would be less than 24 kp j m2, if lhe interpretation of lhe di sponible data is correct) and were sllbstitued by a wooden and shale jacket, attaining apparent1y the unanimity of ali \Vorkmasters about the work 's safety on 1686.

Ncarly forty years after, on 1721 , new maintenance works are necesary, and Joaqun de Churriguera again opens the controversy, when he gives his opinion thal in no way lead should be disposed over, patching the most damagcd areas with sand and lime and laying a bitumiIlous seallayer over the Dome.

Our unfortunale Dome was affected by Lisbon 's earthquake on 1755, which also damaged l he just finished towers, and on 1812, during the Spanish Independence War, as a result of the blasl of San BIas ' powder depot, suspicions about unstability of the dome grew , and church was closed OIl 1817 after a report

by the Architect Francisco de Paula Vega.

At last, a great repair is made on 1845, on which we assume that existing cracks were clogged, and a complete renewal of internaI Dome's decoration ,

higly spoiled by cracki ng, was done.

J. M. IZQ UJERDO BERNALDO DE QUIRÓS I Ancil.'nt domes on Spain 153

Every unfavourable report pleaded as damages' evident guilty to lantern 's excessive weight. Anyway, irom the beginning an apparently underrated aspect worried US. If Dome's stability was as precarious as to alarm ao Architect with the proffesional qualification of Churriguera, how was it able to stand the íormidable shocks caused by Lisbon 's earthquake or Sao Blas' powder depot's

bursting, which would be perceived on Salamanca as a sysmic adion grade VII or VIII? That ~eeUlt:tl iucongruous a.nd, looking for an answer to that qucstion

we began the study.

The Dome is built on a siliceous sandstone with high porosity and water absorption. The lack on resistaoce observed on the stone betweeo dry and

saturated states, loosing five sixths parts of its resistance, is also a remarkable chara.cteristic.

A first checking of the Dome's redesign with a simplified hypothesis was done, in order to roughly evaluate the stone's stress leveI. Results, in spite of the simplifications made 00 those evaluations, were coocludiog: Expeded stresses on the Dome, even on sysmic hypothesis, are low and cao 't explaio the existiug damage. Situation is net: The results of a.n evaluation made by usual methods and with usual desigo hypothesis show ao expected behaviour and the strudure, along the history, persists on haviog a differeot ooe. Obviusly another silent factor had to be there.

It is known that limestooes cao present a certain swell as its hurnidity rises. rcs a commonly underestimated characteristic of that kind of rocks because its value is ussualy of low importaoce and does oot become determinaot for its structural behaviour.

In our case, however, the observance of a coupIe of symptoms on monuments

built in Salamaoca on XVI and XVII centuries forced us to suspect that 00 this particular case the expansivity of stone could have a greater importance thao that usually assigned to, and we opted to study its dimmensional stability in different moisture conditions.

The results were surprising: after 24 hours of water imrnersion, specimens showed deformations that twisted and warped the original prisms, and veins

in a brownish calor sweated out , on wrnch occasionly longitudinal cracks could be seen. Volume expansivity was eigth or eveo ten times higher than rnaximum

expected value following prior bibliography, and internaI stresses deformed 00

a nonreversible way the pieces. Stone had a dimmensional unstability really

remarkable.

'54 STR UCTURAL ANALYS IS OF HISTORICA L CONSTRUCTIONS

Pigure 8- Deformed shape under seism

Figure 9- Dome's stresses under seism

J . 1\1. lZQU IEROü BER NA LDO DE QU1RÓS I ArlÇi..:n\ domes on Spain 255

A conservative estimation on differential strains between two cansecutive

courses showed that local tensile stressess, high enough to be able to be the actual cause of damages were generated.

The master quarriers were perfect connoiseurs, of course, of the stone's lack af resistace 00 high moisture conditions, and 50 they nidged stone in saturated date, and they let it cure once carved prior the placing in its pasitian af each blockj but through storing plenty of times in open air, they could'nt avoid a mise en place with a high moisture content.

Once the stone was placed, it progresively dries, until a dynamic equilibrium between rain and drought periods is reached, but the equilibrium moisture wiII aIIways be lower than primitive one, and a remaining shrinkage will stay.

We established lhat internai stresses generated by this phenomenon are high enough to justi fy the Dome's cracking. Let 's add that dimensional unstability is able lo justify also the historycal behaviour of the whole structure.

First notice about craclUng comes few years after finishing the caver, when somebody observes the bleeding of bitumen employed on water-proofing. Taking away that bitumen and placing a more perfect impermeabilization, cracking grows, until puting in question the whole Dome's stability.

But this behaviour is perfectly predictable starting frem the hypothesis of dimmensional changes: as long as stone drying advances, on a process that takes severa! years to be accomplished, first shrinkage cracks are generated, and those cracks get \Vorse when a beUer impermeabilization is disposed, causing an acccleration of drying and subsequently of shrinkage.

Once was the Dome flanged, cracking seemed to decelerate, but not to stop. Explanation of this phenomenum according to our hypothesis is clear, because, on the first hand stone's thermical inercy is quite different than steel's, so just whcn the flange's action is more necessary, ou drought season and with high temperatures, steel tension delies because of its diferentia! thermical expansion in regard to stone. However, on wet season stone tends to swell and steel to shrink by thermical adion effects, with a differential strain between dome and Range of at least O.15deformations on stone, and puts flanges out of service for

the next drought season.

Accidelltal actions (Lisbon's quake, San Blas's bursting) had not significance

enough, according to our evaluation, to produce by themselves any significallt damage on the Dome, but when acting over elements 011 shown conditions, they

256 STRUCTURAL ANALYSIS OF HI STORICAL CONSTRUCTIONS

must produce ao increase of cracking, as well as a rearraogemeot af lantern's basement ashlar blocks, favoured by dynamical action aloog seism.

Disposal af a new lead cover and new joiot repainting place again the Dame

on suitable condition for a new cracking process, due to moisture incorporated duriog crack repair and on ornamentation restoration , jointly with improvement 00 externaI impt:cwt:auilization.

00 the other hand, the thole has never beeo treated against moisture, and its behavior in froot to swelling has no difference at all with Dome's, aod so cracking on thole can freely develope (with the cardinal structural advantage that , as it works esseotialy in compression, needs oot to develope circunferential stresses for a correct structural behaviour - as Dome needs- and so the effect on this one has a greater transcendency) aod their strains can be transferred to the Dome.

Repair we proposed to spanish Culture Ministry began by trying to give back to the Dome its original capacity of circunferential behaviour, byepoxi grouting (the dryest method) of cracks and unrestrained vertical joiots, on Dome, thole and pendeotives.

lmmediatly after , flutings on the Dome's extrados shall be done by a dry method, and high strength cold-formed (with the object thatonce removed the yield elongation, plastic deformations 00 steel would not be of high magnitude) reioforced bars, which were handly tightcn using steel nipples, and cOlltrolling their elongatioo by mechanical extensometers, ulltiI a final mediurn stress of about 100 MPa was reached.

AI last, flutings were fiUed up with epoxi mortar, and a waterproofing layer, 00 epaxi mortar with fiberglass was disposed, foreseen a ventilation chamber betwcen stone and resin that allow gas interchange on stooe. Restoration was made under direction of ProL Arch. D. Antonio Fernández Alba.

So, the employed method was completely traditional, and similar to tbose previously applied. Maio differences were only 00 the optimal date for working (at the eod of drough season) and 00 a maximum care for avaiding water supply to the Dome, by disposal of a water- proofing cover over scaffolding aod making

every step by the dry way. As to the rest, flange 's capacity was not higher to previous one's, initial stress was even lower to that reached by heating ( the

ancient methad for prestressing), and we bad only the precaution af embeding flanges into the ashlar deep enough for avoiding an its main part the effect af

short-cycle thermical oscilations (day-nigbt).

J. M. lZQU1ERDO BERNALDO DE QU1RÓS I Ancient domes on Sp:ün 257

Figure 10- Dome's Jepair. New binding

Up to now, behaviour has been correct, and at least doubts about Dome's stability (which made once believe than there was no solution but Dome's complete disassembly and subsequent reconstruction, on an estimated budget of about $ 12 million) are now dissipated.

3. PROBLEMS OF GROWTH

Another case on which building's hitorycal study was determinant for finding damage's ultimate cause was Our Savicur's Cathedral, on Saragosse. Church's damages were important enough for fcrcing substituticn cf four columns

on church's zone corresponding to 16th. century ~nlargement. Certainly, immediate causes were on walling's poor quality, on column 's slenderness and on mortar's weathering degradation. But a serious ammount of existing cracks were not, on our opinion, so easily explained.

Building's history study, caracterization of matherials and damage 's pacient inventory lead to us to establish solid hypothesis about structural behaviour of

the whole church able for justifying symptoms.

Existing chronicles of an important disaster on the beginings of 17th. century allowed to us to justify the most alarming damages (placed exactly on immediate

25M STR UCTURAL AN ALYSIS OF I-I ISTORICA L CONSTRUCTIONS

Figure J } - The C lergy 's Dome at plesent time

areas to which collapse was situated by chronicles) , as cracking typology exactly agreed with that expectable by dynamics effects during fali. Area's minutely examination resulted on finding ancient restoration works dated later than collapse, acting as crack's seams, showing that damage's growth had becn null or very smali along the two last centuries, allowing to relativizate the adual problem.

On the other hand, study of constructiol1 stages anel of successives enlargements from primit ive 12th century's Romanic basilica aliow us to establish a consistent hypothesis about damages on church 's wheel window's wali area, at the temple's

bottom.

Due to existing soil characteristics and also considering how massives wal1s and buttrcss are, it was not only possible but very likely that long term differential settlements were produced between primi t ive basilica 's walls , 011 which soil woulel be higly cOl1sollidated along four centuries when enlargernent begall, anel

Tear walls and counterforts .

Damage's symptomatology study on walis and nerves, alIow to establish on an approximate way t he direction and magnitude of settlements, and consequently

to bui lt a mathematical model for vault's stressing.

J. M. IZQ UIERDO BERNALDO DE QUlRÓS I Al1cicll( domes 011 Spain 259

Figure 12- Rebuilded column on Our SII.ViOUI Cathedral (Saragosse)

Results showed that shells themselves were su bmited to moderates stresses, but on settlement hypothesis tensile stresses on determinated areas , with a very neat coincidence with act ually cracked areas, vould appear.

Shells stay stable after cracking, but not columns, on which high load increases were generated, able to justify their buckling.

That's why we reported that the best solution for the actual problem on vaults

was not to act on them, but on walls foundation , and to carry on works aimed on reestablish their continuity,when degradation were unbearable, and mainly structural cosmetic activities. Restoration has been made under the direction of ProL Arch. Angel Peropadre.

4. THE IMPORTANCE OF SMALL DETAILS

The first problem we detied during previous technical studies for t he

restoration of Our Lady of Disperates basilica, on Valence, (proyect by ProL Arch. Ignacio Bosch) was to determinate on a suitable way the actual geometry

of the Dome.

260 $TRUCTUR AL ANALY$ I$ OF HISTORICAL CON$TR UCTIONS

Pigure 13- Enlargement works on the Cathed ral . T he 3rc·buttres lies over an older facade.

Figure 14 - Plant of the primitive Romanic basilica

J. M. lZQUIERDO BERNALDO DE QUIRÓS I Andent domes Oll Sp:l.in 26 [

Figure 15- Plant of the Cathedral at present

Dome's layout is particulary hard as neither suffit nor extrados are accesible, the first oue due to be covered by the internai decorated shell and the other by the glazed tile·roof disposed.

Resistance Dome's layer was 50 defined on an inrurect way. Shape of inner surface, which receive Palornino's Ífescoes was determinated by using photographical photogrammetry, and subsequentially internai sheU thickness and gap between both shells were measured by cores opened 00 the resistance shell.

So, a sector of Dome's cover corresponding to a nerve on ENE orientation was disrnounted , picking up tiles and ceramical board until footing pieces and Dome's extrados were seen.

Then we made several cores 25 mm f through the sheet, {rom lantern's basernent

on six meters foUowing a meridian une, point where we found that rustance between both sheets, resistance and decorated one, was 50 low 50 thick the nervature 's oversize was (about 10/ 15 em), and we stoped due to damage's risk

on decorated sheet. Further verifications were made on upper and lower zones of halí major axis) to confirm that gap between both sheets stay approximately constallt on a horizontal sectioll, on a positive resulto

262 STRUCTURAL ANALYSIS OF HISTORICAL CONSTRUCTIONS

Figure 16- Cracking areas (modelling)

Followjng on, we bad to find a mat hematical medel of Dome 's geometry. At first we tried to fit an ellipsoid, where halfaxis were those of the elliptical basement

and keystone high from basis.

Comparison betwecn theoretical sections 50 obtained and actual measures showed substantial differences , mainly on the crown cap , at about (our meteIS

from Domes keystone, on the arca where the Jantern basement is, and 5 0 the approachal was considered indefensible.

After some unsuccessCuly tentatives, we study t he possible approach by a surface cf the family of bicuadra ti c ellipsoids:

and specifically we tested a parametrical fitting ou t he Corm :

1 M. rZQUrERDO BERNALDO DE QUIRÓS I Ancit;:JlI dOlllçs 0 0 Spain 263

Figu.re 17- Actual cracks on voults

Fitting was surprisingly excellent, with differences lower than 5 em along plotting of theorical and adual surfaces, on resistanee shed as on decorated one, on major halfaxis as on minor one, and so that hypothesis was accepted as the best fitting model for the Dome's geometry.

Beginning with this hypothesis , a mathematical model of both sheets, resistance and decorated one (painted with Palomino's frescoes, whose rescue was the restoration final objetive), hanged this one írom the other. Model had 10,614 degrees of freedom and was performed to evaluate effects of statical and

dynarnica11oads, heating and thermal gradients.

Wind and sysmic analysis was performed on a recurrent period of 500 years.

Thermal variations trough a complete year were measured by therrnocouples placed over tiles, under them and over Dome's extrados, in the gap between sheets, and on internaI sheet intrados.

As final results of performed analysis we found the following interesting conclussions:

• On decorated sheet, final stresses after creep are absolutely non-admissible on

first six meters írom thole, due to unrestrained shell buckling, as on this zone

264 STRUCTURAL ANALYSIS DF HISTORICAL CDNSTRUCTIDNS

Figure 18 -QUI Lady of Disperates (VaJence)

there are not hanging rods.

• Furthermore, Dome's creep produces ellipsiod's flaUering, due to lantern weíght. Rods which hang decorated sheet from resistance one, lack tbeir initial tensile stress due to matherial ueep, and OH certain cases even produces punching on internaI sheet, due to deflection imposed by the lantern weight.

• Differential sun exposal produces very high thermal variations Qver tiles (we even mesured nearly 50 oC on the sunny face in the heart of Jannuary!), but over the Dome differences are much lower, due to system tiles.plaster filieI­boarcl-ventilation ducts-footing pieces higb insulating efficiency. Temperature on Dome's exhados shows practically no short~term (day-night) variation and

time- temperature curve on this point adjust well to tile's three- days moveable mean.

However, stresses on hanging rods were extremely sensitive to diferential displacements on both sheets due to thermal behaviour. Analogously, small differences on temperatures inside chamber (i. e. very small differences on

hanging rod's temperatures) become on great variations on rod stresses.

• Principal modes of vibration are slightly different between both shells, on

J. M. !ZQUlERDO BERNALDO DE QU1RÓS / Ancient domes 00 Spaio 265

Figure 19- Plant and section

Figure 20- Ventilation system undeI tile Iooí

266 $TRUCTURAL ANALYSIS OF HISTORICAL CONSTRUCTIONS

AJUSTE DE LA GEOMETRIA DE LA CUPULA SECCION SEMIEJE MAYOR

Figure 21 - Actua.l shape vs. ellipsoida.l , parabolic & bicuadratic ellipsoide shapes

Figure 22- Dome's flattering due to lantern weight

J. M. IZQU IERDO BERNALDO DE QUIRÓS I Ancient domes on Spain 267

Figure 23- FIe.xuraJ moments induced by rods

a theorical unconstrained hypothesis (i. e. removing hangs). Resonance phenomena are then possibles.

Model answers, then, to every cracking and damage configuration actually existing ou the Dome. On the present time, actual scale model has been built on Valem:e's Polythecnic University, testing different repairing protocols.

5. BEHIND EVERY GREAT COMPUTER ...

Computer is a powerful ally and has become indisp ensable in structural design in very few years, but mathematical models are not , in general , conceived for modelling an ancient building, and our knowledge about long.term behaviour of traditional matherials is ussually very poor.

Neither limes, nor plasters na r bricks used at the present have nothing in common with those employed between 11 th and 17th. centuries. Existing bibliography is h ardly applicable to a particular CMe. When we read, for instance, that certain masonry has a medium strength , in compression, of about

10 MPa with standard deviation af about 3.5 MPa, what are we talkig about?

• WJlich was resjstance and llOmogeinity of ashlar pieces?

268 STRUCTURAL ANALYSIS or HISTORICAL CONSTRUCTIONS

Figu~ 2./- 2nd. mode oí vibIation (principal for d«orated shell)

• In wbich form does strength vary with masonry bond?

• Which were stone's stereotomy and mortar's granulometry?

• Did mortar carry pozzolans incorporated by tIle aggregates? Was it plaster OI

lime? Do we know how much ox's blood or horse's uóne was incorporeated for

makúlg it impermeable?

• How has tIle importance af lacking joint 's matlleriaJ been measured?

Al1d most important: What is the inRuence Df everyone of uppeI circumstances,

and their possible combinations, on strength and e1asticity modulusin short and

long term?

On the present state af the art, theoretical sophistication of model can result sterile. H sct:tmj <i.L least uouLtful ~he vradical rcability of usillg matherial's

elastic.plastic diagrams based on testing made over specimens crafted with actual matherials. The internai behaviour of very thick masonry walls is

hardly modelled by finite-e1ement sistem . Long-tcrm deformations are often determinatcd by joines matherial lack process along centuries, wich is a well­

known phenomellon, but very dificult to evaluate.

J. M. rZQUlEROO BERNALDO DE QUIRÓS I Andent domes 0 11 Sp.:lin 269

Figure 25- 4th. mode af vibration (principal for resistance shell)

50, mathematical modelling is only an additional tool on our toolbox, and not as something magical and definitive able for giving by itself complete sollution to our doubts. !t's too easy to feel impressed for grafic colorfull presentation which, properIy anaized, say nothing new. We will always find a certain mode of vibration whose shape will be coincident with actual deformation, but do the sysmic spectrum have enough energy at that frequency? Is it principal one, or 4th, 5th or even 9th mode? Are we sure that there is not any other clearer explanation of actual phenomena, sueh as matherial anisotropy, loss of the joint material or local differential settlements, producing the same local deformation?

The answer is always in experíence. A practised tecrueian can obtain much more precise results ÍTom an evaluation made on a cellulose tissue that an ali· round theorist one ussing bours of a Ceay IV and producing mountains of paper. Behind eveey great computer tbere is always a small bone's and fiesh' and brain's poor human being.