Seismic Failure Behaviour of Masonry Domes Under Strong Ground Motions

Seismic Failure Behaviour of Masonry Domes Under Strong
Ground Motions
Alemdar Bayraktar (  alemdarbayraktar@gmail.com )
 Karadeniz Technical University: Karadeniz Teknik Universitesi https://orcid.org/0000-0002-8973-9228
Emin Hökelekli
 Bartın Üniversitesi: Bartin Universitesi

Research Article

Keywords: Masonry domes,Damage propagation patterns, Failurebehaviour, Nonlinear seismic response, Strong ground
motion

Posted Date: March 15th, 2022

DOI: https://doi.org/10.21203/rs.3.rs-1378266/v1

License:   This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License

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Abstract
General stability and failure behaviours of masonry domes under static loads have been detailly searched in literature.
However, researchers have devoted little attention to their seismic failure behaviours under strong ground motions. This
study aims a better understanding of seismic failure behaviours of masonry domes with support system including drum
and buttresses using advanced 3D nonlinear numerical simulations under strong ground motions. Four types of
masonry domes built on historical structures are selected such as a dome with circle drum, a dome with circle drum and
buttress, a dome with octagonal drum, a dome with octagonal drum and buttress. The three-dimensional solid and finite
element models of the selected masonry domes are created using isotropic continuum macro modelling technique with
homogenized properties. Concrete Damage Plasticity (CDP) model is chosen for masonry material behaviour. Three
different strong ground motion acceleration records of 1999 Düzce (M = 7.14), 1992 Erzincan (M = 6.69) and 1999
Kocaeli (M = 7.51) earthquakes are selected and matched to the target response spectrum with return period of 475
years in TBEC (2019) using the wavelet algorithm. Static and seismic failure behaviours of the four masonry spherical
dome models subjected to the matched strong ground motion records are compared and evaluated using maximum
principal stresses, damage propagation patterns and failure angles. Failure behaviour angles under strong ground
motions are proposed for spherical masonry domes with support systems and thickness-to-span ratios t/R = 0.092.

1. Introduction
Masonry domes, which are the curved and spherical structural components, have been extensively utilized to cover
spaces of historical structures such as temples, mausoleums, palaces, forts, baths, churches, mosques, etc. They are
subjected to internal and external effects such as changes in the loading configuration, ageing of materials,
environmental degradation, temperature, settlement, experience of several earthquakes, lack of maintenance, etc. Due to
these factors and very low tensile strength of masonry units, they have high seismic vulnerability. Heavy damages and
collapses occured on the masonry domes of historical structures in the past (Pavlovic et al. 2016; Grillanda et al. 2019a
and b; Preciado et al. 2020). Therefore, developing appropriate assessment approaches and analysis methods for
reducing seismic vulnerabilities and preserving the cultural and artistic values of masonry domes are currently a topic of
great interest.

Extensive research has been carried out to determine structural behaviours of historical masonry domes using
theoretical and experimental methods. Theoretical methods such as analytical approaches, simplified procedures,
discrete element method, and micro or macro-modelling based on the finite element method are considered in the
analyses of historical masonry domes. Theoretical, experimental and strengthening studies performed on the historical
masonry domes under static and dynamic effects can be classified as follows: i) theoretical studies under static loads
[Heyman, 1967, 1977; Farshad, 1977; Cowan, 1977, 1981; Kuban, 1987; Oppenheim et al. 1989; Karaesmen, 1993;
Pesciullesi et al. 1997; Bilgin, 2006; Lucchesi et al. 2007; Atamturktur and Boothby, 2007; Milani et al. 2008; Chiorino et
al. 2008; Corradi et al. 2009; Polidano and Fried, 2012; Milani and Tralli, 2012; Bacigalupo et al. 2013; Reyhan et al. 2013;
Baratta, 2013; Ventura et al. 2014; Palmisano, 2014; Foraboschi, 2015; Rovero and Tonietti, 2011, 2014; Fabbrocino et al.
2015; Cavalagli and Gusella, 2015; Como, 2016; Coccia et al. 2016; Pavlovic et al. 2016; Simon and Bagi, 2016; Varma
and Ghosh, 2016; Galassi et al. 2017; Cennamo et al. 2018; Como, 2019; Grillanda et al. 2019a and b; Scacco et al. 2020;
Hejazi and Pourabedin, 2021; Barsotti et al. 2021; Jasienko et al. 2021; Cusano et al. 2021; Nodargi and Bisegna, 2021a,
b, c and d; Sharbaf et al. 2021], ii) theoretical studies under dynamic loads [Atamturktur and Sevim, 2012; Bartoli et al.
2015; Mahdi, 2017; Beatini et al. 2018; Öztürk et al. 2020; Feizolahbeigi et al. 2021], iii) model tests in laboratuary
[Erdogmus, 2008; Zessin et al. 2010; Zessin, 2012; Atamturktur et al. 2012; Li and Atamturktur, 2014; Sorensen and
Erdogmus, 2015], iv) in-situ tests [Aoki et al. 2011; Chiorino et al. 2011; Çalık et al. 2014, 2016, 2020; Bartoli et al. 2016;
Uçak et al. 2016; Ceravolo et al. 2017; Pecorelli et al. 2018], v) strengthening studies [Bloch et al. 2004; Milani et al. 2009;
Milani and Bucchi, 2010; Portioli et al. 2011; Mortezaei et al. 2012; Moeeni and Ghasem Sahab, 2013; Brencich et al.
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2014; Koseoglu and Canbay, 2015; Ottoni and Blasi, 2015; Fraternali et al. 2015; Chmielewski and Kruszka, 2015;
Fabbrocino et al. 2015; Soler-Estrela and Soler-Verdú, 2016; Chiozzi et al. 2017a and b; Panto et al. 2017; Hamdy et al.
2018; Varma et al. 2018; Aşıkoğlu et al. 2019; Aghabeigi and Farahmand-Tabar, 2021; Bayraktar et al. 2022].

It can be understood from the above literature review that the theoretical and experimental behaviors of masonry domes
under static and dynamic loads were investigated in detail. In addition, studies on the failure behaviour of historical
masonry domes under gravity and static loads were also prepared by the researchers. Limit and combined limit-
numerical analysis methods under gravity loads has been widely used for this purpose (Heyman, 1967, 1977;
Oppenheim et al. 1989; Como, 2013, 2016, 2019; Milani et al. 2008, 2009; Milani and Bucchi, 2010; Foraboschi, 2014;
Ventura et al. 2014; Pavlovic et al. 2016; Anania and D'Agata, 2017; Chiozzi et al. 2017a and b; Sharbaf et al. 2021;
Nodargi et al. 2021c and d; Scacco et al. 2022). While there have been numerous studies on the general stability and
ultimate behaviour of masonry domes under gravity loads, researchers have devoted little attention to their seismic
failure behaviour (Zessin, 2012; Grillanda et al. 2019a and b) under strong ground motions. This study aims to
investigates a better understanding of the seismic failure behaviour of masonry domes with different drums and
buttresses using advanced 3D nonlinear numerical simulations under strong ground motions. Seismic failure
behaviours of four masonry dome models subjected to different strong ground motions are evaluated using maximum
principal stresses, damage propagation patterns and failure angles.

2. Failure Behaviour Of Masonry Domes Under Vertical Static Loads
The thickness-to-span ratio (t/R, where t and R are thickness and centerline radius of the dome (Fig. 1a)) and the lateral
springing thrust are the two main parameters for the failure behaviour of masonry domes (Foraboschi, 2014). A dome is
a surface that can be divided into a series of wedges. Meridional and hoop forces occur along the wedges and the
parallel bands (Fig. 1a), respectively (Heyman, 1967, 1997; 2013; Lucchesi, 2007; Foraboschi, 2014; Pavlovic, 2016;
Grillanda et al. 2019a; Zessin, 2012; Como, 2013; Öztürk et al. 2020). The first damages due to vertical static loads occur
along meridians (Fig. 1b). Then, when the meridional stresses exceed the tensile strength of masonry unit, hoop cracks
appear along the hoops as shown in Fig. 1b (Heyman, 1967, 1977; Pavlovic et al. 2016; Como, 2019).

3. Seismic Failure Behaviour Of Masonry Domes With Drum And Buttress
Masonry dome types with circle and octagonal drums, which were widely used in the historical masonry mosques, are
considered to determine seismic failure behaviours in this study. Some views from the masonry domes with circle and
octagonal drums and buttresses are shown in Fig. 2.

Four types of masonry domes built on historal structures are selected to determine static and seismic damage patterns
and failure behaviours under different strong ground motions. These are a dome with circle drum (Dome A), a dome with
circle drum and buttress (Dome B), a dome with octagonal drum (Dome C), a dome with octagonal drum and buttress
(Dome D). The selected dimensions of the dome and drum are plotted in Fig. 3. The masonry domes have the same
geometrical and material properties. Dome materials were chosen as brick, and drum and buttress material as stone.
Masonry system with mortar is taken into account in the models. The radius (R), drum height (h) and thickness (t) of the
selected masonry domes are 6 m, 2.5 m, and 0.55m, respectively. The thickness ratio of the domes (t/R) is 0.092, and
they can be classified as semispherical dome.

The three-dimensional (3D) solid and finite element models of the selected masonry domes shown in Fig. 4 were created
using isotropic continuum macro modelling technique with homogenized properties in Abaqus (2010). Maximum
dimension of each finite element in the models is selected as 0.3m. Hexahedral 8-node linear brick element (C3D8R) with

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reduced integration are used for all finite element models. Total element numbers of Dome A, B, C and D models are
8480, 9616, 9968 and 10480, respectively.

Advanced 3D nonlinear numerical simulations provide the most accurate and reliable seismic assessment of historical
masonry constructions when the nonlinear behavior of the masonry material is properly defined (Valente and Milani,
2019). 3D nonlinear seismic analyses of the selected masonry dome models are implemented by using the Concrete
Damage Plasticity (CDP) model proposed by Lubliner et al. (1989). The uniaxial tensile (σto) and compressive (σcu)
stress behaviours plotted in Fig. 5 are defined as follows,

 σ to = (1 − dt )E 0 (ϵt − ϵplt )(1)
 σ cu = (1 − dc )E 0 (ϵc − ϵplc )(2)
 pl pl
where E 0 is initial modulus of elasticity, ϵ c and ϵ t are the total strain in compressive and tensile conditions, ϵ c and ϵ t
are the equivalent plastic strain in compressive and tensile conditions, and d c and d t are the compressive and tensile
damage parameters (Abaqus, 2010; Bayraktar and Hökelekli, 2020). The brick and stone material properties selected
from the literature for the four masonry dome models are given in Table 1 (Scacco et al. 2022; Bayraktar and Hökelekli,
2020). Strain values of mortar are considered for the failure behaviour evaluations.

 Table 1
 Material properties for masonry units
 Brick masonry units Stone masonry unit

 E (MPa) γ (kg/m3) ν E (MPa) γ (kg/m3) ν

 1500 1800 0.15 2400 2300 0.15

 Uniaxial tensile stress–strain Tensile damage Uniaxial tensile stress–strain Tensile damage
 values parameters values parameters

 σ to(MPa) ϵt
 pl dt pl
 ϵt σ to(MPa) pl
 ϵt dt pl
 ϵt

 0.0480 0.0000 0.000 0.0000 0.2000 0.000 0.000 0.000

 0.0005 0.0007 0.950 0.0007 0.0005 0.007 0.950 0.007

 0.0005 0.1000 0.0005 0.100
The characteristics of the selected three earthquakes, which are 1999 Düzce (M = 7.14), 1992 Erzincan (M = 6.69) and
1999 Kocaeli (M = 7.51) earthquakes, are given in Table 2. The outcropping (original) acceleration records are matched
to the target response spectrum with return period of 475 years in TBEC (2019) using the wavelets algorithm in
SeismoMatch software (URL-5). The outcropping and matched records and spectrums of the selected acceleration
records are plotted in Fig. 6.

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Table 2
 The characteristics of the selected earthquakes (URL-6)
 Event Record Date Rrup Magnitude File Name PGA Shear wave
 Seq. (g) velocity (m/s)
 (km)

 Erzincan 821 13/03/1992 4.38 6.69 ERZINCAN_ERZ_EW.AT2 0.50 352.05

 Kocaeli 1158 17/08/1999 15.37 7.51 KOCAELI_DZC270.AT2 0.32 281.86

 Düzce 1602 12/11/1999 12.04 7.14 DUZCE_BOL090.AT2 0.51 293.57
The matched acceleration records shown in Fig. 6 are applied to the finite element models of the selected masonary
domes in the horizontal direction. The effective duration of the acceleration records is assumed equal to 8s because of
the high computational demands. The Full Newton-Raphson method with the time increment of 0.005s and Rayleigh
damping coefficients are considered in the nonlinear analyses. The nonlinear static analyses of the masonry dome
models are firstly implemented to obtain their static behaviors under self-weight. The minimum (compressive) and
maximum (tensile) principal static stress contour maps of four dome models under self-weight are depicted in Fig. 7.
The maximum values of compressive static stresses in Dome A, B, C and D under self-weight are calculated as 0.204,
0.179, 0.200 and 0.178MPa, respectively. The maximum nonlinear static stress in the Dome A, B, C and D under self-
weight were calculated 0.074, 0.064, 0.079 and 0.055, respectively. The maximum values of the compressive stresses
are concentrated at the bottom region of the drum while the tensile stresses are concentrated in the dome-drum
connection regions. The propagations and values of compressive and tensile static stresses occurring in Dome B and D
models are less than in Dome A and C models. It can be stated that the buttresses reduce the compressive and tensile
static stresses values and propagations on the masonry domes.

Static analysis results obtained from the self-weight of the masonry dome models are considered as initial values of all
nonlinear seismic analyses. The maximum seismic principal (tensile) stress distributions shown in Fig. 8 are plotted at
the end of the selected total duration of the earthquake records. Maximum tensile strength of masonry unit for the dome
models was chosen as 0.048MPa (see Table 1). It can be seen from Fig. 8 that the maximum tensile stresses occurring
in Dome A exceed the limit tensile stress value for the selected three acceleration records. In Dome B, the maximum
tensile stress is exceeded only under Kocaeli earthquake acceleration record. The maximum tensile stresses occurring in
Dome C exceed the limit tensile stress under Erzincan and Kocaeli earthquakes, while those of Dome D exceed the limit
tensile stress under the Düzce and Kocaeli earthquakes. Considering the three selected earthquake records, the
maximum tensile stresses occur under the effect of the Kocaeli earthquake. The buttresses significianlty reduce the
values and propagations of tensile stress in both circular and octagonal drummed domes (Dome B and D).

The maximum tensile failure behaviours of the dome models are plotted in Fig. 9 at the end of the selected duration of
the strong ground motions records. DAMAGET in below figures stand for tensile damage parameter and indicates
maximum tensile damage (failure) level occurring in each element. Blue color represents zero damage (dt = 0), red color
is correlated to full (complete) damage (dt = 1). It can be seen Fig. 9 that the element tensile failure levels in Dome A, B, C
and D models for the acceleration records of Düzce, Erzincan and Kocaeli earthquakes are obtained as 0.95, 0.93, 0.95,
0.95; 0.95, 0.84, 0.92, 0.87; 0.95, 0.90, 0.95, 0.95, respectively. Failure behaviours of hemispherical domes with the
thickness-to-span ratio t∕R = 0.1 for tilting table test and limit analyses under horizontal static forces proportional to self
weight in literature are shown in Fig. 10. It can be seen from Figs. 9 and 10 similar failure behaviours under horizontal
effects are observed between this study results and those of the literature.

In order to evaluate the analysis results more clearly, the elements with a maximum tensile damage rate above 80% are
accepted to be heavily damaged. The percentages of the damaged element numbers are shown in Fig. 11. Considering
the distribution and percentages of elements reaching the maximum tensile damage, the most damage occurs in Dome

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A and C models, and the least damage occurs in Dome B and D models. When comparing four dome models, the least
damage occurs in Dome B model with circular drum and buttresses. The buttresses positively affect the failure behavior
of the masonry dome subjected to strong ground motions and reduces the damage distributions. In addition, it can be
seen from Figs. 9 and 11 that different matched earthquake records affect failure behavior and the number of damaged
elements.

The location and spreading of hoop tension crack regions in Dome A, B, C and D model sections and the angles of the
starting and ending points of the damaged regions to the vertical axis are shown in Figs. 12 and 13 and summarized in
Table 3 for Düzce, Erzincan and Kocaeli earthquake records. For the selected earthquake records, hoop tension crack
regions in Dome A vary between 49–90 degrees, while it varies between 46–77 degrees in Dome B. Failure distribution
regions in Dome C and Dome D sections vary between 46–90 and 43–77 degrees, respectively. When Table 3 is
examined, it is seen that the average interval of hoop tension failure angles of Dome A, B, C and D models range
between 39, 25, 35 and 26 degrees. The failure angles extend to the support region in Dome A and C models while in
Dome B and D models, they occur away from the support region.

 Table 3
 Angles of damaged regions of the dome models
 Dome Model Angle to vert. axis Earthquakes Angle interval

 Düzce Erzincan Kocaeli

 Dome A Start 50o 49o 53o 39o

 End 90o 90o 90o

 Dome B Start 46o 53o 50o 25o

 End 74o 74o 77o

 Dome C Start 46o 61o 46o 35o

 End 90o 80o 90o

 Dome D Start 43o 57o 49o 26o

 End 77o 74o 77o

4. Conclusions
Static and seismic numerical failure behaviours of masonry dome models with different drum and buttresses subjected
to strong ground motion records have been determined using the finite element method in this study. The findings drawn
from the current works based on advanced nonlinear static and seismic analyses of masonry domes with the thickness-
to-span ratio approximately t∕ =0.092 are summarized follows:

 Less damage occurs in masonry domes with circular drum geometry and buttresses compared to octagonal
 masonry domes under static and seismic loads. The buttresses significanlty reduce the tensile stress values,
 propagations and damages in both circular and octagonal drummed masonry domes.
 Earthquake failure behaviors of masonry domes under strong ground motions differs from the vertical static ones.
 Different scaled strong ground motion records affect the failure behaviors of masonry domes and the number of
 damaged elements.

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The angles of the starting and ending points of the damaged regions to the vertical axis of masonry dome models
 with circle drum (Dome A), with circle drum and buttress (Dome B), with octagonal drum (Dome C) and with
 octagonal drum and buttress (Dome D) vary between 49–90, 46–77, 46–90 and 43–77 degrees, respectively.
 The average angle intervals of hoop tension failure regions of Dome A, B, C and D models range between 39, 25, 35
 and 26 degrees, respectively.

To generalize the results on the failure behaviours of spherical and ellipsoidal masonry domes subjected to strong
ground motions, it is recommended that advanced nonlinear seismic analyses should be performed for different
thickness-to-span ratios, such as 0.1, 0.2, in further studies.

Declarations
The authors declare that they have no conflict of interest. The authors declare that no funds, grants, or other support
were received during the preparation of this manuscript.

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Figures

Figure 1

Hoop and meridional forces (a) and cracks(b) of masonry domes under vertical static loads (Heyman, 1997; Como,
2013; Öztürk et al. 2020)

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Figure 2

Some views from the masonry domes with circle and octagonal drums and buttresses (URL-1, 2, 3 and 4)

Figure 3

Dimensions and sections of the selected domes

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Figure 4

3D solid (a) and finite element (b) models of the selected domes with different drumsand buttresses

Figure 5

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Stress-strain curves for Concrete Damage Plasticity (CDP) model (Abaqus, 2010)

Figure 6

The outcropping and matched acceleration records and spectrums of the Düzce (a), Erzincan (b) and Kocaeli (c)
earthquakes

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Figure 7

Minimum (a) and maximum (b) principal stress contour maps of the dome models

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Figure 8

Maximum principal (tensile) stress contour maps of four dome models under different strong ground motions

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Figure 9

Tensile stress damage propagations of the selected masonry domes under different strong ground motions

Figure 10

Failure behaviours of hemispherical domes with normalized thickness t∕R=0.1 for limit analyses under horizontal static
forces (a) and tilting table test (b) (Nodargi and Bisegna, 2021; Zessin, 2012)

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Figure 11

Percentages of damaged elements in brick masonry domes for different strong ground motions

Figure 12

Tensile failure angles occuring in Dome A and B models

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Figure 13

Tensile failure angles occuring in Dome C and D models

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