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Internat ional Journal of Archit ectural

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Structural Design of High Gothic Vault ingSystems in England

Dimitris Theodossopoulos aa

Architecture, School of Arts, Culture and the Environment,Universi t y of Edinbur gh, Edinbur gh, UK

Available online: 08 Feb 2008

To cite this art icle: Dimi t r i s Theodossopoul os (2008): St ruct ural Design of High Got hic Vault ingSyst ems in England, Int ernat ional Journal of Architect ural Heri t age: Conservat ion, Analysis, andRest orat ion, 2:1, 1-24

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STRUCTURAL DESIGN OF HIGH GOTHIC VAULTINGSYSTEMS IN ENGLAND

Dimitris Theodossopoulos

Architecture, School of Arts, Culture and the Environment, University of

Edinburgh, Edinburgh, UK

The introduction of rib systems in High Gothic vaults in England initiated a period of greater

ambition in the design of churches, while subsequent experimentation increased the con-

fidence of the masons in working with more complex forms. This development is discussedthrough the study of the structural behavior and efficiency of vaults at the cathedrals of

Durham, Canterbury, Wells, and Lincoln that represent significant technical innovations.

The collapse mode and safety margin of the original designs is examined, focusing on the

dominant action that results from failure of the buttressing system. The study highlights how

the integrity of the fabric provided by the rib had to be enhanced by refinements of the form

and lateral support. Although the discussion shows a consistent development, each of the

innovations brought individual solutions that this study attempts to treat as options available

to the masons.

KEY WORDS: Gothic architecture, vaults, structural safety, England, structural design

1. INTRODUCTION

The technology of High Gothic stone vaulting in England is marked by the

balance between national characteristics, such as the rib systems, and established

practices imported from France. The period is framed within the groundbreaking

use of ribs in Durham Cathedral in 1093 and in 1192 in Lincoln with the beginning of a

more ornamental approach to construction (Decorated period). Developments were

driven by the introduction of advanced Norman construction technology, which led to

a constant exploration of linearity in the expression of the structural elements, such as

the use of shafts and ribs (Webb, 1965; Stalley, 1999). Churches of the period are also

characterized by less ambitious verticality in the elevation, further combined with a

particular approach to buttressing. Proportions of this layout are often as importantto the strength of the vaults as their form or the reinforcement of their groins with ribs.

This article aims to contribute to the study of the design principles of stone

vaults in England during the High Gothic period Norman (10661190) and Early

English (11751265) through the assessment of their structural efficiency.

Developments in this period follow a more logical and consistent path compared

with the approach of the earlier Romanesque period, but these developments often

appear still as individual solutions with no direct links between them (Hoey, 1987).

International Journal of Architectural Heritage, 2: 124, 2008

Copyright Taylor & Francis Group, LLC

ISSN: 1558-3058 print / 1558-3066 online

DOI: 10.1080/15583050701516484

Address correspondence to Dimitris Theodossopoulos, Lecturer in Architectural Technology,

Architecture, School of Arts, Culture and the Environment, University of Edinburgh, 20 Chambers

Street, Edinburgh EH1 1JZ, United Kingdom. E-mail: [email protected]

Received 15 July 2006; accepted 30 May 2007

1


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The cathedrals of Durham, Canterbury, Wells, and Lincoln represent the major

innovations in a large scale and are examined here as options that were available to

the masons, although an attempt is also made to define relationships and possible

patterns.

Understanding the design process and structural behavior of these systems isgenerated by critically examining geometric and construction data of the main vaults.

Using finite-element (FE) models (Theodossopoulos, 2003), the loads that develop on

critical areas such as the intersections (groins) and springings and the role of the major

structural elements (ribs, webs, buttresses) are assessed to interpret the construction

process and the behavior in service. Further examination is performed by applying the

dominant action of supports spread by means of a nonlinear material and geometric

analysis (NLMGA) that serves as well to assess the origin of the pathology observed in

some cases.

2. DESIGN OF VAULTS IN THE EARLY ENGLISH PERIOD

2.1. Design and Behavior of Large-Scale Cathedrals

English Gothic design resulted from the combination of regional technical

advancements (such as the rib) with established patterns imported from France during

the contemporary spread of Gothic design throughout Europe. England made sig-

nificant contributions to Gothic technology, such as the correct application of the

pointed arch or the innovative use of ribs in a large scale in Durham Cathedral

(10931133), well before the seminal remodeling of St. Denis in Paris (1130). Gothic

design in England is also characterized by balanced horizontal and vertical aspects

and less slender buildings (Figure 1). Such proportions are adequate for the lowlighting angles of the country, which in turn highlight efficiently this balance in the

elevation, so often no large clerestory or wide openings at the walls were required.

The prominence of the linear elements is an important characteristic, but design

was not viewed as a careful arrangement of the elements to withstand the deforma-

tions. For example, structural analyses will show later that the contribution of the rib

in the in-service strength of a Gothic vault is much less significant than the role of the

edge beams in modern shell architecture (Dieste, 1983). The main load-bearing

elements (vaults, arches, buttresses) were most probably proportionally arranged,

and their layout will be discussed in more detail in the case studies.

The stability of this scheme is a common issue in Gothic design and can beillustrated by the deformation of the cross-section in a typical cathedral (Figure 2a),

which together with the analysis of other isolated vaults has confirmed the

sensitivity of the structure to insufficient containment of lateral thrusts

(Theodossopoulos, 2003 and 2004). Excessive deformations can result in spread of

the vaults supports that can destabilize the nave; surveys of heavily deformed cathe-

drals such as Beauvais Cathedral (France) or Vitoria Cathedral (Spain) (Azkarate

Garai-Olaun, 2001) define the safety margins (Figure 2b). A lateral spread of 1/30 of

the span can be considered as the serviceability limit before failure occurs with the

formation of a three-hinge mechanism at the supports and vertices (Figure 3). Smaller

movements have less visible effects due to the use of the more flexible lime mortars,

allowing stresses to redistribute through the masonry. The comparison in Figure 1shows that the less slender lateral walls in England are less subject to instability and, in

2 D. THEODOSSOPOULOS

INTERNATIONAL JOURNAL OF ARCHITECTURAL HERITAGE 2(1): 124


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combination with this smaller scale, the lateral thrusts of the vaults can be lower,

avoiding sophisticated flying buttresses.

The early innovations in Durham achieved construction efficiency, as

described later in text, but direct influences are quite limited (transepts of

Winchester Cathedral, Gloucester Cathedral, Caen Cathedral). Later, the recon-

struction of Canterbury Cathedral (11751184) introduced an unbalanced version of

contemporary French forms, which ultimately due to the prestige of the site

diverted any progress from the structural design advancements in Durham. At the

same time, the native desire for strongly marked linear pattern, highlighted by the

use of shafts and ribs (Webb, 1965; Stalley, 1999), becomes apparent in Canterbury

with the innovative use of dark Purbeck marble shafts.

Gradually in Early English design (11751265) all these innovations and trends

start merging into a more national style. In the subsequent Decorated phase (1250

1370), ribs and elements such as bar tracery will occupy a more prominent place in the

development of the entire structure. The combination of such elements with a less

ambitious scale was central in the English quest to unify the classic division of the

elevation (arcade, triforium, clerestory) that was bestowed by the earlier Romanesque

period (Webb, 1965), improving the structural collaboration between the parts in

elevation and increasing lateral stability (Figure 1). Thickening of upper walls was also

a safe precaution, and in general the structural logic is much less adventurous than intheir French counterparts.

Figure 1. Schematic illustration comparing the nave of a typical English cathedral (Wells, 11861215) and a

French cathedral (Chartres, 11941260).

STRUCTURAL DESIGN OF GOTHIC VAULTS IN ENGLAND 3



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Overall continuity in design characterizes most of the English Gothic phases and

is often combined with a rapid construction program. At the same time, an additive

approach to space is recorded mainly due to the monastic origin of most of thecathedrals, resulting in efficient structural compartmentation between the usually

(a)

(b)

Deformation of lateral wall

B dead load

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6

Normalised displacement (%)

Normalisedheight

B no fl buttr

Vitoria nave N D1

Vitoria nave N G1

Vitoria nave N H1

Aisle spring.

Nave spring.

Figure 2. (a) Schematic illustration of the deformation of the lateral elevation in Burgos Cathedral (Spain),

magnified 200 X (Theodossopoulos 2004), and (b) graph comparing the deformation of the elevation of

Burgos Cathedral (finite element analysis from Theodossopoulos 2004) and Vitoria Cathedral (Spain)

(survey data from Azkarate 2001); lateral displacement is normalized against the span.




6/25

two transepts, the facade, and the deambulatory; such aspects will be discussed in the

case studies. Typically there is emphasis on the separation of the individual parts,

opposite to the French tendency of integration and subordination of the parts to the

greater scheme (Hoey, 1987). This emphasis eventually led to more focused experi-

mentation with the parts, such as the vaults, for example in St. Hughs Choir in

Lincoln. This conceptual compartmentation is further assisted by a diagonal con-struction process, where a few bays are usually erected together and their high vaults

are safely built as they are abutted by a diagonal finishing in the triforium at the last

bay (Figure 4).

Figure 3. Crack pattern in a cross vault (Theodossopoulos, 2003).

Figure 4. Stylistic changes indicating the diagonal addition of vaulted bays in the nave of DurhamCathedral

(after Bilson, 1922).




7/25

Many of these achievements and characteristics can be summarized in the

last two main buildings of the period. Wells Cathedral (11861260) is the building

where a new English gothic style is explored, designed exclusively on pointed

arches. Lincoln (11921280) sets a vocabulary following import of new French trends

that was later implemented in Salisbury (12201284) and challenged in WestminsterAbbey (1254).

2.2. High Gothic Vaults

The development of Gothic vaulting has often been linked to the degree that

the masons understood the role of the rib; however, this archaeological and stylistic

approach misinterprets the design of stone vaults, ignoring the efficiency of the form

of the shell. Historic and experimental evidence (Fitchen, 1961; Theodossopoulos,

2003) shows that the crucial interlocking at the groin was largely facilitated by the

ribs during construction. Geometrically, the complex intersection of the two-barrel

vaults along the groin does not follow a straight line (Figure 5) except at the square

bays of the aisles. In England, rubble masonry was often used to bypass the complex

stereotomy, and overall the ribs served as permanent scaffolding for the formwork

of the shells, improving the quality of the intersection and alignment along the groin

(Figure 6).

A gradual realization of this role made the ribs, together with the shafts of the

piers, a means for a visual unification of the entire structure (concordance). These

linear elements must have also played a role in the overall design of the typical Gothic

scheme (aisle vaults, triforium, clerestory, and high vaults [Figure 2a]), making

possible an abstraction that can guide the design and the establishment of proportionsthat govern the layout (Singleton, 1981). This abstraction often gives the impression of

a structural skeleton, further assisted by the progressive dissolution of the masonry

envelope to let in more natural light and by Late Gothic attitudes where ribs are

transformed into independent stone trusses, supported by an increasing confidence in

the performance of the vaults.

So far as the design of the vaults itself is concerned, straight and visually

uninterrupted longitudinal ridges became the norm in English Gothic high vaults.

L

F

W

R

stilts

Figure 5. Geometry of a quadripartite high vault generated by a constant radius.




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Since all the vertices had to attain the same height as the diagonal rib, it was the latter

that dictated the form of the intersecting shells, which are profiled as a projection of

the diagonal (Figure 5). As the diagonal is often a part of a semi-circle the resulting

incompatibility with the longitudinal shell was resolved by another Durham innova-

tion, the pioneering use of the pointed arch profile. When this arrangement resulted in

narrow clerestory openings natural lighting could improve by placing the formeret

arch on stilts. The main quest was therefore for geometric compatibility and stability

during construction, and the forms are not as structurally efficient as they could have

been using intuitive schemes such as catenaries. It is reasonable to assume however

that the masons developed an intuitive knowledge of the limit state of their structures,

which was later linked to their geometric proportions (Heyman, 1995; Huerta, 2006).

The design and strength of English High Gothic vaults as part of cathedral

design will be discussed next. This work focuses on the achievements of Early English

design in large-scale vaulting and therefore the development between Durham,

Canterbury, Wells, and Lincoln is examined in more detail through structural analysis

of their high vaults. The analysis of the case studies will concentrate on the design and

behavior of those high vaults that represent the original Early English design.

3. STRUCTURAL ANALYSIS OF MAJOR EARLY ENGLISH CATHEDRALS

The vaults were modeled as shell structures using FE analysis both in service

conditions and collapse due to failure of the buttressing system (Figure 2a).

A procedure developed earlier during the study of the collapse of the Holyrood

Abbey Church in Edinburgh (Theodossopoulos, 2003) is followed. The orthotropic

material properties (Table 1) were taken from the Cathedral of Burgos (Spain), which

represents reasonably well a typical gothic masonry bond (Theodossopoulos, 2004).

A constant shell thickness of 250 mm was assigned where not known and the unitweight of the stone masonry was assumed as 24 kN/m3. The support of the vault upon

Figure 6. Construction of the webs around the rebate of the ribs (Fitchen, 1961; author).




9/25

the lateral walls was simulated with beam elements of a substantial thickness in order

to allow the edge to follow the supports outwards rotation. Ribs and transverse

arches were also modeled mainly with beam elements.

The models were analyzed with the FE program Abaqus (Abaqus, Standard

version 6.4, Abaqus Inc., Pawtucket, RI, USA). Failure was simulated as a smeared-

crack approach with nonlinear analysis that used a biaxial failure criterion for

masonry (Sinha, 1997) based on the mechanical properties in Table 1 (Figure 7).

The moments of resistance in the orthotropic directions, Mux and Muy were evaluated

from the corresponding flexural strengths, Fux and Fuy. The criterion was implemented

in Abaqus through a subroutine, allowing loads to shed to the stronger direction of the

material once the weaker axis has failed (Theodossopoulos, 2003). Combinations of

outward spread of supports were applied, simulating unfavorable conditions that may

result from the degradation of buttressing. Failure of the structure occurs when

sufficient cracks develop to transform the structure into a mechanism.

My

Muy

2

Mx

Mux

2

0:75:

Mx

Mux:

My

Muy

2

0:25:

Mx

Mux:

My

Muy 1:0

1

Figure 7. Strength of masonry under biaxial stresses (Sinha, 1997).

Table 1 Mechanical properties considered for the stone masonry of the cathedrals

Material properties

Parallel to bed

joint (x-axis)

Normal to bed

joint (y-axis)

Elasticity modulus, E(N/mm2) 4900 1400Compressive strength fc (N/mm

2) 2 6

Flexural strength, Fu (N/mm2) 0.7 0.2




10/25

3.1. Cathedral of Christ and Blessed Mary the Virgin, Durham (10931133)

Significant innovations such as the integrated use of the rib, the pointed arch,

and the flying buttresses were for the first time applied coherently in large-scale

construction in Durham (Bilson, 1922). Their effect on the design of vaults wasassessed studying a double bay of the high vault that includes supports on the

compound and secondary piers (Figure 8). The groins in both the high and aisles

vaults, as well as the webs at the aisles, were generated by semicircular arches but the

webs at the high vault had to be conveniently pointed to achieve height and radius

similar to the ribs and enable standardization of the voussoirs (cf. Figure 5).

Ribs were used first along the groins of the choir aisle (on a 8- X 5.9-m bay) and

high vaults (on 8- X 9.9-m bays), followed by the transepts and, after the choirs

completion in 1118, the nave (Figure 8). The original choir high vaults were replaced in

1235 during the addition of the Chapel of the Nine Altars at the east end, when

dangerous cracks became visible, but this replacement was most probably a justifica-

tion to modernize the archaic vaults. Therefore only the vaults at the nave currently

belong to the original design, but due to various hypotheses on the form of the choir,

the nave vaults can be viewed as the perfection of the original experimentation and on

those merits they will be examined here. A more detailed discussion on the design and

performance of the first rib vaults along the aisles can be found in (Theodossopoulos,

2006).

The overall scheme of the cathedral is structurally and visually fully articulated

and rational as expressed in the concordance between the ribs and shafts and the

striving for stylistic unity. The formation of the rib by a separate voussoir at the

difficult intersection of the webs along the groins (Figure 6) can be therefore inter-

preted within this frame (Thurlby, 1993). The ribs appear to be integrated to the webduring construction, so it was adequate to simulate them at the FE model with shell

elements.

In terms of structural behavior, the transverse stress pattern (Figure 9) shows

that this new system eventually stiffens a wide area of the vault but there is no evidence

that the masons were aware of the stiffening potential of the ribs in a similar manner to

Figure 8. The high vaults in the nave of Durham Cathedral (11181133).




11/25

the edge beams in modern shell structures (Dieste, 1983). A realization of this potential

could have completely altered the vault form but the ribs are still relatively slender and

their section does not provide sufficient depth at the extrados or even follow an

optimized form-active configuration that would allow stability as self-standing arches.

This detail developed into a permanent guide and formwork that improvedstandardization in the construction process and provided confidence and accuracy

as stone flags were laid to form the webs. The vaults were further secured during this

delicate phase by building them together with their lateral walls resulting in a rapid

construction program, and it is possible the masons felt they had to celebrate the

advancements of the ribs by clearly expressing them through design (Hoey, 1987).

Historicalcritical analysis of the other innovation, the quadrant rampant

arches at the triforium gallery (Figure 10) shows they were mainly devised as a support

of the aisle timber roof (Bilson, 1922; Gardner, 1982). Eventually, however, they

improve the performance of the vault by reducing the lateral spread and the overall

deflection of the vault (curve fb at 1.2 m in Figure 11). The FE model shows the

optimum effect occurs between 1.2 m and 1.7 m, or one-fourth of the rise.

The nave is the work of the second master mason of the original phase (1100

1133), and its performance demonstrates the earlier innovations of the design. The

thrusts of the vaults are tentatively constrained by a combination of pier buttresses

with a thick outer wall that is not disrupted by major window openings. The ribs have

definitely allowed some of the shell thickness, and therefore the thrusts, to be reduced,

even if the masons did not explicitly plan this reduction. Cracks developed, however,

both in the choir vaults (prompting their replacement in 1225) and later in the nave

(Curry, 1981). The movement of the springings due to insufficient bracing or the ill-

fitting of the vault at the bay (as in the transept) was examined in the FE model in two

conditions of outward spread: that of the entire wall and local movement of the minorsupport.

Figure 9. Transverse stress S22 at the intrados of the nave in Durham Cathedral due to self-weight (kN/m2).




12/25

The crack pattern (Figure 12) verifies the failure mechanism usually observed in

gothic vaults: hinges initiate above the supports (zone 1 in Figure 12a) followed by

cracks along the longitudinal vertex (zone 2 spreading to 3), which are sufficient to

cause the collapse of the vault (Theodossopoulos, 2003). Cracks along the wall do not

appear here because the rigid plate of the clerestory wall can follow the movement of

the springing (cf. Figure 2a). The spread of the minor support (that corresponds to the

cylindrical pier of the nave) results in a localized failure around the relatively stiff

conoid pocket (Figure 12b), with the supports cracks developing at the last stages.

Failure occurs at an earlier stage than the full spread of the nave, but is rather local andtherefore less critical.

4.0

3.0

2.0

1.0

0.0

Distance from the wall (m)

Deflection(mm)

Burgos

Durham basefb at 1.2mfb at 1.7m

0 642

Figure 11. Deflection of the transverse vertex of the nave vaults under self-weight from the FE analysis.

Figure 10. Longitudinal section of the nave in Durham Cathedral (Dehio, 1901).




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Maximum deflections concentrate along the longitudinal vertex

(Theodossopoulos, 2006), with low transverse bending stresses (Figure 9) and

major compressive axial forces around the supports. The predominance of the latter

can be attributed to the fact that the pointed profile of the longitudinal barrel

approximates that of the catenary curve (Figure 13) that can be traced for the

span-to-rise ratio L/F 1.85 (Dieste, 1983).The original phase of the building shows the aimed coherence and continuity in

the design but construction progress affected the efficiency of the innovations. The

Cracks at extradosCracks at intrados

1

2

3

21

1

(a)

(b)


3

2

1

23

3

Figure 12. Crack development supportsspread: (a) movement of all supports with failure at 330 mm (1/34of

span), and (b) middle (minor) supports only with failure at 220 mm (1 /50 of span).




14/25

rapid completion of the choir with the innovative yet expensive vaulting was followed

by plans for a cheaper timber ceiling for the rest of the building (Bilson, 1922; James,

1993) but when stone vaults were considered once again, the south transept was

already built up to the springings so the vaults were forcedly inserted. The distorted

geometry resulted in an uneven distribution of the loads on the piers producing deep

cracks within or at the edge of the high vaults as can be seen at both transepts, similar

to Figure 13.Finally, the structural study of the vaults shows it is not straightforward to

establish a relationship between geometric proportions and the structural perfor-

mance of the building as the design priorities stemmed from construction con-

siderations rather than efficiency of the form. The design is square-based, as the

rise of the vault F is equal to half the span L, and therefore the transverse edge

can be inscribed in a rectangle of 1:2 proportions, i.e., made of two squares. The

relationship with the aisles becomes more complicated as the characteristic

cylindrical piers result in an uncomfortable arrangement of the shafts at their

back in the aisles.

3.2. Metropolitan Cathedral of Christchurch, Canterbury

The original Romanesque cathedral of Bishop Lanfranc was damaged by fire in

1174 and a new major cathedral was constructed on the existing foundations in a very

short period (11751184). The study of the construction benefits from the contemporary

account by Gervase that outlines in detail the progress of the works (Pevsner, 1985).

Despite the previous innovations in Durham, little reference exists in the spatial and

technical solutions. These instead were imported from France by the master mason

William of Sens (active between 11751179) but were applied without a full under-

standing by the local masons. Application of the unitary approach of Durham could

have produced a structurally and visually less abrupt junction of the piers with theoverwhelming, archaic sexpartite vaults of the choir (Figure 14). Other characteristics of

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.0 0.2 0.4 0.6 0.8 1.0

Normalised span (x/L)

Normalisedrise(z/L)

Durham transverseDurham catenaryWells tranverseWells catenaryLincoln transverseLincoln catenary

Figure 13. Profiles of the transverse barrel vault in the high vaults of the case studies compared with the

catenary curves corresponding to their span-to-rise ratio (main dimensions are normalized).




15/25

Canterbury include the influential decorative treatment of the main load-bearing ele-

ments where linearity is emphasized by leaner ribs and the use of dark Purbeck shafts.

The sexpartite vault of the choir spans over a 13.1- X 10-m double bay and the

major diagonal rib is semi-circular, resulting in a rise F 7.2 m. The transverse shellhas a pointed directrix, of the same radius as the rib, which facilitates standardization

of the voussoirs, while the intersecting vaults are semi-circular and stilted. The FE

model of the vault was generated based on the geometry published by Dehio (1901)

and shows that deflections under self-weight spread mainly along the longitudinal axis(Figure 15), which is stiffened by the transverse vaults that further result in mainly

compressive stresses in the transverse direction.

The benefits and limitations of the scheme of sexpartite vaults will be discussed

in comparison with the quadripartite vaulting solution of Durham in terms of struc-

tural performance. To assess the effect of the boundary conditions in this scheme, the

lateral displacement associated with a constraint at the base of the vault were plotted

along the entire height of the wall (Figure 16). Outward movement in Canterbury is at

its highest at the top of the stilted supports, approximately two-thirds of the vaults

rise. The minor one spreads less as it is stiffened by the convergence of the skewed

transverse barrels allowing more uniform and lower displacements along the top of the

wall. The walls in Durham deform the opposite way, involving larger portions of thespringings.

Figure 14. The choir of Canterbury Cathedral (Webb, 1965).




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The flying buttresses are semi-hidden in the aisle roof (Kusaba, 1989) and the

elevation is similar to Durham, indicating a tentative design influenced by Durham

but resolved with wider openings. Failure of the FE model due to spread of the

abutments occurred in Canterbury at a high increase of the span, 670 mm or 1/20 of

the transverse span, indicating a more successful form, less demanding for bracing

than in Durham. The stress pattern in Figure 17 shows yield lines at that stage as high

concentration of contour lines. If a uniform spread occurs, then cracks form at 35 mm

around the keystone and the outer, major springings. Hinges above the stilted middle

springing start forming slowly afterwards and the high stiffness of the relatively closely

folded transverse vaults opposes the consecutive failure in the abutments.

Transformation of the cross vault into a mechanism occurs with further longitudinal

cracks on either side of the vertex at 670 mm.

Despite their good performance, sexpartite vaults have not been popular as aprominent feature in England (cf. the smaller bays at the transepts of Lincoln) due to

Figure 15. Deflections at the sexpartite choir of Canterbury Cathedral under self-weight (m).

2

1.5

1

0.5

0

0.5

0.0 0.2 0.4 0.6 0.8 1.0

Normalised length of wall

Lateraldisplacement(mm)

DurhamCanterbury

Minorp

ier

Figure 16. Outward displacement along the edge of the choir vaults in Canterbury and Durham Cathedrals.




17/25

complications in the intersection of the transverse vaults and the need to construct

substantial and relatively unstable ribs at the main diagonals. The integrity of this

element can be compromised by the dimensions of the span and the non-uniform

boundary conditions at the edges of the bay, as the analysis has shown that thrusts at

the mid-support of the bay are higher than in the extremes.

3.3. Cathedral Church of St. Andrew, Wells

Together with Lincoln, this cathedral marks the establishment of confident

English characteristics in Gothic architecture. The entire original design was based

on a uniform use of the pointed arch, which together with the proportions and height

of the building (20.5 m), and the short construction period (11861215) resulted in a

homogeneous structural scheme. John Bilson in his meticulous study of this phase

(Bilson, 1928) has identified a diagonal break in the works at the nave, which appears

to be predetermined, but the scheme was well planned and the execution and behavior

appear uniform today. The choir was later remodeled between 12851345 in

Decorated style, so it is the nave that represents the original Early English project

today (Figure 18).

A representative bay of the nave (Figure 18b) was modeled using Bilsons survey.

A semicircular diagonal rib was considered and the longitudinal vault was generated

using the same radius and rise. The transverse shell has straight vertices and in order to

avoid a very narrow pointed profile that would have blocked light, it had to be stilted.

The rise of the intersecting shells was typically dictated by the diagonal and a system of

proportion 1 :ffiffiffi2

phas been mostly used in the cathedral (Singleton, 1981). Although

the dimensions of the bays are not constant throughout the building (another char-

acteristic of English Gothic), a typical central bay in the nave has the proportion of

span L to width W equal to 2 and it appears that the rise of the vault F is equal to the

diagonal Ds of each of the two squares that form the bay (Figure 18b). This can be

considered as an updating of the square-based system that was earlier used inDurham.

Figure 17. Transverse stresses S22 at the extrados of the sexpartite vault of the choir in Canterbury at

670 mm uniform support movement or 1/20 of span (kN/m2).




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The deflections under self-weight verify the one-way pattern of the scheme, whilethe lateral thrusts need to be balanced almost at one-third of the rise, which is almost

the location of the flying buttress abutment. The stilted cylindrical wall edge provide

transverse rigidity so a more efficient pattern develops, with transverse tensile stresses

around the area of the wall joint and lower bending stresses around the vertices

(Figure 19). Although the vault does not adhere intuitively to the corresponding

catenary profile for L/F 0.65 (Figure 13), the longitudinal vault is mainly in com-pression, further stiffened by the transverse webs.

The benefits of the chosen design and proportions become more evident in the

failure of the FE model due to spread of the abutments. Cracks are indicated as areas

of null stress within a dense contour in Figure 20, due to the smeared crack

approach, and initiate at the clerestory supports and keystone, spreading to thearea around the lower ridge. Failure occurs at 275 mm (or 1/36 of transverse span)

(a)

W

WD

s

D

(b)

Figure 18. The nave of Wells Cathedral (England): (a) photograph of an aspect of the nave (Hoey, 1987),

and (b) plan of a typical bay of the nave (after Singleton, 1981); width L 2W 2 4.82 m.




19/25

following a pattern and sequence similar to Figure 12. This scheme is in agreement

with the established pattern of failure for such structures (Figure 3) and occurs at an

acceptable increase of span (Theodossopoulos, 2004; Ochsendorf, 2006). This is a

further indication that the Early English experimental period has produced a con-

fident vaulting design in Wells, which is in line with similar contemporary develop-

ments at the rest of Europe. Finally, the presence of thin ribs simulated with beam

elements in the FE model has not affected significantly the stress pattern or failureprogress.

Figure 20. Failure of the nave vault in Wells Cathedral due to spread of all the supports; transverse stresses

S22 at the extrados at 91-mm movement (kN/m2).

Figure 19. Transverse stresses S22 at the extrados of the quadripartite vault of the nave in Wells under self-

weight (kN/m2).




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3.4. Cathedral of Blessed Mary the Virgin, Lincoln

Lincoln Cathedral (11921280) has the most advanced design of the period,

developed from the 1174 scheme for Canterbury, and together with Wells defined

the English Gothic expression. The identity of the ribs in design (linear pattern androle in the generation and construction of the vaults) stressed by the use of Purbeck

stone shafts unifies visually the structural scheme of the cathedral (Figure 21).

Remodeling of the Romanesque cathedral (11921280) was triggered by a partial

collapse after an earthquake on April 15, 1185, and started with the new choir

placed on the outer slope of the hill on which the fortified precinct of the cathedral

and castle were built.

The entire building is a project experimenting with new forms and systems that

shaped later the Decorated period and the structural performance of these innovations

is assessed here. The intensity of the design process often disregarded regularity in the

spacing of the bays, as is evident in both directions in the choir; faults (Figure 21a) thatcontinue to appear today (Bailey, 1996) can be attributed to a structural scheme not

(a)

(b)

Figure 21. Vaults of Lincoln Cathedral: (a) tierceron vaults at the nave (12651280), and (b) the skewedcrazy vaults at the choir (11921265).




21/25

fully developed that additionally had to accommodate the fabric loads on uneven soil

conditions.

One of the major advancements is the experimental tierceron vaults that result

from an attempt to visually unify the various structural elements. The so-called crazy

vaults (11921265) of the first part of the choir (todays St. Hughs Choir) explored thepotential of boldly detaching form and elements from their function (Figure 21b) by

making a prominent use of the ribs. As will be shown from the structural performance

of the vaults later, this design liberated them from symmetric arrangements and

generated unorthodox new surfaces that do not result from construction practice

(Kidson, 1986). Forms became more symmetric later, at the first tierceron vaults at

the nave (Figure 21a), which were followed by an eastwards extension of the choir, the

Angels Choir (12651280). All these vaults used ribs in a more extravagant manner to

highlight and strengthen complex junctions like groins and ridges, establishing the

pattern for future similar vaults in England.

FE models of thesetwo main types were generated over a 7- X 12.5-m compartment(Theodossopoulos, 2006) to trace the evolution of the system. Deflections confirmed the

strong one-way behavior of the high vaults, while the stress patterns (Figure 22) show

that the longitudinal vault is mainly in compression and is abutted by the eccentrically

placed transverse vaults. The latter behave similarly to a pointed barrel vault supported

upon diaphragms at the end arches.

Comparison between the two types make clear the benefit of aligning the

transverse vaults in the tierceron vaults as the wide area around the keystone is

under compression forces (Figure 22b). Collapse of the skewed vaults due to sup-

ports outward movement initiates at the springings as well as the keystone between

the offset crowns (zones 1 in Figure 23) at 91 mm, spreading later around the vertex

(zone 2). Failure as a result of the three crack lines occurs at 503 mm or 1/25 of thetransverse span, indicating higher stress reserves than the previous case studies. In

this scheme, however, the transverse webs of the crazy vaults cannot abut each other,

causing a twist at the central section that was probably perceived by the masons

during the construction of the ribs.

Finally, removal from the FE model of the linear elements that represent the

tierceron ribs had a minor effect on the deformation of the vaults, highlighting their

importance as mainly a construction element that guaranteed the continuity assumed by

the FE model along the intersection. In terms of expressing the performance of the vault

in a system of proportions, a different one was followed here, as the span of the bay L

equals two times the height of an equilateral triangle generated by the width of the bay W.This system created a more oblong vault (L/F1.65) that is probably more suitable to thewider transverse span (11.2 m compared with 9.1 in Wells). The form of the intersecting

vaults is determined by the diagonal, and this approach results in a longitudinal shell that

is quite offset from the corresponding catenary curve (Figure 13).

4. DISCUSSION AND CONCLUSIONS

This work has attempted to establish patterns of the structural behavior of

major High Gothic vaulting systems in England. Significant innovations at the high

vaults of the cathedrals of Durham, Canterbury, Wells, and Lincoln have been

analyzed and their evolution was discussed. The FE models considered their idealand final configuration, simplifying the effect of the support conditions or the




22/25

deformations imposed by the rest of the structure. All the cases showed a similar

response to self-weight and spread of the abutments, with a strong one-way beha-

vior. Design based on pointed arches often allowed higher and beneficial compres-

sive forces to develop at the longitudinal vaults, while most of the transverse shells

behaved like independent vaults supported along the ribs that functioned as edge

beams.

This pattern became more evident during the spread of the supports, where a

three-hinge mechanism formed around the longitudinal vertex and above the cleres-tory supports, causing collapse. This behavior is in line with experimental and site

(a)

(b)

Figure 22. Transverse stress S22 at the extrados of the major vault types in Lincoln Cathedral at self weight

(kN/m2) (a) the crazy vaults (St. Hughs Choir), and (b) tierceron vault at the Angels Choir.




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observations (Theodossopoulos, 2003), where the transverse vaults gradually detach

along the groins or the vertex and transform into a barrel vault supported upon

diaphragms and the longitudinal ones behave as a three-pin pointed arch, simply

supported on the diagonal ribs.

The early hinges at the springings is a common issue among the cross vaultsstudied in this work and originate on the high bending stresses developing already

under self-weight. Redistribution of the loads is expected to occur at the flexible lime

mortar joints of the blocks and eventually stability becomes a geometric issue (Huerta,

2006) once mechanisms form as indicated by the FE models.

Sharing of the loads between the transverse and the longitudinal webs depends

highly on the integrity and geometry of the groin. Most of the development of forms in

this period relates to the ribs but it needs to be stressed that they have always been

treated as a construction element. Earlier research (Theodossopoulos, 2003) has

shown that collapse of vaults whose ribs provide a shear connection to the webs

through a stem on their extrados (Fitchen, 1961) occurs after a rupture of the joint

between the intersecting shells. A more accurate knowledge of the geometry of this

junction is required to understand the bond between the shells and the possibility of a

similar rupture.

This study addressed the issue of High Gothic design in England from a

structural and technological point of view. The structural analysis showed a rela-

tively high degree of safety but this is often the result of critically appraising earlier

stages in a cathedral or the removal of failing portions. Based on the results from this

project, next steps will concentrate on the influence of the construction process on

design. Areas to be studied can be the build-up of the stresses at the ribs during

construction and a focus on the construction sequence and development of vaulting

techniques within the same building at each of the case studies examined in thiswork.


1

2

2

1

11

1

Figure 23. Crack pattern of the finite element model of the crazy vaults (St. Hughs Choir) at a uniform

spread of the supports at 503 mm.




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ACKNOWLEDGMENTS

Support in the development of this work was provided by the Humanities

Research Centre at the University of Nottingham. The author would also like to

thank the Chapter in the Cathedrals of Durham and Lincoln, and Professor JohnChilton for access and valuable advice and information.

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