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International Journal of Architectural HeritageConservation, Analysis, and Restoration

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Limit Analysis and the Study of Building Stages inMasonry Structures. Experiences with the GothicApse of Tortosa Cathedral (1345–1441)

Agustí Costa-Jover, Josep Lluis i Ginovart & Sergio Coll-Pla

To cite this article: Agustí Costa-Jover, Josep Lluis i Ginovart & Sergio Coll-Pla (2017): LimitAnalysis and the Study of Building Stages in Masonry Structures. Experiences with the GothicApse of Tortosa Cathedral (1345–1441), International Journal of Architectural Heritage, DOI:10.1080/15583058.2016.1246625

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Limit Analysis and the Study of Building Stages in Masonry Structures.Experiences with the Gothic Apse of Tortosa Cathedral (1345–1441)

Agustí Costa-Jover , Josep Lluis i Ginovart , and Sergio Coll-Pla

Department of Architecture, Universitat Rovira i Virgili, Reus, Spain

ABSTRACTLimit analysis is a commonly used method in the assessment of masonry constructions. This articledeals with the use of limit analysis in order to study the equilibrium conditions during theintermediate building stages of the Gothic apse in Tortosa cathedral (1345–1441). This analysisis typically used in completed structures, but its use in the intermediate building stages mayprovide relevant data beyond the existing historical information. The design of this heptagonalapse entails a greater calculation complexity than the cathedral’s central body. Besides, theintermediate building stages generate asymmetry situations at the pillars. The results obtainedshall enable us to determine the building stages in which the structure’s equilibrium may havebeen endangered, and therefore a temporary thrust countering system was required. Twodifferent graphic statics strategies have been used. These strategies are based on the sameprinciples, but imply a different degree of simplification.

KEYWORDSGothic; graphic statics; limitanalysis; masonry; Tortosacathedral

1. Introduction

In the context of masonry structures, there are currentlymany analysis and prospecting techniques which can pro-vide valuable information to improve knowledge aboutthem, but their implementation can be complex and costlyin some cases. Thewidely known limit analysis techniquesby Heyman (1995) allow to analyze the equilibrium con-ditions of a structure by knowing its materials and geo-metry, as pointed out by authors such as (Block, Ciblac,and Ochsendorf 2006; Boothby 2001; Huerta 2005;O’Dwyer 1999; Ochsendorf 2002). The historical back-ground and the theoretical framework are provided inHuerta (2004, 2008), Roca et al. (2010), andTheodossopoulos and Sinha (2013), who analyze thestate of play regarding the different analysis techniques.

The analysis of masonry structures through thrustlines is particularly appropriate in the case of structureswhich are markedly bi-dimensional. When this is notthe case, some strategies must be adopted so as tosimplify the problem. Thus, we can use the classicalslicing technique (Ungewitter 1890) and also morerecent methods such as the Thrust Network Analysis(TNA) by Block (2009) and Block and Ochsendorf(2007), still under development (Fraternali 2010).

These procedures are commonly used in completedstructures, with all the forces in equilibrium (at least at thebeginning). This article deals with the use of limit stateanalysis in order to study the equilibrium conditions duringthe intermediate building stages of the Gothic apse inTortosa cathedral (1345–1441). The design of this heptago-nal apse entails a greater calculation complexity than thecathedral’s central body. Besides, the intermediate buildingstages generate asymmetry situations at the columns, withonly one vault built in one side. Thus, despite the methodused is well known and of proved validity, the investigationhas enabled us to test the divergences between two differentgraphic statics strategies on a complex masonry structure.These strategies are based on the same principles, but implya different degree of simplification.

A direct forerunner of this research is Lluis iGinovart, Costa, and Fortuny (2015), containing afirst approach to the graphic statics calculation of theapse in order to ascertain the temporary supportingelements used to build it. The underground prospectionof the cathedral using a Ground Penetrating Radar(GPR) identified the old foundation of the so-calledpilar major (main column), a big, multifunctional tem-porary element the existence of which had already been

CONTACT Agusti Costa-Jover agusti.costa@urv.cat Department of Architecture, Universitat Rovira i Virgili, Reus, Spain.Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/uarc.

INTERNATIONAL JOURNAL OF ARCHITECTURAL HERITAGEhttp://dx.doi.org/10.1080/15583058.2016.1246625

© 2017 Taylor & Francis

disclosed in Almuni Balada and Lluis i Ginovart (2011).The GPR also made it possible to identify the remainsof the ancient Romanesque cathedral (1178), the con-struction model of which is proposed in Lluis iGinovart, Costa-Jover, and Coll-Pla (2014b)(Figure 1). The investigation allowed to go a stepbeyond historiography and to provide analytical datawhich validates and enhance current theories.

The assessment is based on the well-known slicingtechnique, which was first used in case studies byWittmann (1879) and Planat (1887) in the late 19thcentury. But it was Karl Mohrmann who, in the thirdedition of Ungewitter’s book, applied this technique ina comprehensive manner to a whole building andestablished rules to decide on the family of cuttingplanes. This book also includes a set of tables showingthe vault’s thrust according to its geometry andmaterial.

Thus, architects in the 20th century were able tostudy the static behavior of vaults in buildings usingsimple graphic statics methods. Classic examples areBenouville’s analysis at Beauvais cathedral (Benouville

1891), Rubió i Bellver’s analysis at Palma de Mallorcacathedral (Rubió 1912) and the study carried out atSankt Martin church in Landshut (Zorn 1933). Other,more recent, examples are found in Mallorca cathedral(Maynou 2001), where Rubió’s calculations (1912) arereproduced by Maynou (this time using a specific soft-ware which calculates the nave’s structural line andobtains an infinite number of valid pressure lines),and in the church of Santa María del Mar inBarcelona (Vendrell et al. 2008), where the graphicstatics results and the finite element calculation resultsare combined in order to analyze the nave’sequilibrium.

Graphic statics also enables an analysis of thebuilding’s history, as in the case of Mª ÁngelesBenito’s study on Avila cathedral (Benito 2011),where she uses this technique to check severalhypotheses on the cathedral’s intermediate buildingstages. In that research, the results from CAD-assisted manual graphic statics are combined withthe results from the tables included in Mohrmann’sedition of Ungewitter’s book (1890).

Figure 1. Floor plan and 3D simulation of the Romanesque and Gothic buildings.

2 A. COSTA-JOVER ET AL.

2. Materials and methods

The building stages of the Gothic apse will be ascer-tained on the basis of existing historical data. Thisinformation is complemented by the underground evi-dence found about the pilar major and the pre-existingRomanesque cathedral, which will help to determinewhich temporary supporting elements were used.

On the other hand, the study of the equilibrium con-ditions using graphic statics techniques involves the geo-metric and constructive definition of the structure. Thegeometric definition is based on a topographic survey bymeans of terrestrial laser scanning (TLS), while the con-structive definition is based on typological references andphysical evidence found by means of test bores.

2.1. Building stages of the apse in Tortosacathedral (1345–1441)

The building chronology of the apse in Tortosa cathe-dral is partially recorded on the Llibres d’Obra (con-struction accounting books) which are kept in thechapter house archive of Tortosa cathedral (ACTo.).This valuable primary source compiles several years ofaccounting records about the construction. The build-ing chronology of the Gothic apse was precisely deter-mined by Josep Lluis i Ginovart (Lluis i Ginovart 2002;Lluis i Ginovart and Llorca 2000) and Victoria Almuni(Almuni Balada 2007; Almuni i Balada 1991) on thebasis of the primary sources and the analysis of thebuilt masonry (Figure 2). In respect of composition,the apse is defined by a heptagonal floor plan and adouble ambulatory covered with ribbed groin vaults.The unit of measure for the apse is the Tortosa span(23.23 cm) (Lluis i Ginovart et al. 2013).

The building of the apse can be divided into threemain stages. First of all, the ring of radiating chapelswas built around the original Romanesque cathedral—which was still in use by then. The chapels were built inorder from the Gospel side (North) to the Epistle side(South). They were covered with square-based ribbedvaults between 1383 and 1424. The keystones of thesevaults are at a theoretical height of 45 spans (10.45 m).

The second stage involved the building of the ninevaults of the ambulatory between 1424 and 1434. Thetwo first vaults have a square base, while the sevenremaining vaults have a trapezoidal base, and theirkeystones are all at a theoretical height of 72 spans(16.72 m). Unlike the radiating chapels, the ambulatoryvaults were built symmetrically from the foot of theapse towards the choir.

Last, the chancel was built between 1435 and 1441.The Clau Major (main keystone) was placed in the apsecenter at a height of 100 spans (23.23 m) and then thevaults were closed, also starting with those located atthe foot of the apse, which are bigger than the rest. Thepilar major presumably helped to balance the thrusts ofthe ambulatory chapels and chancel chapels, and alsosupported the scaffolding system (Figure 3).

2.2. Structure characterization

Two topographic surveys were carried out using TLS. Thefirst survey (2013)1 focused on the indoor spaces, whilethe second survey (2014–2015)2 analysed the outside ofthe apse. Both surveys used very similar devices and offerthe same data acquisition accuracy. The error is the resultof joining the different point clouds, and it is set to a valueof 3 mm. First, the station readings were processed withthe software Cyclone in order to obtain the completepoint cloud of the apse. Second, based on this pointcloud, the software 3DReshaper was used to define amesh (Triangulated Irregular Network). The mesh den-sity was set to an average triangle size of 2.5 cm. Last, the2D-geometry of arches and vaults was defined on thebasis of the 3D mesh. A more detailed explanation canbe found in Lluis i Ginovart et al. (2014a) (Figure 4).

The structure of the apse has a heptagonal floor plan,with a radiating ring of chapels, a second ring with theambulatory, and finally the chancel located in the center.The space is covered with ribbed groin vaults, which arelocated at three different levels in height, topped with a flatroof, and a system of flying buttresses and pinnacles.

The scheme of the structure lines is the same in theoverall apse, but there are two different typologies infunction of the floor plan. The first typology (I),

1Campaign 2013, Josep Lluis i Ginovart, Agustí Costa, Josep M. Puche. Equipment: Leica Scan Station C10. Time-of-flight with amaximum instantaneous speed of 50,000 points/sec and wavelength of 532 nm. Laser class 3 R (IEC 60825–1). Field-of-View: 360°Horizontal, 270° Vertical. The range accuracy of single measurements is: position (6 mm), distance (4 mm), horizontal/vertical angle(60 μrad/60 μrad; 12”/12”). Target acquisition: 2 mm standard deviation up to 50 m. Imaging: built-in camera, 4-megapixels pereach 17° x 17° colour image.

2Campaign 2014–15, Josep Lluis i Ginovart, Agustí Costa, Sergio Coll. Equipment: Leica Scan Station P20, Ultra-high speed time-of-flight enhanced by Waveform Digitising (WFD) technology. Wavelength 808 nm (invisible)/658 nm (visible). Laser class 1 (inaccordance with IEC60825:2014). Field-of-View: 360° Horizontal, 270° Vertical. 3D position accuracy: 3 mm at 50 m, 6 mm at 100 m;linearity error: smaller than or equal to 1 mm. Angular accuracy: 8” horizontal; 8” vertical. Target acquisition: 2 mm standarddeviation up to 50 m. Imaging: built-in camera, 5-megapixels per each 17° x 17° color image.

INTERNATIONAL JOURNAL OF ARCHITECTURAL HERITAGE 3

correspond to the structural lines located in the mouthof the apse, where chapel and ambulatory vaults havesquare floor plan. In addition, thrusts over the pillarsare asymmetric. The second typology (II) correspond tothe rest of the apse, and in this case ambulatory vaultshave trapezoidal floor plan, and thrusts over pillars areperfectly symmetric.

Structural assessment is focused in these two typologies.Themainmeasures of the elements were defined accordingto the first two structural lines, since the geometrical differ-ences with the rest of the structural elements were very littleand irrelevant for calculation purposes. A nomenclaturewere established for each element of the structure(Figure 5).

One of the main construction characteristics of the apsedesign is the definition of a double ambulatory by theremoval of the separating walls between chapels. The struc-ture is solved replacing the wall by a pillar and an arch

which transmits the thrusts of the structure to the peri-meter wall. The solution is completed with two formerarches which define the square floor plan of chapel’s vaults.

In addition, the position of the flying buttressesdraws attention related to the springing of the vaults,since they are lower than the support of the flyingbuttresses. The restoration of the roof which tookplace in 1995 revealed that these elements were partof the water evacuation system.

Thus, at the first roofing level, each pillar in theambulatory receives the thrusts from (1) the archlocated on the same axis, (2) the diagonal ribs, (3) theside arches, and (4) the bonders. At the second roofinglevel, these pillars also support the diagonal arches andthe transverse arch of the ambulatory vault, which areat a greater height (Figure 6). Thus, the pillars mustadapt to changing stresses along the construction pro-cess, with thrust forces alternating on both sides.

Figure 2. Building stages of the apse in Tortosa cathedral (1345–1441).

4 A. COSTA-JOVER ET AL.

In order to define the materials used—particularlyfor the roofing– it is taken as a reference authors suchas Bassegoda (Bassegoda i Nonell 1989), who describethe usual practice in southern Gothic architecture, andalso the test bore made during the writing of the PlaDirector (Master Plan) (Lluis i Ginovart and CostaJover 2013). The vaults springing is solid up to a heightbetween one-third and two-thirds of the rise,

depending on the geometry. These heights werededuced by drawing the structure in 2D and 3D(Table 1).

The filling layer covering the extrados is voidedwith gerres (ceramic jars). On top of that there is a20–40 cm thick lime mortar layer known as trespol,forming a horizontal plane which supports the toproofing (originally built with ceramic tiles). Thislayer arrangement is also detailed in the constructionprocess of one the ambulatory vaults. The verticalstructure, as well as the pinnacles and flying but-tresses, are all built with sandstone masonry. Due tothe lack of experimental data, the specific weight perunit volume of the main materials is taken from bib-liographic references.3

2.3. Calculation strategies

According to the structural function, we can make adistinction between thrust-generating elements (arches,vaults and flying buttresses) and thrust-countering ele-ments (enclosing walls, pillars and pinnacles). The onlysupported element is the top roofing, which rests on thevaults. The existence of auxiliary elements, such as thepilar major, has not been considered in the calculation,since the aim of the assessment is to evaluate whatintermediate stages of construction could be unstable.Current hypothesis about the location and design ofthat auxiliary elements guarantee an easy balance offorces which don’t need to be verified.

Figure 3. The Pilar Major (main column).

Figure 4. Axonometric view of the 3D model obtained of theapse.

3CTE, DB SE-AE; Murcia, J. (2008). Seismic analysis of Santa Maria del Mar church in Barcelona. Master’s Thesis. Advanced Master instructural analysis of monuments and historical constructions. UPC

INTERNATIONAL JOURNAL OF ARCHITECTURAL HERITAGE 5

In the assessment of a completed Gothic masonrystructure, thrusts are typically symmetrical. In thiscase, however, the vaults on the sides were not builtsimultaneously, and several forces are exerted obli-quely on the supporting elements. Three main crosssections were defined for the analysis by means of

two strategies, based both in graphic statics, to cal-culate the pressure lines: (a) characteristic crosssection analysis by means of a spreadsheet(Figure 7, S3), and (b) detailed analysis by meansof manual graphic statics (Figure 7, S1, S2, and S3).

The strategy (a) has allowed to analyse the charac-teristic cross section in the cases where vaults on bothsides of the cross section concerned have already beenbuilt, at the different construction stages. This approachinvolves a higher degree of simplification and does notallow to check the effects of asymmetric situations, butallows to find a wide range of possible solutions.

It is possible to determine the resulting thrustfrom the vaults by likening them to arches andthen calculating the forces exerted on the pillars.

Figure 5. Nomenclature and structural lines typologies.

Figure 6. Scheme of thrusts supported by the pillars.

Table 1. Heights of vault’s extrados layers on each level (mea-sured from the ground level), and specific weight per unitvolume.

H. N1 H. N2 H. N3 Weight

m m m k N/m 3

Heavy filling 8.96 14.00 22.46 24.00Light filling 10.75 16.87 23.42 2.00Trespol 11.15 17.27 23.82 22.00

6 A. COSTA-JOVER ET AL.

Thus, the calculation considers a rectangular surfacewhich is equivalent to the surface of each vault, andthe simplification means to obviate the thrusts ofconcurrent ribs and arches. Each equivalent arch isdivided into 40 portions through vertical cuts, andthen the thrust corresponding to each portion iscalculated. The model is defined by introducingthe geometric properties of the building elements

and the density of the construction materials, infi-nite solutions for the pressure line can be obtained.

As for the strategy (b), we have examined the build-ing stages of each level’s first vaults. The pressure lineswere manually calculated using a CAD software, pro-viding a thorough analysis of each element. It hasallowed to consider the thrusts of transverse sections(Figure 8), which cause an asymmetrical situation when

Figure 7. Structural sections assessed (longitudinal).

Figure 8. Structural cross sections assessed (transverse).

INTERNATIONAL JOURNAL OF ARCHITECTURAL HERITAGE 7

only one vault is built over a column. Thus, the obliquethrust of the vault over its support is verified for themain axes of the element, according to the longitudinaland transverse sections. In this case results ranges werenot calculated, since the structure can be regarded asstable if there is one valid solution.

This approach demands the definition of somehypotheses on the vaults mechanic behavior. The vault’skeystone is higher than the arches keystones in the apseof Tortosa Cathedral, which affects the load distributionon the arches. It can be considered that part of thatweight acts on the transverse arches and side arches(Figure 9) Thus, thrusts of the web of the vault is notconsidered, and its weight is transmitted through archesand ribs. This was the method used by Rubió (1912) tocalculate the load flow in Mallorca cathedral.

The disparity in results between that approach andslice technique were analysed, and concluded that thedifferences between both methods are not significantand they hardly cause any distortion in results from aqualitative point of view. Given a theoretical ribbedvault with 4.80 m sides and a height of 3.8 m at thekeystone, the arches of which have a rise of 3.6 m, theresulting thrust on each corner using these two meth-ods were calculated, considering the same hypothesis ofminimum thrust. Table 1 shows the correspondingnumerical values, with cuts made parallel to the archesin method (A), and loads distributed between archesand ribs in method (B).

The resulting thrusts is practically identical in bothmethods (Table 2). The differences in the vertical and

horizontal components are more apparent but thevalues are still small, being the thrust slightly moreunfavourable in (B). Considering a theoretical mono-lithic pillar with a height of 7.40 m and a ground areaof 1.17 m2, the pressure line is further from the geo-metric center of the pillar’s base in (B), as expected.This difference is 0.12 m, accounting for 8% of thepillar’s total width.

Furthermore, the balance of thrusts in the keystoneof chapel’s vaults is direct, since these are based on asquare floor plan (considering a perfect symmetry of allribs, which is very close to reality). In the case ofambulatory’s and chancel’s vaults, it was necessary toverify the balance in the keystone and to impose in thecalculation that the sum of all thrusts is equal to zero.

Every thrusting element is calculated individually(Figure 10), and the resulting thrust line of every con-curring arch is combined for every support, obtainingthe pressure line for each buttressing element(Figure 11).

3. Results

The calculation models have been defined on the basisof the building stages as established for each level of theGothic apse. Thus, six models are defined for the inter-mediate building stages, and a seventh model for thecross section of the complete apse.

● Level 1: chapel vault (1).● Level 2: enclosing wall (2), pinnacle and flying

buttress (3), ambulatory vault (4).● Level 3: enclosing wall (5), pinnacle and flying

buttress (6), chancel vault (7).

The position of the obtained thrust lines allows oneto evaluate the equilibrium conditions of the structurequalitatively. Main values used for the calculation ofthrusts are also included for each strategy.

3.1. Strategy (A)

The number of considered hypotheses depends onthe number of elements involved in the calculation

Figure 9. Scheme of vaults load distribution over arches andribs.

Table 2. Comparison between the two calculation methods.A B

Vertical component 45.2 kN 42.9 KnHorizontal component 19.2 kN 23.2 KnResulting thrust 49.2 kN 48.8 KnDistance to the center 0.45 m 0.57 m

8 A. COSTA-JOVER ET AL.

and on the equilibrium conditions, with a range of10–14 combinations for each building stage.

Several stable solutions can be found (Figure 12),although in some cases the pressure line is barelycontained within the base of the pillar, meaning thestructure would be at its limit. This is the case ofthe chancel pillar in the building stages (4) and,specially, (5) and (6). (Table 3) shows the maindimensions and weights of the elements used inthe calculation.

3.2. Strategy (B)

A detailed analysis enables to compare the resultswith the spreadsheet, and also to assess the effect ofany asymmetry situations resulting from the vaultsnot being built simultaneously (Figure 13). Aftercalculating the thrusts of each arch and rib(Table 4), the stability of the pillars on their twomain axes was assessed. The differences betweendifferent cross sections are irrelevant, which is why

Figure 10. Example of calculation by graphic statics. Thrusts in the Rib F of Chapel C2.

Figure 11. Example of calculation by graphic statics. Combination of horizontal thrusts in the supports. Vault’s Rib F and Arch T.

INTERNATIONAL JOURNAL OF ARCHITECTURAL HERITAGE 9

the results for the first cross section are shown. Theanalysis reveal minor differences related to previousresults. In addition, the differences between crosssections S2 and S3 are very small and qualitativelyirrelevant, so only results for S3 are shown.

Additionally, the results for the pillars’ transver-sal axis enable to check the negative effect of theasymmetry situation resulting from the vaults notbeing built simultaneously (Figure 14). In the cha-pels’ pillar, the pressure line is outside the middle

third but far from the cross-section limit. As for thepillar located between the ambulatory and the chan-cel, the result is more unfavourable, since the pres-sure line is very close to the cross-section limit.

The magnitude of the weight of the vaults is sosmall in relation to the weight of the counteringsystem that the different hypotheses entail veryminor variations in the location of the pressurelines in the countering system. Table 5 relates theweights of vaults and their corresponding supports,and Table 6 shows the detail of the thrusts obtainedon each structural element.

In addition, the detailed analysis provides infor-mation about the internal equilibrium of the vaults.The chapels have a square base, so all the ribs havethe same thrust. Conversely, the ambulatory vaultshave a trapezoidal base, so the ribs have differentthrusts. The ribs on the side of the enclosing wallgenerate a greater thrust because the supportedvault area is bigger. The same happens in the chan-cel vault, where the two ribs on the side of thecentral body generate a greater thrust than theseven ribs on the opposite side.

Figure 12. Results from the characteristic cross section analysis (S3).

Table 3. Main dimensions of the structural elements in thestrategy (a).

ARCHESSpanm

Risem

WidthM

SpringingM

Arch C 3.90 0.49 9.85 7.35Arch B 4.47 0.66 15.82 12.37Arch Z 5.40 0.40 22.00 16.35Flying Buttress N1 3.90 0.40 14.00 11.17Flying Buttress N2 4.47 0.40 20.40 17.30

SUPPORTS Depthm

Widthm

H. PinacleM

Heightm

Wall 1.65 3.30 20.40 11.15Chapels Pillar 1.80 1.40 14.65 17.27Chancel Pillar 1.66 1.20 - 23.82

10 A. COSTA-JOVER ET AL.

4. Discussion

From a methodological point of view, method (a)revealed that variations on the thrust hypothesis hardlyaffect the resulting pressure line at the base of the

supporting elements. Besides, the more elements areinvolved in the structure, the smaller is the range ofpossible pressure lines at the base. Taking this intoaccount, the comparison between the results obtainedwith both methods is mainly qualitative.

Generally speaking, both approaches provide verysimilar results, although there is some variation(Figure 15). In the complete structure, the main differ-ences lie in the supporting elements of the chapel vault.In model (b), the pressure line at the base of theperimeter wall is further from the middle third thanin model (a). This is mainly due to the abutment’s crosssection being considered, since in model (b) the thrustis divided between the three arches which rest on theperimeter wall. We find that, in order to obtain asolution which is close to the middle third at the base,we must consider the mass of the complete wall, andnot only the mass of each arch support.

On the other hand, model (b) considers the favour-able effect of the thrust generated by the side arches tocounter the chapel vault’s thrust. The stresses combina-tion shows a horizontal component towards the peri-meter wall which is particularly favorable at theintermediate building stages, when the structural sys-tem is not yet complete.

From a structural point of view, the equilibrium ismainly assessed through an analysis of the tippingconditions at the base of the supporting system. The

Figure 13. Detailed analysis results (cross sections S3).

Table 4. Main dimensions of the thrusting elements in thestrategy (b).

. Rise Width Springing

M m m m

C1 Arch F 6.70 3.59 0.15 6,68Arch P 3.84 2.42 0.30 7.43Arch T 4.20 2.63 0.53 7.05Arch C 3.90 2.40 0.49 7.45

C2 Arch F 6.51 3.67 0.15 6.68Arch P 37.16 2.44 0.30 7.43Arch T 4.20 2.75 0.53 7.05Arch C 3.72 2.44 0.49 7.45

G1 Arch J 4.55 3.42 1.00 12.27Arch K 4.32 3.40 1.00 12.27Arch B 4.68 3.45 0.53 12.27Arch M 7.24 4.56 0.15 12.27

G2 Arch G 1.40 1.28 1.00 12.27Arch B 4.68 3.45 0.53 12.27Arch E 3.06 3.89 0.25 12.27Arch D 3.03 3.89 0.25 12.27

Chancel Arch A 0.41 6.51 1.00 16.35Arch X 6.95 7.58 0.33 16.35Arch Y 4.66 6.41 0.33 16.35Arch Z 4.66 6.41 0.33 16.35Flying buttress 3.33 2.82 0.58 11.15Flying buttress 4.07 3.17 0.58 17.27

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TLS survey showed that the vertical elements are per-fectly plumbed, indicating that they have not beensubject to displacement and the structure is perfectlybalanced. This is consistent with the results of theanalysis of the complete structure.

The calculation considered the passive thrust ofthe flying buttresses. A model was performed exclud-ing them, and it revealed that these were not neces-sary to guarantee the equilibrium of the vaults, dueto the low proportion of the structure (9/5 in thechapels, and 9/6 in the ambulatory) and the ratiobetween the thrusts and the weight of the buttressingsystem. These results indicate the function of theflying buttresses, not only as part of the structure,but also as part of the water evacuation system, assuggested their unusual position related to thespringing of the vaults.

The intermediate building stages resulted in some situa-tions where the pressure lines were barely contained withinthe pillars cross section at the base. The vaults thrust againstthe apse perimeter—both the thrust from the chapel vaultsand from the ambulatory vaults—was countered withoutany problem, even though the flying buttresses and thepinnacles had not been built yet. In this respect, the increasein wall thickness at that area is an entirely optimized

solution, since it is located where is necessary to compen-sate the thrusts of the arches.

The thrusts towards the chancel’s center, however, weremore difficult to counter. Pressure lines were barely con-tained in the masonry pillars of the chapels and the ambu-latory, when the upper level have not been built yet. Thepressure lines are very close to the limit at the pillars’ base inall cases, even considering extreme scenarios with a mini-mum thrust from the vaults. It is because the system isbalanced when the structure is completed, and the thrustsof the vaults of one level is balanced with the thrusts of thevaults of the next one through the vertical structure. Theresults are the same for the transverse axis of the pillars,when only the vault on one side has been built. Thisasymmetry situation makes it very difficult to find a stablesolution.

Therefore, according to our interpretation of the results,temporary thrust-countering systems must have been usedin order to ensure the structure’s equilibrium during con-struction work. Thus, the pilar major (main pillar) musthave played a key role in construction, balancing the hor-izontal forces from the vaults by means of props whichmight also serve as supporting elements for the scaffoldingsystem. Besides, the coexistence with the Romanesquecathedral leads to believe that it was also used as a support

Table 5. Ratio between the weight of vaults and columns.Vault C1

kNVault C2

kNVault G1

kNVault G2

kN

Weight of the vault 313.8 283.9 568.1 302.8 -Weight of the pillars N1 895.6 862.4 862.4 862.4 - - - -Weight of the pillars N2 - - - - 1400.6 891.8 891.8 800.4Ratio 0.35 0.36 0.33 0.33 0.41 0.64 0.34 0.38

Figure 14. Transversal cross sections assessed.

12 A. COSTA-JOVER ET AL.

for props and scaffolds during construction, thus reducingthe cost of temporary systems.

Thus, chapel’s vaults should be underpinned towardsthe Romanesque cathedral, while ambulatory’s vaults, andlater chancel’s ones, should use the pilar major. Thisexplains the change in the order of the constructionsequence between the ring of chapels and the ambulatory,previously showed in (Figure 2). The first were builtsequentially, while the later were built with a symmetricalsequence, beginning in themouth of the apse. Thus, thrustsbalanced each other by means of the pilar major located inthe center.

5. Conclusions

The Gothic apse of Tortosa cathedral has been analyzedusing a combination of methods. Historical, geophysical(GPR), and geometric (TLS) data, together with limit ana-lysis data, have enabled to determine the equilibrium con-ditions during the intermediate building stages, thus

complementing our understanding of such a relevant pro-cess as the building of a Gothic cathedral.

The finished structure does not have any equilibriumissues, but during the intermediate building stages the set offorces was not yet complete and may have caused stabilityproblems. Therefore, the pressure lines calculated allowedto verify, with an analytical procedure, the starting hypoth-esis about the need to use temporary thrust-counteringsystems in order to ensure the stability of the whole struc-ture, being the pilar major the main element because itallowed the thrusts from the vaults to be countered.

Thus, pressure lines obtained with the two strategiesshowed equilibrium problems in the base of the pillars.The thrusts of intermediate construction stages, both inlongitudinal and transversal cross sections, causedthrust lines to barely be contained in the masonrylimits. The ambulatory was particularly critical, with ahigher modular proportion of 9/6 and a theoreticalheight of 72 spans (16.72 m) at the vault’s keystone.

Our research has also enabled to find some differ-ences between the methods used to analyze the

Table 6. Detail of the thrusts obtained on each structural element.

C1 Hipothesis 1Res. Hor.1

kNThrust 1

kN Hipothesis 2Res. Hor.2

kNThrust 2

kN

Arch F e. min. 3.2 61.5 e. min. 3.2 61.5Arch P e. máx. 3.0 51.8 e. min. 3.0 49.1Arch T e. min. 4.9 94.5 e. máx. 4.9 86.6Arch C e. min. 1.2 24.8 e. min. 1.2 24.8

C2 Hipothesis 1 Res. Hor.1kN

Thrust 1kN

Hipothesis 2 Res. Hor.2kN

Thrust 2kN

Arch F e. min. 29.7 54.1 e. min. 29.7 54.1Arch P e. máx. 34.4 63.0 e. min. 34.4 63.2Arch T e. min. 49.0 94.5 e. máx. 49.0 86.6Arch T+Wall e. min. 128.7 142.3 e. máx. 128.7 177.9Arch T+Wall+Web L e. min. 152.8 190.2 e. máx. 152.8 178.8Arch C e. min. 17.0 38.4 e. med. 17.0 33.4Arch C’ e. máx. 25.6 51.5 e. min. 25.6 45.9

G1 Hipothesis 1 Res. Hor.1kN

Thrust 1kN

Hipothesis 2 Res. Hor.2kN

Thrust 2kN

Arch J e. min. 33.3 83.8 e. min. 33.3 83.8Arch J+Buttress e.max 238.5 377.1 e.min 238.5 297.1Arch K e.min 47.0 78.4 e.max. 47.0 47.0Arch K+Wall e.max 165.3 244.7 e.min 165.3 215.5Arch B’ e. min. 33.4 83.9 e. min. 33.4 83.9Arch M e. min. 24.9 46.3 e. min. 24.9 46.3Flying Buttress N1 e.min 22.7 27.1 e.min. 22.7 27.2

G2 Hipothesis 1 Res. Hor.1kN

Thrust 1kN

Hipothesis 2 Res. Hor.2kN

Thrust 2kN

Arch G e.min. 16.3 57.3 e.min. 16.3 57.3Arch G+Wall e.max 91.3 212.9 e.max 91.3 212.9Arch B’’ e.min. 24.7 61.1 e.min. 24.7 61.1Arch B e.min. 44.9 10.8 e.min. 44.9 10.8Arch E e.max. 38.0 62.8 e.max. 38.0 62.8Arch D e.min. 26.2 52.4 e.min. 26.2 52.4

Chancel Hipothesis 1 Res. Hor.1kN

Thrust 1kN

Hipothesis 2 Res. Hor.2kN

Thrust 2kN

Arch A e.max. 204.8 316.5 e.max. 204.8 316.5Arch X e.min. 226.3 390.3 - - -Arch Y e.min. 65.3 192.0 - - -Arch Z e.max. 73.6 216.9 - - -

INTERNATIONAL JOURNAL OF ARCHITECTURAL HERITAGE 13

equilibrium conditions of the structure, and the twodifferent strategies has shown the advantages and dis-advantages of each. Strategy (a) makes it possible toquickly obtain an infinite number of possible combina-tions of equilibrium states. Conversely, this procedureimplies a higher degree of simplification and is limitedto the longitudinal section of the structure.

On the other hand, a more detailed analysis bybreaking down each arch’s thrust allows to intro-duce certain nuances which, although in this parti-cular case of study do not involve a significantvariation in results, in other more extreme casescould influence the final assessment of the structure.Besides, this approach allows to introduce asymme-try situations in the calculation and also to verify

equilibrium in the transversal axis of the supportingelements, which is a very unfavorable situation fromthe point of view of equilibrium.

Thus, the transversal thrusts showed the need ofusing a temporary element to guarantee the stabilityunder asymmetry conditions, when only one vault sup-ported by a pillar is built. Furthermore, the perimeterwall is thick enough to compensate the thrusts of thevaults, thanks to the external extra-thickness of thecorners, which allows to build without using any coun-tering temporary element to guarantee the stability. Thecalculation also allowed to verify some aspects aboutthe structure, as the function of the flying buttresses,which are not essential for the equilibrium of thestructure under gravitational loads.

Figure 15. Results for the complete structure. S3. Strategy (a), S1. Strategy (b), S3. Strategy (b) & S3. Strategy (b) without flyingbuttresses.

14 A. COSTA-JOVER ET AL.

ORCID

Agustí Costa-Jover http://orcid.org/0000-0002-6194-3243Josep Lluis i Ginovart http://orcid.org/0000-0001-5957-762XSergio Coll-Pla http://orcid.org/0000-0002-4718-5810

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