Possible Geometric Genesis of a Medieval Cathedral (Alba, Piedmont, Italy

Structural Analysis of Historical Constructions 1

Possible geometric genesis of a medieval Cathedral (Alba,

Piedmont, Italy)

F. Antonino

Architect, Savigliano (Cuneo), Italy

G. Pistone

Politecnico of Turin – School of architecture, Department of Structural Engineering, Turin, Italy

D. Zorgniotti

Architect, Cherasco (Cuneo), Italia

ABSTRACT: Since 1998 extensive studies have been conducted on the cathedral of the city of

Alba, both in order to assess the severity of a number of deterioration phenomena, and hence be

able to assess the current safety margins of the building, and in order to plan possible

strengthening works.

These studies were geared to the identification of the materials and models of behaviour.

A F.E.M. model was also developed as a valuable aid to gain an understanding of the structural

behaviour of the cathedral: the results of the numerical model turned out to be in good

agreement with the experimental findings.

Finally, of special interest was the analogy observed between the geometric genesis that, in all

likelihood, inspired the builders of the cathedral and the schemes that underlie the numerical

model produced in the course of present-day studies.

This article summarises the most important results of the studies conducted on the building so

far.

2 Structural Analysis of Historical Constructions

KEYWORDS: Project genesis, structural behaviour, numerical model, cathedral, geometry,

Alba (Piedmont).

1 FOREWORD

In recent years several studies have been carried out on the cathedral of the city of Alba, the

Church of S. Lorenzo (fig. 1 and fig. 2), in order to evaluate the stability of the structure and to

assess a number of static problems in preparation for the strengthening works designed to

restore the building to acceptable safety conditions.

Fig. 1: Church of S. Lorenzo

The theme was addressed from different points of view, in an effort to understand how an

important monument of architecture comes into being, is preserved thanks to the good qualities

that characterise it and in spite of the defects that undermine its survival, and finally what can

be done to preserve it and hand it down to the future generations while safeguarding, to the

extent feasible, the intimate and phenomenological shape in which it has reached us.

An initial series of investigations focused on the different apparatuses characterising the

building, to obtain indications that might serve as a valuable support for a correct formulation

of the structural diagnosis. Accordingly, the first stage consisted of making surveys of the

F. Antonino, G. Pistone, D. Zorgniotti 3

building and the foundations, identifying the cracking patterns, drilling cores from the masonry

walls for the analysis of its constituent materials, performing flat-jack tests at different points in

the piers and the facade, and monitoring a number of major lesions by means of strain gauges.

The surveys were carried out in parallel with an examination of the historical records, and

the information collected provided an overall picture of the building stages of the complex

through the centuries.

This series of preliminary studies were carried out according to the approach we normally

use to acquire knowledge about an ancient building; however, unlike many important buildings

of the past, whose structural and formal characteristics are relatively familiar, with its imposing

structure, a medieval cathedral is perceived as a silent and somewhat mysterious presence. Its

origins dating back to a remote past, the scarcity of coeval written records illustrating the

design schemes and the construction techniques, mated to a lack of present-day studies

addressing the historical themes through the eyes of a modern engineer, are but some of the

obstacles hindering a full knowledge of the structure.

In other words, we believe that playing an active role in the historical vicissitudes of an

ancient building does not amount solely to interacting with the material it is made of: it also

means to interfere with something deeper, more intimate. To understand its genesis means to

get hold of the key which governs not only its shape, but also its static equilibrium.

Modelling, no matter how refined or even based on a set of highly sophisticated

investigations extended to the smallest details, cannot provide an exhaustive representation of

the micro-cosmos of a construction as fashioned by the countless contingencies that

accompanied its genesis and altered it throughout its history.

Nowadays, the prevailing tendency among scholars studying ancient monuments is to collect

masses of data on a building to produce a static representation by means of numerical models

formulated according to laws compatible with the building’s constituent materials and


elements. This fundamentally analytical procedure is the prerogative of today’s cognitive

approach in the field of structural behaviour, and it should be viewed as a major conquest,

indubitably a powerful interpretation tool.

Nevertheless, such an aseptic and automated interpretation method, by itself, entails some

risks: albeit loyal to the details and the constitutive laws of the materials, it only provides a

model of the real building and may omit some of the fundamental aspects characterising the

behaviour of a structural body, due to the simple fact that the investigation failed to highlight

them.

Going back to the problem of how to gain an insight into the creative genesis of a building,

an effort to penetrate the conceptual outlook that generated a construction may help us identify

the criteria that guided the building process, i.e., their scope and their limits, as well as the

intrinsic errors present in them. At this point, a subsequent analytical approach as is typical of

contemporary science might yield fruitful results, in that it becomes possible to correct possible

deficiencies of the modelling process and to assess the results obtained.

Fig. 2: Interior view of cathedral

2 THE STATICS OF ANCIENT BUILDINGS

A thorough knowledge of a structure to be restored requires an understanding of the ancient

construction methods and the techniques used during the building process, the identification of


the historical period and the schools of thought that affected the development, the

transformations and the peculiarities of the individual structure. All this information constitutes

an indispensable tool to be able proceed correctly in the analyses, the diagnostics and the

choice of the restoration methods to be applied.

Surveying a monument or a building means much more than merely recording the numerical

data collected in the field. When we look at the architectural works of the past we must try to

identify the principles that determined their proportions. Otherwise, all we get is a set of

measurements: extremely accurate ones, as made possible by present-day technologies and

tools, but having no critical significance.

Any attempt to interpret an ancient structure with the tools of present-day structural analysis

without trying to understand or even gain a glimpse of the laws and principles that went into its

design and construction, by applying acritically the design and checking methods based on

elastic or elasto-plastic assumptions may produce uncertain or outright illusory results.

The number of structures created by mankind, throughout its history, without worrying about

the possibility of their being made of materials having an elastic behaviour exceeds by far the

number of structures erected since this behaviour was identified.

Before the eighteenth century, the strength of materials (in the modern acceptation of the

term) was not regarded as a scientific discipline: bearing capacity was basically entrusted to the

stability of the structure and the latter was essentially correlated with a modality of collapse due

to a loss of equilibrium, equilibrium being guaranteed first of all by the mutual relationships of

the various components of a building, i.e., more precisely by the geometry of the construction.

This explains why the architects of the past used small-scale models of the structure to be

erected, and this applied in particular to buildings of a certain importance, especially when

dealing with unique, unprecedented works, such as the Dome of S. Maria del Fiore in Florence:

the stability of the small-scale model was expected to ensure the stability of the full-scale

construction.


All the foregoing considerations and the questions arising from them persuaded us to adopt a

different approach to the study of the Cathedral of S. Lorenzo, to be conducted in parallel with

the method dictated by modern engineering science.

Though we do not claim that this approach will necessarily lead to scientific conclusions we

believe it is worth summarising the information and the principles that inspired the

investigation carried out along these lines (in parallel with studies more strictly associated with

structural engineering). We were driven by a desire to penetrate the impenetrable mystery

surrounding the construction of the Cathedral, by a desire to learn that imposed itself

obsessively on our judgment.

"Naturally good men desire to know" (Leonardo da Vinci)

In order to get across our state of mind at the beginning of this “alternative” study we

reproduce below a passage that expresses it quite aptly:

Invited by Holmes to the umpteenth crime scene, Watson had gotten there late and seeing

Holmes kneeling with his nose almost grazing the floor asked him “Mr. Holmes, what are you

looking for?”, without taking his eyes off the floor, Holmes replied “I’ll tell you as soon as I

find it, my dear Watson”.

Thus, in the spirit of someone who does not know what to look for, but knows he might find

something, we started out full of enthusiasm and inspired by many readings, and began to

question the Cathedral, which, after centuries of silence, began to tell us something.

3 MATHEMATICS AND GEOMETRY IN MEDIEVAL CATHEDRALS

3.1 The School of Chartres and geometry

The school was founded in 1020 by the bishop of Chartres, Fulbert, a disciple of Gerbert (pope

in 998 with the name of Sylvester II).


The School of Chartres, where future priests were educated, had an immense influence on the

construction of the cathedrals of western Europe. At the school, they also studied and recorded

the principles of Christian thought, which, in the construction of the cathedrals, found

expression through symbols, shapes, statues, numbers and special colours designed to get

across the Christian message to the – largely illiterate – populations of the Middle Ages.1

The science that was called "geometry" during the Middle Ages played a decisive role in

determining the dimensions of the floor plan and the cross sections of a cathedral, as well as in

the construction of the elements it was composed of.

A majority of Gothic builders did not spell out the symbolic meaning of their projects, but

acknowledged that geometry was the foundation of their art of building. Starting from a single

basic dimension, the Gothic architect developed all plan and elevation measurements according

to strictly geometric criteria and by adopting specific regular polygons as “modules”.2 The

masters of Chartres were obsessed with mathematics, which was viewed as the trait-d’union

between God and the world, the magic tool that could unveil the secrets of both.

Thierry of Chartres, the most influential representative of the School, tried to explain the

mystery of the Trinity through a geometric demonstration. The identity of the three Persons, in

his view, was represented by the equilateral triangle.

In the late twelfth century, Alain de Lille described the creation of the world, indicating God

as the skilled architect (elegans architectus) that built the cosmos as his royal palace by

composing and harmonising the variety of things created by means of the “subtle chains” of

musical consonance.

During the same period, the first Gothic cathedrals began to be built and the "musical

proportions" used by Alain’s divine architect were regarded by medieval architects as the

closest you could get to perfection.

1 Gout (2001)

2 Von Simson (1988)


It is necessary to take into account the influence exercised on the followers of the School of

Chartres by the Augustinian philosophy of beauty, conceived in musical terms and as the

enjoyment of an eternal symphony. In his De Musica, Saint Augustine defines music as the

“science of proper modulating”, designed to link together several musical units according to a

module, a measure, enabling the relationship to be expressed in simple mathematical ratios. The

informative principles of proper modulating and its evaluation are mathematical and hence,

according to Saint Augustine, apply equally well to the visual arts. Music and architecture are

both generated by the number. Saint Augustine uses architecture and music to demonstrate that

the number, as borne out by the simplest proportions that are based on “perfect” ratios, is the

root cause of all aesthetic perfection.

The application of "perfect proportions" was achieved through rigorous geometric reasoning

to define the requirements to be fulfilled in terms of both stability and beauty: Gothic

architecture seems to indicate that all static problems were in actual fact solved by resorting to

purely geometric methods.3

By submitting to geometry, the medial architect felt he was imitating the works of his divine

master.

The cathedral was conceived by the medieval man as an image or a symbol of the cosmos,

designed in an attempt to reproduce the structure of the universe. By designing a sanctuary in

conformity with the laws of a harmonious proportion, an architect was not only imitating the

order of the visible world, he was also providing an indication of the perfection of the future

world.4

The same concept of proportions was upheld by Viollet-le-Duc in his Architecture

Raisonnée when he claimed that in architecture, the proportions are established first of all

based on the laws of stability and the laws of stability arise from geometry. A triangle is a fully

satisfactory figure, it is perfect, in that it gives the most exact idea of stability. The Egyptians,

3 Von Simson (1988)

4 Von Simson (1988)


the Greeks started from this and medieval architects, later on, did nothing else. By means of the

triangles firstly they established the rules of proportions, so that proportions would be

governed by the laws of stability. [...] To obtain harmonic ratios in the composition of the

orders, medieval architects made use of triangles.5

3.2 The influence of Vitruvius: the circle, the triangular and the quadrangular systems

The modular coordination of geometry discussed above had been clearly expressed by

Vitruvius in his De Architettura when he speaks about the six categories of architecture:

ordinatio [...], dispositio [...], eurythmia, symmetria, decor and distributio [...].

Ordinatio: consists of the right proportion and measure of the individual parts of a work, taken

separately and in their relationship of proportion and symmetry with the whole. It is founded

on quantity [...]. But quantity is nothing but the assumption of a unit of measure of the work

itself and the harmonic realisation of the latter in its entirety in relation to the individual parts

that compose it. [...]

Eurythmia: is the fine looking and harmonic exterior appearance offered by the various parts

in their combination. This is achieved when the components of a building retain the harmonic

proportion of height and width relative to length and reflect an internal symmetry.

Symmetria: is the harmonic agreement between the parts of a single work and the

correspondence between the individual elements and the overall image of the figure. As in the

human body the eurhythmic characteristic lies in a symmetrical relationship between the foot,

the hand, a finger and the other limbs, the same must be in the creation of an architectural

work. [...]6

The influence of Vitruvius in the Middle Ages included the interpretation of geometric

figures. The geometric figures of medieval cathedrals were mostly dominated by the triangular

and quadrangular systems introduced by Vitruvius. In the Middle Ages, Vitruvius’ writings

5 Viollet-le-Duc (translation 1981)


were viewed as a sort of manual of construction techniques, were transcribed in many

languages and were available for consultation in the libraries of many cathedral schools.7

Of essential importance were the concepts associated with the divider and the circle that can

be generated and how "half the diameter of every circle divides the entire circumference into

six equal parts": this was the principle of the triangular geometric system. In medieval times,

the circle was a symbol associated with divinity and founded on it, so much so that Pope

Sylvester II in his De Geometria explained that the centre and the circle are correlated, the

former as the origin of everything (and geometry in particular) and the latter as the visible

manifestation of this origin.8

By linking the points obtained on the circumference at the points of subdivision into six

equal parts we get a regular hexagon. If we plot the diagonals and link the end points, we get

two intersecting triangles that form the six-pointed "Star of David": this is the “triangular

system”. If we divide the circle into four equal parts by plotting a vertical and a horizontal line

passing through the centre and link the points on the circumference we get the "quadrangular

system".

3.3 The influence of Plato: the regular pentagon and its golden section

A figure that appears in medieval cathedrals is the regular pentagon. It was one of the most

mysterious figures in the history of geometry as borne out by the fact that the master builder

revealed the design of the construction to a small group of trusted collaborators. From a

detailed analysis of the construction we find that it is an application of the so-called "golden

section", already applied by the Greeks and the Egyptians and held in great esteem in the

Middle Ages. Plato defined it as the "the most beautiful of all ratios". Plato explains that

6 Vitruvio (translation 1990)

7 Gout (2001)

8 Gout (2001)


triangles and squares are not original figures of the Creation, but rather faces of regular

polyhedrons that symbolise the elements from which the entire universe was created.

The symbols of the "Sky" (the spiritual principle that exists eternally) were the tetrahedron

(Fire) having 4 triangular faces and the octahedron (Air) having 8 triangular faces, the symbol

of the "Earth" (the temporary principle of matter) was made of a solid part, the cube. with 6

square face (symbolising the Earth) and a liquid part, symbolised by the icosahedron (Water)

having 20 triangular faces.

In book 55, Plato adds: "And since there was a fifth body left, God used it as a symbol of the

universe, which he painted it with all manners of shapes", and that’s all he said of the

dodecahedron, the regular polygon having 12 pentagonal faces: all other forms can be derived

from its diagonals.9

4 THE CATHEDRAL OF S. LORENZO: TRANSFORMATIONS AND GEOMETRY

4.1 The concept of cathedral

The Cathedral (from Latin cathedra) was the Church that housed the official chair or throne of

the bishop of a diocese. The Church was a symbol (from Greek symbàllein = putting together)

and as such, embodied, in its different constituent elements a variety of meanings, e.g., the plan

that symbolises the Cross, or the direction of the main axis from east to west that paralleled the

movement of the sun, which in its turn was perceived as a symbol of life and Christ, the

beginning and the end (A and Ω) and hence the life of everything. The Church also bore witness

to the growth and the commercial power of the city.

In the Middle Ages, the Cathedral was an altogether different place from the way we

conceive it nowadays, it was an integral part of the daily life of the community.

The areas, buildings and fixtures of the cathedral did not belong to the bishop and were

under the jurisdiction of the Chapter, but the sanctuary proper was under the direct sovereignty

9 Gout (2001)


of the bishop. Nowadays we might find the animation and the activities that went on in a

medieval church rather improper: people could eat and sleep in church, get together and speak

loudly instead of whispering, or even talk business or discuss other matters that had nothing to

do with religion. Town representatives met in the church to resolve on municipal affairs and in

many cases the construction of a cathedral made it unnecessary to build a town hall. Many

cathedrals, and churches in general, were erected on sites that had been occupied by pagan

temples, sites that had been selected because they were viewed as sacred places.10

The construction process and the subsequent expansion and strengthening interventions

always involved the entire population: in this connection, it is interesting to note that, with

reference to the Cathedral of S. Lorenzo, Pozzetti underscored the importance of the works

performed on the cathedral in the nineteenth century and recalls can. Stella collecting big stones

from the Cherasca torrent "and making them pass from hand to hand along a long line of men

and women all the way to the road"; or the catholic women of Alba making a carpet for the

cathedral (1870), "a long and painstaking job". 11

4.2 Historical background

We have no certain knowledge about the origins of the cathedral, about its initial shape and size

or the time it was founded, though it certainly dates from before the tenth century. It is believed

that, as was often the case, it was erected on the ruins of an ancient pagan temple, dedicated to

Apollo, but there are no documents to confirm this hypothesis.

The reconstruction of the cathedral, between 1486 and 1517, was prompted by an initiative

by bishop Andrea Novelli. In 1624 it was announced by "expertis fabriis murariis" that the

church was in danger of crumbling "in tegulis et voltis" and, in fact, a few years later the vaults

of the nave and some beams of a chapel collapsed.12

In the 1650’s, through the intervention of

10

Gimpel (1982) 11

Morgantini (1988) 12

Morgantini (1988) 13

Morgantini (1988)


Monsignor Paolo Brizio, the vaults were reconstructed, the foundations of various chapels were

built and the entire structure was restored.

In 1789 access to the cathedral was barred for fear of a total collapse and a series of works

were performed to repair the roof and parts of the building. 13

The cathedral was reopened to the public in 1791.

In 1832 the vault of the choir and the portion of the vault atop the big arch required urgent

repairs but there is reason to believe that by 1834 nothing had been done: in 1844 we learn from

the records that they removed the snow from the roof, deemed too weak to withstand the load.

On 24 July 1863, the definitive Restoration Committee was convened upon the initiative of

Andrea Formica, and on 24 February 1864, Count Edoardo Arborio Mella was entrusted with

disegno and site management tasks.

In a letter of 20 May 1865, Mella signed his cenni di schiarimento (explanatory notes) to the

restoration drawings submitted, in which he pointed out [...]static damage to the wings, due to

an unwise round arch reconstruction of the nave, instead of using pointed arches as envisaged

originally [...]. Equally damaged is the apse. [...] The effects of the damage [...] include cracks

in the choir and the displacements of the nave walls and those of the wings [...]. In the same

document, Mella advised against the reconstruction of the vaults, both in view of the excessive

costs entailed and because the construction of the new chapels would have increased the

inclination of the vaults.14

A meeting with the participation of Mella, Busca – the mayor of Alba – and engineer Peyron

was held before January 1866. It was acknowledged that it is necessary to reconstruct the

semicircular part of the choir [...], it is advisable to construct the vaults of the nave so as to

reduce their thrust on the aisles, and to secure with strong spurs and piers resting on solid

foundations the outer walls of the aisles along the sides of the building, which are susceptible

to considerable settlements, especially the wall on the south. [...]

14

Morgantini (1988)


On 11 June 1866, the ancient arch wall behind the chapels of S. Bovo and S. Luca began to

be demolished and in the course of 1867 the works were limited to the side chapels.

In September 1867 they came to the conclusion that [...] the foundations of the choir were

unusable in that they were insecure having given rise to the cracks in the apse [...].

In the summer of 1868 [...] Having removed the last column to the left before the choir to

reduce the load on the arcade, a wall was found, divided in many poorly connected parts and

without foundations, displaying settlements and cracks that made it necessary to halt all works

and lighten the load atop to proceed with proper underpinning and then reconstruct it partly

[...].15

In February 1869 the columns and the side vaults were scraped down and on 4 April it was

decided to demolish them, in view of their rundown conditions, and it was also decided to

remove all the keys from the nave.

In December 1869, the vault of the crypt was reconstructed with a reinforcing arch to support

the main altar.

In August 1872 the rose window in the facade had been opened and the two side windows

had been modified.

It is believed that the works performed between 1866 and 1872 included: replacement of the

six lateral chapels with new polygonal ones; construction of the eight lateral buttresses topped

by small spires; restoration of the vertical alignment of the side walls through the construction

of a covering layer of new bricks; demolition of the walls bearing down on the arches of the

side vaults, replaced with a demi-arch resting on the main vault (serving as a buttress);

demolition of the church vaults (save for those of the two main chapels), which were

subsequently reconstructed in the shape of pointed vaults; reconstruction of the roof and

creation of six large round windows in the nave; demolition of the apse, the entire vault of the

presbytery and the crypt. New foundations were dug for the apse, which was reconstructed in

polygonal form. The inner walls were also modified to improve their verticality. The interior of


the church was coated with lime plaster; the three vaults of the pronaos were demolished and

reconstructed.

4.3 Current structural set-up

In its present-day configuration, the structural set-up of the cathedral consists of a nave and two

aisles, with chapels opening on either side, two of which, the ones flanking the new altar, are

much bigger than the others. The presbytery terminates in a polygonal apse, with five ogive

shaped windows

The dimensions of the building are imposing, with a plan developing over a length of ca 47

m, in the main section, and a further 23.50 m in the presbytery; the building is 36 m wide and

its height, at the ridge of the intrados of the nave vaults, is 23 m from floor level.

The front part of the cathedral has an entrance portico topped by a volume which is part of

the building and accommodates the pipe organ. Thus, the facade is perforated at the base by the

three arches of the porch, matching the width of the nave and aisle, while the rest of the facade,

up to the top is quite compact: only a central rose window and the two ogive shaped windows

on its sides break the massive surface of the masonry façade.

The interior of the main section of the church is characterised by the slender four-lobed

columns supporting the system of ogee vaults marked by diagonal and contour ribs.

4.4 The equilateral triangle inside the cathedral: chance or geometric precision?

As can be inferred from this brief historical description, when we started to gather “knowledge”

about the Cathedral, the structure was markedly affected by the reshaping performed between

1866 and 1872 by Edoardo Arborio Mella, an architect from Vercelli who was active in

Piedmont primarily as a restorer of medieval buildings.

15

Morgantini (1988)


At this point we should explain some aspects and the spirit of the analysis performed, which

required an introduction to the world of medieval cathedrals and especially an introduction to

the geometry and the symbolism underlying their conception.

One of the primary goals of this study was to reconstruct, to the extent feasible, the mental

processes that inspired the substantial interventions performed by Mella on the cathedral during

the second half of the nineteenth century. This was deemed necessary not only in view of the

intrinsic interest of the problem, but also because knowing the ideas that went into the static

conception of a system is, as mentioned in the introduction, the most powerful key to the

interpretation of its static behaviour.

The history of restoration tells us how the approach to restoration works evolved over time,

from the initial notion of maintaining the ancient configuration, to the debate fuelled by the

different outlooks of Eugène-Emmanuel Viollet-le-Duc and John Ruskin, or the experiences

and theories formulated in Italy in the nineteenth century (Camillo Boito, Alfredo D’Andrade,

Luca Beltrami, Edoardo Arborio Mella, etc.), up to the modern notion of restoration and

conservation.

Nor should we overlook one of the aspects that best characterise the culture of Piedmont in

the nineteenth century: a neo-Gothic taste that became increasingly popular throughout the

century and found expression in many medieval pageants.

Mella’s drawings and the relative documents16

, backed up by extensive writings by architect

Filippo Morgantini, shed light on the works of the architect from Vercelli.

A plan found in the archives of Vercelli17

illustrates the major interventions performed by

Mella and makes it possible to identify some pre-existing structures dating back to 1486, as

well as the conditions of the cathedral after 1486 and in 1868. From figs. 3, 4 and 5, all of them

produced with reference to Mella’s original plan, it is possible to gain a better understanding of

these interventions.

16

AIMAV and AIBAV 17

AIBAV (N. 700)


Fig. 3: Situation in 1486.

The zones identified by the red lines correspond to the portion ascribed to Novelli.

No corresponding sections were found (other than those reflecting the intervention actually

performed) leading to a clear interpretation of the works carried out during the reconstruction

of the vaults.

Fig. 4: Situation after 1486 Fig. 5: Restoration works by Mella,

1868


Some insight into the conditions of the building before the reconstruction of the vaults is

provided by Mella: on 20 May 1865, when he signed his cenni di schiarimento (“explanatory

notes”) to the drawings submitted, he stated First of all the static damage to the wings, due to

an unwise round arch reconstruction of the nave, instead of using pointed arches as envisaged

originally [...] and went on to say the current height of the vaults by lowering slightly the

impost capitals might, if desired, make it possible to restore with little effort the existing round

arches to a pointed form [...] 18

Mella was well-versed in the Gothic architecture of Piedmont and our intention was to

familiarise with his way thinking and to try to identify the guide lines and the geometry of the

project and to determine whether what he did for the cathedral had really interpreted, as

Viollet-leDuc might have said, what the Gothic builders in all probability would have

conceived if they had been in charge of the works.

Trying to grasp the inner significance of the situation and confirming our view as to the need

for identifying a module and the role of geometric figures, we engaged in a search for the

central figure and tried to address the same issue that was faced by the architects involved in

the construction of the Duomo of Milan: i.e., the square or the equilateral triangle.

Different attempts were made, and the most significant results achieved so far are briefly

summarised below.

Starting from the plan of the cathedral and focusing on the portion that can be safely ascribed

to Novelli (fig. 3), i.e., the portion between the apse and the narthex, enclosed between the two

arms of the transept, we can isolate the nave and the two aisles that develop in length over four

bays. By measuring the width of the nave and the aisles at each bay, relative to the midspan of

each element we get:

18

Morgantini (1988)


Left side

aisle

Nave Right side

aisle

Average width

of the aisle

Nave to aisle

width ratio

683 1129 710 697 1.62

679 1130 701 690 1.64

678 1126 693 686 1.64

The measurements taken during our survey were expressed in cm, whereas the unit of

measure used at the time probably was the trabucco (a local unit), but our goal was to work out

a module, a basic ratio that would have generated all the other dimensions. As the reader will

have surely realised, this ratio comes very close to the divine proportion, the golden number

(1.6180339887...). At this point, someone might object that in our case this ratio is not

complied with down to the second decimal place, but we believe this tiny difference would not

have been much of a concern for medieval builders.

From this initial consideration we deliberately left out the outer span, that of the narthex, for

two reasons: first of all, Novelli registers the construction of four bays starting from the apse

and, secondly, in spite of several attempts, we were unable to establish a correlation between

the size of the narthex and the module identified. This suggests that the narthex may have been

built at a later stage, while the probable reasons why it was based on a different module will be

discussed later on.

Having defined the dimensions of the nave and aisles, the cross-section of the church (fig. 6),

was analysed, again with reference to the portion that can be ascribed to Novelli.

On this cross-section, starting from the midline of the exterior walls, we plotted a line at an

angle of 60° so as to define an equilateral triangle, which, in addition to being comprised within

the cross-sectional area, marks the heights at the keystones of the main arches of the nave.

Now, by plotting lines that run parallel to the sides of this equilateral triangle and making them

pass through the intersection between the horizontal and the basic midspan axes identified

before, we find the imposts of the arches of the aisles and the impost of the arches of the nave.


Fig. 6: Cross-sectional view

In addition to having symbolic meanings as mentioned above (the Holy Trinity), the triangle

facilitated greatly the design process, doing away with the need to take measurements, since all

one had to do to produce a perfect equilateral triangle was to define a module and apply it to the

three sides. It should be noted that, as mentioned before, Mella found that the nave vaults had

been reconstructed around 1624 in “round arch” form.

Fig. 7: Longitudinal section

Fig. 8: Longitudinal section

He demolished them and rebuilt them in pointed arch form, probably with reference to the

old impost.

Having defined the dimensions of the nave and aisles and the height of the church, we

studied the dimensions of the longitudinal section. In this case too, the equilateral triangle can

be used to define the positions of the columns delimiting the nave. Starting from the apse, let us


plot a straight line at a 60° angle until it ideally meets the straight line marking the imposts of

the main arches defined before. By doing so, we obtain another module marking the distances

between one column and the next (fig. 7).

It should be noted that the use of the equilateral triangle makes it possible to verify the

vertical alignment of a column during its construction. To obtain perfect verticality it was

sufficient to stretch a rope from the apex of the triangle to the exact mid point of the base side.

The alternative method would have consisted of using plumb lines, but errors of vertically

might have been made with this method in view of the considerable height of the column.

The considerations developed so far found some confirmation in the planimetric portion

ascribed to Novelli. At this point, using Mella’s drawing, an attempt was made to identify a

certain correspondence even in the earlier portions of the building, i.e., those from 1486 (fig. 3).

Taking into account the two modules determined previously, that is, the distance between

one column and the next in the longitudinal direction and between a column and the outer wall

in the transverse direction, we plotted parallel lines in either direction and the figure obtained

was another equilateral triangle (fig. 8).

As pointed out before, there was one last unknown element to be investigated: the narthex.

It can't be certain that this had been constructed by Novelli, who spoke of four bays in this

description. Yet there are no records attesting its construction in later years. From some records

we learn that the narthex was extensively repaired in 1588, and that the nave was reconstructed

in 1624, probably with pointed arch vaults. This suggested the hypothesis that the nave vaults

had to be demolished due to the excessive off-plumb of the facade, caused at the time, as it is

now, by the thrust of the cross vaults, a component of which acted on the facade.

Nowadays it is clear that the flattening of the pressure curve in one arch is matched by an

increase in the thrust on the piers and, conversely, a steeper curvature gives rise to resultants on

the piers with a predominantly vertical component. Ancient builders were not aware of pressure

lines, but, as claimed by Viollet-le-Duc, used this theory instinctively.


Viollet-le-Duc also relates an extremely simple geometric formula to determine the thickness

of the columns (fig. 9) in relation to the thrust to be carried. It basically consists of dividing a

half-circle into three equal parts: the direction of the thrust will be given with sufficient

accuracy by the direction of two lateral segments (as shown in fig. 10). As can be readily

grasped, pointed arches will be matched by slenderer shoulders.19

Fig. 9: Determination of the strength needed

for the masonry shoulder to withstand the

thrust of an arch or vault. Image from Viollet-

Le-Duc (translation of 1981)

Fig. 10: Construction of the narthex in the

cathedral of S. Lorenzo

In our case it was probably deemed necessary to maintain the keystone of the arch at the

same height as the keystones of the nave, which entailed a need to reduce the span so that it

would fall within the outer shoulder of the facade. The modification of cross vaults into pointed

vaults was probably dictated by the need to eliminate the thrust on the facade, since a pointed

vault has its directrix parallel to facade and takes its trust on the lateral piers. This arrangement

involved an increase of trust on the side walls of the nave, to the point that in 1864 Mella

19

Viollet-le-Duc (translation of 1981)


measured an off-plumb of 30 cm. No documents by Mella were found registering off-plumbs of

the facade, though he was constantly engaged in considerable reshaping works.

Mella’s return to the cross-vaults generated again the thrust on the facade, as will be

discussed in a paragraph below.

The module, the geometry and the ratios identified were a product of chance or the

expression geometric precisions? Perhaps French theorist Jean Villette was faced with this very

question when he accounted for the positions of the four main columns at the crossing of the

nave and transepts in the Cathedral of Chartres.

Fig. 11: The distance between the four columns at the crossing between the nave and the

transepts in the Cathedral of Chartres was established by inscribing a triangle and a

square in a circle with a radius of ca 40 roman feet. Two of the piers rise in the

connectons of the circle with the square, two in the connections of the traiangol with the

square. Image taken from Gout (2001)

He inscribed a square and an equilateral triangle in a circle and discovered that the columns

rose exactly at the points of intersection of the two figures. The subtitle to his work was Hasard

ou Stricte Gèometrie?20

(fig. 11)

A geometric system was advocated by major architects of the twentieth century, including Le

Corbusier, who demonstrated the application of a measuring method based on the “golden

section” in his Le Modulor.

20

Gout (2001)


5 THE CATHEDRAL OF S. LORENZO: INVESTIGATIONS AND STRUCTURAL

ANALYSES

5.1 Structural survey and cracking configuration

The geometric data was supplemented with information of a structural nature, indispensable for

the creation of a correct numerical model.

This applies in particular to the elements of decisive importance from the standpoint of

stability: the intersections between the nave partitions and the façade and the transverse links

between the nave vault arches and the longitudinal partitions between the nave and the aisles.

The damage configuration was determined by taking into account two main aspects: the

system of cracks proper and the set of permanent deformations that appeared in the building

over time.

The system of cracks includes a number of major lesions cutting the building in the

transverse direction in the proximity of the facade. In particular, cracks are observed in the

outer walls of the two aisles, in the second span starting from the facade: these cracks are seen

to become progressively wider, going from the bottom to the top of the walls; under the roof,

cracks up to several centimetres wide can be observed in the vertical walls. These lesions

marked the aisle partition walls in the bays closer to the facade: inside the church the cracks are

small, but below the roof they appear wide and slanted.

Finally, the vaults of the first bay adjacent to the façade are cut by lesions set at different

slants, which, as a whole, reproduce the detachment of the front part from the body at rear.

Approximately in the centre of the church, in the fourth bay from the facade, there is a lesion

marking a clear separation between the transverse arch and the vault; it is the innermost

instance of the system described above.

The permanent deformations produced in the building over time can be summarily described

as a translation of the upper part (vaults, aisle partitions, roofing) relative to the base; the

direction of this translation is toward the square with an off-plumb of up to 41.9 cm. The same


fate has befallen the piers inside the building, albeit to a lesser extent (ca 10 cm, with a peak of

11.4 cm in one or the central piers).

Smaller, but non negligible, transverse deformations are also observed in the interior piers; in

this case, the displacements suggest a predominant southward component.

Fig. 12: Principal system of cracks identified by the red lines in part of plan

Fig. 13: Principal system of cracks identified by the red lines in part of longitudinal

section

The picture outlined so far indubitably represents the vicissitudes of the building through the

centuries. It should be noted, however, that since the values determined with these

measurements may be due to the numerous repair works performed in the past to attenuate the

misalignments present at the time, the present-day situation might not reflect the displacements


that actually occurred in a distant past. In this connection, consider the reconstruction of the

vaults or the construction of the buttresses on the façade.

5.2 Surveys of the foundations

Measurements were made both on the ground where the building rises and on the foundations

of the isolated buttresses supporting the facade.

The overall picture that emerges has revealed, under layers of fill and soil reworked to a

depth of 3.90 m, the presence of a thick layer of silty clay followed by a plastic layer of clayey

silt, of poor to low consistency and containing with a modest quantity of fine sand, down to a

depth of 8.50 m. This is followed by sand, mostly medium-fine and weakly silty, reaching down

to below 10 m of depth.

Though it may regarded as largely stabilised, this situation can be deemed as partly

accountable for the damage conditions described in the previous paragraph, with special regard

to the rotation of the facade, despite the contribution of other factors such as the thrust of the

vaulted system, to be discussed in greater detail later on.

This situation was confirmed by the boreholes drilled in the foundations of the facade, which

revealed soils having rather poor mechanical properties.

5.3 Core drilling, quality and state of preservation of the masonry

The conditions and quality of the constituent materials of the masonry were documented by

drilling cores and then inspecting the faces of the holes with TV probes.

The masonry of the facade was seen to be characterised by an inner core enclosed by walls,

about 25 cm thick: it is made of bricks, brick fragments, stones and pebbles of different sizes

embedded in lime mortar, without any big voids, but having limited strength.

A significant phenomenon is the presence of many cracked bricks, although at present these

cracks do not add up to lesions of appreciable size. Their presence is particularly noticeable in


the outer courses and was probably caused by crushing, indicating a state of compression and

bending in the masonry, in good agreement with the displacements of the masonry masses

occurred in the past.

In better conditions are the inner piers: the cores revealed they are made of brickwork free of

voids, firmly packed, set in mortar displaying good bond, and having excellent mechanical

properties.

5.4 Flat-jack tests

Tests with flat-jacks were performed in order to determine the compressive stress in some

portions of the brickwork.

The tests included 4 tests on two inner piers, 9 on the piers supporting the facade internally

and externally: 1 core was drilled inside the church from the wall closing the pronaos.

The tests revealed high compressive stresses, especially in the piers, of between 0.9 and 3.0

MPa. Stress intensity is not homogeneous throughout the section, with a peak in the proximity

of the facade and the lowest value on the opposite side, confirming the existence of a rather

severe state of compression and bending in the vertical elements.

The tests performed on the facade yielded values of similar intensity in the wall sections at

ground level, with a peak of 3.1 MPa in one of the central pier.

5.5 Monitoring a number of appreciable lesions

Some cracks have been monitored since 1999 by means of strain gauges, by recording the

measurements at regular intervals. The cracks were present both in the aisle and nave vaults and

in the partitions above them.

The plan in fig. 14 shows the positions of the strain gauges and the diagrams produced with

the readings (fig. 15) give the displacements as a function of time.


Some points, e.g., the ones corresponding to strain gauges nos. 5, 6, 7, appear to be

quiescent: during the time interval considered, maximum displacements at such points did not

exceed 2 - 3 tens of a millimetre.

Fig. 14: Arrangement of the strain gauges

However, the other strain gauges reveal a tendency of the faces of the cracks to diverge; nos.

1, 2, 3, 4 display positive residual displacements ranging from 3/10 mm to just under 2 mm. In

particular, high values were recorded for nos. 2, 4, which are not adjacent to the facade, but at

some distance from it (see the plan with the arrangement of the strain gauges). A possible

explanation of this phenomenon may consist in the fact that, now that the detachment of the

parts close to the facade has occurred, crack propagation is going on in an area farther within.

This finds clear confirmation in the numerical calculations.


Fig. 15: Diagrams of strain gauge readings

5.6 Numerical analysis of the structure

The complexity of the structure made it necessary to perform a series of numerical analyses

taking into account different assumptions regarding the restraint conditions that might have

occurred through the centuries, also in relation to the various historical transformations. To this

end, models had to be produced by taking into account the spatial behaviour of the building. A

short description of the final model deemed most significant is given below.

The construction of the various models was facilitated by the considerations on the geometry

of the cathedral formulated above, which made it possible to introduce considerable

simplifications, without, however, deviating from the reality of the construction.

Finite elements models were developed, reproducing the materials in the elastic and linear

field. The modelling process was performed on the basis of the surveys conducted according to

modalities selected precisely to this end.

Mechanical properties

Mechanical characterisation is based on the results obtained on the cores and the observation of

the faces of the holes drilled in the masonry; the parameters assumed were determined by


analogy with tests conducted on elements having similar construction features and dating back

to the same period. They are21

:

E = 2500 MPa; µ = 0.15; γ = 180 KN/m3

Finite elements used

The finite elements used were: shell elements for the vaults and solid elements for everything

else.

Restraints

The structure is restrained at the base. Even the bell tower constitutes a stiff restraint. For the

models reproducing only the main section of the church, the connection with the apse also

consists of a fixed joint.

Actions considered

Only dead weight was taken into account, on account of its predominance compared with any

live loads the building may be subject to.

The seismic behaviour of the structure was not analysed.

Size of the models

The models of the main section of the church use 13246 elements, linked by 18833 nodes,

resulting in a system with 112,998 degrees of freedom.

21

Pistone (2001)

Pistone (1993)

Pistone (1991)

Pistone (1988)

Pistone (1982)

Pistone (1979)


5.7 Results of the analyses

First of all, it should be noted that the model expresses the most important aspects of the

structural behaviour of the building and is useful to obtain a full picture of stress and strain

conditions.

The salient aspects of the behaviour of the cathedral are summarised below.

1) The deformed configuration is characterised by a slight projection of the nave towards the

square, mated to a widening of the cross-section at the centre of the building in the transverse

direction, which, however, is a much less noticeable (fig. 16).

2) These displacements, affecting the central core of the church, are countered by a number

of stiffer zones.

Laterally, albeit weakened by the presence of the chapels, the exterior walls serve as blocking

elements, in as much as they constitute a substantially rigid partition in their own plane;

accordingly, they are able to retain in place the facade that is locked into them.

In the back of the church, the presbytery and the apse, on account of their massive and close

configuration, prevent all movements.

In the transverse direction, the chapels of the transepts, of a much bigger size than the others,

prevent any significant displacement.

3) The foregoing displacements find full confirmation in the verticality measurements taken

on the walls and the inner piers. It should be noted that the displacements determined with the

numerical model cannot be correlated with the values measured, which are much greater: this is

obviously due to the material employed in the model, which is perfectly elastic and tension

resistant.

Nevertheless, the theoretical representation obtained is of fundamental importance in the

interpretation of the experimental data.


4) As a result of this behaviour, major tensile stresses can be observed in the nave partitions

and, in greater detail, in the figures reproducing the connections of the facade to the body

behind (fig. 17).

Fig. 16

Fig. 17

Particularly significant are the images based on the model, in which the facade is connected to

the partitions behind it (fig. 18).

Fig. 18

A confirmation of the results supplied by numerical modelling comes from the cracks that cut

the building in the transverse direction in the proximity of the facade and can be observed both

in the partitions and in the vaults. Needless to say the explanation lies in the inability on the

part of the masonry to withstand tensile stresses.

It should be noted how in actual fact the appearance of cracks releases the facade from the body

behind it.


5) Another significant aspect stemming from this behaviour lies in the compression and

bending forces generated in the facade and the internal piers.

In the facade, compression peaks of 0.8 MPa are reached, compression values in the piers are

even higher.

The validity of the model is also borne out by the results of flat-jack tests which yielded

compressive stress values of up to 3.0 MPa in the façade, and of 3.1 MPa in the west side of the

piers. The correspondence between the numerical values of experimental and modelling data is

only qualitative, precisely because the real material is influenced by countless factors

associated with the present-day conditions of the structure, while the modelling material is ideal

and distributes its properties in a regular fashion over its constituent elements. However, the

consistency between the model and the real situation is clearly revealed by this comparison.

Less pronounced is the correlation that can be established between the compression and

bending stresses in the model and in the real structure in the transverse direction, because of the

great scatter in the data obtained both from pier verticality measurements and from flat-jack

tests.

6) In general, however, the transverse thrust. which is the root cause of the stresses and

strains mentioned above, is effectively countered by the resistant mechanisms of the present-

day structures.

The downward migration of the loads to the ground is illustrated in the images (figs. 19-20).

Fig. 19


Fig. 20

From these figures it can be seen that the principal compressive stresses follow a diagonal

trajectory slanted by approximately 45° to the horizontal: in this case, compression isostatics

can move unobstructed across the lateral walls of the SS. Sacramento and S. Teobaldo transept

chapels, which constitute continuous diaphragms specially suitable to convert – locally, where

it is most needed – into struts countering the thrusts coming from the upper masses.

Between the facade and the presbytery, the system countering the thrusts is more complex, but

substantially effective. In short: the thrust of the vault on the nave descends via the rampant

arches onto the aisle tops, and, through the system of vaults and arches of the aisles, is

transferred to the outer walls of the cathedral; in the meantime, the thrust of the aisles has also

come into play, but here the thrust acting in a slanted direction is effectively opposed by the

boxed structures of the minor chapels, which provide a sufficient degree of stiffness and are

able to transfer the actions all the way to the ground.

In reality, this system has but limited effectiveness, as demonstrated by the transverse

displacements observed in the piers, and as borne out by the deformation configurations of the

sections in the numerical model. Nevertheless, it is able to ensure equilibrium conditions.

Particularly effective are the rampant arches of the aisles: the deformed configurations

mentioned above, as well as the axonometric views of the deformation configurations of the

building, show that such arches counter the tendency of the nave partitions to move outwards.


Transverse displacements, in fact, are greater at the impost of the vault of the nave, whilst

higher up, the partitions are seen to move back towards the inside of the building.

This behaviour can be rated as real, since, various possible conditions of local restraint between

the component elements were tested with the numerical models, and this state of affairs was

invariably confirmed.

7) No substantial problems are observed in the apse section: if the vaults, with their thrust,

give rise to centrifugal strains in the supporting and surrounding masonry system, such strains

are effectively countered by the system of buttresses characterising the back section. The

strains and stresses arising from this situation are limited, as demonstrated by the numerical

model.

5.8 Guide lines for the strengthening project

Based on the information obtained from the strain gauge readings and the results of the

numerical model, it can be concluded that hinges have been progressively forming at the bays,

both in the nave partitions and in the vaults, and especially at the connections between the

vaults and the main transverse arches.

Strain gauge nos. 1 and 3, adjacent to the facade, reveal residual strain values lower than

those recorded at nos. 2 e 4 positioned farther inside; however, they only register the readings

relating to the faces adjacent to the lesion, but since they are part of the same masonry mass,

they both move by the same extent as lesions 2 and 4.

The first hinge probably formed at the apse and was soon followed by the hinges at the

façade, as is also evidenced by the numerical model. This process continued with the formation

of additional hinges accompanied by continuous forward displacements towards the façade.

The total collapse of the nave might occur if, for the lack of further resources, the first hinge

converted into a carriage, or the local collapse of the first vault adjacent to the façade could be

caused by excessive relative displacements.


The studies conducted on the geometric-structural genesis of the cathedral, the physical-

mechanical analyses and the numerical models highlighted some defects intrinsic to the

construction of the building, the most important of which are:

- the thrust of the internal system of arches and vaults onto the facade, which, in the past,

prompted major corrections to the original structure of the buildings, such as the construction

of the imposing buttresses on the facade;

- the resulting compression and bending stresses in the piers in the main section of the church;

- weakness of all the load bearing masonry walls, thick but characterised by poor bond and low

strength.

The strengthening works recommended, some of which have already been completed, have

been conceived to remedy these shortcomings; in particular, as a further aid flanking the

nineteenth century buttresses, a system must be devised for anchoring the facade to the rest of

buildings by means of an appropriate set of ties. In the authors’ view, this system should be

designed to improve the efficacy of the behaviour of the building and its overall response,

instead of just linking the facade to the body behind it.

The other two problems identified must be addressed in a more punctual matter, so as to

improve the mechanical performance of the piers and the masonry making up the lower parts of

the facade.

ACKNOWLEDGMENTS

For the materials supplied and their helpful collaboration, the authors wish to thank in particular:

Mr. Paolo Franchetti (AIBAV Archivio Istituto Belle Arti - Vercelli)

Mrs. Rosso (AAMV Archivio Arborio Mella - Vercelli)

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ARCHIVE SOURCES

AAMV Archivio Arborio Mella, Vercelli

AIBAV Archivio dell'Istituto Belle Arti di Vercelli

Possible Geometric Genesis of a Medieval Cathedral (Alba, Piedmont, Italy

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