structures. For that purpose, a Generalized Matrix ... · Due to geometric and material complexities found in ancient historic ... with a complex geometry, with curved unidimensional

A numerical model for the study of ancient

masonry structures

C. Molins, P. Roca

Department of Construction Engineering, Universitat Politecnica

de Catalunya, Gran Capita s/n Barcelona, Spain

Abstract

A numerical model specifically developed for the structural analysis of

ancient structures is presented together with its application to the study

of some examples of masonry buildings in Spain. The model, based

on a Generalized Matrix Formulation, has been designed to deal with

multiple structures consisting of curved members with variable crosssection. Among the described examples are two buildings laid out by

the architect Antoni Gaudi at the beginning of the century.

1 Introduction

Due to geometric and material complexities found in ancient historicbuildings, conventional calculation methods developed for modern

structures often prove to be inadequate or impractical when applied intheir analyses. The limitations encountered include: (1) the need to deal

with a complex geometry, with curved unidimensional or bidimensional

members like arches and domes; (2) the incomplete knowledge of thebehaviour of traditional materials, like masonry or wood; (3) the multiplesources of actions (such as chemical processes, environmental effects,...);

(4) and long-term effects due to rheology and cyclic or sustained loads,

which may progressively alter the initial geometry.

In an attempt to make available more specific tools, a numerical

model has been developed for the study of ancient brick or stone masonry

Transactions on the Built Environment vol 15, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509

276 Dynamics, Repairs & Restoration

structures. For that purpose, a Generalized Matrix Formulation was

adopted, including nonlinear geometric effects. Modal vibration analysis

was also implemented in order to allow for dynamic studies. The

techniques adopted to simulate the geometry and the mechanical aspectswere designed focussing on both the accuracy and the numerical efficiency

in the treatment of such structures.

The special properties of this formulation have been used for the analysis

of some complex ancient spatial structures such as the three examples

presented in this paper. The first one is the Crypt of Colonia Guell, built

by the architect A. Gaudi near Barcelona. The second one is a vault

of a gothic church in Morella, Spain. Finally, the study of the stability

of an entire building supported by a system of load bearing walls, also

corresponding to a construction by A. Gaudi, is presented.

2 Numerical model

The structural analysis of systems composed of curved members such as

the arches, diaphragms or nervures which may be found in many ancient

structures, is commonly carried out using the finite element method with

iso-parametric type with displacements as unknowns. It may be seen that,due to the deficiencies of the method in the description of the internal

equilibrium of the elements, accurate results of internal forces are onlyobtained when a considerable amount of individual elements are used in

the geometric discretization. However, for structural systems composed

of unidimensional curved members, it is possible to establish analytical

generalizations of conventional matrix methods based directly on exact

equilibrium. Although the practical use of these matrix formulationswas limited in the past by the large volume of mathematical operations

required, recent developments in digital computers make this point less

critical nowadays, while the aspects of accuracy and versatility gain

renewed interest.

The analytical model used in the studies presented is directly based on

a Matrix Generalized Formulation specifically developed to treat ancientbuildings consisting of multiple structural systems with curved, variable

cross section members. The formulation, initially based on the workof Baron* for static linear analysis, has been extended to nonlineargeometric and modal vibration analyses. Relevant aspects of the resulting

method are the following: (1) Automatic generation of complex geometries

throughout the length of the element. Three cross sections, having

arbitrary shapes, are to be given at three respective points of the axial


Dynamics, Repairs & Restoration 277

curve of each element. (2) Each different cross-section is defined as

a composition of elementary trapezoids, where each of them may be

associated to a different type of material. (3) Specific devices are includedto model load bearing or shear walls as equivalent systems of linearelements according to the method proposed by Kwan%. (4) Nonlinear

geometric analysis based on an updated Lagrangian formulation, thus

allowing the treatment of cases involving instabilty phenomena of arches

or other curved elements. (5) Modal dynamic analysis based on the

formulation of a consistent elementary mass matrix which objectively takes

into account the distribution of mass and stiffness throughout the element.

Constitutive equations for brick or stone masonry at the macro-modeling

level are now being implemented in the general model so that an integrated

nonlinear geometric and material nolinear analysis method will availablein a short time.

Before its systematic use for the study of existing buildings, the model,

implemented in the computer program CRIPTA, was checked through the

analysis of a series of simple and multiple systems of curved members, for

which analytical or experimental results were available. The comparisons,described by Lopez-Almansa et al.̂ and Molins et al.̂ , showed the very

satisfactory level of accuracy and numerical efficiency which are achieved

even for geometrically complex structures.

Figure 1. Numerical model built for the Crypt of the Colonia Guell



3 Cripta de la Colonia Giiell

A more detailed description of this building and its static analysis by GMFhas been previously presented by Roca et alA The Crypt of the Colonia

Guell is the only part actually built of what was to have been a church

in Sta. Coloma de Cervello, Barcelona, Spain. The general form of the

floor plan of the Crypt is oval, with a star-shaped outline, and it measures

26 x 63 m. The materials used were mainly brick and stone masonry. The

slab roof is supported by a skeletal system, which consists of a hierarchy

of ribs, arches and oblique columns (Figure 1). The structure was laid by

the architect Antorii Gaudi based on a three-dimensional model made of

strings in order to foresee a suitable anti-funicular type of equilibrium.

Figure 2. Numerical model built for the Crypt of the Colonia Guell

Due to this special design, optimum equilibrium should be only reached

at the finished configuration of the building, including the nonexistingupper main church. Although the Crypt seems stable now, it does not

remain intact; several crack patterns can be seen both in the ribs andupper slab, which implies a distribution of forces other than that which

was planned. In order to reach a better understanding of the presentstate of equilibrium, several analyses were performed using the techniquedescribed above. The arches, ribs and diaphragms were modelled taking

into account their actual curved geometry and cross section variations; in

addition, solid undeformable elements were introduced to simulate the

massive capitals where columns, arches and ribs connect. The lead joints



on the central columns, at their junction with the pedestal and the capital,

were treated alternatively as completely fixed or rotationally free hinges.

When the existing part of the structure was studied subjected tothe vertical dead load (Figure 1,2.), a satisfactory correlation was found

between the analytical prediction of high tension levels and the cracks

observed in the structure. All these cracks matched the prediction,

which nevertheles showed many other potentially cracked zones that were

apparently intact. The building was shown to be stable under the dead

loads at present affecting it. This is so in spite of the fact that these

loads were the cause, at some time, of the existing damage. The analysis

also showed that, owing to their much larger sectional dimensions, thedeformations of the columns and the perimetral wall are very small in any

case, so that the equilibrium of arches and ribs is not affected by the factthat the devised global structural system is not completed.

4 Vault of the choir of Morella church

The second example presented consists of a vault built during the XVI

century inside an existing gothic church in Morella, Spain, to shelter thechoir (Figure 3). The vault, having a 10.7 x 10.2m̂ span, is supported at

its bottom corners on four of the main columns of the church.

Figure 3. Choir of MorellaChurch. Geometry.

Figure 4. Shape of the First andSecond Natural vibration modes

The aim of the analyses was in this case to know the actual riskof collapse due to the large deflections visible today (about 20 cm at

mid span) in all the nervures as well as in the main perimetral arches.



The studies included, first, a nonlinear geometric analysis upon the

today deformed geometry and, second, a complementary modal vibration

analysis. The latter was also considered informative because of the

possibility to correlate the predicted frequencies and movements with

experimental measurements (Figure 4).

For the nonlinear geometric analysis, an approach to an ultimate state of

equilibrium was considered by placing perfect hinges at the bottom fibers

of the springings as well as at the top fibers of the mid-span sections

of arches. The obtained results show the choir to be stable under its

actual condition. However, the main arches are introducing intense thrusts

against the supporting columns, which should have deformed considerably

despite its robustness and caused some damage to the upper main vaults,

as some actually existing cracks in them.

5 Casa Botines, Leon

The capability of the present model to deal with buildings consisting of

systems of load-bearing two-dimensional walls is illustrated through this

example. The Casa Botines, built by A. Gaudi in 1891 in Leon, has an

irregular plan limited by four fagades having lengths of 35, 28, 25 and20 m and height of 22 m (five floors). The structure is composed of (1)

the perimetral facade wall, made of limestone masonry and rubier, with

a total thickness varying from 1.0 m at the base to 0.45 at the crown,

(2) the inner load bearing walls, made of brick masonry, which have only12 cm thickness, (3) the bottom columns used to create a more diafane

underground and ground levels, which support the inner wall system, and

(4) the floor slabs made of masonry vaults supported on steel or wood

beams (Figure 5).

In this case, the structure was studied under the effect of a moderate

earthquake of MSK degree V of intensity which, according to the Spanishseismic code?, must be considered in the location for design purposes.

Due to the weak support given by the floor slabs of the building in the

developing of an optimal resisting mechanism under horizontal forces,

the need to introduce a certain type of reinforcing was clear (Figure 6).

However, the use of such a detailed model allowed the characterization

of the actual deficiencies of the original building and to define a set ofvery light reinforcing devices which permitted the preservation of most

of the original configuration and materials but ensuring adequate safety.

The reinforcing solution proposed was to provide a rigid diapfragm at the

upper floor level, consisting of a light steel grillage. The efficiency of this



Figure 5. Numerical model for Casa Botines

Figure 6. Stress intensities and deformed shape in case of a

transversal earthquake



technique was appraised through numerical simulation.

6 Conclusions

A numerical model and its application to static and dynamic analysis of

ancient spatial structures formed by curved elements with variable cross

section has been presented.The appropriateness of the formulation for dealing with ancient

structures has been illustrated through the following examples: the Crypt

of the Colonia Giiell, built by the architect A. Gaudi near Barcelona, the

choir vault inside a gothic church in Morella, Spain, and the Casa Botines

in Leon, Spain, also designed by the architect A. Gaudi.

Aknowledgements

The numerical tools used in the study were developed as a part of the

research project SEC93-1160, funded by the Comission of Science and

Technology of the Spanish Government.

References

1. Baron, F. Matrix Analysis of Structures Curved in Space, Journal of the

Structural Division ASCE, 1961, Vol. 87, N°. ST3.2. Kwan, A.K.H. Analysis fo coupled wall/frame structures by frame method

with shear deformation allowed. Proc. Inst. of Civil Eng., Part II, 1991.3. Lopez-Aimansa, P., Casas, J. R., Molins, C., Serra, I. Numerical

Simulation of Dynamic Behaviour of Early Demolded Reinforced

Concrete Beams, Computer Modelling of Concrete Structures, Pineridge

Press, 1994.4. Molins, C., Roca, P., Mari, A. R. Una formulacion matricial generalizada:

(I) analisis estatico. Hevista Int. de Metodos Numericos en Ing., 1994,

Vol. 10, N°.4.5. Molins C., Roca P., Bar bat, A. H. Una formulacion matricial

generalizada:(II) analisis dinamico. Revista Int. de Metodos Numericos

en Ing, 1995, Vol.11, N°.l.6. Roca P., Gonzalez, A., Gonzalez, J. L., Casals, A. Studies of Gaudi's

Cripta de la Colonia Guell. Proc. of IABSE Symposium on Structural

Preservation of the Architectural Heritage, Rome, 1993.

7. Norma Sismorresistente P.D.S.- 1. Decreto del Ministerio de Planincacion

del Desarrollo 3209/1974 de 30 de Agosto.


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structures. For that purpose, a Generalized Matrix ... · Due to geometric and material complexities found in ancient historic ... with a complex geometry, with curved unidimensional