Seismic Analysis of the Church of Kuño Tambo

Transcription

Sara Zanotti
Seismic Analysis of the
Church of Kuño Tambo (Peru)
2015
Seismic Analysis of the Church of Kuño Tambo (Peru)
DECLARATION
Name:
Sara Zanotti
Email:
sarazanotti@alice.it
Title of the
Msc Dissertation:
Supervisor(s):
Nuno Mendes
Paulo B. Lourenço
Josè Veira de Lemos
Year:
2014-2015
I hereby declare that all information in this document has been obtained and presented in
accordance with academic rules and ethical conduct. I also declare that, as required by these rules
and conduct, I have fully cited and referenced all material and results that are not original to this
work.
I hereby declare that the MSc Consortium responsible for the Advanced Masters in Structural
Analysis of Monuments and Historical Constructions is allowed to store and make available
electronically the present MSc Dissertation.
University:
University of Minho
Date:
13/07/2014
Signature:
___________________________
Erasmus Mundus Programme
ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS
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This page is left blank on purpose.
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ACKNOWLEDGEMENTS
First, I would like to express my gratitude to my supervisor, PhD. Nuno Mendes, for the unvaluable
support from a personal and professional point of view, in encouraging my studies and guide me in
the work, sharing his knowledge and his point of view during the research.
I would like to thank Prof. Lemos for share with me his deep knowledge on Discrete Element
Approach and offer me a prompt answer whenever I encountered difficulties in my work; I would
never reach these results without his constant suggestions. A gratitude to Prof. Lourenço for giving
me the chance to be part of the project that gave me the opportunity to enrich my knowledge.
I’m grateful to the Itasca Consulting Group for accepting my application and the project proposal
for the Itasca Educational Partnership: it was an incredible opportunity for me to learn the discrete
element approach applied to historical construction.
I would like to thank the Peruvian Seismic Retrofitting Project to get me the chance to approach the
seismic analysis of an historical adobe construction and give me the opportunity to be part of this
unique project.
I’m grateful to Eng. Giorgos Karanikoloudis and Eng. Federica Greco for their support and the
interesting discussion about Kuño Tambo Church. Thank you to all my SAHC collegues for the
experience shared, especially to my flatemates Maria and Alessio for the enjoyable time shared and
their support even in difficult times.
I could not be here without the support of my parents and friends, always ready to give me moments
of relieves and talks, in particular to Cristina and Peter. I will be always grateful to Gianni and
Maria for their unconditional love, even far away; you will be always part of my life.
Thanks especially to Diego, for his patience and his constant presence.
Lastly, I express my gratitude to the Erasmus Mundus Consortium for the financial support.
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ABSTRACT
Adobe constructions have presented high vulnerability to seismic actions and advanced studies are
required to assess their behaviour and design proper strengthening techniques. The adobe structures
present several challenges, such as the low mechanical properties and their high variability, mainly
for historical constructions. The Discrete Element Method (DEM) can represent as an appropriate
technique for these structures, due to its ability to study the typical large displacement of adobe
construction and be able to represent the weak joints between units and mortar. Furthermore, few
studies are available in the literature for large and complex structures due to the high computational
effort required by DEM.
A study on the seismic vulnerability of the Kuño Tambo Church in Cuzco (Peru) based on the DEM
was carried out. Pushover analysis were performed with the software 3DEC, supported by the
programming language FISH, for the implementation of user defined variables and functions, the
UDEC and a mathematical software for the implementation of the Voronoi algorithm for rubble
stone masonry.
The study involved a sensitivity study on the out-of-plane behaviour of the most deformable wall as
identified through preliminary analysis based on the Finite Element Method. The model of the wall
was prepared with unitary length, similar to the plain stress formulation. In the sensitivity analysis,
several parameters and aspects were evaluated, namely: (a) block deformability; (b) influence of
adobe pattern; (c) influence of the representation of rubble stone masonry; (d) influence of the
inelastic properties of the materials. Thus, a comprehensive literature review on the mechanical
properties of the materials was carried out to overcome limitations due to their uncertainty in
historical constructions. Furthermore, the analysis was compared with the limit analysis based on
the kinematic approach, aiming at validate the results obtained and to evaluate the difference
between of results obtained from a complex model with DEM and the results obtained from a
simplified analysis.
Based on the parametric analysis performed on the previous studies, the overturning of the façade of
Kuño Tambo Church was analysed. The out-of-plane collapse mechanism was studied due to the
crack pattern observed in situ and since it is one of the most commonly failure mechanisms
observed. The corner joint between the façade and the transversal walls was also analysed. Thus, a
3D detailed model involving a large number of blocks was created, aiming at evaluating the
response of the façade built with massive walls. The model was calibrated based on dynamic
identification tests.
The analysis showed that DEM is powerful tool for evaluating the collapse mechanisms of the
structure, which is confirmed by the similar crack pattern observed in situ. The values of the
ultimate load are in accordance with the limit analysis performed. The results of the DEM analysis
are highly sensitive to the type and the size of the blocks for large and complex structures. Thus, the
blocks should be carefully modelled in order to simulate all the possible collapse mechanism
involved in the dynamic behaviour of the structure. The simplification of the adobe unit using rigid
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blocks instead of deformable blocks provides a satisfactory representation. Furthermore, this
simplification can be convenient for the dynamic analysis and simulation of large and complex
structures. Concerning the collapse mechanism of the facade, the results showed that the
consideration of inelastic joint with Coulomb friction behaviour, instead simplified elastic
connection, is needed for a more conservative prediction of the out-of-plane collapse of the façade.
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RESUMO
Os edifícios de alvenaria de adobe apresentam elevada vulnerabilidade sísmica, sendo necessário
desenvolver estudos avançados para avaliar o seu comportamento dinâmico e para o
dimensionamento de técnicas de reforço adequadas. Um dos principais desafios dos edifícios de
adobe está associado aos valores reduzidos das suas propriedades mecânicas e à sua dispersão,
sobretudo em construções históricas. O Método dos Elementos Discretos (MED) pode-se apresentar
como uma técnica apropriada para este tipo de estruturas, uma vez que permite estudar os grandes
deslocamentos típicos dos edifícios de adobe e permite representar as juntas com baixa resistência
entre unidades e argamassa. Além disso, a bibliografia apresenta poucos estudos de estruturas
complexas e de grandes dimensões, devido ao esforço computacional exigido pelo MED.
Este trabalho apresenta um estudo sobre a vulnerabilidade sísmica da Igreja de Kuño Tambo em
Cuzco (Peru), baseado no MED. A análise não-linear estática foi realizada com recurso ao
programa de cálculo automático 3DEC, o qual é suportado pela linguagem de programação FISH,
para implementação das variáveis e funções definidas pelo usuário, ao UDEC e a programas de
matemàtica para implementação do algoritmo de Voronoi para a alvenaria irregular de pedra.
O estudo envolveu a análise de sensibilidade sobre o comportamento para fora do plano da parede
mais vulnerável, que foi identificada através de um estudo preliminar baseado no Método dos
Elementos Finitos. O modelo da parede foi preparado com largura unitária, semelhante à
formulação de estado plano de tensão. Vários parâmetros e aspetos foram analisados na análise de
sensibilidade, nomeadamente: (a) a deformabilidade dos blocos; (b) a influência do aparelho da
alvenaria de adobe; (c) a influência da representação da alvenaria irregular de pedra; (d) a influência
das propriedades inelásticas dos materiais. Assim, foi efetuada a ampla revisão bibliográfica sobre
as propriedades mecânicas, tendo por objetivo ter em consideração as incertezas existentes nas
construções históricas. Além disso, a análise foi comparada com a análise limite baseada na
abordagem cinemática, tendo por objetivo validar os resultados obtidos e avaliar as diferenças entre
os resultados obtido através de um modelo complexo desenvolvido com base em MED e os
resultados obtidos através de uma análise simplificada.
Foi ainda realizado um estudo sobre o comportamento para fora do plano da fachada da Igreja de
Kuño Tambo, tendo em consideração a análise de sensibilidade efetuada anteriormente. O
mecanismo de colapso para fora do plano da fachada foi analisado devido ao padrão de fendilhação
observado no local e uma vez que este tipo de mecanismo corresponde ao típico mecanismo de
colapso observado para este tipo de estruturas. Foi também estudada a junta do cunhal entre a
fachada e a parede ortogonal. Por último, foi preparado um modelo DEM 3D detalhado envolvendo
um número significativo de blocos, tendo por objetivo avaliar a resposta da fachada construída com
paredes massivas. O modelo foi calibrado de acordo com as propriedades dinâmicas estimadas
através de ensaios de identificação dinâmica.
As análises demonstraram que o MED é uma ferramenta crucial para a avaliar os mecanismos de
colapso da estrutura, o que é confirmado pelo padrão de fendilhação observado no local. Os valores
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da carga de colapso encontram-se de acordo com o valor obtido através da análise limite. Os
resultados das análises MED são muito sensíveis ao tipo e dimensões dos blocos para estruturas
complexas e de grandes dimensões. Assim, os blocos devem ser cuidadosamente modelados,
permitindo a simulação de todos os mecanismos de colapso englobados no comportamento
dinâmico da estrutura. A simplificação das unidades de adobe utilizando blocos rígidos, ao invés
dos blocos deformáveis, apresenta-se como uma simplificação aceitável. Além disso, esta
simplificação pode ser conveniente para a análise dinâmica e para análise de estruturas complexas.
No que diz respeito ao mecanismo de colapso da fachada, os resultados demonstraram que a
consideração do comportamento inelástico da junta com a lei de atrito de Coulomb, ao invés da
ligação elástica simplificada, é necessária para uma previsão mais conservadora do mecanismo para
fora do plano da fachada.
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RIASSUNTO
Le costruzioni in adobe presentano una significativa vulnerabilità alle azioni sismiche e uno studio
specifico a riguardo è richiesto al fine di stabilire il loro comportamento e progettare
appropriatamente le soluzioni di rinforzo adatte. Le maggiori difficoltà relative alla costruzione in
adobe riguardano le scarse proprietà meccaniche e la loro elevata variabilità, specialmente per le
costruzioni storiche. Il Metodo agli Elementi Discreti (DEM) può rappresentare una tecnica
appropriata, in quanto in grado di rappresentare gli ampi spostamenti che caratterizzano il
comportamento delle costruzioni in adobe, nonché le caratteristiche meccaniche dell’interfaccia
malta-mattone. Inoltre, pochi studi sono a disposizione riguardanti lo studio di strutture complesse a
causa del considerevole sforzo computazionale che viene richiesto nell’applicare il metodo DEM.
Uno studio sulla fattibilità dell’applicazione del metodo DEM per grandi strutture in adobe viene
qui presentato, dove il caso studio della Chiesa di Kuño Tambo in Cuzco (Perù) viene analizzato.
Analisi pushover vengono condotte mediante l’utilizzo del software 3DEC, supportato dal
linguaggio di programmazione FISH, per l’implementazione di funzioni definite dall’utente, nonché
del software UDEC and di un software matematico per l’implementazione dell’algoritmo Voronoi
per la modellazione di una muratura disorganizzata.
La ricerca condotta si è focalizzata su uno studio paramentrico del comprtamento fuori piano del
muro identificato, in analisi preliminare agli elementi finiti, come il più deformabile. Il muro è stato
modellato tenendo in considerazione una lunghezza unitaria (stato di sforzo piano9. Nell’analisi
parametrica, differenti fattori e aspetti furono valutati, quali: (a) deformabilità dei blocchi, (b)
influenza della disposizione dei mattoni in adobe, (c) influenza del metodo di modellazione della
muratura disorganizzata, (d) influenza delle proprietà inelastiche dei materiali. Pertanto, uno studio
bibliografico riguardante le proprietà meccaniche dei materiali fu affrontato per colmare le
incertezze relative alle proprietà meccaniche dei materiali. L’analisi fu poi confrontata con il
metodo dell’analisi limite con approccio cinematico per validare i risultati ottenuti e stabilire la
differenza in termini di risultati derivanti da un modello complesso (DEM) e da un’analisi
semplificata.
A seguito dell’analisi parametrica condotta, lo studio del comportamento fuori piano della facciata
della Chiesa di Kuño Tambo fu analizzato. La scelta dello studio di tale meccanismo è basata a
seguito dell’analisi del quadro fessurativo della Chiesa e considerando che il ribaltamento fuori
piano della facciata è uno dei meccanismi più comuni osservati nelle costruzioni in adobe a seguito
di un evento sismico. L’interfaccia tra la facciata e le murature trasversali viene analizzato. Infine,
un modello dettagliato basato sull’utilizzo di un considerevole numero di blocchi fu creato, al fine
di valutare l’influenza dell’organizzazione dei mattoni in adobe nella risposta sismica della facade
costitutita da una muratura massiva. Il modello fu calibrato basato su test di identificazione
dinamica condotti in situ.
L’analisi ha mostrato che DEM è applicabile per la valutazione del meccanismo di collasso della
struttura: che rappresenta il quadro fessurativo osservato in situ. I valori del carico ultimo sono in
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accordo con l’analisi limite condotta. I risultati dell’analisi DEM sono altamente influenzati dalla
scelta e della dimensione dei blocchi che costituiscono la rappresentazione di strutture complesse.
Pertanto, la scelta deve essere adeguatamente condotta al fine di non escludere alcun possibile
meccanismo di collasso. L’assunzione del mattone di adobe come rigido piuttosto che deformabile
può essere applicata, in modo da consentire di effettuare analisi dinamiche, non consentite se
blocchi deformabili vengono utilizzabili, e se complesse e grandi strutture vengono modellate con il
metodo DEM. Riguardo al meccanismo di collasso della facciata, i risultati hanno mostrato che
l’inserimento di un giunto inelastico (implementato mediante la legge Mohr-Coulomb) in
corrispondenza dell’intersezione di due pareti ortogonali è preferibile alla rappresentazione di un
giunto elastico, al fine di ottenere un risultato in termine di collasso ultimo più conservativo.
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TABLE OF CONTENTS
1
2
INTRODUCTION ....................................................................................................................... 1
1.1
Motivation ............................................................................................................................. 1
1.2
Scope of the thesis and objectives of the work ..................................................................... 3
THE ADOBE CONSTRUCTION IN PERU ............................................................................... 5
2.1
Adobe construction technique ............................................................................................... 5
2.2
Mechanical properties ........................................................................................................... 7
2.3
Collapse mechanism of unreinforced adobe buildings........................................................ 12
2.3.1
Damage survey............................................................................................................. 12
2.3.2
Experimental tests ........................................................................................................ 20
2.4
3
THE KUÑO TAMBO CHURCH ............................................................................................. 27
3.1
Historical overview ............................................................................................................. 27
3.2
Description Of the building ................................................................................................. 30
3.2.1
General description ...................................................................................................... 30
3.2.2
Interventions................................................................................................................. 37
3.2.3
Damage survey............................................................................................................. 40
3.3
Material properties .............................................................................................................. 47
3.3.1
Adobe masonry ............................................................................................................ 47
3.3.2
Rubble stone masonry .................................................................................................. 49
3.3.3
Adobe-stone interface .................................................................................................. 50
3.3.4
Timber elements ........................................................................................................... 50
3.3.5
Summary of the mechanical properties ........................................................................ 51
3.4
4.
Numerical modelling of adobe construction ....................................................................... 24
Dynamic identification test ................................................................................................. 52
3.4.1
Introduction .................................................................................................................. 52
3.4.2
Description of the test setup ......................................................................................... 53
3.4.3
Experimental results ..................................................................................................... 56
OUT-OF-PLANE BEHAVIOUR OF THE WESTERN WALL ............................................... 59
4.1
Introduction ......................................................................................................................... 59
4.2
The Discrete Element Approach ......................................................................................... 59
4.3
Description of the numerical model .................................................................................... 63
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4.3.1
Geometry of the portion of the building analysed ....................................................... 63
4.3.2
Geometry of the elements ............................................................................................ 63
4.3.3
Block and contact representation ................................................................................. 65
4.3.4
Material properties ....................................................................................................... 66
4.3.5
Application of the load of the roof ............................................................................... 68
4.4
Methodology ....................................................................................................................... 69
4.5
Results and discussion ......................................................................................................... 70
4.5.1
Out-of-plane behaviour of the western wall ................................................................ 70
4.5.2
Comparison between deformable and rigid blocks modelling approach ..................... 71
4.5.3
Influence of the adobe pattern on the results ............................................................... 73
4.5.4
Influence of the representation of rubble stone masonry ............................................. 75
4.5.5
Sensitivity analysis on the mechanical properties of the materials .............................. 77
4.6
5
Limit analysis ...................................................................................................................... 80
OUT-OF-PLANE BEHAVIOUR OF THE FAÇADE .............................................................. 84
5.1
Geometry of the model ........................................................................................................ 84
5.2
Material properties .............................................................................................................. 88
5.3
Load application .................................................................................................................. 89
5.4
Model calibration ................................................................................................................ 90
5.5
Pushover analysis ................................................................................................................ 93
5.5.1 Assessment of the inelastic properties assigned to the corner between orthogonal and
transversal walls ......................................................................................................................... 93
5.5.2
Methodology ................................................................................................................ 94
5.5.3
Results and discussion ................................................................................................. 95
6
CONCLUSION AND FUTURE WORK ................................................................................ 103
7
REFERENCES......................................................................................................................... 106
APPENDIX A: LIMIT ANALYSIS CALCULATIONS ................................................................ 113
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LIST OF FIGURES
Figure 1: (a) Huaca Pucclana, one of the earliest pyramidal construction (Cancino, 2009) and (b)
view of the archaeological remains of the city of Caral (Civilizaciòn de Caral, 2015) ....................... 1
Figure 2: (a) View of the remains of the city of Chan-Chan (2015) and (b) typical casona in Lima
(the first floor is built with adobe masonry and the second floor presents quincha walls (Cancino,
2009) .................................................................................................................................................... 2
Figure 3: View of the Iglesia de San Pedro de Carabayllo, built between 1571 and 1632 and
partially reconstructed after the 1687 and 1746 earthquake (Carabayllo, 2015) ................................. 2
Figure 4: (a) Adobe construction performed in Perù (Trek Earth-South America Photos, 2015) and
(b) adobe house in Lima (Blondet et al., 2011) ................................................................................... 5
Figure 5: (a) Erosion at the base that may have been caused by salts deposited by rising
groundwater and/or rainwater (Preservation of Historical Adobe Construction, 2015) (b) Structural
work at Mission Santa Cruz, where it is possible to observe the stone base course that isolates adobe
masonry from the soil (Sanchez, et al., 1990) ...................................................................................... 6
Figure 6: Construction of (a) rammed earth wall (Tarque, 2011) and (b) quincha panel (Bandeo,
2015) .................................................................................................................................................... 6
Figure 7: Diagram which relates Diagonal Compression Strength (D.C.S.) of adobe masonry and
compressive strength of adobe cubes of the six soil types tested (the size fraction analysis is shown
in the table) - (Vargas, et al., 1986) ..................................................................................................... 7
Figure 8: Diagonal compressive strength versus volumetric shrinkage of the brick (Vargas, et al.,
1986) .................................................................................................................................................... 8
Figure 9: Correlation between Young's Modulus and compressive strength of unreinforced adobe
bricks (Caporale et al., 2015) ............................................................................................................. 10
Figure 10: (a) Compression test on adobe prism; diagonal compression test on adobe wallets
(Tarque, 2011) (b) horizontally tested (Blondet and Vargas, 1978) and (c) vertically tested (Vargas
and Ottazzi, 1981) .............................................................................................................................. 11
Figure 11: The map with the location of the historical earthen heritage buildings surveyed in 2007
by the GCI team (Cancino, 2009) ...................................................................................................... 13
Figure 12: Damage observed in adobe buildings in case of (a) Roof supported by the façade and (b)
Roof supported by the transversal walls (Tarque, 2011) ................................................................... 14
Figure 13: Typical collapse mechanisms observed in adobe buildings after the Northridge
earthquake .......................................................................................................................................... 15
Figure 14: (a) Collapse of the entire gable-end wall and (b) Collapse of the upper portion of a gableend wall .............................................................................................................................................. 16
Figure 15: Main failure mechanism of adobe constructions according to Dowling (Dowling, 2004)
............................................................................................................................................................ 17
Figure 16: Damage pattern of adobe buildings located at the corner, after the 2007 Pisco earthquake
(Tarque, 2011) .................................................................................................................................... 17
Figure 17: Diagonal cracking on an unreinforced adobe wall due to in-plane shear (Blondet, et al.,
2008) .................................................................................................................................................. 18
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Figure 18: Detail showing (a) damage in the adobe walls at the Hacienda Arona y Montalvan
caused by beetle attack and (b) weak mortar adhesion strength at the Church of Coyallo, Canete, in
October 2007 (Cancino, 2009) ........................................................................................................... 19
Figure 19: Church of Chilca with different materials: a concrete frame with infill was observed in
October 2007 (Cancino, 2009) ........................................................................................................... 19
Figure 20: Adobe walls (a) with window opening (Blondet et al., 2005) and (b) without openings
(Blondet et al., 2008).......................................................................................................................... 20
Figure 21: Hysteretic curves and crack pattern of the adobe walls subjected to cyclic tests
(a) Wall 1, (b) Wall 2 and (c) Wall 3 (Blondet et al. 2005; 2008) ................................................... 21
Figure 22: Plan and elevations of the adobe structure subjected to dynamic test (Tarque, 2011) ..... 22
Figure 23: View of the adobe module before and after each phase in the dynamic test (Blondet et al,
2005; Tarque, 2011) ........................................................................................................................... 23
Figure 24: Modelling strategies for masonry structures: (a) Masonry sample; (b) Detailed micromodelling; .......................................................................................................................................... 24
Figure 25: Maximum tensile strains results of the model developed by Tarque (2011). .................. 25
Figure 26: Seismic behaviour of a typical vernacular adobe house modelled with the use of Discrete
Element Method (Furukawa, et al., 2010) ......................................................................................... 26
Figure 27: Location of the Kuño Tambo village (Images on Bing's website, 2015) ........................ 27
Figure 28: Overview of Kuño Tambo village (Kuño Tambo, 2015). ................................................ 28
Figure 29: (a) Ideal plan for an Andean reducciòn (from Juan de Matienzo, Gobierno del Perù,
1567) and (b) reduccion plan scheme based on Matienzo proposal (adapted from Juan de Matienzo,
Gobierno del Perù, 1567) ................................................................................................................... 29
Figure 30: (a) View of Kuño Tambo Church and its relationship with the Plaza (Cancino et al., 2012
); (b) the main road and the bell tower (Kuño Tambo, 2015) ............................................................ 29
Figure 31: Plan and section of Kuño Tambo Church (2015) ............................................................. 30
Figure 32: (a) Orthographic elevation of frescos of the South façade (b) view of the frescos located
on the inner wall of the façade (Percy, et al., 2013) .......................................................................... 31
Figure 33: View of the choir loft, constructed with wood (Cancino et al., 2012 ) ............................ 31
Figure 34: View and schematic representation of the rubble stone foundation – East wall (Cancino
et al., 2012, drawing Mirna Soto, for the GCI) .................................................................................. 32
Figure 35: (a) View of the façade (southern wall) and (b) view of the northern wall ....................... 33
Figure 36: View of the lateral wall (a) from the interior and (b) from the exterior (Kuño Tambo,
2015) .................................................................................................................................................. 33
Figure 37: View of (a) the baptistery (south view) and (b) sacristy (north view) ............................. 34
Figure 38: Isometric drawing of the baptistery, which shows the lack of connection between the
baptistery and the eastern wall of the main nave (Cancino et al., 2012) ........................................... 34
Figure 39: Plan and overview of the connection typology ................................................................ 35
Figure 40: (a) View of the roof of the main nave (Cancino et al., 2012 ); (b) Half-lap joint at the
intersection of roof rafters (Cancino et al., 2012 ) ............................................................................. 36
Figure 41: Lack of connection between (a) the southern wall of the baptistery and the eastern wall
of the main nave and (b) the northern wall of the baptistery and the eastern wall of the main nave 37
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Figure 42: Alteration of the walls: (a) an infilled door on the main façade; (b) infilled window along
the northern wall ................................................................................................................................ 38
Figure 43: a) The Church of Rondòcan, in which it is possible to observe the similar design with
Kuño Tambo Church (www.maps.google.it); b) Intact quincha arch located in the interior of the
Church of Rondòcan (Cancino et al., 2012 )...................................................................................... 38
Figure 44: (a) Remains of the quincha arch in Kuño Tambo Church (Cancino et al., 2012 ); (b)
Location (in red) of the quincha arch in Kuño Tambo, with the function to separate the presbytery
from the nave (Ph.D. Eng. G. Karanikoloudis drawings, adapted).................................................... 39
Figure 45: (a) 3D view of Kuño Tambo Church with the location of the tie beams (Ph.D. Eng. G.
Karanikoloudis drawings) and (b) Wood cross anchors at the south end of the west lateral wall ... 39
Figure 46: Loss of vertical mortar joints in some portion of the adobe wall and local loss of adobe
units: (a) Current state of the roof along the main nave, protected by a provisional structure
(Cancino et al., 2012); (b) The loss of adobe on the top of the wall has led to a local failure in the
connection between the wall and the rafter. A wood post is installed to temporarily support the
rafter end of the roof. ......................................................................................................................... 40
Figure 47: Vertical cracking at the corner where the southern façade intersect the lateral walls:
(a) south-eastern corner, from interior; (b) from exterior; (c) south-western corner (interior view) . 41
Figure 48: Crack pattern observed along the baptistery: (a) southern wall; (b) eastern wall;
(c) northern wall ................................................................................................................................ 42
Figure 49: Main damages observed at the western wall of Kuño Tambo Church ............................. 43
Figure 50: Main damages observed at the eastern wall of Kuño Tambo Church .............................. 44
Figure 51: Main damages observed at the southern wall of Kuño Tambo Church ........................... 45
Figure 52: Main damages observed at the northern wall of Kuño Tambo Church ............................ 46
Figure 53: Granulometric curve for: a) Historical adobe units, b) Historical mortars. (SRP, 2014) . 48
Figure 54: Shear-compression tests on adobe-adobe triplets: (a) View of the test performed;
(b) Shear-compression curve (SRP, 2014) ........................................................................................ 48
Figure 55: Shear-compression tests on adobe-adobe triplets: (a) View of the test performed;
(b) Shear-compression curve (SRP, 2014) ........................................................................................ 50
Figure 56: The Output-only identification technique (Ramos, 2007) ................................................ 52
Figure 57: Classification of output-only identification methods (Ramos, 2007) .............................. 53
Figure 58: View of the accelerometers positioned: (a) Reference sensor, positioned at the bottom of
the west wall, between the infilled opening and the quincha arch pillar; (b) Sensor 2 for setup 3,
positioned on the façade, at the south-east corner of the window; (c) Sensor 4, for setup 1,
positioned on the gable end roof of the northern wall ....................................................................... 54
Figure 59: Dynamic identification setup 1 performed in Kuño Tambo Church (in red is highlighted
the reference sensor) .......................................................................................................................... 54
Figure 62: Stabilization diagram: (a) Stochastic Subspace Identification Method; (b) Enhanced
Frequency Decomposition Domain Method ...................................................................................... 56
xv
Figure 63: Experimental mode shapes of the first four modes of the structure (SSI Method) .......... 58
Figure 64: Application to discrete element method to large and complex structures: (a) Geometry
and first mode shape of a wall, (b) Torre de Relógio in Azores (Lemos, 2000). .............................. 61
Figure 65: Application to discrete element method to model rubble masonry specimen tested in
laboratory (Milosevic et al., 2012) ..................................................................................................... 62
Figure 66: Application to discrete element method to model rubble masonry: (a) Masonry arch
bridge (Bicanic et al., 2002), (b) Out-of-plane behaviour of a masonry wall (Lemos, 2009) ........... 62
Figure 67: Identification and geometry of the model ........................................................................ 63
Figure 68: Modelling of adobe masonry wall: (a) Geometry of the masonry; .................................. 64
Figure 69: Modelling of rubble stone masonry: (a) Geometry of the rubble stone masonry base
course; (b) Implementation of the Voronoi polygons considering as data set of points the centroid of
the rubble stone that constitutes the masonry .................................................................................... 65
Figure 70: Geometry of the model (a) Model 1 with rigid block model, (b) Model 2 with deformable
block model ........................................................................................................................................ 66
Figure 71: Interface model code (Idris, et al., 2009).......................................................................... 67
Figure 72: Application of the roof load: (a) Vertical load applied as additional mass density; (b)
Horizontal load ................................................................................................................................... 68
Figure 73: Out-of-plane behaviour of the western wall: (a) Capacity curves; (b) Collapse
mechanisms ........................................................................................................................................ 70
Figure 74: Displacement history – Model 2 (Rigid blocks model).................................................... 71
Figure 75: Displacement history – Model 1 (Deformable blocks model) ......................................... 71
Figure 76: Pushover capacity curves for Model 1 (deformable blocks model) and Model 2 (rigid
blocks model) and collapse mechanism (magnification factor adopted for deformable blocks is
equal 15 and for rigid blocks is 10).................................................................................................... 72
Figure 77: Adobe masonry pattern of the western wall: a) Pattern observed in situ; b) Structural
prospection (Cancino et al., 2012 ) .................................................................................................... 73
Figure 78: Geometry of the three possible masonry pattern .............................................................. 73
Figure 79: Pushover capacity curve – Sensitivity analysis based on the influence of adobe masonry
pattern and collapse mechanism (displacement magnification factor adopted is equal to 10) .......... 74
Figure 80: Crack pattern of the wall, considering the three different adobe masonry patterns ......... 74
Figure 81: Adoption of simplified approach for the implementation of rubble masonry in large and
complex structures: (a) Adoption of a regular block pattern with bed joints; (b) Construction of
random Voronoi polygons (average length 0.40 m) generated with UDEC software ....................... 75
Figure 82: Horizontal force vs. displacement curves for regular and Voronoi block patterns (Lemos
et al., 2009)......................................................................................................................................... 75
Figure 83: Sensitivity analysis based on different rubble stone masonry modelling approach ......... 76
Figure 84: Sensitivity analysis for cohesion adobe units ................................................................... 78
Figure 85: Sensitivity analysis for the friction angle value ............................................................... 78
Figure 86: Sensitivity analysis for cohesion of stone ........................................................................ 79
Figure 87: Sensitivity analysis for friction angle of stone ................................................................. 80
Figure 88: Collapse mechanisms assumed in the limit analysis ........................................................ 81
xvi
Figure 89: Fracture line assumed for the fracture line method. The fracture line was divided
assuming the different material properties ......................................................................................... 82
Figure 90: Geometry of the partial model of the façade .................................................................... 84
Figure 91: (a) Lack of connection between the baptistery and the eastern wall; (b) Crack along the
southern wall of the baptistery. The crack does not follow the adobe step pattern ........................... 85
Figure 92: Geometry of the model ..................................................................................................... 86
Figure 93: Voronoi polygons created in UDEC software to represent the rubble masonry stone
course of the western wall .................................................................................................................. 87
Figure 94: Crack observed on the southern wall of the baptistery. The configuration does not follow
the step pattern of the mud mortar ..................................................................................................... 87
Figure 95: Geometry of the model. In red are indicated the rigid blocks that act as architraves ...... 88
Figure 96: Application of the dead load from the roof and definition of the surface region ............. 90
Figure 97: System for the definition of the system of equation ......................................................... 90
Figure 98: First mode shape of the structure in the out-of-plane direction of the façade .................. 92
Figure 99: Construction corner joint between the façade and the transversal walls .......................... 93
Figure 100: Reference point assumed for the analysis ...................................................................... 94
Figure 101: Capacity curves for the outward mechanism of the façade ............................................ 95
Figure 102: Collapse mechanism for the Model 1 ............................................................................. 96
Figure 103: Collapse mechanism for the Model 3 ............................................................................. 96
Figure 104: Vertical cracks at the corner between transversal walls and the façade ......................... 96
Figure 105: Damage pattern observed: (a) Y-displacement of the Model 2 (0.34 g), (b) Damage
survey of the building ........................................................................................................................ 97
Figure 106: Detail of the damage at the corner of the Model 2 (0.34 g), .......................................... 97
Figure 107: X-displacement of the Model 2 (seismic coefficient 0.34 g) ......................................... 98
Figure 108: Collapse mechanism of the façade with the assumption of elastic connection at the
intersection ......................................................................................................................................... 98
Figure 109: (a) Simulation of the stiffness of the whole wall (western wall); (b) Deformation of the
western wall with the insertion of the elastic springs ........................................................................ 99
Figure 110: Pushover analysis with elastic joints at the corner transversal walls-façade –
Comparison of the results ................................................................................................................ 100
Figure 111: Collapse mechanism o the Model n. 2 with elastic springs .......................................... 100
Figure 112: Capacity curves for the inward mechanism of the façade ............................................ 101
Figure 113: Collapse mechanism for the (a) Model 1; (b) Model 3 ................................................ 102
Figure 114: Collapse mechanism for the inward collapse of the façade of the Model 2 ................. 102
Figure 115: Ongoing construction of the full model adopting 3DEC software ............................... 104
xvii
xviii
LIST OF TABLES
Table 1: Experimental data available in literature regarding the compressive strength, the Young’s
Modulus and the tensile strength of the adobe units. ........................................................................... 9
Table 2: Experimental data available in literature regarding the compressive strength, the Modulus
of Elasticity and the tensile strength of the adobe masonry ............................................................... 12
Table 3: Results from compression test of adobe units in Kuño Tambo Church, Peru (SRP, 2014) 47
Table 4: Experimental results from shear-compression tests of adobe-adobe triplets (SRP, 2014) .. 48
Table 5: Experimental data available in literature regarding the compressive strength, the Modulus
of Elasticity and the tensile strength of the stone units ...................................................................... 49
Table 6: Experimental data available in literature regarding the inelastic properties of stone-stone
interface .............................................................................................................................................. 49
Table 7: Experimental results from shear-compression tests of adobe-stone triplets ........................ 50
Table 8: Summary of the elastic properties of the materials.............................................................. 51
Table 9: Summary of the inelastic properties of the materials .......................................................... 51
Table 10: Frequencies and damping ratio of the first four modal shapes – SSI Method ................... 57
Table 11: Frequencies and damping ratios of the first four mode shapes – EFDD Method .............. 57
Table 12: MAC for the SI Method and EFDD Method ..................................................................... 57
Table 13: Elastic properties of the deformable block model ............................................................. 67
Table 14: Elastic properties of the rigid block model ........................................................................ 67
Table 15: Inelastic properties of the deformable and rigid models .................................................... 68
Table 16: Cohesion and friction angle values adopted during the sensitivity analysis...................... 77
Table 17: Cohesion and friction angle values adopted during the sensitivity analysis...................... 79
Table 18: Comparison of the numerical methods adopted – Overturning of the western wall in the
direction X+ ....................................................................................................................................... 82
Table 19: Comparison of the numerical methods adopted – Overturning of the western wall in the
direction X- ........................................................................................................................................ 82
Table 20: Elastic properties for rigid block model ............................................................................ 88
Table 21: Inelastic properties for the rigid block model .................................................................... 89
Table 22: Vertical load applied to the model to represent the dead load received from the roof
system................................................................................................................................................. 89
Table 23: Definition of the upper and the lower bound of the variable (keq) chosen to (Douglas, et
al., 1982) perform the calibration....................................................................................................... 91
Table 24: Elastic properties of the model after the calibration .......................................................... 91
Table 25: Calculation of the modulus of elasticity after the calibration of the model ....................... 92
Table 26: Ultimate load capacity for the outward overturning of the façade .................................... 95
Table 27: Ultimate load capacity for the outward overturning of the façade and elastic corner ....... 99
Table 28: Ultimate load capacity for the inward overturning of the façade .................................... 101
xix
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xx
1 INTRODUCTION
1.1 Motivation
Adobe construction is the traditional technique used in the coastal areas of Peru throughout its
history, mainly due the availability of the soil which constitutes the adobe blocks and the
affordability of the material. Adobe construction was largely used along the Peruvian coastal areas
both in dry weather condition, where earth was mixed with cane reeds and straws, and in rainy
weather condition, where stones were used as foundation to isolate the adobe masonry from the soil
(Argumedo, 1990).
Since 2000 BCE, complex buildings were constructed in Peru with adobe technique. One of the first
example of the earth used for complex construction techniques was the construction of the pyramids
(Figure 1a), erected between 2000 BCE and 500 CE. Most of them were built up with adobe for the
exterior, earth and stone for the interior; the base of the building was constituted by rocky material
(Williams, 1982). In the city of Caral (Figure 1b), located at 200 km north of Lima and one of the
oldest city in America (3000 BCE), the adobe construction is combined with the use of other two
techniques (rammed earth and quincha (wattle-and-daub)), with the addition of stone masonry for
the construction of the pyramids (Tarque, 2011).
(a)
(b)
Figure 1: (a) Huaca Pucclana, one of the earliest pyramidal construction (Cancino, 2009) and (b) view of the
archaeological remains of the city of Caral (Civilizaciòn de Caral, 2015)
Between 100 CE and 800 CE, during then Mochica civilization, adobe blocks were widely used for
both ceremonial buildings and residential housing. Remains of two huge temples are visible
nowadays, namely the Huaca del Sol temple and Huaca de la Luna temple. An example of the
extensive use of earth material in Perù in ancient time is Chan-Chan (Figure 2a), one of the
history’s earliest city built by the Chimus around 850 CE. Nowadays, archaeological remains of the
city, which lasted till the conquest of the Inca Empire, are visible nowadays. The city was built with
adobe brick smoothed on the surface in order to allow the carving of intricate drawings. The
residential housing foundation were built in stone, combined with the adobe construction
techniques, and the quincha walls present a limited number of openings (Williams, 1982).
However, the use of quincha walls was particularly used during the Spanish Viceroyalty (1535-
1
1821). Architectural styles and construction techniques changed due to the European influence, but
adobe was continuously used, since it was a material largely adopted even by the Spanish
civilization. In the religious buildings constructed during the Spanish colonialism, the use of
quincha construction technique was applied for the realization of domes, pillars, lanterns and vaults
(Cancino, 2009), in which wooden trusses and ring beams were connected by cane reed across the
roofing system (domes, vaults or flat roof). The roof of the churches, built in the colonial period, is
made of wooden trusses which are covered by cane reeds (Auxiliadora, 1990) which follows the
direction of the longitudinal axes of the vault, as in the case of Kuño Tambo Church. Therefore, the
Spaniards adopted the construction techniques developed by the Incas and the ancient Peruvian
cultures in order to build cathedrals, government palaces, estates (haciendas), and urban residences
(casonas), many of which are still standing in Peru and are part of the earthen building heritage of
the Country (Cancino, 2009).
(b)
(a)
Figure 2: (a) View of the remains of the city of Chan-Chan (2015) and (b) typical casona in Lima (the first floor is built
with adobe masonry and the second floor presents quincha walls (Cancino, 2009)
After the 1764 earthquake, only twenty-five houses of three thousand houses did not collapsed and
ten thousand deaths were registered (Cancino, 2009). As a consequence, technical recommendation
were reported by D. Luis Gaudin, after the observation of the damages caused by the earthquake, in
order to improve the seismic resistance of the buildings. D. Luis Gaudin was a mathematician
involved in assessing recommendations to improve seismic resistance to earthquake In particular,
he suggested the use of mud, cane and adobe as construction materials instead of lime, brick and
walls, and the use of quincha walls, mainly for the construction of buildings composed by more
than one storey (Figure 2b). Moreover, thicker adobe walls were recommended and the addition of
lateral buttresses for the adobe churches.
(b)
(a)
Figure 3: View of the Iglesia de San Pedro de Carabayllo, built between 1571 and 1632 and partially reconstructed after
the 1687 and 1746 earthquake (Carabayllo, 2015)
2
The use of thicker walls, placed in the two principal direction of each house, was widely used for
the construction of earthen dwellings during the Colonial period. Some of them still exist nowadays
and have a slenderness ratio lower than 6 (Tarque, 2011). Furthermore, the rooms have symmetrical
distribution and similar dimension. Therefore, it is possible to observe more attention on the
construction procedure aimed to reduce the seismic vulnerability of the buildings. The presence of
adequate space or square to gather the population in case of disasters, the limitation about the height
of the towers and about the construction of bow windows were also recommended. Other
recommendations were developed during the Spanish Viceroyalty, mainly based on the damages
observed during the earthquake. For example, the dome of the Church of Santo Domingo in Lima,
constructed with the use of wood, cane and lime, was the only one that had survived without any
damages for the 1687 earthquake. After this event, the first document that regulates the construction
of earthen construction during the Spanish colonization recommended the use of wood, mud and
cane for the construction of the roofing structure.
Since the Republican Period, earthen dwellings increased. However, on 1868 (August 13), an
earthquake of magnitude 9.0 hit the coastline at the Peru-Chile border, followed by a tsunami. Most
of the construction built in adobe were completely destroyed. As a consequence, and after another
destructive earthquake in 1908, the state banned the construction of urban housing in adobe and
quincha, recommending the use of brick, masonry and reinforced concrete. However, earthen
construction were still very much used in the rural areas, due to the availability of the material and
the limited cost (Cancino, 2009). In 2008, the population that lived in earth dwellings was
approximately 43% (INEI, 2008).
Historical buildings made in earthen material play a great importance for the societies, presenting a
relevant importance not only for the historical significance but for the actual role that they play in
the Peruvian community (Fonseca Ferreira, et al., 2012). For this reason, a research project named
“Earthen Architecture Initiative – Seismic Retrofitting Project in Peru” (SRP) has been carried out
by Getty Conservation Institute with the aim to assess the seismic vulnerability of Peruvian
historical construction. Four prototypes buildings were selected by the SRP project as representative
for each typology. Kuño Tambo Church is one of the four buildings selected as representative of
historical Peruvian adobe construction. The interest was growing observing the involvement of the
community to restore the Church. Kuño Tambo was nominated as national monument for the
originality, authenticity and preservation of earthen structures and construction techniques (Cancino
et al., 2012).
1.2 Scope of the thesis and objectives of the work
The work of the thesis corresponds to the seismic analysis of Kuño Tambo Church based on the use
of Discrete Element Method (DEM). The DEM is a suitable method for the analysis of adobe
construction due to the possibility to simulate the weaker interface between adobe unit and mud
mortar, which characterize this type of construction, and the possibility to represent large
movements, that adobe masonry can undergo (Fonseca Ferreira, et al., 2012).
3
However, the use of discrete element was always applied to relatively small and regular structures.
The Kuño Tambo Church is a massive structure characterized by high complexity in terms of
geometrical configuration. As a consequence, the use of DEM applied to large structures requires
the use of macroblocks that are no equal of the real geometry of the structural constituents of the
walls. The reliability of the use of DEM is mainly characterized by the correct geometrical
representation of the constituents (or a reasonable simplification of the masonry pattern) and a
proper material characterization (Lemos, 2007). For this reason, Chapter 2 presents the assessing
the Peruvian adobe material properties and understanding the seismic behaviour of adobe
construction, in order to, on the view of large and complex structures, make the correct
simplification of the geometrical pattern. Consequently, in the Chapter 3 the characteristics of Kuño
Tambo Church, in terms of historical overview, architectural and structural configuration, damage
pattern and alteration are analysed. Furthermore, a dynamic identification test performed in May
2015 were used to estimate the dynamic properties of the structure.
Adobe material is characterized by a brittle behaviour: as a consequence, a small movement can
generate vertical cracks on the corner. The walls are separated from the transversal walls and the
main conditions that controls stability are the rocking behaviour and the slenderness of the wall
(Tarque, et al., 2012). Thus, the out-of-plane behaviour of the walls corresponds to an importance
aspect for the seismic analysis of adobe construction. For this reason, Chapter 4 presents the out-ofplane behaviour of the western wall, in order to assess the collapse mechanism and the influence
that the masonry pattern and the material properties has in terms of collapse mechanism and
ultimate load.
Finally, the Chapter 5 presents a study on the overturning of the main façade and its interaction of
the transversal walls to study the possible collapse mechanism of the façade, which is one of the
main principal collapse mechanism observed in Peruvian adobe construction. The choice to analyse
the overturning of the façade was also based on the damage pattern observed in Kuño Tambo
Church, namely the vertical cracks located between the façade and the transversal walls. A complex
model with the use of 6886 blocks was prepared with the aim to assess the influence of the adobe
masonry pattern on a massive structures and have a realistic representation of the damage pattern. A
first phase was necessary to assess the geometrical characteristic of the model. The discrete element
method was, in fact, rarely applied to large structures with blocks of small dimensions. The results
are mainly discussed in order to analyse the collapse mechanisms of the structure, the applicability
of the discrete element method for such large and complex adobe structures and the reliability of the
partial model constructed to study the overturning mechanism of the façade.
4
2 THE ADOBE CONSTRUCTION IN PERU
2.1 Adobe construction technique
Adobe is a Spanish term that derives from the Arabic world atob, which means mud brick dried in
the sun (Auxiliadora and Alvarenga, 1990), The adobe masonry is the denomination of the
construction technique composed by plain-earth brick and mud (or lime) mortar joints. The blocks,
made in plain earth, are sun-dried after they have been given shape, by hands or casted in suitable
wooden or steel forms. A plastic mixture of clay, sand and straw is compressed into them in order to
drive out the air. The moulding is usually made on the ground on a drying surface that can be
covered with straw (Houben, 2008). Clay is the most important component that characterizes the
soil used for adobe construction, since it can increase the strength and the adhesion between the
particles. On the other hand, clay causes drying shrinkage of the soil. For this reason, straw is
usually added in Peruvian adobe construction to control the micro cracking of the mortar. In
alternative to straw, other additives can be added (for example the addition of a coarse sand). The
presence of sand in the selection of the soil adopted is also important for the compressive strength
of the material (Auxiliadora, 1990). Adobe structures allow do-it-yourself construction and often
the blocks are made from local soil in a homeowner’s yard or nearby. Traditionally, in Peru, the
mud was left rest one day. The procedure consists in leaving the soil with water before preparing
the mud that was used to make mortar and bricks. This can improve the integration and the
distribution of water with the clay particles and enhance the cohesive properties (Blondet et al.,
2011).
(a)
(b)
Figure 4: (a) Adobe construction performed in Perù (Trek Earth-South America Photos, 2015) and (b) adobe house in
Lima (Blondet et al., 2011)
Adobe bricks have to be completely cured before their use. Mud is the most commonly material
used for mortar layers in adobe construction. Its use can result in a more homogenous final result.
The mud can be used similar to the one adopted for the bricks or with different proportion, in order
5
to make it more cohesive and workable. Adobe construction can be vulnerable to water erosion and
salt deterioration (Figure 5a). Thus and since ancient times, a provisional method was adopted to
avoid the phenomena. Before constructing the adobe masonry, stone, brick or wood can be used to
prevent the infiltration of soil water on the walls (Figure 5b). As external finishing, it is possible to
find a mud plaster or a mud mixed with small quantity of cement and lime in order to protect the
wall against the rain.
(b)
(a)
Figure 5: (a) Erosion at the base that may have been caused by salts deposited by rising groundwater and/or rainwater
(Preservation of Historical Adobe Construction, 2015) (b) Structural work at Mission Santa Cruz, where it is possible to
observe the stone base course that isolates adobe masonry from the soil (Sanchez, et al., 1990)
Similar to adobe, other two earth construction techniques are used in Latin America, namely tapial
(rammed earth) and quincha (wattle-and-daub). The constructions can be built combining both
techniques, as combining adobe and quincha walls. Rammed earth construction involves a mould
for the construction of a packed earth wall (Tarque, 2011). Quincha consists in a wooden panels
with cane reeds, which corresponds to a technique able to create an earthquake resistant framework.
(b)
(a)
Figure 6: Construction of (a) rammed earth wall (Tarque, 2011) and (b) quincha panel (Bandeo, 2015)
In general, earthen construction were used in Peru for more than thousand years. Till nowadays,
earthen construction is used in the rural areas, with a loss of the quality due to the lack of
6
experience of the workmanship and the changing in the architectural layout, which is progressively
more similar to the one used for the brick masonry architecture.
2.2 Mechanical properties
The low cost, the local availability and the possibility of self/owner-made with unskilled labour are
the main reasons why adobe construction are widely used in developing countries, such as Latin
America, Middle East, North and South of Africa. Additional advantages might be the good thermal
inertia and acoustic properties. On the other hand, adobe construction, especially the earthen
heritage buildings, are massive and can develop large inertia forces during an earthquake.
Additionally, the adobe is characterized by low tensile strength and brittle behaviour. Thus, the
construction can collapse without any warning due to its brittle behaviour. The seismic capacity of
the adobe construction was observed each time the earthquake occurred in region where earthen
construction are abundant, in which social and economic losses are recorded, as in the case of the
1970, 1996, 2001 and 2007 Peru earthquakes (Varum, et al., 2014).
The type of soil used for construction is influent on the values of the mechanical properties of the
material. For this reason, it is difficult to assess standard values regarding the mechanical
characteristic of the adobe masonry and its constituents. The type of the soil has a great influence on
the seismic performance of adobe masonry. Vargas et al. (1986) studied the characteristics of six
types of soil extracted from six zones where adobe construction is traditionally used in Peru.
Physical, chemical and mineralogical characteristics with the seismic strength of adobe masonry
made with these soils were analyzed. The tests demonstrate that the clay minerals present in the soil
are the most influent to the strength of adobe masonry, namely that higher is the content of clay,
higher is the dry compressive strength. However, concerning the diagonal compressive strength of
the masonry, it was observed that the presence of clay reduces the strength due to the high levels of
micro-cracking caused by drying shrinkage (Figure 7).
Figure 7: Diagram which relates Diagonal Compression Strength (D.C.S.) of adobe masonry and compressive strength
of adobe cubes of the six soil types tested (the size fraction analysis is shown in the table) - (Vargas, et al., 1986)
The degree of micro-cracking of the soil due to the drying shrinkage influences strongly the
strength of adobe masonry. In Figure 8, the inverse relation between volumetric change of adobe
bricks versus the diagonal compressive strength is presented. The rate of drying is even correlated
to the level of micro-cracking, showing that if the drying process is slow, less cracks can be
7
developed. For this reason, bricks dried in shadow can develop less fissures. In addition, the water
transfer between mortar and brick can occur very rapidly.
The quality of the workmanship can play an important role in the realization of strong adobe
masonry (Vargas, et al., 1986). For example, the strength can be improved by putting the adobe
brick in the water for 10 minutes before putting mortar or the use of drying mortar can reduce the
level of micro-cracking and improve the strength of adobe masonry.
Figure 8: Diagonal compressive strength versus volumetric shrinkage of the brick (Vargas, et al., 1986)
Table 1 summarizes the main experimental tests available in literature performed to obtain the
compressive strength, the modulus of elasticity and the tensile strength of the adobe units. In 1978,
twelve adobe brick units of different dimension (0.2x0.4x0.8 m and 0.3x0.6x0.08 m) were prepared
to obtain the axial compressive strength (Blondet, 1978). The bricks, dried under the sun for two
weeks, were made of clay soil mixed with straw and fabricated by only one person to reduce the
variability. The compressive strength estimated range between 1.20 to 1.80 MPa (average 1.44
MPa). Six specimens were tested after one year and six specimen after one month, but the value of
the compressive strength did not vary substantially over the time. The modulus of elasticity was not
evaluated, due to the brittleness of the material. It is noted that the adobe performs an elastic
behaviour for a very small range. As a consequence, it is difficult to estimate it accurately and
obtain reliable results. Later, Tarque (2011) estimated the value of 230 MPa for Young’s modulus
of adobe units.
As already highlighted by Silveira et al. (2013), high variability of results can be observed when the
testing adobe specimen are extracted by existing construction, even when the specimen are
extracted from the same building, due to the usual lack of production control. In Portugal, an
experimental campaign was conducted to evaluate the mechanical properties of existing adobe
construction in the region of Aveiro, selecting eight different houses. Cylindrical specimens were
extracted and tested under uniaxial compressive loads. The mean compressive strength range
between 0.66 MPa and 2.15 MPa (1.32 MPa is the global mean value). The high variability is, in
fact, expressed by the high standard deviation (Silveira, et al., 2013). High dispersion of results was
observed for the modulus of elasticity, which ranges between 87 and 448 MPa.
8
Table 1: Experimental data available in literature regarding the compressive strength, the Young’s Modulus and the
tensile strength of the adobe units.
Location
References
Compressive strength fcb
[MPa]
mean value
range
Blondet et al., 1978
1.48
1.2 - 1.8
SRP_UPCP, 2011
1.08
0.43 - 1.84
Silveira et al., 2012
0.95
0.66 - 2.15
0.58
0.23 - 1.02
Mechanical properties - Adobe units
Young´s Modulus
[MPa]
Perù
Silveira et al., 2013
185.94
51 - 448
0.19
0.12 - 0.4
13214
7609 - 25000 a
0.16
0.03 - 0.28
Bricks
Extr. in situ
Cubes
Extr. in situ
Cylinders
Extr. in situ
Cylinders
Extr. in situ
Cubes
F. in lab.
Cylinders
0.81 - 3.07
Varum, 2006(b)
1.1
147.3
F. in lab.
Piattoni, 2010
5.15
94
F. in lab.
Bricks
Extr. in situ
Cubes
F. in lab c
Cubes
Fratini et al, 2011
Illampas et al.,
2014
Bouhicha et al.,
2005
Gavrilovic et al.,
1998
Meli, 1980
Adorni et al., 2013
315
115 - 650
0.33-1.87
59
15.2- 87.2
0.72 -2.44
166
89-287.3
0.29-1.56
1.15
0.60 - 1.75
1.54
0.52 - 2.47
0.99
0.76 - 1.41
0.17-0.40
Extr. from
ind. Brick
Extr. from
ind. Brick
Extr. from
ind. Brick
4.10 - 5.10
1.18
0.27
1.00
0.10 - 0.30
Meli, 2005
Turkmenist
an
Not achieved due to the brittleness of
the material
F. in lab.
1.71
Liberatore et al.,
2006
Mexico
range
Varum, 2006 (a)
1.32
Algeria
mean value
Geometry of
the
specimen
0.28 - 1.21
0.50
Cyprus
Range
Unit
typology
0.54
Portugal
Italy
mean value
Tensile strength
[MPa]
0.51 - 1.57
0.94
0.24 - 1.33
1.10 - 2.50
F. in lab.
0.12 - 0.32
Extr. in situ
Cubes
Prisms
Cylinders
Bricks
0.20- 0.43
192.61
54.7 - 289.1
0.20
1) F. in lab = the specimen are built up in laboratory
2) Extr. in situ = the specimen are extracted from existing buildings
3) Extr. from ind. brick = the specimen are extracted from industrial adobe unit
a
High value due mainly to the methodology of calculation of E, where the deformation suffered by system and interfaces was not taken into account
b
the new bricks are obtained adopting the earth of the original bricks
The same heterogeneity was observed when uniaxial compressive test was performed on adobe
cubes extracted from historical earthen Peruvian prototype buildings. The compressive strength is
relatively constant for each site. However, the compressive strength on Ica Cathedral and Kuño
Tambo adobe units showed lower and scattered values in comparison to the mean value proposed
by Blondet (1978). The mean compressive strength for adobe units extracted respectively for Kuño
Tambo Church and Ica Cathedral were 0.71 MPa and 0.59 MPa. Several attempts were made to
calculate the modulus of elasticity. However, it was not possible due to the brittleness of the
material (SRP_UPCP, 2011).
9
Based on the experimental data available in literature, Caporale et al. (2015) derived, through robust
regression analysis, a correlation between the compressive Young’s Modulus, the compressive
strength and the tensile strength of the adobe units. In particular, the mean compressive Young’s
Modulus of unreinforced adobe bricks may be estimated through the compressive strength value as
follows:
with R2 equal to 0.91 (R2 corresponds to the coefficient of determination). An analogous correlation
was also implemented by Silveira et al. (2012) based only on units extracted from existing adobe
construction in Aveiro, Portugal:
Figure 9: Correlation between Young's Modulus and compressive strength of unreinforced adobe bricks (Caporale et
al., 2015)
The mechanical properties of the adobe masonry were also evaluated through a research conducted
by the Pontificia Universidad Catòlica del Perù, in which several tests were performed on adobe
masonry specimen with the aim to assess the mechanical properties of adobe construction (Blondet
and Vargas, 1978; Vargas and Ottazzi, 1981). Later, several tests were performed by the same
University with the aim to provide a material characterization of historical earthen prototypes in
Perù (SRP_UPCP, 2011).
In 1978, a total of 89 adobe prisms were built considering different slenderness ratio (1:1, 1:1.5,
1:2, 1:3, 1:4 and 1:5) and constructed with the use of mud mortar to estimate the compressive
strength of adobe masonry. Other 31 specimen were tested using mud mortar with the addition of
cement and gypsum. The compressive strength for prisms of slenderness 1:4 was evaluated between
0.8 and 1.2 MPa, depending on the specimen age. The results has shown that the variability of the
slenderness ratio doesn’t considerably affect too much the compressive strength value and similar
values were obtained (Blondet and Vargas, 1978; Vargas and Ottazzi, 1981). A modulus of
10
elasticity E, indirectly computed from full adobe wall tests, equal to 170 MPa was estimated. Other
authors, always referring to Peruvian adobe, suggests a value of 180 MPa (Ottazzi et al., 1989).
Lower and scattered values were obtained performing axial compression test on eight adobe piles
using existing adobe units from the original walls of the Peruvian historical earthen prototypes. The
value ranges between 0.46 MPa (Ica Cathedral) and 0.76 MPa (Casa Jr. Ancash). The modulus of
elasticity range between 48.7 MPa (Casa Welsch) and 106.4 (Casa Jr. Ancash). In all the specimen,
the type of failure was the typical vertical cracking running through the units and mortar
(SRP_UPCP, 2011).
Diagonal compression tests were carried out by Blondet and Vargas (1978) to obtain the tensile
strength ft of the adobe walls. Ten square wallettes of 0.6x0.6x0.2 m were built with the use of
0.2x0.4x0.08 m adobe bricks. The load was applied at the opposite corner of the wallettes and
diagonal deformation were measured to evaluate the shear modulus (G). Other seven panels were
tested by Vargas and Ottazzi (1981) with the same objective. The mean value of the tensile strength
obtained by Blondet and Vargas was 0.03 MPa and 0.026 MPa by Vargas and Ottazzi. The mean
value of the shear modulus estimated by Blondet and Vargas was 31.8 MPa and 39.8 MPa by
Vargas and Ottazzi.
(a)
(b)
(c)
Figure 10: (a) Compression test on adobe prism; diagonal compression test on adobe wallets (Tarque, 2011)
(b) horizontally tested (Blondet and Vargas, 1978) and (c) vertically tested (Vargas and Ottazzi, 1981)
The main results obtained are summarized in Table 2 and compared with the value obtained by the
research made at the University of Aveiro in Portugal (Varum, 2006a), at the Universidad de los
Andes in Colombia (Yamin et al., 2004) and at the Polytechnic University of Marche in Italy
(Quagliarini, 2010). As expected, the results obtained by tests performed from existing adobe
construction showed lower values in comparison to the ones available in literature and obtained by
specimen prepared in laboratory.
11
Table 2: Experimental data available in literature regarding the compressive strength, the Modulus of Elasticity and the
shear strength of the adobe masonry
Mechanical properties - Adobe masonry
Location
Perù
References
Blondet et al.,
1978
Vargas and
Ottazzi, 1981
SRP_UPCP,
2011
Colombia
Yamin et al.,
2004
Compression strength fcb
[MPa]
Young Modulus E
[MPa]
mean
Range
value
range
0.83
0.73 - 0.96
31.8
0.95
0.73 - 1.22
39.8
11 - 90.7
17.65
2.90 -49.16
0.39 - 0.76
48.7 - 106.4
.
30.5
Italy
Quagliarini et
al.,2010
1.2; 0.77
40; 26
Portugal
Varum, 2006(a)
1.16
Real scale masonry
c
Scale 1:5
mean
value
0.77 - 1.57
186.00
0.03
range
Shear Modulus G
[MPa]
mean
range
value
mean value
.58
a
Shear strength [MPa]
0.01 - 0.05
F. in lab.
Extr. in situ
a
0.03
18
0.0
18.7 c
0.11
Unit
typology
F. in lab.
0.06; 0.12
95 - 250
Density
[kN/m³]
F. in lab.
F. in lab.
F. in lab.
0.05 - 0.19
32
10 - 57
F. in lab.
2.3 Collapse mechanism of unreinforced adobe buildings
2.3.1
Damage survey
Adobe construction was widely used during all the history in Peru for the construction of both
residential and religious buildings. After the discovering that the wet earth can gets harder as soon
as it dries and has a high compressive strength, earth becomes to be used as a construction material.
The availability of the material in Peru combined with the economic aspect and its thermal
properties are the causes of the wide use of earth construction in that area (Tarque, 2011).
Unfortunately, Peru is one of the world’s most hazard-prone regions. The Pisco earthquake of
August 15, 2007, caused significant damages on the earthen buildings and on the historical heritage.
Damage surveys were conducted with the aim of better understand seismic damages and the
influence of external factors that can affect the seismic performance of the adobe structures
(Cancino, 2009).
Within two to six days after the Pisco Earthquake, a group pf researchers of the Pontificia
Universidade Catolica del Perù visited six cities, namely San Clemente, San Miguel, Pisco,
Guadalupe, Ica, Pachacutec (Blondet, et al., 2008). The survey was conducted observing both the
residential and the historical constructions. Regarding the adobe dwellings, differences in
construction techniques were observed during the survey comparing the old and the new buildings.
The oldest houses have a slenderness ratio less than 6 and quincha panels were used to create
independent environment or for the construction of the second floor, above a floor built in adobe
construction (the technique was introduced after the Gaudin’s recommendation, after the 44
12
earthquake). The modern adobe buildings observed has a high slenderness, absence of quincha
walls and interior rooms without a roof.
Figure 11: The map with the location of the historical earthen heritage buildings surveyed in 2007 by the GCI team
(Cancino, 2009)
Regarding the historical earthen heritage in Peru, the National Institute of Culture reported that,
after the 2007 earthquake, 32% of the historical and cultural monuments in Ica have completely
collapsed, 23% were under strong risk of collapsing, 26% were under moderate risk and only 19%
showed minor damage to the structure (Blondet, 2008). Getty Conservation Institute (GCI) team
conducted a survey on October 30 and 31, 2007, on significant historical earthen heritage buildings
in Peru which showed after the earthquake significant damages (Figure 11). The historic buildings
visited can be placed in two categories: ecclesiastical adobe buildings with quincha domes and
vaults (typical building typologies constructed during the Spanish Viceroyalty) and estates adobe
buildings with a courtyard and side chapel also with quincha vaults (Cancino, 2009).
Due to the usual weak connections between the façade and the orthogonal walls in adobe
constructions, the most observed damage is the overturning of the walls and the roof’s collapse.
During the surveys, it was observed that the first vertical cracks appear on the corner producing the
breaking of the adobe blocks and their falling. This can lead to the separation between the walls and
consequently to the out-of-plane collapse of the walls (Blondet, et al., 2008). The direction of the
roof has a decisive influence on the seismic vulnerability of the building. It was observed that if the
wooden joists were supported by the façade, the wall collapses and causes the out of balance of the
timber element. As a consequence, also the roof collapses (Figure 12a). On the other hand, if the
roof is perpendicular to the façade wall, the roof did not falls (Figure 12b).
13
(a)
(b)
Figure 12: Damage observed in adobe buildings in case of (a) Roof supported by the façade and (b) Roof supported by
the transversal walls (Tarque, 2011)
Tolles et al. (1996) described the main types of damage observed in adobe buildings after the
Northridge Earthquake in California (1994). Similar collapse mechanisms were later observed by
the GCI (Getty Conservation Institute) team in Peru during the damage survey in 2007. The extent
of the damage in the adobe masonry walls is summarized in 1996 by Tolles et al. (1996) as function
of:




The severity of the ground motion amplitude;
The geometry of the structure, i.e., the configuration of the adobe walls, roof, floors,
openings, and foundation systems;
The presence and the effectiveness of seismic retrofit measures;
The conservation state of the building at the time of the earthquake.
The individual block characteristics, the building location and building design, the quality of
construction and its maintenance influence the capacity of adobe constructions to withstand
earthquakes (Dowling, 2004). The geometrical characteristics of the walls influence also the seismic
response of the buildings. During the survey, it was observed that the thicker the walls are, easier it
is for the wall to remain stable. A proper connection between the transversal walls and the addition
of buttresses are characteristics that can help the structure to increase its seismic performance
(Cancino, 2009). Furthermore, the use of quincha walls can decrease the seismic vulnerability of
the structure. The wooden frame and the cane structure can resist to the tensile stresses. The
collapse can occurs when the wooden elements suffers of decay or they disconnected each other.
Furthermore, the canes cannot work structurally if they are extremely compressed so that they
cannot perform their structural function. In addition, the presence a good connection between roof
and adobe walls can significantly improve the overall performance of the building (Tolles et al.,
1996).
14
For example, the out-of-plane wall stability is influence by the main following factors (Tolles et al,
1996):






Slenderness of the wall. A very thick wall, with a slenderness ratio less than 3.5 is
recommended to prevent the overturning. However, the sliding can occur at the base before
the overturning;
Connection between the walls and the roof and/or the floor system. It was observed that
even if a bond beam or a partial plywood diaphragm is present, it can stabilize better the outof-plane behaviour;
Load bearing or not load bearing walls. The presence of a vertical load at the top of the
walls can stabilize the overturning by bearing down on the raised corner;
Distance between intersecting walls;
Condition at the base of the walls. In several adobe buildings, basal erosion or excessive
moisture content at the base was observed. The first condition is critical, since it reduces the
bearing area, while the second condition reduce the strength. Furthermore, a weaker adobe
can be observed if the wall at the base is subjected to continuous wet-dry cycles. The lack of
maintenance, as it will be discussed later, can be significant regarding the seismic
vulnerability of the building itself.
Mechanical characteristic of the adobe, which is material able to resist to compressive forces
but very weak to tensile forces. The stresses absorbed by the adobe walls during the
earthquake exceed the wall’s tensile strength. As a consequence, the building can dissipate
the energy released by the earthquake through crack formation. This divides the walls into
separated and isolated blocks that pound against each other until the structure collapses
(Cancino, 2009).
Figure 13: Typical collapse mechanisms observed in adobe buildings after the Northridge earthquake
(Tolles et al., 1996)
15
During the 2007 survey conducted by the GCI team in Peru, different behaviour were observed
between the ecclesiastic buildings and the estate-worker residences. The slender adobe walls of the
residential buildings were unable to resist to the rocking movement. For this reason, as soon as the
vertical crack appears and detaches the transversal walls, the walls collapse entirely due to
instability. The absence of an efficient intersection between the transversal walls and the lack of
anchorage of the roof joists to the top of the walls caused the damage on these kind of structures.
On the other hand, the ecclesiastic buildings are usually massive structures. For this reason, when
they move independently they can remain standing. However, damages along the quincha
ceiling/roof system are observed, which can be related to the independent movement of the
longitudinal nave walls, which cannot be hold and accommodated by the flexible roofing/ceiling
system.
One of the most common failure in churches occur between the towers and the main nave, when
they are connected each other. Another common damage observed was the separation between the
façade and the longitudinal walls, showing the lack of connection between the transversal walls
(Blondet, et al., 2008).
One of the most observed out-of-plane failure in the walls involves the gable-end wall, which is
usually tall and thin and not connected to the structure at the floor or at the roof levels. In addition,
gable end walls are not load bearing walls, favouring the overturning. In California, two main types
of collapse of the gable-end walls were observed in adobe buildings, namely the overturning of the
entire wall (Figure 14a) and the partial collapse of the upper level (Figure 14b). In other buildings,
only severe cracking without collapse were observed.
(a)
(b)
Figure 14: (a) Collapse of the entire gable-end wall and (b) Collapse of the upper portion of a gable-end wall
(Tolles et al., 1996)
The out-of-plane movement can occurs only the upper part of the wall panel (Figure 15c), due to
bending actions that causes the splitting and the crushing, generating vertical cracks in the upper
part of the wall (Dowling, 2004).
16
Figure 15: Main failure mechanism of adobe constructions according to Dowling (Dowling, 2004)
During the 1994 survey, no significant damage was observed due to the mid-height, out-of-plane
flexural damage because of the relatively low slenderness ratio that characterizes the adobe walls.
This collapse mechanism was observed only in the case of slender walls restrained by a bond beam
on top.
If the adobe building is located on the corner (Figure 16), a possible damage observed is the
presence of diagonal and vertical cracks or the collapse of both the façade and the roof. The
diagonal cracks extend from the top to the base forming a “v” crack pattern shape (Blondet, 008).
Figure 16: Damage pattern of adobe buildings located at the corner, after the 2007 Pisco earthquake (Tarque, 2011)
17
In terms of shear failure, diagonal cracks were observed (Figure 17), even if most of the walls not
subjected to overturning were able to withstand the earthquake’s in-plane actions (Blondet, et al.,
2008). Due to the weak properties of the mortar, the diagonal cracks follows the step patterns along
the joints. Due to the stress concentration at the opening corners, the diagonal cracks usually starts
at the corners of doors or windows.
Figure 17: Diagonal cracking on an unreinforced adobe wall due to in-plane shear (Blondet, et al., 2008)
The severity of the damage can increase if the seismic movement continues. The cracked part will
break in separate pieces and collapse independently in an out-of-plane overturning (Tarque, 2011).
The mechanism described are summarized by Dowling (2004) and is based on the damage pattern
observed after the severe earthquakes in 2001 in El Salvador.
The vulnerability of the adobe construction is not only influenced by the constructive technique
adopted (weak intersection of the walls, direction of the wooden joists), but even by the material
with whom the adobe masonry was built up. The adhesion observed between the adobe and the
mortar was poor and weak. Furthermore, the soil used for the preparation of the adobe blocks along
the coastal area was composed mainly by sand with a poor content of clay material, which is the
soil component that, with the interaction with the water, allow the particles to be chemically bond.
As a consequence, the material used for construction would easily crumble when scratched with a
nail (Blondet, 2008).
As already observed by Tolles et al. (1996) during the Northbridge damage survey in 1994, the lack
of maintenance of the adobe buildings has a relevant influence on their structural performance.
After the 2007 earthquake in Peru, the seismic behaviour was affected by the loss of structural
integrity and the lack of connection between the adobe walls and the roofing system, due to the
decay of the material which constitutes the ceiling system. At the church of Hacienda San José de
Nazca, damage in the wooden framework due to the termite was observed, which contributed to the
partial collapse of the building.
Damages caused by beetle attack and moisture on adobe walls can significantly reduce the strength
of the material, favouring the collapse of the wall deteriorated (Figure 18). Water causes damages
on the adobe walls not only reducing the strength of the material, but by eroding the portion of the
18
adobe wall. This can be due to the surface runoff or by water falling from the roof. Another source
of the presence of moisture can be the capillarity action that can diffuse water in the surface and
then evaporates. In the last case, the soluble salts conducted by the water can fracture the adobe, as
soon as the water evaporates and the salts crystallize along the surface. The strong difference
between adobe and other masonry material is the considerable strength reduction when the material
becomes wet. The repeated dry/wet cycles can also affects the strength of the material, due the
passage between the wet to dry state break the bonding between the particles, influencing the
structural performance of the building (Tolles et al., 2006).
(a)
(b)
Figure 18: Detail showing (a) damage in the adobe walls at the Hacienda Arona y Montalvan caused by beetle attack
and (b) weak mortar adhesion strength at the Church of Coyallo, Canete, in October 2007 (Cancino, 2009)
Furthermore, the earthen buildings of cultural significance experienced in Peru several earthquakes.
Some of the damages caused by the previous seismic events were not repaired. As a consequence,
the presence of unrepaired structural cracks and weak mortar decrease the structural performance of
the buildings, causing further cracking. Damages can become extensive even for moderate ground
shaking if the pre-existing cracks are present. As for all the masonry construction typologies, the
addition and the insertion of different building materials or an improper retrofitting technique can
significantly change the structural performance of the building (Figure 19). Reinforced concrete
frames with infill of adobe or fired brick had a significant effect on the seismic resistant of adobe
construction during the 2007 earthquake (Cancino, 2009).
Figure 19: Church of Chilca with different materials: a concrete frame with infill was observed in October 2007
(Cancino, 2009)
19
2.3.2
Experimental tests
After the May 31, 1970, earthquake, with an epicentre located near the coast of central Peru, an
experimental research conducted by the Pontificia Universidad Catòlica del Perù was carried out,
aiming at better understanding the seismic behaviour of Peruvian adobe constructions and study the
structural influence of possible retrofitting techniques. Cyclic and dynamic tests on unreinforced
adobe walls were performed.
Three I-shape unreinforced adobe walls were tested by Blondet et al. (2005) and Blondet et al.
(2008) and the results were discussed later by Thorpe (2011). The three walls (Wall 1, Wall 2, Wall
3) present the same geometrical characteristics. The main wall is 3.06 m long, 1.93 m high and 0.3
m thick. The Wall 1 presents a central 0.6x0.4 m window opening (Figure 20a) and the Wall 2 and
Wall 3 do not present openings.
(a)
(b)
Figure 20: Adobe walls (a) with window opening (Blondet et al., 2005) and (b) without openings (Blondet et al., 2008)
The adobe bricks used for the construction had dimensions 0.13x0.10x0.30 m and 0.13x0.10x0.22
m. The composition of the adobe bricks was soil, coarse sand and straw in proportion 5/1/1 and for
mud mortar 3/1/1. However, no information about the composition of the adobe blocks used for the
construction of the second and third walls is reported (Tarque, 2011). A reinforced concrete beam
was built at the base of the walls as a foundation, and on top with the objective to provide the selfweight of a typical Peruvian roof, constituted by wooden beams, cane, straw, mud and corrugated
zinc sheets (Tarque, 2011). The load was applied horizontally at the top concrete beam with a servo
hydraulic actuator. Steel and wooden plates were placed at the contact of the actuator with the
crown beam in order to avoid a concentrated load. The walls showed a similar x-shaped diagonal
cracks and horizontal cracks in the transversal walls. As expected, the two walls without openings
had larger initial stiffness and they resist 25% more in comparison to the one with opening. As
already observed in the previous subsection referring to the damages observed after the 2007 Pisco
earthquake, the diagonal cracks in the first wall appears at the corner of the window opening
(Figure 21).
20
(a)
(b)
(c)
Figure 21: Hysteretic curves and crack pattern of the adobe walls subjected to cyclic tests (a) Wall 1, (b) Wall 2 and
(c) Wall 3 (Blondet et al. 2005; 2008)
Blondet et al. (2005, 2006) performed a dynamic test of a typical vernacular adobe building, in
order to consider simultaneously the in-plane and out-of-plane actions in the walls. The seismic
demand is represented by a displacement signal applied at the base of the structure and
corresponding to the record of a Peruvian earthquake occurred in 1970 (Magnitude M W 7.9,
maximum intensity XI in MMI). The geometry is shown in Figure 22. The adobe bricks and the
mud mortar used for construction of the module had a soil/coarse sand/straw proportion of 5/1/1
and 3/1/1 and the adobe brick dimension were 0.25x0.25x0.07 m. A mud plaster with proportion
3/1/1 was applied externally. The lintel of the opening is made of cane rods and mud. The roof
corresponds to the traditional Peruvian wooden roof and its configuration does not allow the
realization of a rigid diaphragm. Four lateral wooden beams (two over the walls and that connects
the perpendicular walls) are connected to the walls only with mud and steel nails. Wooden joints
and clay tiles are placed on the beams.
21
Figure 22: Plan and elevations of the adobe structure subjected to dynamic test (Tarque, 2011)
(Dimensions in meters)
The adobe structure was subjected to three levels of displacement signal that corresponds to 0.3, 0.8
and 1.2 g. Each level represents respectively a frequent, moderate and severe earthquake on the
adobe buildings (Tarque, 2011). The response of the adobe module during each phase can be
summarized in:
-
-
-
First phase (0.3 g): Slightly diagonal and vertical cracks at the walls parallel to the seismic
action (right and left walls). The wooden beams start to loose connection with the walls but
no relative displacement was observed between the structural elements (Figure 23b);
Second phase (0.8 g): Complete vertical cracks appear at the wall corners. Thus, the walls
start to behave separately, in accordance with the damage pattern surveyed after the 2007
Pisco earthquake. Diagonal cracks were incremented in width and new cracks appear
(Figure 23c). In the perpendicular walls, cracks due to vertical and horizontal bending were
observed in front and rear wall with the starting of rocking of masonry blocks, which behave
independently. The roof is completely separated from the walls (Tarque, 2011);
Third phase: The perpendicular walls collapsed and the parallel walls, which supported the
roof, were completely cracked and became instable (Figure 23d).
The greatest relative displacement and total acceleration responses were mainly obtained at the
walls perpendicular to the movement, due to the rocking behaviour (Tarque, 2011).
22
(a) Adobe module before Phase 1
(b) Adobe module after Phase 1
(c) Adobe module after Phase 2
(d) Adobe module after Phase 3
Figure 23: View of the adobe module before and after each phase in the dynamic test (Blondet et al, 2005; Tarque, 2011)
The structural behaviour observed during the experimental test was in accordance with the damage
pattern already observed in adobe buildings caused by earthquake in the past. The behaviour is
influence by the low tensile strength of the material and by the absence of rigid diaphragm. As a
consequence, vertical cracks along the intersection of the walls appears and each wall can move
independently. The accelerometers and the displacement transducers placed on top of the walls have
recorded different measurements for each wall as soon as the vertical cracks appear. Another
important observation was the effect of the plaster on the seismic behaviour of the structure.
Torsional movement was observed due to the application of the plaster on the right wall, which
behaved stiffer and thicker in comparison to the left wall (Tarque, 2011).
23
2.4 Numerical modelling of adobe construction
In order to understand how the adobe structures behave under different level of ground motion, the
need to assess a reliable modelling approach is necessary in order to assess their seismic
vulnerability. However, only some works have been published regarding modelling of adobe
structures (Tarque, 2011).
Adobe, classified as an unreinforced masonry material, is a composite and non-homogeneus
material, formed by earth brick and mortar joints. As a consequence, the behaviour is affected by a
high level of complexity. Unreinforced masonry is an orthotropic material due to the presence of the
mortar layers, which are usually weaker and softer than the bricks. Regarding unreinforced masonry
structures, several methods are available for numerical analysis, for example limit analysis, finite
element method and discrete element method (Roca et al., 2010). Three main finite element
modelling approach can be used to analyse unreinforced masonry structures, depending on the level
of accuracy and simplicity required (Lourenco, 1996) namely:
-
-
Detailed micro-modelling: Units and mortar joints are represented as continuum elements
while the unit-mortar interface is represented as a discontinuum element;
Simplified micro-modelling: In this case the units are expanded including the geometrical
space of the mortar joints and represented by continuum elements, while the behaviour of
the mortar-joint and of the unit-mortar interface lumped into an average interface,
represented by a discontinuum element. As a consequence, the accuracy of the detailed
micro-modeling is lost because the Poisson ratio of the mortar cannot be taken into account.
The approach is similar to the discrete element method, introduced by Cundall & Strack
(1979) in the area of rock mechanics, and then applied even to masonry structures (as it is
presented in § 4.2);
Macro-modelling: Units, mortar and unit-mortar interface are smeared out in the continuum
and the masonry is treated as a homogenous and isotropic continuum material.
Figure 24: Modelling strategies for masonry structures: (a) Masonry sample; (b) Detailed micro-modelling;
(c) Simplified micro-modelling; (d) Macro-modelling (Lourenço, 1996)
The first approach is not only computational intensive, but it requires a detailed knowledge of the
material properties (elastic and inelastic), the correct disposition of units and joints, the anisotropy
24
and the geometry of the units. On the other hand, macro-modelling approach is less accurate but, at
the same time, it can considerably reduce the time consumed and allows the use of a user-friendly
mesh generation. It can be applied when the structure is composed by solid walls with sufficient
large dimension that the stress distribution can be considered uniform (Lourenço, 1996). The choice
of the modelling strategy to be adopted can be driven by several factors (i.e. complexity of the
structure, time consuming, characteristics of the material, accuracy of the results, and objective of
the analysis).
Adobe masonry is constituted by mortar and units made with the same material constituents. For
this reason, Tarque (2011) applied the macromodeling approach for adobe constructions. The study
was carried out in two phases. First, the material parameters not derivable from experimental results
were calibrated through using both micro-modelling and macro-modelling approaches, based on the
experimental results of an adobe walls subjected to cyclic tests (Blondet et al., 2005). In the second
phase, typical vernacular adobe building tested previously on a shaking table (Blondet et al., 2005)
with the macro-modelling approach was used. Good agreement with the experimental results in
terms of failure pattern and seismic capacity was obtained. However, the macro-modelling
approach, did not present the physical separation of the rigid blocks that allows an independent
rocking behaviour of the walls after the formation of the vertical cracks, as it was observed in the
experimental test and during the damage surveys (Figure 25). Furthermore, Tarque observed that,
even if the mud mortar and the adobe bricks are usually made with the same constituent materials,
the adobe bricks and the mud mortar has not usually of the same age of the bricks. As a
consequence, the dry process can generate weak zones between the bed and the head contact joints
(Tarque, 2011).
Figure 25: Maximum tensile strains results of the model developed by Tarque (2011).
Although the following numerical study was carried out for Iranian buildings, Furukawa et al.
(2010) studied the collapse of a typical Iranian adobe construction through the use of Discrete
Element Method (DEM), after observing that the main failure mechanism of adobe buildings is
mainly characterized by vertical separation of the walls that can be represented through the
25
separation of bricks interconnected by mortar layers. Furthermore, the representation of the
masonry as a an assemblage of distinct blocks (the adobe units) where joints are modelled as
contact surfaces between different blocks, affords the model of non-linear behaviour, including
large displacements both in static and dynamic ranges (Roca et al., 2010).
Due to the use of DEM, Furukawa et al. (2010) could correlate the outcome of the model with the
typical failure pattern observed in the damage investigation survey discussed in the literature
(Mahmoud et al., 2005; Zahrai and Heidarzadeh, 2007). In contrast with Tarque (2011), using the
DEM method was possible to observe in time the failure process of the building, including the
rocking behaviour of the wall consequent the their physical separation (Figure 26), assess a time
history of damage index and correlate it with the PGA (Peak Ground Acceleration). However, the
computational effort associated to this type of modelling approach, limit the number of blocky
elements. For this reasons, the analysis of complex adobe structures is not yet approached through
this method.
Figure 26: Seismic behaviour of a typical vernacular adobe house modelled with the use of Discrete Element Method
(Furukawa, et al., 2010)
26
3 THE KUÑO TAMBO CHURCH
3.1 Historical overview
The Kuño Tambo, also named as Iglesia de Santiago Apóstol, is the church of the Comunidad
Campesina Kuño Tambo, a rural village of 500 inhabitants that is located in the province of
Acomayo, at the southeast of the city of Cusco (Figure 27).
Figure 27: Location of the Kuño Tambo village (Images on Bing's website, 2015)
The Church was continuously used by the community since its construction in the seventeenth
century and it is an important gathering and religious place for the community members. For this
reason, although initially was not registered as a national monument at the beginning of the Seismic
Retrofitting Project directed by the Getty Conservation Institute in Los Angeles, the interest
demonstrated by the community to express the desire to restore the church has led to the Cusco
regional office of the Ministerio de Cultura del Perù to begin the process to nominate the entire
27
town, included the Kuño Tambo church, as national monument due to its originality, authenticity
and preservation of earthen structures and construction techniques (Cancino et al., 2012).
Kuño Tambo Church was in fact, like the entire village, constructed with adobe mud brick walls.
The first reference of the village of Kuño Tambo appeared in a document in 1577, where the four
suyos (the regions) of Cusco were cited: Cocno, the earliest name of Kuño Tambo, was listed as one
of the villages belonging to the suyo Condesuyo (Cancino et al., 2012).
It is one of the typical village constructed by the Spanish Viceroyalty in order to organize the
indigenous culture in a more political manner (Cancino et al., 2012). In particular, Kuño Tambo
village (Figure 28) was constructed under the governance of Viceroy Francisco de Toledo, who,
during his governance, emitted a reform that establish to resettle the Indian villages, in order to
establish over the population a direct control and facilitate the church’s Christianization. Through
the reorganization of the villages in a more rational manner, Toledo could more easily impose
effective labour, tax and religion over the Andean population (Hosne, 2013). Before the reform, the
Native Americans lived in sparse and small villages, which were difficult for the Spanish
Authorities to oversee. During the Toledo Viceroyalty, starting from 1567, more than 1000 Indian
villages called reducciones were constructed. The term derived from the word reducir (to reduce),
as the purpose of the reform was to reduce and consolidate the scatter and smaller villages. In this
way, The Primer Concilio Provincial Limense of 1552 (Constituciòn 2) stated the necessity of the
construction of the Church inside each reducciones, large enough to hold all the community and it
has to be adorned by art and paintings to give it the dignity required to an ecclesiastic building.
Figure 28: Overview of Kuño Tambo village (Kuño Tambo, 2015).
The reducciones were organized in grid, which corresponds to a model for creating Christian order
in a world where antisocial and wild chaos prevail in the countryside (Matienzo, 1567). Juan de
Matienzo, who in 5
wrote “Government of Perù”, focused on the physical layout of the
reducciones, sketching a model plan (Figure 29). The plan specified a village of five hundreds
households, organized in square blocks which surrounded a plaza. Matienzo specified that the
Church has to occupy one block in front of the plaza, as visible in Kuño Tambo village
(Mumford, 2012). The other square blocks around the plaza would mainly house a municipal
hospital, the house of the Corregidor de indios (the mayor of the village), a jail and an inn for
28
travelling Spaniards (Figure 29).The native people would live in blocks far from the plaza, each of
one would have been divided in four house lots (Mumford, 2012).
Governor
Mayor
Inn for
travelling
Spaniards
Prison
Reverend house
Square
Church
Council
house
Hospital
Hen
house
(b)
(a)
Figure 29: (a) Ideal plan for an Andean reducciòn (from Juan de Matienzo, Gobierno del Perù, 1567) and (b) reduccion
plan scheme based on Matienzo proposal (adapted from Juan de Matienzo, Gobierno del Perù, 1567)
According to the documents of the the Acomayo Parish Archive, the Kuño Tambo church, named
as el Templo de Santiago Apòstol de Cunotambo, was built in 1681. Before that, the village was
included in the parish of San Juan de Quihuares. Usually, the churches were constructed with the
façade directly facing the main plaza. Kuño Tambo Church, however, has a different orientation
(Figure 30). Probably, it was built over an existing temple which did not have any relationship with
the new plaza constructed after the Toledo reform. The Third Constitution indicated that old
temples should be destroyed, but if the location was appropriate, the new church should be built in
the same place (Cancino et al., 2012).
(a)
(b)
Figure 30: (a) View of Kuño Tambo Church and its relationship with the Plaza (Cancino et al., 2012 ); (b) the main road
and the bell tower (Kuño Tambo, 2015)
29
3.2 Description Of the building
3.2.1
General description
Kuño Tambo Church was constructed following the typical pattern of the Churches of Indian
villages. Rural churches in Indian villages has to typically include an atrio (a walled forecourt), a
sotacoro (the area under the choir loft), the choir loft, the single nave, a presbytery with altar, a
sacristy, storage for ecclesiastical furniture, and a cemetery. A bell tower, separated by the main
church, has to be constructed nearby.
The church, a one-storey building, is part of a larger complex, which include a bell tower not
structurally connected to the building (Figure 30), but separated by the main church through the
main road. Probably, the bell tower and the church were previously connected by a church yard,
now not present due to later modification of the road and the construction of new buildings in
between. The building seems to be constructed in the same time (Cancino et al., 2012). It is
constituted by the main nave, a rectangular space oriented north-south, a baptistery and a sacristy,
adjacent to the eastern wall and located respectively on the north and on the south part of the wall.
The main nave is divided in five different sectors: a sotacoro, a choir loft, a nave, a presbytery and
an altar (Figure 31).
CHOIR
LOFT
Choir Loft
Elevation +3.80
Altar Elevation +2.00
Presbitery Elevation
+0.80
ALTAR
NAVE
SOTACORO
Entry Elevation
-0.24
0
SECTION C-C
B
1.81 1.66
3.25
Infilled
opening
0.93
15.98
1.29
1.71
ALTAR
1.91
ALTAR
NAVE
2.26
PRESBITERY
2.83
C
MAIN
ALTARPIECE
Pulpit
C
11.08
5M
A
11.80
1.72
2.66
Nave Elevation
0.00
PRESBITERY
7.90
1.30
18.13
ALTAR
1.59
4.71
2.20
5.97
BAPTISTERY
2.61
6.08
2.09
2.25
2.69
1.28
6.88
2.21
0.85
6.01
SACRISTY
2.95
Infilled
opening
7.10
B
A
3.43
3.44
0.60
0
5M
N
FIRST FLOOR PLAN
Figure 31: Plan and section of Kuño Tambo Church (2015)
30
During investigations carried out by the Getty Conservation Institute in July 2010, human bones
were found beneath the floor level in the southeast corner of the church (Cancino et al., 2012 ). The
Kuño Tambo Church was constructed following the architectural indication for the realization of
the rural churches to be built in the rural reducciones. A staircase, that cut the southern adobe wall
of the baptistery, allows the access to the wooden-frame choir loft located above the sotacoro and to
a wooden balcony present on the façade (Figure 35).
Internally, the walls are covered with a mud plaster whose width can reach 60 mm and decorated
with decorative paintings, in particular at the east, south and west wall of the main nave (Figure 32).
The change in the floor level, designed with fired bricks, and wooden pilasters define the separate
space of the churches which are devoted to different functions. Wood in the Church is widely used
for the roof construction, the choir loft construction (Figure 33), the realization of the doors and
monumental altar piece, and railings, which are used to separate the altar, the nave and the
presbytery.
(a)
(b)
Figure 32: (a) Orthographic elevation of frescos of the South façade (b) view of the frescos located on the inner wall of
the façade (Percy, et al., 2013)
(b)
(a)
Figure 33: View of the choir loft, constructed with wood (Cancino et al., 2012 )
31
All the walls are built with the use of adobe bricks on a base course. The base course is composed
by rubble stone masonry with mud mortar. The size of the rubble stone masonry varies in size,
some stone exceed 0.64 m in width and the mud mortar joints vary in width from 20 to 60 mm
(Cancino et al., 2012 ). The base course varies in height (the medium height is between 1.20 to 1.50
m), following the natural slope of the site. The church was, in fact, constructed on a natural rock
outcropping. Compacted clay fill layers were used to level the site. As a consequence, the
foundation sits directly on the natural rock or on the compacted clay fill. Along the east wall, where
the height of the rubble stone masonry course reaches the maximum level (on the southeast corner a
height of 3.5 m was registered), is possible to observe natural portions of the rock outcropping
below the stone base course. The interior floor level, as a consequence, does not match with the
exterior bottom of the base course. In some parts the interior floor is higher than the bottom of the
base course (Figure 34). In other parts of the structure the base course is higher than the floor level
and the exterior grade, and rocky soil is visible from the exterior.
Figure 34: View and schematic representation of the rubble stone foundation – East wall (Cancino et al., 2012, drawing
Mirna Soto, for the GCI)
The total width of the rubble stone masonry matches with the width of the adobe walls which bears
on top. The walls are load-bearing mud-brick construction set in an English bond pattern and laid in
a mud mortar. The typical brick measured has a dimension of 0.70 m long x0.35 m wide x0.2 m
high. The average thickness of the joints is 15 mm. The thickness of lateral walls of the main nave
varies between 1.60 m to 1.90 m and has a maximum height of 6.60 m, measured from the visible
top of the rubble stone masonry base course. The façade, located at the south, and the north wall
have a thickness respectively of 1.90 m 1.30 m and a height at the gable end of 8.70 m. The façade
has a symmetrical configuration. Only the presence of an opening, located at the eastern part of the
wall as an access to the wooden balcony, breaks the symmetry. An arched opening was built as the
main entrance of the Church. Above the wooden balcony, a niche that contains a religious statue
and a rectangular opening are present. The longitudinal walls extends beyond the façade to act as
buttresses (Figure 35a). The north wall is constituted by an uninterrupted adobe wall with gable end
(Figure 35b).
32
(b)
(a)
Figure 35: (a) View of the façade (southern wall) and (b) view of the northern wall
The lateral walls present several openings and adobe masonry buttresses. . An arched opening is
present in the centre of each lateral walls, probably used before as additional entrances and now
infilled with adobe masonry (Figure 36). Other openings are present along the lateral walls and
serves as windows to illuminate the main nave.
(a)
(b)
Figure 36: View of the lateral wall (a) from the interior and (b) from the exterior (Kuño Tambo, 2015)
The baptistery has a thickness that ranges between 0.60 m and 2.00 m. The southern wall varies in
thickness due to the presence of the staircase that gives access to the choir loft (Figure 31). The
height from the top of the stone base course is 5.70 m at the gable end and 3.80 m at the eaves. Only
one opening is located in correspondence of the gable end and symmetrical to the wall. The walls of
33
the sacristy are the most slender and have an average thickness of 1.10 m and their height is 5.70 m
at the gable ends and 3.80 m at the eaves. The southern wall and the eastern wall present an
opening and a niche, respectively (Figure 37).
(a)
(b)
Figure 37: View of (a) the baptistery (south view) and (b) sacristy (north view)
Several portion of the walls are externally unfinished and, as a consequence, it is possible to
observe the brick pattern of the adobe walls. Internally, the walls are covered with mud plaster, for
which the thickness ranges between 20 and 60 mm. One layer of 1-2 mm of thick painted gypsum is
presented in some portion of the walls (Cancino et al., 2012 ).
All the walls are well connected each other, through the realization of overlapping mud bricks, with
the exception of the baptistery, which was constructed adjacent to the eastern wall of the main nave
(Figure 38). The southern and the northern walls are not perfectly connected to the lateral main
nave walls. Regarding the buttresses placed along the main nave, all are connected to the nave walls
with overlapping bricks, with the exception of additional buttresses located at the eastern and the
western part of the walls (Figure 39).
Figure 38: Isometric drawing of the baptistery, which shows the lack of connection between the baptistery and the
eastern wall of the main nave (Cancino et al., 2012)
34
WIII.1– East wall
WIII.2 – East wall
(Cancino et al., 2012 )
B.11 – East wall
B9 – B10
WII.2 – North wall
B.5 – B.4
Plan and structural details (Fonseca Ferreira, et al., 2012)
Figure 39: Plan and overview of the connection typology
B.3 – B.2
35
The roof is constructed with timber elements, with par y nudillo method. The par y nudillo is a
traditional method that consists in two rafters with diameter equal to 0.20 m, which are joined
through a collar tie (Figure 40a). The rafters are joined together by half lap cuts and are tied
together with leather straps and wrought iron nails (Figure 40b). The collar tie is nailed and
strapped with leather, but is not joined by half lap cuts. The par y nudillo trusses are connected to
the lateral walls through the use of 0.17 x 0.17 m wooden plates, which are not continuous along the
all wall. This means that they are not built as bond beams, but they have only the function to
support the ends of the roof rafters (Figure 41). The end of the rafters has, in fact, 90 degree cut,
allowing to rest on the wooden beams, which are embedded approximately 0.60 m into the walls.
The roof located on top of the main church consists in 47 par y nudillos. The roof of the baptistery
and of the sacristy is constructed with the same method, but in an orthogonal direction, with
separate gable roof but with similar slope (8:12 approximately). They consist in 6 par y nudillos
elements. Unlike the main nave, the rafters sits directly on the mud bricks walls, without the use of
wooden plates. Due to the use of this traditional method, all the roof of the church are unidirectional
and transfer the loads on the eaves. The lateral forces transmitted by the inclined roof (the
approximate slope is 8:12) is transferred directly to the lateral walls. Above the rafters there are the
Kur Kur canes, which are thin canes without voids in the centre, which are tied together and to the
rafters through the use of ropes. Above the canes, there is a mud and straw layer that supports and
adheres the clay roof tiles.
Six tie beams are present along the main nave of approximately 0.20 m of diameter and are fixed to
the walls through the use of timber anchors (Fonseca Ferreira, et al., 2012). One additional tie is
present in the baptistery.
(a)
(b)
Figure 40: (a) View of the roof of the main nave (Cancino et al., 2012 ); (b) Half-lap joint at the intersection of roof
rafters (Cancino et al., 2012 )
36
(a)
(b)
Figure 41: Lack of connection between (a) the southern wall of the baptistery and the eastern wall of the main nave and
(b) the northern wall of the baptistery and the eastern wall of the main nave
3.2.2
Interventions
Although the buildings of Kuño Tambo village were all constructed at the same time, the church
has been subjected to several alterations through the time. In some parts, the repointing technique
was applied in order to consolidate the portion of the walls affected by the wind erosion. In fact, the
bricks are more deteriorated than the mortar joints. The walls are, during the years, subjected to
several numbers of alterations, mainly in relation to the window and doors openings, some of them
infilled with lateral bricks (Figure 42). As previously mentioned (§3.2.1), the two lateral doors
located at the centre of the longitudinal walls of the main nave are now infilled with adobe masonry.
No connection was built between the infilled masonry and the original wall. Probably, an opening
that had the function to connect the choir loft with the exterior balcony, were constructed along the
façade. At the present time, the door is infilled.
37
(b)
(a)
Figure 42: Alteration of the walls: (a) an infilled door on the main façade; (b) infilled window along the northern wall
As it is possible to observe in Figure 39, at the west lateral wall, two earthen buttresses collapsed
and only the presence of a rubble stone masonry course remains; probably the causes of their
collapse is due to the instability of the base course; the buttresses were connected to the west
longitudinal wall, as visible observing the remains of the mud bricks of the buttresses into the wall
construction. At the east lateral wall, another buttress collapsed or was dismantled; in this case, the
buttresses were not connected to the east wall.
In the internal part of the church, it is possible to observe two mud brick piers, which are probably
the remains of a quincha arch that was constructed in order to separate the nave from the
presbytery. A similar quincha arch without damage is observed in the church of the nearby
Rondòcan village, which has a similar design of Kuño Tambo church (Figure 43). The causes of the
collapse or of the dismantling of the arch are not documented. However, the pier exhibits cracking,
probably related to the out-of-plane movement of the west wall, which exhibit an outward
displacement.
(a)
(b)
Figure 43: a) The Church of Rondòcan, in which it is possible to observe the similar design with Kuño Tambo
Church (www.maps.google.it); b) Intact quincha arch located in the interior of the Church of Rondòcan
38
The roof system was replaced throughout the history, as it possible to observe in Figure 44, the
current roof was reconstructed after the collapse of the quincha arch. The previous roof had
probably the shape to accommodate the arch, so probably it was taller than the existing one. When
the roof was substituted, vegetal ropes were used to connect rafters and iron nails to connect the
rafters with the collar ties, instead of the original leather straps. In addition, three wooden keys were
installed at the south ends of the eastern and western walls with the intent to connect the rafters with
the mud brick walls. Probably, the wooden keys were installed in substitution of a seventh tie beam
that collapsed or was removed (Figure 45).
(a)
(b)
Figure 44: (a) Remains of the quincha arch in Kuño Tambo Church (Cancino et al., 2012 ); (b) Location (in red) of
the quincha arch in Kuño Tambo, with the function to separate the presbytery from the nave (Ph.D. Eng. G.
Karanikoloudis drawings, adapted)
(a)
(b)
Figure 45: (a) 3D view of Kuño Tambo Church with the location of the tie beams (Ph.D. Eng. G. Karanikoloudis drawings)
and (b) Wood cross anchors at the south end of the west lateral wall
39
3.2.3
Damage survey
Even if the walls are considered in fair/good condition, some damages and local decay of the walls
are visible. In some parts, it is possible to observe the loss of some vertical joint. This is mainly
visible on the northern wall of the sacristy and at the southern side of the lateral walls, when they
belongs to the main façade (Figure 50-E.9). In the upper part of the wall, a loss of adobe units was
observed. The sources of the local erosion of the adobe walls on top are attributed to the water
infiltration due to the lack of maintenance of the roof. The local decay of the adobe affect, in some
parts of the structure, the connection between the roof and the wall (Figure 46b). Not only the
connection between the walls and the roof are sometimes weak or non-existent, but all the roof is in
poor condition. Some of the connection between the rafters and the collar ties are failed and some of
the structural elements are deformed. The covering of the roof is in poor condition and allow water
infiltration, causing not only the erosion of the adobe bricks at the top of the walls, but even the
detachment and the loss of the interior plaster. At the present, the roof is protected by the mean of a
provisional structure (Figure 46a).
The base course is, in general, in fair good condition. However, in some portion of the building,
there are some localized losses of stones (Figure 50-E5, Figure 51-S4) and mortar (Figure 50-E6,
Figure 52-N3), probably due to improper site drainage, erosion, excavation of the rock outcropping,
that may have contribute to the damage observed (Cancino et al., 2012 ). The main consistent
damages are observed along the eastern walls and the baptistery.
(a)
(b)
Figure 46: Loss of vertical mortar joints in some portion of the adobe wall and local loss of adobe units: (a) Current
state of the roof along the main nave, protected by a provisional structure (Cancino et al., 2012); (b) The loss of adobe
on the top of the wall has led to a local failure in the connection between the wall and the rafter. A wood post is
installed to temporarily support the rafter end of the roof.
Kuño Tambo Church, constructed in the seventeenth century, experienced several earthquakes
throughout the history, including the 1950 Cusco earthquake (Mw 6.0), approximately 35 km to
northwest, the 1943 Yanaoca earthquake, approximately 35 km to the southeast, and the 1913
Abancay earthquake, approximately 120 km to the west. It was possible that the church was
subjected even to more earthquakes in the past, as the 1746 Lima and 1687 Lima (Mw 8.5)
40
earthquakes, approximately 600 km to the northwest (Cancino et al., 2012 ). Probably, some
vertical cracks observed at the intersection between the façade and the longitudinal walls could be
attributed to seismic events. In fact, in Kuño Tambo church vertical cracks appears at the corners
where the southern façade intersects with the buttresses and the longitudinal walls. The presence of
the vertical cracks at the corner are frequently observed after seismic events on adobe construction,
due to the lack of connections between the façade and the transversal walls (§ 2.3.1). The
mechanical characteristics of the adobe masonry and the lack of proper confinement and
connectivity between the structural elements influence the seismic resistance of the construction.
On the other hand, the northern façade does not exhibit the same vertical cracking at the corner, but
only local and minor cracking.
(b)
(c)
(a)
Figure 47: Vertical cracking at the corner where the southern façade intersect the lateral walls:
(a) south-eastern corner, from interior; (b) from exterior; (c) south-western corner (interior view)
Several vertical cracks are observed along the eastern, the northern and the southern walls, near the
southern corner. The settlement of the base course, due to the rising damp and the erosion of the
base course, could probably the cause of the appearance of the crack. In alternative, the cracks
could be have appeared due to seismic events. The presence of the staircase that gives access to the
choir loft reduces the width of the wall. As a consequence, the two part of the wall can behave
independently due to a seismic event, causing the appearance of the vertical crack along the
southern wall. The second crack is located at the intersection of the eastern and the southern wall,
which, as previously mentioned, is a typical damage observed in adobe construction after a seismic
events due to the low connectivity between the walls. The presence of a gable-end-wall, not load
bearing tall walls not connected to the structure, favourishes the overturning and the separation of
the orthogonal walls. Other minor cracks are observed close to the opening located along the gableend wall.
41
(c)
(b)
(a)
Figure 48: Crack pattern observed along the baptistery: (a) southern wall; (b) eastern wall; (c) northern wall
Based on external visual inspection conducted in May 2015, the main external damages observed
can be classified in:
a) Efflorescence, mainly localized below the roof on the adobe walls, which is the result of the
rainwater which soaks into the masonry surface;
b) Deteriorated mortar joints, localized both in correspondence of adobe masonry walls and
rubble stone masonry. As a consequence, it is possible to observe in localized zone that the
repointing with the use of mud mortar was already applied as an intervention technique;
c) Major loss of structural material, localized in particular at the base and at the top of the wall;
d) Structural cracks, mainly due to past seismic activities or soil settlement.
42
Figure 49: Main damages observed at the western wall of Kuño Tambo Church
43
Figure 50: Main damages observed at the eastern wall of Kuño Tambo Church
44
Figure 51: Main damages observed at the southern wall of Kuño Tambo Church
45
Figure 52: Main damages observed at the northern wall of Kuño Tambo Church
46
3.3 Material properties
The material properties of the Kuño Tambo Chuch were determined based on the experimental
campaign conducted for the Seismic Retrofitting Project by the Pontificia Universidad Catòlica del
Perù. The campaign was conducted with the aim to provide specific historical material
characterization in order to understand the structural behaviour of the four historical earthen
building prototypes in Peru that were selected for the study (SRP, 2014), including Kuño Tambo.
The information are then integrated with experimental results available in literature.
As already previously mentioned, the structure is composed by walls and buttresses built with
adobe masonry, which sit on a rubble stone base course. The base course was built with stone and
mud mortar. The roof, the tie beams and the architraves were built with timber elements.
3.3.1
Adobe masonry
Adobe masonry is prepared using mud bricks and mud mortar. In order to characterize the
compressive strength of the adobe units fc,b, four cubic specimen of approximately 13 cm by side
were carved and tested in uniaxial compression. The average compressive strength value obtained is
0.71MPa.
Table 3: Results from compression test of adobe units in Kuño Tambo Church, Peru (SRP, 2014)
Uniaxial compressive strength of adobe units – Kuño Tambo Church
Unit
1
2
3
4
Dimensions
[cm]
13.3 x 13.0 x 14.3
13.6 x 13.8 x 14.7
13.4 x 13.2 x 14.5
13.5 x 13.5 x 14.6
Area
[cm2]
172.9
187.7
176.9
182.3
Maximum load
[kN]
14.61
12.23
8.51
15.90
Compression strength
[MPa]
0.84
0.65
0.48
0.87
Average compression strength
[MPa]
0.71
The modulus of elasticity was not evaluated through experimental test. Thus, it is estimated based
on the correlation between compressive strength fc,b and modulus of elasticity Eb of adobe units
developed by Caporale et al. (2015), where Eb = 160 fc,b (114 MPa) obtaining a results between the
range available in literature (§ 2.2). No experimental tests were conducted to assess the mechanical
properties of the mud mortar. However, a granulometric analysis was performed for historical
adobe units and mortar. It is noted that for the Kuño Tambo Church the granulometric analysis was
conducted only for adobe units. The tests show that the granulometric composition of historical
adobe units and mortar is very similar, which may imply that that it was usual practice to use the
same material for units and mortar (SRP, 2014). For this reason, the value is in accordance with the
one proposed by Tarque (2011) for Peruvian mud mortar, equal to 80 MPa, close to the one
calculated for the adobe units.
47
(b)
(a)
Figure 53: Granulometric curve for: a) Historical adobe units, b) Historical mortars. (SRP, 2014)
In order to obtain the friction angle and the cohesion for adobe masonry, fifteen triplets were tested
in shear/compression, in which twelve were prepared using new adobe units and three triplets were
constructed using mortar with straw and existing units extracted from Ica Cathedral (Figure 54). All
the specimens failed at the interface of the mortar with adobe, confirming the presence of a weakest
link in the adobe masonry. The results obtained are similar and the small discrepancy could be
attributed to the addition of straw to the mortar mixture in the test conducted using existing units
(SRP, 2014):
Table 4: Experimental results from shear-compression tests of adobe-adobe triplets (SRP, 2014)
Shear- compression test for adobe-adobe triplets
Cohesion c [MPa]
0.037
0.044
Modern adobe
Ica Cathedral adobe
Friction angle ft [rad]
0.60
0.50
Experimental tests to obtain the tensile strength at the interface between adobe units and mortar
joints were not carried out. Thus, the value is taken in accordance with the one proposed by Tarque
(2011), in which the tensile strength, ft,a = 0.01 MPa, was obtained calibrating a numerical model
based on the experimental results of an adobe walls subjected to cyclic tests (see Section 2.3.2).
(a)
(b)
Figure 54: Shear-compression tests on adobe-adobe triplets: (a) View of the test performed; (b) Shear-compression
curve (SRP, 2014)
48
3.3.2
Rubble stone masonry
The experimental tests conducted by the Pontificia Universidad Catòlica del Perù was not
performed to obtain the mechanical properties of the rubble stone masonry that constitutes the base
course of the walls. For this reason, the values of Young Modulus Es was evaluated based on test
available in literature. Regarding the eastern region of Cuzco, the quarries used to extract stone
since the antiquity comprise the late Pleistocene Rumicolca Formation. Although there is a variation
in the appearance of the rock, they are very similar in geochemistry and much of the stone quarried
was igneous, including andesite, diorite, granite and rhyolite (Ogburn, et al., 2013). Not many tests
are available in literature regarding the mechanical properties of the stone used in the eastern region
of Cuzco as building material. Three stone specimen extracted from the Cuzco Cathedral were
tested to evaluate the compression strength fc,s and the modulus of elasticity Es (Olarte, et al., 2011),
which are in the range of the values available in literature for granitic stones (Figure 55). The
density of rubble stone mortar is assumed equal to 19 kN/m3, in accordance with the
recommendation of the Italian Code (LLPP, 2009). The modulus of elasticity of the mud mortar is
evaluated as 80 MPa, in accordance with the value proposed by Tarque (2011).
Table 5: Experimental data available in literature regarding the compressive strength, the Modulus of Elasticity and the
tensile strength of the stone units
Location
Mechanical properties – Stone
Compression strength fc,s
Young Modulus Es
[MPa]
[GPa]
Reference
Mean value
Perù
(Cuzco)
Portugal
Olarte et al. (2011)
Range
45.0
Vasoncelos (2005)
Mean value
19.4
69.2
Granite stone - Cuzco Cathedral
11.0 – 63.8
35.2 - 148.5
Sentivel and Lourenço (2009)
Typology
Range
20.2
Rubble granite masonry
The value of cohesion, friction angle and tensile strength of the interface between mud mortar and
stone are not available in literature. The value of the cohesion, friction angle and tensile strength
were obtained from the literature, where a limestone mortar or hydraulic lime mortar is used in the
experimental tests. However, a parametric analysis was conducted to assess the influence of the
variability regarding the seismic behaviour of the structure.
Table 6: Experimental data available in literature regarding the inelastic properties of stone-stone interface
Inelastic properties for stone-stone interface
Location
References
c [MPa]
tanФ [rad]
Vasoncelos (2005)
0.359
0.63
Sentivel and Lourenço (2009)
0.100
0.40
Binda et al. (1994)
0.330
0.74
ft
[MPa]
Granite stone; limestone mortar; test on triplet
Portugal
Italy
Comments
0.05
Modeling of a rubble stone masonry-validation
of experimental test (Vasoncelos, 2005)
Sandstone; Hydraulic lime mortar
49
3.3.3
Adobe-stone interface
Shear-compression tests were carried out on eleven triplets stone-adobe-stone by the Pontificia
Universidad Catòlica del Perù, aiming at evaluating the cohesion and the friction angle of the
interface between stone and adobe unit. Eleven specimen were constructed with two stone units,
one intermediate adobe unit and the mortar used was composed by mud. The stone units used in the
tests were extracted from another historical Peruvian building construction (Hotel El Comercio) and
the adobes and the mud mortar are new. The results of the cohesion and friction angle obtained
from experimental results are presented in Table 7. The tensile strength, and in correspondence of
the adobe stone interface, was assumed equal to the value used for adobe-adobe interface (ft = 0.01
MPa).
Table 7: Experimental results from shear-compression tests of adobe-stone triplets
Shear-compression test for adobe-stone triplets
Cohesion c [MPa]
Hotel El Comercio stone – New adobe and mortar
0.065
(a)
Friction angle ft [rad]
0.45
(b)
Figure 55: Shear-compression tests on adobe-adobe triplets: (a) View of the test performed; (b) Shear-compression curve
(SRP, 2014)
3.3.4
Timber elements
The roof, the tie beams and the architrave of Kuño Tambo Church are built with timber elements.
However, due to the lack of connection between the rafters and the adobe walls, and due to the
deformability of the roof system, the roof was considered in the numerical model of the structure
only as a vertical load (see Section 4.3). No information are available regarding the effectiveness of
the tie beams and the effectiveness of the connection with the adobe walls. Furthermore, observing
the mode shape of the structure based on the dynamic identification tests performed (see Section
3.4), the tie beams do not have a high influence on the dynamic behaviour of the structure. For this
reason, the mechanical properties of the roof and of the tie beams were not evaluated. Regarding the
architrave, they were considered in the numerical model as rigid blocks (see Section 4.3). Thus,
only the density of the timber elements were considered.
50
Due to the absence of characterization of the timber elements in Kuño Tambo Church, the material
properties of Sapele timber were adopted, based on the experimental tests performed on two other
historical constructions (Ica Cathedral and Hotel El Comercio). The density of Sapele timber is
assumed equal to 4 kN/m3.
3.3.5
Summary of the mechanical properties
The mechanical properties that considered for the analysis of the structure are summarized in the
Table 8 and Table 9. The values proposed were assumed as the reference values. However, due to
the lack of knowledge of some mechanical properties and due to the variability of the experimental
values available in literature, a parametric analysis was conducted, aiming at evaluating the
influence of the parameters on the behaviour of the structure (see Chapter 4).
Table 8: Summary of the elastic properties of the materials
Units and mortar
Young’s Modulus
[MPa]
Elastic properties
Poisson ratio ν
Shear Modulus G
[MPa]
Density γ
[kN/m3]
Adobe units
114
0.25
46
19*
Stone units
19400
0.25
4850
19*
Adobe mortar
80
0.25
20
-
*The value of the density refers to the adobe masonry. The modelling approach that will be used, the discrete element method,
approximate the masonry as a composite material where the units are expanded including the geometrical space of the mortar
joints.
Table 9: Summary of the inelastic properties of the materials
Inelastic properties
Cohesion c
Friction angle ϕ
Tensile strength ft
[MPa]
[rad]
[MPa]
Adobe-adobe interface
0.044
0.500
0.010
Stone-stone interface
0.100
0.400
0.050
0.065
0.450
0.010
Type of interface
51
3.4 Dynamic identification test
3.4.1
Introduction
The necessity of preservation of historical construction combines with the need of acquire a deep
knowledge regarding the structural assessment of the structure and its seismic response, with the
use of less invasive investigation techniques. Furthermore, the complexity of historical construction,
subjected to damage, structural and architectonical alterations, lead to obtain difficulties in
achieving reliable and realistic results. However, the use of minor or non-destructive techniques (for
example dynamic, sonic, and flat jack tests) may be profitably used to obtain an advanced
knowledge of historic masonry buildings. For example, the dynamic identification test is one of the
most powerful tool to estimate the parameters related to the dynamic behaviour (Lorenzoni, 2015).
Output-only identification techniques, is widely applied in the context of historical masonry
structures to estimate dynamic parameters. This type of identification technique is based on ambient
vibration. The structure is excited by ambient vibration, such as wind, traffic or even vibration
caused by persons (Lourenco, et al., 2012), in order to obtain the main modal parameters (natural
frequencies, mode shapes and damping ratios). The main assumption is the consideration that
ambient excitations (uk) are a stationary Gaussian white noise stochastic process in the frequency
range of interest (Figure 56) (Ramos, 2007). Thus, the structural response can be measured only
with the use of high sensitive sensors. In this way, the tests can be performed without causing
damages on the structures. A numerical model can be then prepared to simulate the structure and
based on the results of the performed dynamic tests, the numerical model can be updated to match
the experimental values obtained in terms of frequencies and mode shapes.
Figure 56: The Output-only identification technique (Ramos, 2007)
In the output-only identification techniques, the response (yk) (Figure 56) includes the modal
contribution of the ambient forces, the contribution of the structural system and the contribution of
the noise signals from undesired sources (Ramos, 2007). Thus, the identification techniques must
have the capability to identify the desired information, separating them from the noise. The
techniques adopted can be divided in two main groups: (a) time domain methods; and (b) frequency
domain methods (Figure 57). Frequency domain methods are the most user friendly and faster to
process. However, difficulties are encountered for evaluating and to identify close frequency values,
due to the frequency resolution based on Fast Fourier Transformation process. The time domain
method is based on model fitting by correlation functions or time history series of every measured
point in the time domain. In general, the processing of results with the Stochastic Subspace
Identification Method (SSI), which deals with time series processing, is more complex than with
52
frequency domain. However, the time domain methods are more robust and, thus, are able to
estimate with more accuracy the modal parameters even with high frequency resolution. As a
consequence, the SSI method is considered one of the most accurate methods to estimate the modal
parameters (Ramos, 2007).
Figure 57: Classification of output-only identification methods (Ramos, 2007)
3.4.2
Description of the test setup
The dynamic identification tests were performed on Kuño Tambo Church in May 2015 to estimate
natural frequencies, mode shapes and damping ratios of the structure. The position of the
accelerometers are determined based on the results of a preliminary damage survey and numerical
analysis (Karanikoloudis, et al., 2015). The dynamic identification tests were carried out through
three setups with one reference accelerometer and three accelerometers positioned in the area of
interest (Figure 58). The accelerometers were placed along the main nave and the baptistery, which
are the areas that show more significant damages (the sacristy was not investigated).
The transducers used during the tests correspond to piezoelectric accelerometers with a sensitivity
of 10 V/g and a frequency range of 0.15 to 1000 Hz (measurement range ± 0.5 g). The sensors were
connected by a data acquisition board of 24-bit resolution, which has the function to record the
signals given by the accelerometers.
53
(a)
(b)
(c)
Figure 58: View of the accelerometers positioned: (a) Reference sensor, positioned at the bottom of the west wall,
between the infilled opening and the quincha arch pillar; (b) Sensor 2 for setup 3, positioned on the façade, at the southeast corner of the window; (c) Sensor 4, for setup 1, positioned on the gable end roof of the northern wall
Figure 59: Dynamic identification setup 1 performed in Kuño Tambo Church (in red is highlighted the reference sensor)
54
55
3.4.3
Experimental results
The results obtained were processed in ARTeMIS software (SVS - Structural Vibration Solution
A/S, 2013) . Two different methods were used to process the data and obtain the dynamic properties
of the structure, namely: (a) the Enhanced Frequency Decomposition Domain Method (EFDD)
(Figure 62a), a frequency based method; and (b) the Stochastic Subspace Identification Method
(SSI) (Figure 62b), a time domain method. The first four mode were estimated. In general, the
higher modes do not present significant influence in the dynamic behaviour of civil engineering
structures and the calibration of a numerical model based on many modes is difficult (Lourenço et
al., 2012). The value of the frequencies and damping ratios of the first four modes obtained from the
SSI and EFDD Methods are presented in Table 10 and Table 11, respectively. In the EFDD method
is difficult to identify the frequency values of higher modes. Although it is a faster method, the
results may be inaccurate after the first three mode shapes, due to the difficulties of recognize the
frequency peak value. The SSI method allows to obtain more accurate results. The inaccuracy of the
EFDD method after the first three modes is confirmed observing the Modal Assurance Criterion
(MAC) values between SSI and EFDD method (Table 10). For this reason, the modes obtained
from the SSI method were adopted and they are presented in Figure 63.
(a)
(b)
Figure 62: Stabilization diagram: (a) Stochastic Subspace Identification Method; (b) Enhanced Frequency
Decomposition Domain Method
56
Table 10: Frequencies and damping ratio of the first four modal shapes – SSI Method
SSI Method
Mode
Mode 1
Mode 2
Mode 3
Mode 4
Frequency [Hz]
1.6
2.2
2.7
3.0
Std. Frequency [Hz]
0.006534
0.001343
0.02002
0.04897
Damping Ratio [%]
0.70
3.67
0.63
3.92
Std. Damping Ratio [%]
0.326
0.951
0.493
0.991
Table 11: Frequencies and damping ratios of the first four mode shapes – EFDD Method
EFDD Method
Mode
Mode 1
Frequency [Hz]
1.6
Std. Frequency [Hz]
0.0000047
Damping Ratio [%]
0.30
Std. Damping Ratio [%]
0.005
Mode 2
Mode 3
Mode 4
2.1
2.7
3.2
0.0002281
0.0000488
0.0007383
0.27
0.20
0.15
0.015
0.002
0.038
Table 12: MAC for the SI Method and EFDD Method
SSI Method
EFDD
Method
Mode 1
Mode 2
Mode 3
Mode 4
Mode 1
0.9982
0.6357
0.3555
0.03239
Mode 2
0.6762
0.9497
0.3945
0.1153
Mode 3
0.3363
0.2889
0.9492
0.05818
Mode 4
0.5581
0.3915
0.8604
0.05078
The natural frequencies of the modes obtained from the SSI Method ranges between 1.6 Hz to
3.0 Hz. The standard deviation is very low and this indicates that the values were well estimated.
The damping ratio ranges between 0.7 and 3.9%. It is noted that the damping ratio is a very
sensitive parameter and difficult to estimate experimentally, mainly in masonry structures (Mendes,
2012).
The first and the second mode correspond to the first transversal and longitudinal global modes,
respectively. However, they are affected by the presence of damages along the corner. In the first
mode the west wall, due to the presence of the crack in the corner, moves independently from the
rest of the structure and the eastern wall presents a higher displacement than the baptistery walls in
east-west direction, which can be related to the lack of connectivity between the two portions of the
building. The mode 3 correspond to a combined mode, involving out-of-plane movement of the
façades and a second curvature of the longitudinal walls. The mode 4 is local and characterized by
the simple overturning of the main façade (Figure 63). Observing the mode shapes of the structure,
it can be observed that the tie located along the main nave might not play a significant contribution
on dynamic properties of the structures and the two longitudinal walls moves independently,
57
showing the lack of effectiveness of the wooden ties. Furthermore, it is concluded that the mode
shapes are influenced by the damage present in the structure.
Mode 1 (1.6 Hz)
Mode 2 (2.2 Hz)
Mode 3 (2.7 Hz)
Mode 4 (3.0 Hz)
Figure 63: Experimental mode shapes of the first four modes of the structure (SSI Method)
58
4. OUT-OF-PLANE BEHAVIOUR OF THE WESTERN WALL
4.1 Introduction
In the present study, the seismic behaviour of Kuño Tambo Church is evaluated, based on Discrete
Element Method (DEM). The use of the discrete element method can be very effective for the study
of the damage patterns and the collapse mechanisms of adobe constructions. However, the structure
analysed is a large and complex structure, which requires computational effort when discrete
element is chosen as method for the seismic analysis.
As already mentioned by Lemos (2007), the study of large and complex masonry structures requires
the adoption of larger blocks size, because a limit of block number is required due to computational
demanding of the method. Furthermore, the use of rigid blocks is preferred instead of a deformable
one, to reduce the computational effort. As a consequence, the use of larger and rigid blocks
requires a preliminary study to ensure that the assumptions adopted are able to represent the real
behaviour of the structure analysed.
Based on these considerations, in the first phase of the seismic analysis of Kuño Tambo Church the
out-of-plane behaviour of the walls, considering a unitary slice, was analysed. The study of the outof-plane behaviour of the walls is very important in the analyses of adobe construction. The
response of the adobe construction to seismic events is governed by the brittleness of the material,
so that a very small movement of the walls can cause the separation of the walls. Due to the lack of
rigid diaphragm and a proper horizontal or vertical confinement, the walls behave separately and
the only aspects that can control stability are the rocking behaviour and the slenderness of the wall
(Tarque, et al., 2012). For this reason, the analyses of the independent behaviour of the walls is of
particular interest during the analysis of adobe construction. In particular, a unitary portion of the
western wall was taken into account. The model was prepared take into account the real dimension
of the units that constitutes the walls. Furthermore, the approach of an initial simple model allows to
investigate the role of the pattern during the collapse mechanism due to the overturning of the wall.
The adoption of a simple model constructed with real scale blocks allows to conduct a parametric
analysis based on uncertainties related to the mechanical properties and the influence of the
approximation that are required to be adopted when a more complex part of the structure are be
analysed with discrete element approach.
4.2 The Discrete Element Approach
The discrete element method, or distinct element method, was proposed by Cundall in 1971, as a
numerical approach in the field of rock mechanics to represent jointed rock mass as an assembly of
rigid blocks, for the study of slopes in hard rock (Lemos, 2007). The approach was later extended to
other engineering application, where the study of the contact between particles or rigid blocks was
required (Roca, et al., 2010).
59
Masonry structures, which are essentially composed by discrete units separated by mortared or dry
joints, are analysed by mean of the discrete element approach. The method allows to represent the
material as an assembly of distinct bodies, the units, interacting along the boundaries through
contact surfaces, which represents the masonry joints. The use of the discrete element approach
allows, as a consequence, to study the non-linear behaviour of the material, including phenomena of
sliding and total separation of the blocks, with the possibility to study large displacements of the
structure and updating the changing in the geometry, until its structural failure. The main
applications for structures involve the safety assessment of the masonry buildings, modelled either
in quasi-static or dynamic process (Lemos, 2007).
The blocks may be assumed, in the representation of masonry, as rigid or deformable. An internal
Finite Element (FE) mesh can be represented to better take into account the deformability of the
material or in situations of stress concentrations. However, the representation of masonry as a rigid
blocks is widely used, in particular where the collapse failure mechanism is analysed. In this case,
the deformation of the structure is concentrated in the elastic contacts. Rigid block analysis has, in
fact, a significant computational advantage, mainly when the adoption of explicit time-stepping
algorithms are applied with the aim to study the large motion of the structure and a three
dimensional representation of the structure is required to achieve a realistic simulation of the
seismic response of the structure (Lemos, 2007).
The application of discrete element approach is widely used for the analysis of simple blocky
structures. Arch structures were frequently studied with DEM to assess the collapse loads and its
mechanism (Gilbert, et al., 1994). Other studies applied the method on arches with the aim to
validate experimental tests (Broccoli Bati, et al., 1995) or demonstrate the applicability of different
formulations. An interesting application of the method was performed by Tran et al. (2014) to study
the mechanical behaviour of a stone arch bridge during the phase of formwork removal. Discrete
element method allowed to determine the specific displacement pattern that was possible to observe
in situ, not achievable with the use of the Finite Element Method (FEM). The seismic analysis of
drum columns or classical temples is widely investigated with this approach. Temples and columns
were built by an assemblage of stone blocks, which are well approximate by the use of rigid blocks,
due to the high stiffness of the material. The discrete element approach was, for example, a
powerful tool adopted by Psychiaris et al. (2003) to evaluate the seismic performance of a columnarchitrave group of the Parthenon and the effect of the proposed restoration.
The discrete element approach could well represent the behaviour of simple blocky structures, as
masonry arches, stone bridges, drums and columns, brick panels, arch and pillars, due to the simple
geometry and the good approximation of the structure as an assemblage of rigid blocks. However,
the study of more and complex structure is of difficult application. The high computational efforts
needed by the method, mainly for dynamic analysis, limit the number of block elements. Thus, for
large structures, a simplified representation has to be adopted, for which the blocks cannot represent
the size of the real unit, but they are larger than the real ones. The approximation adopted has to
ensure that the model is able to represent the fundamental modes of the structure and its failure
mechanism, including the correct representation of the real crack pattern (Lemos, 2007). Lemos et
60
al. (2009) studied the impact of the use of different pattern adopted on masonry walls under out-ofplane loading. The use of joint offset brick wall pattern may overestimate the wall strength
especially when the block size are larger than the real one. For this reason, different formulation
based on random block generator, are proposed as an alternative. Model checking and calibration
under assumption of elastic contact is always suggested when seismic studied are conducted on
larger and simplified blocks. An interesting application to large and complex structure was
proposed by Lemos (2009). The out-of-plane behaviour of a wall was analysed through the use of
macroblocks. The wall modes were calculated with the assumption of elastic contacts and the
elastic supports were adopted to consider the effect of the surrounding structure (Figure 64a).
Furthermore, due to the use of macroblocks, any sizeable offset between blocks might overestimate
the real interlocking. Thus the blocks were modelled with no offset. Another interesting application
of the discrete element method to large and complex structure was performed by Lemos et al.
(2000), in which the seismic behaviour and the possible structural restoration of a masonry bell
tower (the Torre do Relógio in Azores) was studied. Even in this example, not every joint of the
masonry structure were simulated, but macroblocks were modelled, including enough discontinuity
surface to examine the dominant modes of deformation and failure (Figure 64b).
(a)
(b)
Figure 64: Application to discrete element method to large and complex structures: (a) Geometry and first mode shape of
a wall, (b) Torre de Relógio in Azores (Lemos, 2000).
The application of the method to such a complex structure has highlighted, as confirmed by the
same authors, the potential of the method, but also the practical difficulties related to the necessity
to obtain detailed information on structural features (geometry and material properties) and seismic
action. In order to assess the reliability of the model, only global agreement with the crack pattern
observed confirms the reliability of the results.
Alexandris et al. (2004), carried out a study based on the discrete element method to evaluate the
seismic behaviour of typical structures of traditional Cypriot architecture. Due to the complexity of
the structures studied, a macro-modelling approach was necessary. Even in this case, the calibration
of the model was considered of primary importance. Discrete element method was considered a
61
powerful tool able to capture the collapse mechanism and the damage patterns of the buildings.
However, it requires higher amplitude of base excitation to collapse, attributed to the presence of
perfectly squared blocks.
The difficulties to represent the exact geometry of the structural units that constitutes the masonry
becomes problematic even when is necessary to model the rubble masonry with discrete element
method. Milosevic et al. (2012) modelled with discrete element approach a rubble stone masonry
wall specimen, in order to calibrate the non-linear properties of the material based on experimental
results. The specimen was modelled as a group of randomly sized polygonal blocks, generated by
an automatic joint generator (Figure 65). The numerical results showed a good agreement in terms
of damage pattern and ultimate load. A similar approach was applied by Bicanic et al. (2002) to
study the collapse mechanism of a rubble masonry arch bridge (Figure 66a). An alternative as a
representation of the rubble masonry was derived from soil mechanics, where rocks are represented
with the use of Voronoi polygons (Galindo Torres, et al., 2007). Lemos et al. (2009) studied the
out-of-plane failure of the wall studying the influence of different masonry pattern, considering the
method suitable for the representation of rubble stone masonry (Figure 66b). An interesting result
was achieved assessing that the three different Voronoi pattern implemented does not significantly
affect the out-of-plane behaviour of the wall.
Figure 65: Application to discrete element method to model rubble masonry specimen tested in laboratory (Milosevic et
al., 2012)
(a)
(b)
Figure 66: Application to discrete element method to model rubble masonry: (a) Masonry arch bridge (Bicanic et al.,
2002), (b) Out-of-plane behaviour of a masonry wall (Lemos, 2009)
62
4.3 Description of the numerical model
4.3.1
Geometry of the portion of the building analysed
A preliminary study of the overturning of the western wall was studied, considering a portion of the
wall with unitary length. The western wall has a variable height due to the presence of a natural
outcropping where the church was constructed. The maximum total height is increasing in the
southern direction from 6.0 m to 7.9 m. The height is variable from the interior to the exterior due to
the presence of a slope to allocate the wooden rafters. The portion of the wall analysed was chosen
as representative of the middle portion of the wall. It has an internal total height of 6.45 m and an
external height of 6.95 m. The base course has a constant height of 0.85 m (Figure 67). The width
of the western wall is equal to 1.71 m, including the presence of the plaster. In the analysis, the
presence of the plaster was not considered and the structural width was estimated (1.64 m). The
adobe blocks has a dimension of 0.70x0.35x0.20 m. The mud mortar layers between the adobe units
has a medium height of 0.015 m, which was adopted for the analysis. The stone masonry that
constitutes the rubble masonry has variable dimensions, with a maximum width of 0.70 m. The mud
mortar joints between the stone units has a variable height. An average height of 30 mm was
considered.
Figure 67: Identification and geometry of the model
4.3.2
Geometry of the elements
The geometrical representation of the structure is an important and influential task when the discrete
element is adopted. For this reason, an accurate study of the geometry of the structure and of its
element is a fundamental preliminary step that anticipate the construction of the model.
The model used for the analysis is composed by prismatic block elements, modelled with discrete
element software 3DEC (Itasca Consulting Group, 2013). As previously mentioned (see
63
Section 4.2), masonry are modelled as distinct units that interacts through contact surfaces. Thus,
the masonry units are modelled as zero-thickness joints and the geometry of the units expand to
include the width of the joints. As a consequence, the adobe units are modelled as prismatic
elements of 0.715x0.315x0.215m and the mud mortar between the units are considered as zerothickness joints (Figure 68).
Brick Brick
Unit Unit
MortarMortar
0.5675
0.20
0.20
0.21
0.56
0.21
0.70
0.56
0.70
0.70
(a)
0.70
0.70
0.35
0.35
0.2150
0.5675
Joint Joint
0.7150
0.7150
0.3575
0.3575
0.2150
0.70
(b)
Figure 68: Modelling of adobe masonry wall: (a) Geometry of the masonry;
(b) Geometry of the model (DEM approach)
If the adobe units can be represented with respect of the exact geometry of the block units, more
difficulties can occur for representing the geometry of the rubble stone masonry base course. Two
dimensional Voronoi polygons were adopted as a technique to represent the behaviour of a rubble
stone masonry.
The Voronoi diagrams is, in mathematics, a partitioning of a plan into regions based on distance to
points in a specific subset of the plan. Specifying a set of points, it is possible to construct a
Voronoi diagram taking pairs of points that are close together and draw an equidistant line between
the points considered. The data set of points, called the Voronoi points, adopted to construct the
Voronoi polygons are the centroid of the stone that constitutes the sectional morphology of the
rubble masonry (Figure 69). For each point it is so possible to define the Voronoi polygon
associated with it as a set of points that are closer to the given Voronoi point than to any other one:
where V(p) is the Voronoi polygon associated to the Voronoi point p in any other Voronoi point q
and the function fdist is the euclidian distance between two points in the plane (Galindo Torres, et al.,
2007). Thus, the lines that constitutes the boundaries of the polygons are equidistant to the set of
points and the vertices are equidistant to the closest three Voronoi points. The Voronoi polygons are
generated with the use of a mathematical software. The coordinates of the Voronoi vertex are then
imported in 3DEC software to construct the block units.
64
Stone
Mortar
P.1
P.3
P.2
P.4
P.9
P.7
P.5
P.8
P.6
P.13
P.12
P.10
P.14
P.15
P.11
P.18
P.20
P.19
P.17
P.16
P.21
a)
P.22
P.23
P.24
Geometry of the rubble masonry
Unit
Joint
P.1
P.3
P.2
P.4
P.9
P.7
P.5
Modification of
the
original line to avoid
unrealistic local failure
of the stone
0.5
P.8
P.6
P.14
P.12
P.10
P.13
P.11
P.17
P.18
P.15
P.19
P.20
P.16
P.21
P.22
P.23
P.24
0
0
0.5
1
b)
1.5
Voronoi polygons
Figure 69: Modelling of rubble stone masonry: (a) Geometry of the rubble stone masonry base course;
(b) Implementation of the Voronoi polygons considering as data set of points the centroid of the rubble stone that
constitutes the masonry
4.3.3
Block and contact representation
As discussed before, in discrete element method there are two possibility for the representation of
the blocks:
a) Rigid blocks, assuming that the units behave as rigid bodies and the deformability is
concentrated in the elastic contacts. The method is suitable for the evaluation of the collapse
mechanism of stone masonry failure, because stones are characterized by stiff blocks. As a
consequence, it can be assumed that mechanism of sliding and rotation are concentrated
along the joints;
b) Deformable blocks, where the blocks are discretized into a FE mesh to take into account of
the block deformability. The method is suggested for weaker materials, in order to take into
account of the deformability of the units.
The stone that constitutes the base course can be modelled as rigid blocks, assuming that sliding and
the rotating mechanism are concentrated along the mud mortar joints. Adobe material is a weak
material characterized by a low modulus of elasticity. For this reason, the modelling approach
considering deformable block units is preferable. However, the model is a preliminary stage for the
65
study of a more large and complex structures, which, due to the computational efforts, requires the
use of rigid blocks. However, the comparison between deformable and rigid blocks was discussed,
in order to compare the two modelling approach in terms of ultimate load and damage pattern.
Thus, two different models were analysed (Figure 70), namely:
a) Model 1: The blocks are deformable and modelled with an internal FE mesh with an edge
equal to 0.20 m.
b) Model 2: The block are modelled as rigid. The faces were triangulated into sub-contacts in
order to increase the precision of the results obtained. In particular, a radial triangulation
with central node and radial edges;
(a)
(b)
Figure 70: Geometry of the model (a) Model 1 with rigid block model, (b) Model 2 with deformable block model
4.3.4
Material properties
The material properties of Kuño Tambo Church were obtained through an experimental campaign
conducted for the Seismic Retrofitting Project by the Pontificia Universidad Catòlica del Perù. The
data was integrated based on experimental tests available in literature review. The values are
summarized in § 3.3.5.
When deformable blocks are taken into account, the material properties assigned to the units are the
density, the Young’s Modulus and the shear modulus. When rigid blocks are considered, only the
mass density is applied to the blocks and all the deformability is concentrated to the joint stiffness.
Regarding the joints, 3DEC uses a soft contact representation. The contact stiffness is represented
by two springs in normal and shear direction, relating contact stresses with relative block
displacements (Figure 71) (Lemos, 2007).
66
Figure 71: Interface model code (Idris, et al., 2009)
When rigid blocks are adopted and the deformability is concentrated only along the joints, the
normal stiffness is calculated as an equivalent stiffness kn,eq (Tran, et al., 2014):
(4.2)
where kn,b is the normal stiffness of the block, calculated as a ratio between the Young’s Modulus of
the block Eb and its height Lb/2, and the stiffness of the joint is calculated as the ratio between the
Young modulus of the mud mortar Ej and its height Lj. Since the Poisson ratio is assumed equal to
0.25, the shear stiffness is assumed as 40% of the normal stiffness calculated. A viscous damping
was applied to the blocks, which is suitable for static analysis.
The deformable blocks are represented as elastic, in which the non-linear properties of the mortar
are only represented by the non-linear characteristic assigned to the joints. 3DEC uses a MohrCoulomb constitutive model. The joint cohesion, friction angle and tensile strength were defined
based on experimental tests and integrated with tests available in literature. Even in the case of rigid
block model, the inelastic properties are assumed localized only in the joints. A summary of the
mechanical properties of deformable and rigid model is presented in Table 13, Table 14 and
Table 15.
Table 13: Elastic properties of the deformable block model
γ [kN/m3]
E [MPa]
G [MPa]
kn [GPa/m]
ks [GPa/m]
Deformable blocks model – Elastic properties
Stone units
Adobe units
19
19
19400
114
776
46
Stone – stone interface
Adobe – adobe interface
2.67
5.33
5.33
1.07
2.13
2.13
:
Table 14: Elastic properties of the rigid block model
γ [kN/m3]
kn [Gpa/m]
ks [Gpa/m]
Rigid blocks model – Elastic properties
Stone units
Adobe units
19
19
2.60
0.52
0.94
1.04
0.21
0.37
67
Table 15: Inelastic properties of the deformable and rigid models
c [MPa]
ϕ [rad]
ft [MPa]
4.3.5
Rigid blocks and deformable blocks model – Inelastic properties
0.100
0.044
0.400
0.500
0.050
0.010
0.065
0.450
0.010
Application of the load of the roof
As discussed in the previous section (§3.3.4), due to the insufficient connection between the rafters
that constitutes the wooden roof and the adobe walls and due to its high deformability, the roof was
considered only as a weight on top of the walls. Due to its shape, the roof transfers horizontal load
to the structure. In order to take into account of the mass of the roof even during the pushover
analysis, the load is applied as an additional mass density localized on the top of the roof, while the
horizontal load is applied as horizontal stresses (Figure 72). The vertical load applied, converted in
additional mass density, is equal to 6.15 kN/m2 and the horizontal load is equal to 3.12 kN/m2. The
calculation values are in accordance with the ones used in the preliminary report (Karanikoloudis, et
al., 2015).
(a)
(b)
Figure 72: Application of the roof load: (a) Vertical load applied as additional mass density; (b) Horizontal
load
68
4.4 Methodology
Pushover analysis is a static non-linear numerical procedure where a structure is, in general,
subjected monotonic load increasing the amplitude. Applying the horizontal load, the structure
starts to exhibit non-linear behaviour through degradation of stiffness and redistribution of internal
forces (Nayeri, 2012). Eurocode 8 (Comitè Europèen de Normalisation, 2004) two load patterns for
the seismic analysis, namely:
1) A “uniform” pattern, based on lateral forces proportional to mass regardless of elevation
(uniform response acceleration);
2) A “modal” pattern, proportional to lateral forces consistent with lateral force distribution in
the direction under consideration determined in elastic analysis through the lateral force
method or modal response spectrum analysis.
The pushover analyses were performed considering a “uniform” pattern distribution. It is noted that
3DEC software has not yet implemented the possibility to apply the lateral load according to the
first mode shape.
First, a pushover analysis was conducted to assess the influence of the use of deformable blocks and
rigid blocks on the results. This first step is conducted with the aim to understand which could be
the influence of the application of the rigid blocks on the construction of a more complex model.
Then, the influence of the adobe rick pattern was analysed, due to uncertainties related to section
morphology of the wall. Furthermore, the assessment of a more reliable modelling approach for
rubble masonry stone was carried out and the influence of the pattern on the results conducted.
Lastly, a sensitivity analysis was carried out, aiming at evaluating the influence of the uncertainties
related to the material properties on the results.
The analysis was performed applying monotonically the horizontal load, proportional to mass, with
an increasing step size equal to the 1% of the total mass. The software 3DEC models a nonlinear
system that evolves in time, as a consequence the interpretation of the results in order to assess their
reliability may be more difficult than a conventional finite element program. For this reason, three
main indicators were monitored to assess the state of the numerical model for a static analysis, in
order to check the stability of the results:
1) Significant low values of the maximum unbalanced forces in comparison to the magnitude of
the internal forces that act in the whole model. The unbalanced forces are the forces
accumulated at each centroid, in case of rigid blocks, or at each gridpoint, in case of
deformable blocks, at equilibrium must be almost equal to zero. In other words, the forces that
act on the sides of each blocks must balance each other. (Comitè Europèen de Normalisation,
2004)
2) Significant low values of velocity of the blocks in relation to the displacement of the system
monitored;
3) Stable value of displacement after each step. The history of the displacement of the system is
monitored during the cycles to check that a stable value is obtained at the end of each step.
69
When a stable value is not obtained during the cycles implemented in each step, this means
that the results might be unstable or that the structure is due to collapse.
4.5 Results and discussion
4.5.1
Out-of-plane behaviour of the western wall
The out-of-plane behaviour of the western wall was evaluated based on the pushover analysis in the
two principal direction. As it is possible to observe in Figure 73, the maximum horizontal
acceleration is equal to 0.14 g in the positive direction (+X) and 0.23 g in the opposite direction
(-X). The lowest load capacity of the wall occurs in the direction of the horizontal force that is
transmitted by the roof to the adobe wall. In both cases, the collapse mechanisms corresponds to a
rotational rigid body due to the presence of high shear forces at the base.
Pushover analysis
Overturning of the western wall
0.25
Seismic coefficient [g]
Horizontal acceleration [g]
0.2
0.15
+X direction
-X direction
0.1
0.05
-X
+X
0
-0.1
-0.05
0
0.05
0.1
0.15
displacement[m]
[m]
Displacement
(a)
-X direction
+X direction
(b)
Figure 73: Out-of-plane behaviour of the western wall: (a) Capacity curves; (b) Collapse mechanisms
70
4.5.2
Comparison between deformable and rigid blocks modelling approach
A comparison between the results obtained with pushover analysis for Model 1 (deformable blocks
model) and Model 2 (rigid blocks model) is presented in the Figure 74 and Figure 75. The history of
the displacement of two points located at the top of the wall is presented, showing that model moves
with the same magnitude along the unitary width. Thus, the results were analysed base on one
reference point, namely the point at the top of the wall.
1 2
Pushover analysis
Displacement history - Rigid blocks model
0.1
0.09
Displacement [m]
0.08
0.07
0.06
0.05
0.04
0.03
Ref. Point 1
0.02
0.01
Ref. Point 2
0
0
50000
100000
150000
200000
250000
300000
Step [n.]
Figure 74: Displacement history – Model 2 (Rigid blocks model)
1 2
Pushover analysis
Displacement history - Deformable blocks model
0.07
0.06
Displacement [m]
0.05
0.04
0.03
0.02
Ref. Point 1
0.01
Ref. Point 2
0
0
100000
200000
300000
400000
500000
600000
700000
Step [n.]
Figure 75: Displacement history – Model 1 (Deformable blocks model)
71
The results showed that the difference in terms of ultimate load is equal to 1.5%, in which the lower
value was obtained for the model with rigid blocks (Figure 76). The damage and collapse patterns
are similar in both cases. It consists in a rigid body rotational due to high shear forces at the base of
the wall. The deformed shape is affected by the different deformability of the elements. In case of
rigid block model, the failure for the rigid block models, is sudden, due to the higher stiffness of the
elements involved. The first cracks are observed along the adobe-adobe interface, characterized by
low cohesion and low tensile strength, and it propagates at 45 degrees till the base of the wall.
Pushover analysis
Comparison between deformable and rigid blocks model
0.14
[g]
Horizontal
coefficient
Seismicacceleration
[g]
0.12
0.1
0.08
0.06
Deformable
blocks
0.04
0.02
Rigid blocks
Deformable blocks
Rigid blocks
0
0
0.02
0.04
0.06
0.08
Displacement
[m]
displacement
0.1
0.12
0.14
0.16
[m]
Figure 76: Pushover capacity curves for Model 1 (deformable blocks model) and Model 2 (rigid blocks model) and
collapse mechanism (magnification factor adopted for deformable blocks is equal 15 and for rigid blocks is 10)
Observing the elastic range, a small difference can be observed due to the irregular geometry of the
Voronoi pattern. Due to the irregular geometry of the blocks, more difficulties are encountered to
calculate the exact stiffness on the joint in the case of the rigid block model, in which even the
deformability of the blocks have to be taken into account.
The seismic analysis was evaluated mainly in terms of ultimate load and collapse mechanism. In
these terms, the results are very similar if deformable or rigid blocks are considered. Due to the less
computational efforts of the rigid blocks analysis, the out-of-plane failure of the west wall will be
conducted adopting rigid blocks for stones and adobe.
72
4.5.3
Influence of the adobe pattern on the results
Due to uncertainties related to the section morphology of Kuño Tambo Church, three different
configuration were analysed to assess the influence of the possible patterns on the behaviour of the
wall.
As previously mentioned, Kuño Tambo Church is constituted by adobe blocks disposed as an
English Bond pattern (Figure 77) (Cancino et al., 2012 ).
(a)
(b)
Figure 77: Adobe masonry pattern of the western wall: a) Pattern observed in situ; b) Structural prospection
A configuration analysed are based on a preliminary study conducted on the possible sectional
morphology for masonry units disposed as an English bond pattern (Nash, 1966). Furthermore, and
since the buttresses on the side of the façade are considered the prolonging of the lateral walls, the
adobe pattern of the observed on the southern side of the buttress were taken into account
(§ Pattern 1). The three configuration selected are presented in the Figure 78.
Figure 78: Geometry of the three possible masonry pattern
73
The three cases similar collapse mechanisms for the wall, which an overturning of the wall is
observed (Figure 79). However, the different distribution of the units that compose the masonry has
a slightly influence on the value of the collapse load. Pattern 1 and Pattern 3 have more similar
pattern. As a consequence, the crack pattern observed is the same and, thus, the value of the
ultimate load, which is equal to 0.136. On the other hand, the crack pattern correspondent to the
model with Pattern 2 is different and it includes a larger area (Figure 80). Thus, the collapse load is
higher and equal to 0.139. The difference is limited and only a percentage of 2.2% of difference in
terms of ultimate load is observed.
Pushover analysis
Sensitivity analysis based on different adobe unit pattern
0.14
[g][g]
acceleration
Horizontal
coefficient
Seismic
0.12
0.1
0.08
Pattern 1
0.06
Pattern 2
0.04
Pattern 3
0.02
Pattern 1
Pattern 2
Pattern 3
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
displacement
Displacement
[m] [m]
Figure 79: Pushover capacity curve – Sensitivity analysis based on the influence of adobe masonry pattern and collapse
mechanism (displacement magnification factor adopted is equal to 10)
Figure 80: Crack pattern of the wall, considering the three different adobe masonry patterns
74
4.5.4
Influence of the representation of rubble stone masonry
As previously discussed, the correct geometrical representation of the structure and its constituents
is fundamental to achieve reasonable results with discrete element method. The difficulties to
represent the exact geometry of rubble masonry walls require the adoption of schematization of the
masonry is necessary. The reference model adopted the use of Voronoi polygons adopting as
reference point the centroid of the stones that constitute the masonry. However, on the view of the
preparation of a complex structure, the need of a simpler and faster modelling approach is
necessary. For this reason, two different type of simplification were evaluated (Figure 81):
1) Adoption of a regular block pattern with unit blocks;
2) Adoption of larger Voronoi polygon, created through a random Voronoi generator.
The realization of Voronoi polygons starting from the centroid of the masonry unit as a reference
point is time consuming for the preparation of large and complex structure. One solution could be
the realization of 2D Voronoi polygons from a random generator. The polygons are generated with
the use of the software UDEC (Itasca Consulting Group (b), 2013). Then, a fish function was
constructed in order to export the coordinates of each blocks and import them in 3DEC software.
An average length of 0.40 m was assumed, which is almost two times larger than the average stone
length of the reference model (Model 2).
(b)
(a)
Figure 81: Adoption of simplified approach for the implementation of rubble masonry in large and complex structures:
(a) Adoption of a regular block pattern with bed joints; (b) Construction of random Voronoi polygons (average length
0.40 m) generated with UDEC software
It is noted that the use of random Voronoi polygons was already studied by Lemos (2009) for the
analysis of the out of plane failure of masonry walls prepared with irregular fabric and without the
presence of regular bed joints (Figure 82).
Figure 82: Horizontal force vs. displacement curves for regular and Voronoi block patterns (Lemos, et al., 2009)
75
The analyses were performed considering three different random patterns with an average length of
1 m (Figure 83). The results show that the use of Voronoi polygons presents results in terms of
deformability and collapse mechanism more similar to a masonry pattern with imbricated joints
with the smallest offset, since it does not consider any interlocking effect. The three different
patterns present similar capacity curves. However, it is noted that the study was carried out
considering the uniform shape along the thickness. In the present study, the results showed that the
adoption of a larger random Voronoi pattern, presents similar results in terms of collapse load to the
reference model (0.136) and the use of a regular block pattern with bed joints presents higher
collapse load (0.145; +6.6%), due to the interlocking of the elements.
The use of larger and random Voronoi polygons might offer a more realistic value of collapse load
instead of the use of a regular block pattern. The use of macroblocks, as already observed by
Alexandris (2004), might overestimate the collapse load value. However, the use of random
Voronoi polygons can approximate better than the regular block pattern, in terms of ultimate load.
In the present study, the Voronoi pattern was adopted to model the rubble masonry pattern along the
thickness.
Pushover analysis
Sensitivity analysis based on different rubble stone masonry modelling approach
0.14
Horizontal
[g]
coefficient [g]
Seismicacceleration
0.12
0.1
0.08
Regular block pattern
0.06
Reference model
(Model n. 2)
0.04
0.02
Regular block
pattern
Reference
model
Random Voronoi
polygons
Voronoi polygons random generator edge 0.4 m
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
displacement [m]
[m]
Displacement
Figure 83: Sensitivity analysis based on different rubble stone masonry modelling approach
76
4.5.5
Sensitivity analysis on the mechanical properties of the materials
The mechanical properties are calculated based on test campaign conducted by the Pontificia
Universidad Catòlica del Perù and integrated on values based on literature review. A sensitivity
analysis were performed taking into account the mechanical properties available in literature.
Furthermore, the analysis was performed on the view of the construction of large and complex
building.
Adobe-adobe unit interface
A sensitivity analysis was performed based on the cohesion value of the adobe-adobe interface.
Two typologies of tests were carried out by the Pontificia Universidad Catòlica del Perù on an
adobe-adobe triplet (§ 3.3.1). The first was conducted using existing adobe units and straw mud
mortar and the second was conducted using new adobe units and mud mortar without the addition
of the straw. The two results has a variability of the 20% in terms of cohesion and friction angle. In
addition, a third test was conducted on an adobe-stone triplets to assess the mechanical properties of
the adobe-stone interface, for which the specimen failed along the adobe-mortar interface. For this
reason, the value of cohesion and friction angle of the results obtained testing the adobe-stone
interface were analysed as an upper bound in terms of cohesion and friction angle of the adobeadobe interface. The results has difference in percentage from the reference value of 50% in terms
of cohesion and 11% in terms of friction angle (Table 16).
Table 16: Cohesion and friction angle values adopted during the sensitivity analysis
Adobe masonry
Lower value
Reference
Upper value
Cohesion
c =0.84 c0 (0.039 MPa)
c0 = 0.045 MPa
c = 1.5 c0 (0.068 MPa)
Friction angle
Φ = 0.90 Φ0
Φ0 = 0.50 rad
Φ = 1.2 Φ0
The results showed that the variability of the properties in terms of cohesion and friction angle
doesn’t significantly influence the out of plane failure of the western wall. Adopting the three
different cohesion values, we obtain the same ultimate load equal to 0.136g. The maximum
variability obtained varying the friction angle is equal to 3% and it is in correspondence of the
lower value of the friction angle adopted.
77
Pushover analysis
Sensitivity analysis - Cohesion - Adobe-adobe interface
0.14
[g] [g]
acceleration
Horizontal
coefficient
Seismic
0.12
0.1
0.08
c = 0.85 c0
0.06
c0 = 0.045 MPa
0.04
c = 1.5 c0
0.02
c = 0.85 c0
0
0
0.02
0.04
0.06
c = 1.5 c0
c0 = 0.045 MPa
0.08
0.1
0.12
0.14
0.16
displacement
Displacement
[m] [m]
Figure 84: Sensitivity analysis for cohesion adobe units
Pushover analysis
Sensitivity analysis - Friction angle - Adobe-adobe interface
0.14
acceleration
Horizontal
[g][g]
Seismic coefficient
0.12
0.1
0.08
frϕangle=0.9
= 0.9 ϕfr0 angle0
0.06
frϕangle=0.5
0 = 0.5 rad
rad
0.04
frϕangle=1.2
= 1.2 ϕfr0 angle0
0.02
ϕ0 = 0.9 ϕ0
ϕ0 = 0.5 rad
ϕ0 = 1.2 ϕ0
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
displacement[m]
[m]
Displacement
Figure 85: Sensitivity analysis for the friction angle value
78
No investigation were conducted to evaluate the mechanical properties of the rubble stone masonry.
The material properties considered were implemented based on values available in literature. For
this reason, a sensitivity analysis was performed to assess the influence of the variability of the
values available in literature. In particular, the variability of the cohesion value and of the friction
angle were evaluated (Table 17).
Table 17: Cohesion and friction angle values adopted during the sensitivity analysis
c0 = 0.1 MPa
Φ0 = 0.4 rad
c =0.7 c0 (0.07 MPa)
Φ = 0.90 Φ0
c = 3 c0 (0.3 MPa)
Φ = 1.2 Φ0
The results showed that, even the variability of the cohesion and friction angle adopted doesn’t
significantly vary the collapse mechanism and the ultimate load, equal to 0.136g. Only in
correspondence of the higher friction angle (ϕ = 1.2 ϕ0) we have a variability of 1.5%.
In general, the sensitivity analysis showed that the collapse mechanism of the western wall, namely
the out-of-plane failure, is governed by the tensile strength of the material and not by the cohesion
and friction angle of the materials. However, if the sliding is involved, the cohesion and the friction
angle can play a more relevant role and the variability could be higher.
Pushover analysis
Sensitivity analysis - Stone cohesion
0.14
acceleration [g]
Horizontalcoefficient
[g]
Seismic
0.12
0.1
0.08
c=0.7*c0
c0=0.1 MPa
0.06
c=3*c0
0.04
0.02
c = 0.70 c0
c0 = 0.10 MPa
c = 3 c0
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Displacement [m]
displacement
[m]
Figure 86: Sensitivity analysis for cohesion of stone
79
Pushover analysis
Comparison between deformable and rigid blocks model
0.14
[g][g]
Horizontal
coefficient
Seismicacceleration
0.12
0.1
0.08
ϕ = 0.9 ϕ0
ϕ0 =lower
0.4 rad
Rigid blocks
ϕ = 1.2 ϕ0
0.06
0.04
higher
0.02
ϕ = 0.9 ϕ0
ϕ0 = 0.4 rad
ϕ = 1.2 ϕ0
0
0
0.02
0.04
0.06
0.08
Displacement
[m]
displacement
0.1
0.12
0.14
0.16
[m]
Figure 87: Sensitivity analysis for friction angle of stone
4.6 Limit analysis
The discrete element method is a powerful tool to evaluate the crack pattern and the mechanism of
collapse of masonry structures. The results obtained with this method are compared with a simple
and hand calculation approach, widely used to evaluate the possible collapse mechanism of
masonry structure, namely the limit analysis based on the kinematic approach. The method is
particularly suitable for buildings characterized by poor connection between the structural vertical
and horizontal elements, where is possible to modelled the structure as an array of rigid blocks. As
already explained before, Kuño Tambo Church is characterized by the presence of a wooden roof,
which is not properly connected to the adobe walls: Furthermore, the seismic events and the lack of
maintenance of the buildings have caused several cracks between the structural elements. The use
of kinematic analysis is, thus, particularly suitable for the safety evaluation of the building analysed.
In this study, the simple overturning of the western wall was evaluated to compare the two method:
the limit analysis and the discrete element approach.
Three main hypothesis are usually adopted for the analysis of masonry structures with the limit
analysis based on the kinematic approach:
a) Masonry has zero tensile strength;
b) Masonry has infinite compressive strength;
c) Absence of sliding between blocks;
80
d) The factor that allows to activate the mechanism can be calculated through the virtual work
principle.
The infinite compressive hypothesis of the masonry can be reasonable assumed due to the fact that
masonry crushing has minor relevance in the response of masonry structures (Roca, et al., 2010).
However, when very shallow arches, pillars, towers or massive vertical structures are studied, the
compressive strength might have a relevance in the results. Kuño Tambo Church was built with
massive adobe walls. The western wall, which was analysed, has a slenderness ratio approximately
equal to 4. For this reason, the compressive strength was taken into account, assuming a value of
3 MPa. The results were compared with the limit analysis considering infinite compressive strength.
Different failure mechanism were assumed (Figure 88):
1) Overturning of the wall considering the rotational point located at the base (K1 and K3);
2) Overturning of the wall considering the rotational point located at the interface between
adobe and stone unit (K2);
3) Overturning of the wall considering the crack pattern according to the collapse mechanism
observed during the discrete element method (K4).
The non-linear kinematic analysis was performed, aiming at estimating also the ultimate
displacement of the structure.
K1
σc =
∞
K2
σc =
∞
K3
σc = 3 MPa
K4
σc =
∞
Figure 88: Collapse mechanisms assumed in the limit analysis
The results were compared with the fracture line method (FLM), which is an ultimate load
method suitable for laterally load walls. Applying this method, the damage occur only along the
fracture line. The parts of the structure divided by the fracture line rotate as rigid bodies. The
fracture line considered in this case is assumed in accordance with the crack pattern observed
during the analysis implemented with discrete element method. Every portion of the wall
divided by the fracture line is in equilibrium under the action of the external forces and the
81
reaction along the fracture line and supports. The internal work was calculated considering the
contribution of the tensile strength, according to the material involving the fracture (Figure 85).
ft = 0.01 MPa
f t = 0.05 MPa
Figure 89: Fracture line assumed for the fracture line method. The fracture line was divided assuming the different
material properties
Table 18 and Table 19 present the results of the limit analysis. The calculation developed to obtain
the results are presented in the Annex A.
Table 18: Comparison of the numerical methods adopted – Overturning of the western wall in the direction X+
K1 [σc = ∞]
K2 [σc = ∞]
K3 [σc = 3 MPa]
K4 [σc = ∞]
FLM [ft,ad = 0.01 Mpa,
ft,st = 0.05 Mpa]
Overturning of the western wall – Direction X+
Ultimate load
Ultimate load
[g]
[g]
Δ [%]
KA
DEM
0.16
0.14
+ 16%
0.22
K2 > K1
0.13
0.14
+ 6%
0.15
0.14
+ 7%
0.19
0.14
+36%
dk,0 [m]
dk,u [m]
0.64
0.63
0.49
0.59
0.26
0.25
0.20
0.24
-
-
Table 19: Comparison of the numerical methods adopted – Overturning of the western wall in the direction X-
K1 [σc = ∞]
K2 [σc = ∞]
K3 [σc = 3 MPa]
K4 [σc = ∞]
FLM [ft,ad = 0.01 MPa,
ft,st = 0.05 MPa]
82
Overturning of the western wall – Direction XUltimate load Ultimate load
[g]
[g]
Δ [%]
KA
DEM
0.26
0.23
+ 12%
0.32
K2 > K1
0.22
0.23
+ 3%
0.24
0.23
+ 5%
0.29
0.23
+25%
dk,0 [m]
dk,u [m]
0.81
0.77
0.69
0.78
0.32
0.31
0.27
0.31
-
-
The limit analysis based on the kinematic approach shows that the lower value, in both direction,
under the assumption of rigid block and infinite compressive strength, is obtained with the
configuration that assumes the crack in accordance with the one observed applying the discrete
element method. The fracture line method presents a load capacity significantly higher than the
capacity obtained from the reference DEM model. As already observed by Roca et al. (2010), the
assumption of rotation of rigid blocks at the base with infinite compressive strength might be
questionable for massive structures, as in the case of Kuño Tambo Church, since it might
overestimates the ultimate load of the structures. The analysis of massive walls assuming the
compressive strength takes into account the crushing, which can lead to obtain acceptable results.
83
5 OUT-OF-PLANE BEHAVIOUR OF THE FAÇADE
During the damage survey conducted in Peru after the 2007 Pisco Earthquake, one of the most
damage mechanism observed was the separation between the longitudinal walls and the main
façade (Blondet, et al., 2008), due to the lack of connection between the transversal wall. For this
reason, the study of the seismic behaviour of Kuño Tambo was analysed considering the behaviour
of the main façade, which exhibit separation cracks with the transversal walls.
The use of Discrete Element Method was applied to study the collapse mechanism of the southern
façade. The applicability of the method adopted is considered suitable for adobe construction, since
is able to predict large displacement that characterized the behaviour of adobe buildings. However,
the use of discrete element when large and complex structures are studied requires the
simplification of the structures in macroblocks, due to the high computational effort that the
modelling of the single structural unit would require. On the other hand, adobe buildings are built
with mud mortar and adobe unit characterized by the same material properties. Thus, the crack
pattern observed in the buildings does not always follow the mortar step pattern but cracks are
observed along the adobe unit. For this reason, the applicability of the discrete element approach
with the use of a sophisticated model that adopts a large number of blocks was studied, in order to
understand the suitability of the approach to the structure analysed.
5.1 Geometry of the model
In the partial model prepared to study the seismic behaviour of the southern façade, the interaction
with the transversal wall is studied. To take into account of the influence of the western and eastern
walls, the partial model presents also the two walls with an influence of almost 10 meters (Figure
90).
Figure 90: Geometry of the partial model of the façade
84
The façade has a constant thickness of 1.91 m, while the transversal walls has a thickness variable
from 1.59 m to 1.71 m. Due to the lack of connection between the baptistery walls and the eastern
wall, the out-of-plane behaviour of the façade was evaluated without taking into account the effects
of the baptistery walls (Figure 91a).
The lateral buttresses are a prolongation of the lateral walls and were constructed well connected to
the southern façade through an overlapping of the English bond pattern, adopted for the realization
of all the adobe walls that constitutes the building.
As already mentioned before, the adobe walls are constituted by adobe units characterized by the
same material properties of the mud mortar between them. As a consequence, the crack pattern
observed does not follow the step pattern of the mortar joints (Figure 91b). Furthermore, the main
cracks were observed along the masonry walls.
(b)
(a)
Figure 91: (a) Lack of connection between the baptistery and the eastern wall; (b) Crack along the southern wall of the
baptistery. The crack does not follow the adobe step pattern
The definition of the geometry plays a fundamental role in the results that can be obtained by
discrete element modelling and, as a consequence, on the reliability of the model constructed.
The definition of the block size and the realization of the geometry of the model was constructed
based on a study of the adobe pattern of the walls and on the damage mechanism observed in situ.
As already mentioned in Chapter 4.1, the needs of simplification of the masonry pattern with the
adoption of larger block size is necessary in the application of discrete element method for large and
complex structure. Furthermore, the study aims to evaluate the role of the adobe pattern and of the
connection between the walls on the response to seismic events.
For this reason, the block size adopted during the construction of the model are essentially two:
1) Rectangular blocks 0.7x0.35x0.2 cm, which represents the exact geometry of the adobe
units. The size and the disposition is respectful of the real geometry;
2) Macro Voronoi Blocks for the representation of the rubble stone masonry.
85
The model is constitute by 6886 blocks (Figure 92), which is in general a limit number for the
realization of a complex model with the use of discrete element method that can give reasonable
computational effort.
The choice to represent the rubble stone masonry block with macroblocks and the adobe walls with
the use of real dimension blocks is driven by the consideration that the main cracks due to seismic
events were observed along the adobe walls. As a consequence, the dimension of the block is
driven, as suggested by Lemos (2009), by the need not to avoid possible collapse mechanisms.
Smaller blocks that can represent closer the reality were adopted for adobe units, while macroblocks
were adopted to represent the rubble stone masonry course. The representation of the blocks as rigid
is necessary to reduce the computational efforts.
The walls are massive and constituted by the disposition of several blocks (from 3 to 5). The
representation of the exact geometry of the adobe units aims to evaluate the influence of the pattern
along the longitudinal development of the wall and along the thickness on the seismic behaviour.
Furthermore, the realization of the exact pattern of the adobe units allows the representation of the
interlocking between the orthogonal walls. Several considerations can be made to understand how
significant is the connection of the walls and how, on the contrary, due to the brittleness of the
adobe units, the interlocking of the walls does not play a significant role.
Figure 92: Geometry of the model
The rubble masonry stone base course is represented through the use of Voronoi polygons. In this
case, the geometry is not carefully represented but the use of macroblocks is necessary to decrease
the computational effort. As already discussed in Chapter 4, the use of random Voronoi generator is
86
necessary for large and complex structures. The choice to simplify the rubble stone masonry walls
with Voronoi polygons instead of regular masonry walls, as was observed in Chapter 4, was
conducted not to overestimate the collapse load of the structure due to the interlocking of the block
pattern. The random Voronoi polygons were prepared with the use of the software UDEC (Figure
93). The edge size assumed is equal to 0.70 m, which is almost the maximum stone dimension
observed in situ. The Voronoi polygons are two dimensional, in which the direction is parallel to the
direction of the load that was applied to perform the pushover analysis. The coordinates were then
imported in 3DEC and some modification were applied manually to avoid the local failure of the
stones located at the border of the wall due to the presence of unrealistic inclined mortar joints.
Figure 93: Voronoi polygons created in UDEC software to represent the rubble masonry stone course of the western
wall
The arched structure was simplified through the use of larger macroblocks due to the impossibility
to represent small element size and the absence in the software to represent curved elements (Figure
94). The wooden architraves were prepared through the use of continuous rigid blocks elements
(Figure 95).
Figure 94: Crack observed on the southern wall of the baptistery. The configuration does not follow the step pattern of
the mud mortar
87
Figure 95: Geometry of the model. In red are indicated the rigid blocks that act as architraves
5.2 Material properties
Since the blocks are modelled as rigid, the elastic and inelastic properties are concentrated in the
joints. A Mohr Coulomb constitutive model was adopted to define the inelastic behaviour of the
structure.
The material properties that are implemented in the model are the same that are presented in
Chapter 3 for the definition of the overturning of the western wall, applying rigid block model.
However, the elastic stone properties are resized considering that larger Voronoi blocks are
implemented. The material properties adopted are presented in the Table 20 and Table 21.
Table 20: Elastic properties for rigid block model
γ [kN/m3]
kn [GPa/m]
ks [GPa/m]
88
Partial model of the façade – Elastic properties
Stone units
Adobe units
19
19
1.36
0.52
0.54
0.21
Architraves – timber
4
0.80
0.32
Table 21: Inelastic properties for the rigid block model
Partial model of the façade – Inelastic properties
0.100
0.044
0.400
0.500
0.050
0.010
c [MPa]
ϕ [rad]
ft [MPa]
0.065
0.450
0.010
The definition of the geometry of the model starts from the realization of larger blocks that then has
to be discretized in small rigid block that represents the constituent of the structure. For this reason,
vertical joint all across the adobe wall are present (Figure 95). The vertical joint were assumed as
rigid in order to guarantee the continuity of the blocks modelled. If the joints are located at the
corner of the walls, they were assumed elastic (with the same stiffness of the joints but with infinite
values of cohesion and tensile strength), to allow a certain deformability at the corner.
5.3 Load application
The roof was considered only as a load applied to the structure, due to the lack of connection and its
deformability. The horizontal forces transmitted to the walls were also taken into account. The
lateral walls receive in fact not only the vertical load of the roof but even the horizontal forces due
to the morphology of the rafters. The roof proceed along the whole lateral walls. Thus, the load is
applied even to the façade, but no horizontal forces are transmitted. The tie are not taken into
account. The dynamic identification demonstrates that the role of the tie is not significant for the
dynamic properties. Furthermore, no information are available about their effectiveness. The
vertical load applied on the structure presented in the Table 22. The value are selected in
accordance with Karanikoloudis and Lourenço (2015).
Table 22: Vertical load applied to the model to represent the dead load received from the roof system
Dead load applied from the roof system
Lateral walls
Façade
Vertical load [kN/m2]
6.15
2.33
Horizontal component [kN/m2]
3.12
-
In the previous model, the vertical load was apply as an increased density located at the top of the
wall, in order to take into account of it in the pushover analysis conducted (Figure 96). In this case,
more difficulties were encountered due to the presence of the gable end that constitutes the façade.
For this reason, surface regions were assigned in the model where the vertical load has to be
applied. Then, boundary stresses were assigned to the surface region: A programming language
FISH, which enables to user define variables and function in 3DEC, was used to include as
horizontal loads applied to the structures, the contribute of the vertical load from the roof.
89
Figure 96: Application of the dead load from the roof and definition of the surface region
5.4 Model calibration
The discrete element model was updated, correlating the second experimental mode shape of the
structure, which involves the overturning of the walls with consequent inward movement of the
lateral walls (Figure 63). A modal analysis under linear elastic behaviour was performed and the
natural frequency obtained was equal to 2.72 Hz. It is noted that the natural frequency of the second
mode based on the dynamic tests performed (§ 3.4) is equal to 2.20 Hz.
The calibration of the model was carried out by the method proposed by Douglas and Reid (1982),
in which the frequency is estimated by means of:
where
are the variables to calibrate and
and
are constants. The
constants can be obtained through a system of equation, where a base value , an upper bound
and a lower bound
of the variables to calibrate are necessary to be defined:
Figure 97: System for the definition of the system of equation
90
The stiffness of the adobe-adobe interface unit was taken as a reference value. The variable selected
for the calibration of the model was the normal stiffness of the adobe-adobe joint. The upper and the
lower bound were taken from the results of the compressive tests conducted on adobe specimen in
Peru (Table 23). The value of the compressive strength ranges from 0.48 to 0.84 MPa. Assuming
the correlation between compressive strength and Young Modulus proposed by Caporale et al.
(2015), where Ecb = 160 fcb, the Young’s Modulus were calculated taking into account the lower
value and the higher value of compressive strength obtained during tests. As a consequence, the
equivalent stiffness can be calculated to define the lower bound and the upper bound of the value.
Table 23 presents the values for the parameters.
Table 23: Definition of the upper and the lower bound of the variable (keq) chosen to (Douglas, et al., 1982) perform
the calibration
Lower bound
Reference value
Upper bound
E,lb = 0.7 Eref
Eref
Eu,b = 1.2 Eref
Eb [MPa]
80
114.08
140
Lb [m]
0.1
0.1
0.1
Ej [MPa]
56
80
96
Lj [m]
0.015
0.015
0.015
kb [GPa/m]
0.80
1.14
1.40
kj [GPa/m]
3.73
5.33
6.40
keq [GPa/m]
0.36
0.52
0.63
The decreased value obtained performing the calibration test is equal to 0.768. From the decreased
value obtained, the adobe-stone interface and the stone-stone interface were decreased of the same
factor. A modal analysis were performed again with the value updated of the stiffness presented in
the Table 24.
Table 24: Elastic properties of the model after the calibration
kn [GPa/m]
ks [GPa/m]
Partial model of the façade – Elastic properties – Model calibrated
1.04
0.40
0.62
0.42
0.16
0.25
The natural frequency of the analysis conducted with the updated value is equal to 2.22 Hz, with a
percentage of difference of 0.9%. The mode shape is represented in the Figure 98.
91
Figure 98: First mode shape of the structure in the out-of-plane direction of the façade
The value of the stiffness of the model calibrated is consistent with the granulometric analysis
performed by the Pontificia Universidade Catòlica del Perù (Figure 53), that showed the same soil
composition both for mortar and unit. The value calibrated, which corresponds to 0.396, is exactly
the value obtained assuming the same Young Modulus for both mortar and unit (Table 25).
Table 25: Calculation of the modulus of elasticity after the calibration of the model
Values calibrated
E,lb = 0.7 Eref
Eb [MPa]
85
Lb [m]
0.1
Ej [MPa]
85
Lj [m]
0.015
kb [GPa/m]
0.85
kj [GPa/m]
5.67
keq [GPa/m]
0.396
The value assumed for the Young’s Modulus of the adobe unit and of the mortar is 85 MPa, which
is included in the range available in literature review (§ 2.2). Applying the correlation of Caporale
et al. (2015), the compressive strength of adobe unit is equal to 0.53 MPa.
92
5.5 Pushover analysis
5.5.1
Assessment of the inelastic properties assigned to the corner between orthogonal and
transversal walls
The study was conducted in order to assess all the possible collapse mechanism of the structure and
the collapse load associated to the failure mode.
The studied involved the possible collapse mechanism related to the interaction between the façade
and the transversal wall. The damage pattern of adobe buildings observed during the surveys
conducted after the seismic events has in fact assessed that one of the weakest link of the building
was the corner between orthogonal walls. However, the corner of the church was constructed with
overlapping adobe units, with the aim to guarantee connection between the orthogonal walls.
The construction method of the model required the realization of prismatic elements that constitute
each wall that has to be divided in rigid blocks that represent the constituents of the masonry. As a
consequence, the blocks that, due to the effective connection of the walls, belong to both
intersecting walls are cut by an initial model construction joint. To deal with this problem, it is
necessary to assign elastic and inelastic properties to the construction joint (Figure 99).
Figure 99: Construction corner joint between the façade and the transversal walls
Assigning to the corner the same inelastic properties of the interface between the unit and the
mortar could underestimate the ultimate load of the local failure. It would mean no connection
between the walls. On the other hand, the representation of the corner as rigid could be too stiff and
overestimate the ultimate load and do not give reliable results in terms of damage pattern and
collapse load. For this reason, the corner façade-transversal walls was represented in two different
configuration:
1) Inelastic joint with the characteristic of the adobe unit, which characterize the corner. No
experimental tests were performed to investigate the value of tensile strength and cohesion
of the adobe unit. For this reason, the value of the tensile strength was assigned based on the
tests conducted by Silveira et al. (2012) and Adorni et al. (2013) based on existing adobe
units, respectively in Portugal and in Turkmenistan. The tensile strength ft was assumed
0.12 MPa, which corresponds to the lower bound of the values obtained during the tests.
The cohesion c was calculated based on the tensile strength as ft = 1.5 c, and equal to 0.18
MPa (Model 1).
93
2) Elastic joint: The corner has the same stiffness of the joint that constitutes the mud mortar
but with infinite cohesion and tensile strength (Model 2). This allow to present
deformability along the joint but continuity of the adobe pattern represented;
The results are compared with the assumption of the lack of connectivity between the transversal
walls (assuming the characteristic of the mud mortar), to assess the differences in terms of capacity
and collapse mechanism if the connection between the walls were absent (Model 3).
5.5.2
Methodology
A pushover analysis was performed to assess the possible collapse mechanism of the façade,
considering the interaction with the orthogonal walls. As already mentioned, 3DEC software has
not implemented yet the possibility to conduct a pushover analysis according to the first mode
shape of the structure. As a consequence, the pushover analysis was conducted with a load
distribution proportional to the masses. It is noted that the Kuño Tambo Church is constituted by
massive structures with very low slenderness ratio. Thus, the pushover analysis proportional to
masses could be the most interesting for the study of the collapse mechanism of Kuño Tambo
Church.
The convergence of the results are conducted with the same control parameters applied for the
analysis of the out of plane of the western wall, namely low relative out-of-balance force, low
velocity and the control of a stable displacement at the end of each cycle (§ 4.4). The results were
analysed assuming as a reference point the point placed at the top of the façade (Figure 100).
Reference point
Figure 100: Reference point assumed for the analysis
94
5.5.3
Results and discussion
Outward overturning of the facade
The results of the pushover analysis (capacity curves and maximum horizontal acceleration)
conducted for the three models are presented in Figure 101 and Table 26. As it can observed, the
presence of an overlapping connection between the walls can significantly improve the ultimate
load of the structure. Furthermore, the results were compared with the existing crack pattern.
Pushover analysis
Assessment of the properties of the corner transversal walls-facade
4
3.5
3
2.5
2
Model n. 1
1.5
Model n. 2
1
Model n. 3
0.5
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
displacement
[m]
Displacement
[m]
Figure 101: Capacity curves for the outward mechanism of the façade
Table 26: Ultimate load capacity for the outward overturning of the façade
Model 1
0.24 g
Ultimate load capacity – Outward overturning of the façade
Model 2
0.34 g
Model 3
0.15 g
The collapse mechanism of the Model 1 and 3 correspond to the overturning of the façade (Figure
102; Figure 103). The façade rotates like a rotational body around the base. The crack starts from
adobe masonry and it propagates at 45 degrees till the base of the wall. A similar mechanism was
already observed studying the overturning of the western wall. The length of the lateral walls
assumed in the model, equal to almost one third of the real length, is sufficient in these cases to
represent the stiffness of the overall wall.
Observing the crack pattern of the building (Figure 104), vertical cracks are visible along the corner
with the transversal walls inward and outward the wall. This means that the overturning of the
façade, predicted in Model 1 is a possible collapse mechanism, as already observed during the
historical damage survey.
95
Figure 102: Collapse mechanism for the Model 1
Figure 103: Collapse mechanism for the Model 3
Figure 104: Vertical cracks at the corner between transversal walls and the façade
96
Assuming the collapse mechanism as elastic (Model 2), the damage pattern is similar observing the
displacement in Y direction in Model 1 (orthogonal to the direction of the load) (Figure 108a) and
the damage pattern observed on site (Figure 105b). However, in the interior wall, the crack is
predicted in the model not along the corner in correspondence of the inner wall of the façade, but
along the transversal walls (Figure 106). This might mean that the simulation of elastic joints might
be too stiff to represent the reality. Discrete element approach approximate, in fact, the adobe unit
as rigid and so the interlocking at the corner is too stiff if the adobe unit are represented as rigid and
the intersection corner as elastic.
(a)
(b)
Figure 105: Damage pattern observed: (a) Y-displacement of the Model 2 (0.34 g), (b) Damage survey of the building
Figure 106: Detail of the damage at the corner of the Model 2 (0.34 g),
97
The ultimate load of the Model 2 is higher than the one predicted in the case of Model 1 and 3, due
to the simulation of stiffer connection at the corner. Thus, due to the assumption of infinite cohesion
and tensile strength, the mechanism involve the lateral walls (Figure 108). The collapse starts with
the collapse of the western wall, which is characterized by a greater height and a higher
deformability in comparison to the eastern wall. In particular, the failure of the western wall starts
with a local failing of the stones located at the base of the eastern wall and close to the large
opening that brings to the baptistery (Figure 107).
Figure 107: X-displacement of the Model 2 (seismic coefficient 0.34 g)
Figure 108: Collapse mechanism of the façade with the assumption of elastic connection at the intersection
The collapse mechanism observed could be influenced by the simplification adopted in the local
model, where the stiffness of the transversal walls are represented considering only one third of
their length. High deformation are present at the end of the wall, which could probably not represent
the real behaviour of the whole wall. To overcome this difficulty in the consideration of a local
98
model, a rigid block was constructed adjacent to the transversal wall and an elastic joint that
simulates the deformability of the wall was implemented, with the following characteristics:
kn=Emasonry/(Lw,wall-Lw,mod)
ks=0
where kn is the normal stiffness, ks is the tangential stiffness, Lw,wall is the total length of the western
wall and Lw,mod is the length of the wall considered in the model. The tangential stiffness is equal to
zero in order to allow the wall to settle gravity (Figure 109).
From the deformation, it is possible to observe the effect of the spring on the deformation of the
wall (Figure 109b). The simplification has the limit that does not allow, with the realization of the
rigid block, the deformation of the wall in the opposite direction of the application of the horizontal
load due to vertical loading, but is a good approximation to observe if, taking into account of the
additional stiffness, the collapse behaviour can change. The difference on ultimate load is 6%, in
which inserting the elastic spring that simulate the deformability of the walls the collapse load is
higher, as expected (Table 27 and Figure 110). In terms of failure mechanism, the collapse involves
a rotational movement of the façade and part of the transversal walls (Figure 111).
(a)
(b)
Figure 109: (a) Simulation of the stiffness of the whole wall (western wall); (b) Deformation of the western wall with
the insertion of the elastic springs
Table 27: Ultimate load capacity for the outward overturning of the façade and elastic corner
Ultimate load capacity - Outward overturning of the façade – Elastic corner
Model 2
Model 2 with elastic spring
0.34 g
0.36 g
99
Pushover analysis
0.4
Horizontal
coefficient [g][g]
Seismicacceleration
0.35
0.3
0.25
0.2
Model n. 2 - with
elastic spring
0.15
0.1
Model n. 2
0.05
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
displacement
Displacement
[m] [m]
Figure 110: Pushover analysis with elastic joints at the corner transversal walls-façade – Comparison of the results
A FEM model of Kuño Tambo Church was constructed for a preliminary seismic analysis of the
structure (Karanikoloudis, et al., 2015). It was observed that the same diagonal cracks were
predicted even in the FEM model, which adopted a macro-modelling approach. In terms of ultimate
load, the seismic coefficient predicted with FEM model is equal to 0.27 g. The difference in terms
of ultimate load is affected by several factors. The FEM model represents the whole structure and
the inelastic behaviour is characterized by a total strain base crack model, in comparison to Discrete
Element Approach which adopts a Mohr-Coulomb friction law concentrated at the joints between
the rigid blocks. Furthermore, the adoption of Discrete Element Approach to large and complex
structure requires the realization of larger and squared blocks, which, as already observed by
Alexandris et al. (2004), might overestimate the capacity of the structure.
Figure 111: Collapse mechanism of the Model n. 2 with elastic springs
100
Inward mechanism of the façade
The ultimate load of the inward mechanism is lower than the one predicted for the outward collapse
mechanism (Figure 112 and Table 28). However, this value could be affected by several factors,
namely the asymmetrical configuration of the façade through the section and the simplification
adopted in the modelling of discrete element method. The use of larger Voronoi polygons and one
possible masonry pattern can significantly influence the behaviour of the structure. Furthermore, to
validate the results, the need of model the same structure with different dimension of Voronoi
polygons and different masonry pattern has to be adopted.
For the Models 1 and 3, the inward mechanism is characterized by the overturning of the façade
(Figure 113). In these models, the length of the transversal walls represented is enough to represent
the stiffness of the transversal walls. For Model 2, the local model is not reliable to represent the
stiffness of the transversal walls and the collapse mechanism involves the failure of the transversal
walls, which does not represent the real behaviour of the structure (Figure 114).
Pushover analysis
0.4
0.35
0.3
0.25
0.2
0.15
0.1
Model n. 1
Model n. 2
0.05
Model n. 3
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
displacement
Displacement
[m][m]
Figure 112: Capacity curves for the inward mechanism of the façade
Table 28: Ultimate load capacity for the inward overturning of the façade
Ultimate load capacity – Inward mechanism
Model 1
0.19 g
Model 2
0.37 g
Model 3
0.14 g
101
(a)
(b)
Figure 113: Collapse mechanism for the (a) Model 1; (b) Model 3
Figure 114: Collapse mechanism for the inward collapse of the façade of the Model 2
102
6 CONCLUSION AND FUTURE WORK
The seismic analysis of the Kuño Tambo Church in Cuzco (Peru) was carried out based on Discrete
Element Method (DEM). Being one of the first researches where the method is employed on such
large and complex structures, the work was focused on a sensitivity analysis of a significant unitary
length of the western wall, characterized by high deformability, followed by the study of the out-ofplane behaviour of the facade (3D model) and its interaction with the transversal walls. The main
conclusions can be summarized as follows:
-
-
-
-
-
-
Rigid blocks can be suitable than the deformable blocks for the analysis of adobe
construction in terms of failure mechanism and ultimate load. The elastic and inelastic
properties, in case of rigid blocks, are concentrated only in the joints. In terms of ultimate
load, the difference between the use of deformable and rigid block is 1.5%, in which the
lower value is represented by the rigid blocks. This brings the advantage of less
computational efforts in the simulation of large and complex structures;
The influence of adobe pattern does not present significant influence on the failure
mechanism of the wall. A slightly difference can be observed when the distribution of the
unit allow the formation of a larger cracked area. The three possible configuration analyzed
showed a difference in terms of ultimate load of 2.2%. However, different behaviour was
observed in terms of collapse mechanisms and higher ductility is observed for the pattern
that allow the formation of a larger cracked area;
Voronoi polygons prepared with the use of a random generator algorithm provides better
simulation in terms of ultimate load and collapse mechanism, in comparison with the model
with regular stone masonry pattern, which results in a higher collapse load due to the block
interlocking;
The sensitivity analysis based on uncertainty related to the mechanical properties of the
materials (cohesion and friction angle of adobe-mortar and stone-mortar interface) showed
no significant variance in terms of ultimate load value for the out-of-plane mechanism of the
wall. As a consequence, the maximum horizontal acceleration of the western wall due to
out-of-plane mechanism is equal to 0.14 g. The maximum variability is about 3%;
The results obtained with DEM were validated through the limit analysis based on the
kinematic approach, which is considered a simple and powerful tool for the study of the
possible collapse mechanisms of masonry structures characterized by low connectivity
between structural elements. The result range is in agreement to the values calculated with
DEM. The most vulnerable mechanism obtained with limit analysis corresponds to the
collapse mechanism obtained with DEM. The assumption of infinite compressive strength in
kinematic analysis leads to overestimate the ultimate load for massive structures while
consideration of the compressive strength leads to more conservative results;
The stiffness value of the adobe-adobe interface obtained by the calibration of the partial
model of the structure based on dynamic identification tests is in agreement to the
granulometric analysis, in which the adobe unit and mud mortar are prepared with the same
material, and exhibit similar modulus of elasticity;
103
-
-
-
-
The connection of the façade to the transversal walls has an important role on the out-ofplane collapse of the main façade. According to the results, modelling the joint assuming the
inelastic properties of the adobe unit is more conservative than simplifying it as elastic. In
the case of elastic joint, the failure mechanism does not only involve the failure mechanism
of the façade but even part of the transversal walls. As a consequence, the increment in
terms of maximum horizontal acceleration is approximately equal to 0.12 g (an increase
of 50%);
The results obtained demonstrated that the increase of the connectivity of the wall can play a
significant role in terms of seismic resistance of adobe constructions. Strengthening
solutions that improve the connectivity of the wall are thus important to reduce the seismic
vulnerability of these structures;
The analysis showed that the DEM characterized by a large number of blocks is suitable for
adobe construction. However, careful attention has to be paid on the choice of the dimension
and the geometry of the blocks, in order not to overestimate the ultimate load and allow all
the possible collapse mechanisms involved;
The use of the partial model can be suitable for the study of the outward overturning of the
façade. However, the inward overturning should be studied through more extensive
modelling of Kuño Tambo Church, in order to take into account the real stiffness of the
transversal walls.
Regarding the partial model of the façade, the future research on the seismic analysis of Kuño
Tambo Church can involve the modelling of different Voronoi random pattern of different size to
assess, even for large and complex structures, the influence of the rubble masonry modelling,
approached with the use of macroblocks. Furthermore, the baptistery, which is constructed adjacent
to the main body of the church, can be analysed separately to understand the possible collapse
mechanism and the causes of the existing cracks present along the southern, eastern and northern
walls.
The analysis with a complete model is should be also carried, aiming at evaluating the global
behaviour of the structure.
Figure 115: Ongoing construction of the full model adopting 3DEC software
104
The study of the global behaviour should be performed by using macroblocks, due to the high
complexity of the structure. The interconnectivity of the walls can be assess considering the
inelastic properties of adobe unit in the presence of a proper interlocking between the orthogonal
walls. When the walls are only adjacent, as in the case of the eastern wall-baptistery wall
connection, the inelastic properties of the mud mortar can be assigned. The model should be
calibrated based on the dynamic properties obtained from the dynamic identification tests. The
existing ties should be also modelled to assess the influence and the interconnectivity between the
structural elements. Furthermore, a non-linear dynamic analysis with time integration can be
conducted in order to evaluate the seismic behaviour of the church and to perform a comparison
with respect to the results obtained from the pushover analysis. Finally, the simulation of the
existing crack pattern can be inserted to evaluate the difference in terms of capacity for the actual
condition, since the structure already present damage caused by several earthquakes in the past.
105
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112
APPENDIX A: LIMIT ANALYSIS CALCULATIONS
The calculation of the limit analysis with kinematic approach conducted are here presented. The
figure below indicates the load involved in the calculation. Four different configuration are taken
into account. The description of the method applied is described in Chapter 4.
Froof
Hroof
F1
P1
Proof
F2
P2
Y
F3
P3
X
LIMIT ANALYSIS – DIRECTION X+
1)
A) CASE K1 – DIRECTION X+
LIMIT ANALYSIS
Volume [m³]
Density [kN/m³]
P [kN]
x centroid [m]
δX
y centroid [m]
V1
0.42
γ
19
P1
7.95
x1
1.09
y1
6.61
δx.1
0.95
V2
9.17
γ
19
P2
174.18
x2
0.80
y2
3.65
δx.2
0.52
V3
0.30
γ3
19
P3
5.61
x3
0.82
y3
0.43
δx.3
0.06
σc [MPa]
0
Proof
10.73
x4
0.82
y4
6.70
δx.4
0.96
Hroof
5.36
x5
1.64
y5
6.70
t1
0.00
Htotal
6.95
δr
1.00
h.f
3.84
δx.f
0.55
///Linear kinematic analysis
///Non Linear kinematic analysis
Ms [kNm]
125.32
M* [kNm]
18.82
θ [rad]
0.17
Mr [kNm]
761.64
e* [m]
0.93
θ [deg]
9.57
2
a0 [m/s ]
1.74
dk.0 [m]
0.64
a0/g
0.18
d.0 [m]
0.64
du [m]
0.26
α0
0.16
113
B)
CASE K2 – DIRECTION X+
LIMIT ANALYSIS
Volume [m³]
Density [kN/m³]
P [kN]
δX
x centroid [m] y centroid [m] - 0.85
x centroid [m]
V1
0.42
γ
19
P1
7.95
x1
1.09
y1
5.76
δx.1
0.94
x1
1.09
V2
9.17
γ
19
P2
174.18
x2
0.82
y2
2.80
δx.2
0.46
x2
0.80
Proof
10.73
x4
0.82
y4
5.85
δx.4
0.96
x4
0.82
Hroof
5.36
x5
1.64
y5
5.85
x5
1.64
t1
0.00
Htotal
6.10
δr
1.00
t1
0.00
h.f
3.09
δx.f
0.51
Ms [kNm]
128.99
M* [kNm]
18.15
θ [rad]
0.21
Mr [kNm]
595.33
e*[m]
0.92
θ [deg]
11.78
a0 [m/s ]
2.30
dk.0 [m]
0.63
a0/g
0.235
d.0 [m]
0.63
du [m]
0.25
α0
C)
2
0.22
LIMIT ANALYSIS
Volume [m³]
Density [kN/m³]
P [kN]
x centroid [m]
δX
y centroid [m]
V1
0.42
γ
19
P1
7.95
x1
1.093
y1
6.61
δx.1
0.95
V2
9.17
γ
19
P2
174.18
x2
0.80
y2
3.65
δx.2
0.52
V3
0.30
γ3
19
P3
5.61
x3
0.82
y3
0.425
δx.3
0.06
σc [MPa]
3
Proof
10.73
x4
0.82
y4
6.695
δx.4
0.96
Hroof
5.36
x5
1.64
y5
6.695
t1
0.13
Htotal
6.95
δr
1
h.f
3.84
δx.f
0.55
Ms [kNm]
99.06
M* [kNm]
18.82
θ [rad]
0.13
Mr [kNm]
761.65
e* [m]
0.93
θ [deg]
7.35
2
a0 [m/s ]
1.37
dk.0 [m]
0.49
a0/g
0.140
d.0 [m]
0.49
du [m]
0.20
α0
114
0.13
D)
LIMIT ANALYSIS
Volume [m³]
Density [kN/m³]
P [kN]
x centroid [m]
y centroid [m]
δX
V1
0.42
γ
19
P1
7.95
x1
1.093
y1
6.61
δx.1
0.95
V2
8.71
γ
19
P2
165.55
x2
0.80
y2
3.78
δx.2
0.54
V3
0.30
γ3
19
P3
5.61
x3
0.21
y3
0.524
δx.3
0.08
Proof
10.73
x4
0.82
y4
6.695
δx.4
0.96
Hroof
5.36
x5
1.64
y5
6.695
t1
0
Htotal
6.95
δr
1.00
h.f
3.97
δx.f
0.57
///Non linear kinematic analysis
Ms [kNm]
114.9744436
M* [kNm]
18.10
θ [rad]
0.15
Mr [kNm]
753.0585772
e* [m]
0.94
θ [deg]
8.61
α0
0.15
a0 [m/s2]
1.60
dk.0 [m]
0.59
a0/g
0.163
d.0 [m]
0.59
du [m]
0.24
2)
LIMIT ANALYSIS – DIRECTION X-
A) CASE K1 – DIRECTION XLIMIT ANALYSIS
Volume [m³]
Density [kN/m³]
P [kN]
x centroid [m]
δX
y centroid [m]
V1
0.42
γ
19
P1
7.95
x1
0.55
y1
6.61
δx.1
0.95
V2
9.17
γ
19
P2
174.18
x2
0.82
y2
3.65
δx.2
0.52
V3
0.30
γ3
19
P3
5.61
x3
0.82
y3
0.43
δx.3
0.06
σc [MPa]
0
Proof
10.73
x4
0.82
y4
6.70
δx.4
0.96
Hroof
-5.36
x5
/
y5
6.70
t1
0.00
Htotal
6.95
δr
1.00
h.f
3.84
δx.f
0.55
Ms [kNm]
196.5
M* [kNm]
18.82
θ [rad]
0.21
Mr [kNm]
761.6
e* [m]
0.93
θ [deg]
12.19
2
a0 [m/s ]
2.72
dk.0 [m]
0.81
a0/g
0.28
d.0 [m]
0.81
du [m]
0.32
α0
0.26
115
B)
CASE K2 – DIRECTION XLIMIT ANALYSIS
Volume [m³]
Density [kN/m³]
P [kN]
x centroid [m]
δX
y centroid [m]
V1
0.42
γ
19
P1
7.95
x1
0.55
y1
6.61
δx.1
0.95
V2
9.17
γ
19
P2
174.18
x2
0.82
y2
3.65
δx.2
0.52
V3
0.30
γ3
19
P3
5.61
x3
0.82
y3
0.43
δx.3
0.06
σc [MPa]
0
Proof
10.73
x4
0.82
y4
6.70
δx.4
0.96
Hroof
-5.36
x5
/
y5
6.70
t1
0.00
Htotal
6.95
δr
1.00
h.f
3.84
δx.f
0.55
Ms [kNm]
196.5
M* [kNm]
18.82
θ [rad]
0.21
Mr [kNm]
761.6
e* [m]
0.93
θ [deg]
12.19
2
a0 [m/s ]
2.72
dk.0 [m]
0.81
a0/g
0.28
d.0 [m]
0.81
du [m]
0.32
α0
C)
0.26
CASE K3 – DIRECTION XLIMIT ANALYSIS
Volume [m³]
Density [kN/m³]
P [kN]
x centroid [m]
δX
y centroid [m]
V1
0.42
γ
19
P1
7.95
x1
0.547
y1
6.61
δx.1
0.95
V2
9.17
γ
19
P2
174.18
x2
0.82
y2
3.65
δx.2
0.52
V3
0.30
γ3
19
P3
5.61
x3
0.82
y3
0.43
δx.3
0.06
σc [MPa]
3
Proof
10.73
x4
0.82
y4
6.70
δx.4
0.96
Hroof
-5.36
x5
/
y5
6.70
t1
0.13
Htotal
6.95
δr
1.00
h.f
3.84
δx.f
0.55
Ms [kNm]
170.20
M* [kNm]
18.82
θ [rad]
0.18
Mr [kNm]
761.64
e* [m]
0.93
θ [deg]
10.30
2
a0 [m/s ]
2.36
dk.0 [m]
0.69
a0/g
0.24
d.0 [m]
0.69
du [m]
0.27
α0
116
0.22
D) CASE K4 – DIRECTION XLIMIT ANALYSIS
Volume [m³]
Density [kN/m³]
P [kN]
x centroid [m]
y centroid [m]
δX
V1
0.42
γ
19
P1
7.95
x1
0.55
y1
6.61
δx.1
0.95
V2
8.71
γ
19
P2
165.55
x2
0.80
y2
3.78
δx.2
0.54
V3
0.30
γ3
19
P3
5.61
x3
0.21
y3
0.52
δx.3
0.08
Proof
10.73
x4
0.82
y4
6.70
δx.4
0.96
Hroof
-5.36
x5
1.64
y5
6.70
t1
0.00
Htotal
6.95
δr
1.00
h.f
3.97
δx.f
0.57
///Non linear kinematic analysis
Ms [kNm]
182.41
M* [kNm]
18.10
θ [rad]
0.20
Mr [kNm]
753.06
e* [m]
0.94
θ [deg]
11.36
α0
0.24
a0 [m/s2]
2.54
dk.0 [m]
0.78
a0/g
0.26
d.0 [m]
0.78
du [m]
0.31
3) FRACTURE LINE ANALYSIS
Fracture line method direction X-
Fracture line method direction X-
Geometry
Geometry
H.tot [m]
6.95
L [m]
1.64
Load [kN] b.x [m] b.y [m]
H.tot [m]
L [m]
6.95
1.64
Load [kN]
b.x [m]
b.y [m]
P1 [kN]
174.18
0.82
3.87
P1 [kN]
174.18
0.82
3.87
Proof [kN]
10.78
0.82
6.70
Proof [kN]
10.78
0.82
6.70
Hroof [kN]
5.36
-
6.70
Hroof [kN]
-5.36
-
6.70
ft [kN/mq]
l [m]
ft.adobe-adobe
10
0.96
1.17
0.73
ft.stone-stone
50
1.26
0.43
0.73
b.y [m] angle [rad]
ft [kN/mq]
l [m]
b.y [m]
angle [rad]
Ft,adobe-adobe
10
0.96
1.17
0.73
ft.stone-stone
50
1.26
0.43
0.73
W.ext,1
107.37
W.ext,1
107.37
W.ext,2
16.10
W.ext,2
26.43
W.int
4.07
W.int
4.07
α
0.19
α
0.28
117

Seismic Analysis of the Church of Kuño Tambo

Transcription

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