Effect of buttress form on transverse seismic resistance of high masonry walls

In historical masonry structures having vault, dome and high walls, for safety against gravity, wind and earthquake loads, buttresses are crucial elements. This study aims to investigate the effect of buttress form on the transverse seismic resistance of high masonry walls. For this purpose, a real historical masonry structure covered by a barrel vault, and has a simple and regular plan and elevation shape was considered. From a slice of this structure, a calculation model and a basic model were created. For the masonry, macro-modelling approach was used. Nonlinear static analysis method was adopted, and finite element modeling and calculations were carried out using the Abaqus program. Firstly, on the calculation model the effect of buttress depth on transverse seismic resistance was investigated, and it was seen that the resistance is almost linearly related to the buttress depth. Then, two groups of analyses were performed on the models obtained by mounting different buttress forms to the basic model. In the first group, the volumes of the buttresses were kept equal, and in the second group their base depths and thicknesses were taken equally. At the end of these analyses, by comparing the base shear forces obtained for the models, the effectiveness of buttress form on the transverse seismic resistance has been determined and evaluated. The results revealed that the model with curvilinear concave buttresses in the first group analyses and the model with semi-cylindrical buttresses in the second group analyses have the highest seismic resistance. 750 f c  11738 MPa and f t = 0.06 f c  1 MPa in the study. The Poisson's ratio was taken as 0.20, which is widely used for brittle materials.

mosques, cathedrals, castles, caravanserais, missions etc, we see that high walls of these structures had been propped by buttresses almost every time. This shows indisputably that ancient builders were well aware of the importance of buttresses. This is also evident from the fact that most historical building design rules that have survived today mention intensively about the buttress design (Huerta, 2010).
This study deals with the effect of buttress form on the transverse (perpendicular to longitudinal direction) seismic resistance of high masonry walls which are frequently encountered in building type historic masonry structures. In the second section buttress forms encountered in historical structures reviewed. Third section gives a literature review about the works on masonry buttresses. In the fourth section which forms the main body of the study, the method is given, and by considering different buttress forms, transverse seismic resistance of a real historical masonry structure is investigated. In this section, the results obtained are also discussed. After presenting the conclusions in the fifth section, the study is finished with references.

Buttress types in historic masonry structures
Besides some secondary factors such as maintenance, restoration and strengthening, many historical masonry structures have reached present day from the past, thanks primarily to the wisdom, deep foresight and experience of ancient builders. These structures are living examples of correct understanding of the structural behavior by their builders. It can be said that the buttresses are one of the principal indicators of this correct understanding.
To the authors' knowledge, there is no precise information about when and where the buttress was used for the first time. However, it can be seen from many scientific studies that the buttress was used in many buildings in Mesopotamia, the cradle of the oldest civilizations. Over time, the buttress was used in various structures of all civilizations.
There are close definitions for the "buttress" in the literature. Based on these definitions and its function, the buttress can be defined simply as "a projection built up against a wall or a structure, to form additional strength or furnish support", (Smith, 1880). Buttresses have been used in almost all building type structures, castles, dams, bridges, retaining walls, etc. in the past.
It is seen that in most of the historical structures, buttresses are original, i.e. they had been built together with the walls of the structure. However, in some cases, it is understood from historical documents and/or on-site inspections that they were added to the structure later. The famous Hagia Sophia in İstanbul, whose some buttresses were added later by the great Ottoman architect "Mimar Sinan", can be given as an example to these structures.
It is observed that the buttresses had been positioned in three ways with respect to the walls in the building type historical masonry structures. On the outer face of the wall, which is the most common situation, on the inner face and, some part on the outer face and the other on the inner. In the vast majority of structures, these buttresses are perpendicular to the walls. But, for the corners of intersecting walls, there are also examples where the buttresses had been settled diagonally.
When investigated it is seen that, in historical masonry structures there are a wide variety of forms for the buttresses. In choosing of buttress form, builders may have considered the following factors: i) requests of building owners, ii) aesthetics, iii) construction simplicity, iv) height of construction, v) tradition, vi) climate conditions, vii) seismicity and viii) wind effect.

Previous works on masonry buttresses
Because of their primary role for the safety of most of the building type historical masonry structures, various investigations have been carried out in the past on sizing, behaviour, capacity and collapse of masonry buttresses. In this section, summaries of a good part of these studies are presented briefly.
By making a concise interpretation of successful material minimalism in Gothic architecture, Bartholomew (1840), with a leaning human posture model, explained the association of the flying buttress, the pinnacle and the external buttress trio in supporting the structure, Fig. 2.
Ungewitter and Mohrmann (1890, 1892) wrote a very detailed two-volume book on the construction and behavior of Gothic cathedrals. They used generally graphic static analysis techniques and gave a wide coverage to both classical and flying buttresses. It can be definitely said that, the book is one of the most comprehensive works that deals with the buttresses in such detail. Sanabria (1982) engaged in the mechanization of structural design in the 16th century. He commentated on the Spanish architect Rodrigo Gil de Hontãnón's 16th century master masons' formulas concerned with the safe dimensioning of buttresses. Mark (1984) in his fascinating book investigated the structural elements of Gothic cathedrals.
He used the photoelastic modeling technique of structural mechanics to solve historical arguments such as whether the flying buttresses hold the roof or are simply decorative, the ornate pinnacles on the buttresses are structurally necessary or completely aesthetic, and whether the ribs of the vaults hold the ceiling, (Url-5).
In his PhD thesis (1990) and in his another work (2004) Huerta gives vast information about the sizing rules of buttresses during the medieval and Renaissance eras. It is understood clearly from the Huerta's these two precious works that the ancient builders had a rather high level of awareness about the importance of buttresses for the safety of the whole structure.
In his famous book "The Stone Skeleton: Structural Engineering of Masonry Architecture", Heyman (1995) provides a comprehensive and at the same time intuitive understanding especially of historical masonry structures. In this respect, he also explained the behavior of some structural elements, including flying buttresses, and gave knowledge with various examples. Ochsendorf et al. (2004) investigated the collapse of masonry buttresses under concentrated lateral loads. They determined that, a fracture forms at the collapse state and this decreases the overturning resistance of buttress significantly. And therefore, they pointed out that, the conventional analysis which assumes that a masonry buttress acts monolithically to resist lateral loads is unsafe, and the possibility of a fracture at the collapse state must be considered in the design and assessment of masonry buttresses. Ochsendorf and De Lorenzis (2008) considering a number of parameters, such as sliding, finite compressive strength and leaning, investigated the collapse condition of rectangular masonry buttresses subjected to a concentrated inclined load. García and Meli (2009) aimed at to find out whether the 16th century Mexican convent church builders followed structural rules stated in the building treatises from that period, when dimensioning the structural elements, especially the walls and buttresses which counteract the thrust of the vaults. Huerta (2010) performed a comprehensive investigation on the safety of masonry buttresses.
Within this context, he outlined the development of buttress design since the 16th century, and explained the main approaches used. In the study, the effect of the buttress form in terms of stability against overturning is also briefly examined.
De Lorenzis et al. (2010aLorenzis et al. ( , 2010b in their two companion studies, taking into account fracturing prior to overturning and possible sliding, carried out detailed collapse analyses of trapezoidal and stepped masonry buttresses. (2015) studied on the fracturing of rectangular masonry buttresses and towers when subjected either to a concentrated oblique force at their head or to lateral inertial loading due to earthquake ground motion. For the inertial loading, the equivalent static analysis procedure was adopted, and a uniform and an inverted triangular load distribution were considered. Karimi et al. (2016), through library studies and, descriptive and analytical research method, investigated the typology and developments of buttresses in Iranian architecture. They examined 32 archaeological sites from prehistoric and historical periods. The buttresses in these sites were examined and recorded in terms of their features such as position, form, material, technical function and ornamentation. They came to the conclusion that, with the increase of architects' awareness about the structural importance of buttresses over time, they had emphasized more and more the technical aspect of this element in their structures. Fuentes (2018) after accurately surveying the Mallorca cathedral, she used the theoretical framework of limit analysis to analyse the distortion and sliding movements observed at some flying buttresses of the cathedral. It was concluded from the study that, flying buttresses have a fundamental role in the stability of the structure, although some of them experienced various problems over time. This study investigates the transverse seismic resistance of high masonry walls in case of they having various buttress forms such as rectangular, trapezoidal, stepped, combined, curvilinear or semi-cylindrical. In this way, the effects of buttress form on the transverse seismic resistance of the

Introduction of the structure and its calculation model
In this study, it is intended to make analyses on the models of a real, i.e. existing historical masonry structure. The structure considered is Şarapsa Han (Inn) in Alanya (Antalya, Turkey), Fig. 3. It is a stone masonry structure and involves most of the features suitable for the purpose of the study. These features are that; it is a building-type historical masonry structure, it has a simple and regular plan and transverse section shape, has long walls supported by buttresses placed at fairly equal intervals, and its cover is a vault.   Taking advantage of the mentioned regular plan shape and simple transverse section features of the Şarapsa Han, instead of the whole structure, only one slice from the main part (accommodation part) of it can be modeled for analyses. As can be seen from the Fig. 4, this slice consists of the wall portions facing each other among the center lines of two neighbour bays, vault portion connecting them, and two buttresses. The model of the structure, generated on this basis, is given in Fig. 6a. The Han is simple and regular, but its aspect ratio, i.e. height / transverse length ratio, without including the buttresses, is calculated as 6.75 m / (8.90 + 2 × 1.70) m  0.55, Fig. 6a − elevation. As is known, simply, structures which have aspect ratios less than 1 are called as "squat structure" and those with aspect ratios greater than 1 are as "slender structure". Thus, our structure is a squat structure. This is an undesirable feature for the purpose of the study. Because in a squat structure under horizontal loads collapse may occur with shearing (Makris and Alexakis, 2015). This type of collapse is not included within the scope of the present study. The effect of buttress form on transverse seismic resistance of squat masonry structures is being privately investigated by the authors in an ongoing study. Thus, here only the bending collapse case is investigated. Accordingly, the real model of the structure has been modified in three respects to see the effect of the buttress form on the transverse seismic resistance of the high masonry walls. The first modification is that, the height difference between the north and south of the model has been eliminated, and the heights of the wall and buttress on both sides have been equalized. The second change is that, the height has been increased reasonably without exaggeration. The new height value of the buttresses and walls was chosen as twice the average of the original heights of the buttresses. And the third modification has been to rotate the buttresses 90 degrees, to use them around their strong axes. Except for these three changes, all the features of the real model have been left as they are. The model obtained in this way and used in the calculations is shown in Fig. 6b. The aspect ratio of this model without considering the buttresses is 0.98 ( 1). Thus, although not much, nonetheless a model that can be called as "slender" has been obtained. Hereafter, this model will be called as "calculation model". Of course, as the buttress form changes (triangle, stepped, curvilinear etc.) the buttresses of the model will be in that form.

Material properties of the structure, modeling of masonry and assumed material model for the analysis
As for the materials of the Şarapsa Han, it is observed that, although there are also various other stones in the texture of the structure, mostly Isparta stone, which is a local stone, constitutes its body.
Mortar used in the structure is Horasan Mortar. This mortar had been used extensively in Byzantine, Seljuk and Ottoman structures. While generally dressed stone had been used in the buttresses of the Han, the walls of it had been built of dressed and roughly shaped stones. The portal, the door wall of masjid, and arches of the vaults are from cut stone. In the vaults, generally rubble stone and fill materials had been used.
A literature search has shown that there is no specific work, in-situ testing or other, that deals with the mechanical properties of the materials of the Şarapsa Han. The authors themselves also did not have the opportunity to determine these features of the structure. However, a reliable approximation of material properties can be based on master's thesis of Korkmaz (2019)  There is a tremendous literature, that is still expanding, about the behaviour and modelling of masonry structures under static and dynamic loadings. Here, it is not possible and necessary to enter this subject in detail. There are mainly two approaches to model the masonry structures. These are the 'micro-modelling' or 'two-material approach', and the 'macro-modelling' or 'homogenized/equivalentmaterial approach'. In the micro-modelling, the finite element discretization follows the actual geometry of both the units (stone or brick) and mortar joints, adopting different constitutive models for the two components. Whereas, the macro-modelling assumes that the masonry structure is a homogeneous continuum to be discretized with a finite element mesh, which does not copy the wall texture, but complys with the method's own criteria. Aiming to produce results at global level and keeping computational work at a manageable degree, this modelling approach finds a middle way between accuracy and simplicity (Illampas et al. (2020), Giordano et al. (2002), and Angelillo (2014)).
In this study, the macro-modelling approach is adopted, and from hence the material of the structure is assumed to be homogeneous. As explained above, Şarapsa Han does not have a texture consisting entirely of dressed stones and with regular joints between them. On the contrary, it is a structure in which irregular texture is dominant, and therefore there is a certain degree of homogeneity and isotropy. Hence, the assumption made for the material of the structure is rational.
In the light of the above explanations, now it should be determined the homogenized material properties of masonry texture of the Şarapsa Han. There are some formulas in the literature for the calculation of the compressive strength and the elastic modulus of the masonry depending on the values of its constituents. Here, the expressions in Tomaževič's book (1999) have been adopted. In this book, which is about the earthquake resistant design of masonry buildings, the following relationship that is based on Eurocode 6, is given to determine the compressive strength of the masonry: in which K is a constant (in MPa 0.10 ) between 0.40 and 0.60 depending on the texture, and fs and fm are the compressive strength of stone and mortar, respectively. Taking K as an average value of 0.50 and using the values in Table 1  Poisson's ratio was taken as 0.20, which is widely used for brittle materials.
Regarding the unit weight, , of the masonry, it is obvious that this depends on the unit weights and volume fractions of its materials. It is impossible to know exactly the volume fractions, only an estimate can be made in this regard. In a historical masonry structure such as under consideration, 75%, 20% and 5% volume fractions are reasonable values for stone, mortar (including filling material) and voids, respectively. Then the unit volume weight of masonry can be calculated by using of the following expression: Here s, m, v and s, m, v show the volume fractions and the unit weights of stone, mortar and voids, respectively. Using unit weight values in Table 1, taking v = 0, and adopting above estimated volume fractions,  is calculated as 23.37 kN/m 3 .
Thus, values of all necessary material properties of the structure have been determined. These are presented in Table 2 all together. The stress-strain, σ-ε, graphic formed artificially for use in the calculations is shown in Fig. 7. The behavior of plain concrete, which is brittle like masonry material, has been taken as basis. For compression, Hognestad's (1951) modified model, and for tension, Massicotte et al.'s (1990) model have been used. In this study, for the calculations Abaqus finite element program was employed. The nonlinear material behavior explained above was introduced to the program, and the option to consider the effect of nonlinear geometry changes was activated. In Abaqus, the concrete damaged plasticity (CDP) model provides an opportunity for modeling of concrete and other quasi-brittle materials such as masonry. Details of the model can be found in user manual of Abaqus. The model uses a damage parameter, d, to take into account degradation in modulus of elasticity of the material due to damage. In various studies on masonry structures, it is seen that the value 0.95 is often used as an upper limit for this parameter. In the present study, the value 0.87 was used for this parameter.

Fig. 7
The artificially formed uniaxial σ -ε graphic of the homogenized masonry material of the Han When using the CDP model, other required parameters are; the dilation angle ψ, the ratio of initial equibiaxial to initial uniaxial compressive strength fb0/fc0, the eccentricity ϵ, the Kc parameter, and the viscosity parameter μ. For the analyses, the adopted values of the parameters are given in Table 3. These values are either the default values in the user manual of Abaqus or the values that are compatible with those in the relevant literature (see for example to Valente and Milani, 2016).

Buttress forms considered for the structure
As already stated, main purpose of the present study is to investigate the effect of buttress form on transverse seismic resistance of high masonry walls. In this context, a compilation has been made from the buttress forms encountered in historical masonry structures reviewed in Section 2. The selected forms are shown in Fig. 8 as lined up on a wall. As can be seen from the figure, there are, among these forms, the most common types (Fig. 8a, b, c, h), the stepped types (Fig. 8d1, d2), the combined types (Fig. 8e1, e2, e3), the less common types (Fig. 8f1, f2), and the rare types (Fig. 8g1, g2,   g3). In total, fourteen buttress forms have been considered. It can be said that the variety of forms is rich enough.
These are given in Table 4. It should be noted that, the dimensions perpendicular and parallel to the wall, of a buttress are named as "depth" and "thickness", respectively.

A summary of the method
It can be said that, it is not an easy task to determine the seismic resistance of historic masonry structures. Because as is known, there are many factors that cause this. The fact that brittle failure occurs in masonry material under seismic loads, due to its nature, and that the earthquake is a highly complex event are only two of them.
In this study, nonlinear static analysis method, also known as pushover analysis method, was preferred as the analysis method. Its suitability for the purpose of the study and its simplicity are the main factors in its preference. It will be appreciated that if a complex method such as the time history analysis method is used, the effects of the buttress form on the seismic resistance of the walls may not be distinguished as easily as in a simple nonlinear static analysis.
As it is known, the nonlinear static analysis method is a quasi-static method. With the method, it is possible to make a general seismic assessment with less computational labor, time and cost by completely neglecting the dynamic effects of the earthquake or approximating them in a quasi-static manner. Although more approximate, the method is powerful and practical, and is among the primary tools used for evaluating masonry structures, (DeJong (2009), Endo et al. (2016)).
In nonlinear static analysis method, dynamic earthquake loading is typically simplified as a quasi-static loading in one of the following two ways: ⁕ The first way is to apply a progressively increasing unilateral acceleration to the modelled structure up to the exceedance of the maximum lateral resisting capacity and the development of a failure mechanism. This way is equivalent to subject the structure to a progressively increasing unidirectional horizontal ground motion. This option conservatively ignores the fact that actual ground motions only occur for a short period of time, and also neglects the possibility of resonant amplification, (DeJong (2009), Illampas et al. (2020)).
⁕ The second way involves applying monotonously increased horizontal forces, adhering to a predefined distribution pattern along the height of the structure to approximate the effects of an earthquake. In this procedure, the relative magnitudes of the forces are distributed with preference towards the top of the structure, accounting for amplification caused by dynamic resonance effects (DeJong, 2009).
Either way, at the end of the analysis, the relationship between base shear force and control node displacement is plotted. In this study, the first way has been used.

Calculation procedure
The aim is to determine the effect of buttress form on the transverse seismic resistance of high masonry walls. This will be achieved by performing nonlinear static analysis both in the calculation model of the structure, Fig. 6b, and in each new model obtained by changing only the buttress form, Fig. 8, of this model. By comparing the capacity curves or seismic coefficients obtained from these analyses, "the effect of form on resistance" will be understood. Here, while presenting the calculation procedure, Illampas et al. (2020) has been taken as example by virtue of the fact that the similarity of the structure type and the same calculation method used.
To perform nonlinear static analysis on a model, a uniformly applied, unidirectional, mass proportional loading pattern has been considered. Loading has been progressively increased up to the exceedance of the maximum lateral resisting capacity and the development of a failure mechanism.
As mentioned above, such a loading is equivalent to subject the model to a gradually increasing unidirectional horizontal ground motion. It is illustrated schematically in Fig. 9  The nonlinear static analysis has been carried out in two consecutive numerical steps. In the first step, their own weights have been applied to the models, and in the second step, monotonically increasing lateral forces have been incrementally enforced on them at rather small time intervals over a fictitious period of 1 second. It should be stated that there is no physical meaning of "time" in the analysis and that the computed responses are not time-dependent. Here, "time" is a scalar parameter used by the Abaqus for determining load variation during the solution process. In the analysis, the effect of nonlinearity, both material and geometric, was taken into account.

Fig. 9
A schematic representation of the increasing lateral forces and failure mechanism occuring in triangular buttressed model due to the gradually increasing unidirectional horizontal earthquake ground motion For each model, result of the above summarized analysis has been presented as a capacity curve, i.e. the curve of base shear force vs lateral displacement of a preselected control node. Base shear force has been determined as the sum of the horizontal reaction forces computed at the nodes on the base section of the model, and the middle node on the ridge of the vault has been chosen as the control node. And, maximum seismic coefficient cn, Fig. 9, which represents the largest transverse seismic force to which the model can resist, has been obtained as: = Maximum base shear force Self-weight of the model ⁄ = max model ⁄ Thus, the analysis for the model is completed. By comparing the maximum capacities or seismic coefficients obtained for the models, the effect of the buttress form on the transverse seismic resistance of the walls is determined.

Performed analyses
To create the models, firstly a "basic model" consisting of two opposite walls, vault and interior arch was formed. Then, by adding each buttress form in Fig. 8 in the middles of the outer surfaces of the walls of this model, the models having different buttress forms were obtained. When forming the finite element mesh of the models (basic and other), C3D8R (8-node linear brick, reduced integration, hourglass control) solid element was used. Different mesh sizes were tested, starting with the coarse ones, until further improvement in the results were achieved by refining the meshes and stable results were reached. In addition, in nonlinear static analysis kinetic energy was controlled and it was checked whether the inertia effects remained at negligible levels. To save time, calculations were performed on halves of the models. Moreover, the bottom cross-sections of the models were fixed, i.e., all the degrees freedom of all nodes in these sections were restricted.
Here, besides the beginning analyses that examine the effect of buttress depth on seismic capacity, two groups of analyses are performed in which the effect of the buttress form on the capacity is investigated.

Beginning analyses: Investigation of the influence of buttress depth
As the depth of a buttress increases, the moment of inertia about bending axis of its cross-sections and thus the stiffness contribution to the wall it supports will also increase. It is obvious, then, that the "buttress depth" is one of the basic parameters for the seismic resistance of a building-type masonry structure. Here, the influence of this parameter on transverse seismic capacity will be examined. The investigation will only be carried out on the rectangular buttressed model.
In this context, firstly the "calculation model" shown in Fig. 6b, whose buttress depth is 2.20 m, has been analysed and base shear force versus control node displacement graphic, that is the capacity curve has been obtained. Then, the non-buttressed version of this model, i.e. the "basic model" has been analysed. In addition, calculation model's modified versions which have buttress depths of 1.10 m, 3.30 m and 4.40 m have been calculated. These depth values are respectively half, one and a half times and two times of the buttress depth value in the calculation model. The results of the calculations are presented in Fig. 10a. The increase in transverse seismic capacity with the increase in buttress depth is rather clear. Therefore, as expected, the buttress depth has a great influence on the seismic capacity of a building-type masonry structure. Figure 10b presents the variation of maximum seismic capacity depending on the buttress depth. It is seen that there is nearly a linear relationship between capacity and buttress depth.
As expressed in the Section 2, the most common buttresses in historical masonry structures are rectangular, triangular and trapezoidal buttresses. The capacity curves of the models having these buttresses are presented in Fig. 11. It can be seen from the figure that the transverse seismic resistance of the triangular buttressed model first and then the trapezoidal buttressed one is saliently greater than that of the rectangular buttressed model.

Fig. 11
Capacity curves of rectangular, triangular and trapezoidal (Type 1) buttressed models Capacity curves belonging to the Type 1, Type 2 and Type 3 trapezoidal buttressed models, which their buttresses have 5.05%, 3.36% and 10.09% slopes, respectively, Table 5, are given in Fig.   12a. It can be seen that the seismic resistance increases considerably as the buttress slope decreases.
In other words, the more oblique the buttress, the greater the seismic resistance. The triangular and rectangular buttresses are limit states of the trapezoidal buttress. Comparison of the seismic capacities of the models having these buttresses is presented in Fig. 12b. As can be seen, the seismic resistance of the model having trapezoidal buttresses with less slope (Type 2) is close to that of the triangular buttressed model, and the resistance of the model having trapezoidal buttresses with higher slope (Type 3) is close to that of the rectangular buttressed model. Stepped rectangular buttresses are a very common buttress type especially in European historical architecture. Here, only two-and three-stepped versions of this buttress have been considered, Figs. 8d1, d2. Capacity curves of the models having these buttresses are plotted in Fig.   13a. It is evident that, provided that heights and depths of its parts are arranged harmoniously, a stepped buttress will approach a triangular buttress, when the number of its steps increases. And, if there is no steps, then it becomes a rectangular buttress. This situation can be clearly seen in Fig. 13b where the capacity curves of the models with stepped buttresses and the models with triangular and rectangular buttresses have been shown together. were considered, Fig. 8f1 and f2. The capacity curves of the models having these buttresses, and again for comparison, curve of the rectangular buttressed one, are given in Fig. 15. It is seen that both models have much higher seismic resistance than the rectangular buttressed model. Among the models with different buttress forms that we took into consideration, it was determined that the curvilinear concave buttressed model has the highest transverse seismic resistance. With a seismic resistance of 5520.29 kN, this model has a 63% higher resistance than the rectangular buttressed model which has a resistance of 3382.91kN. When historical masonry architecture is reviewed, it is seen that these curvilinear concave buttresses had been mostly preferred in the upper parts of the structures. This situation can be seen, for example, in the Cuetzalan del Progreso Church in Mexico, the San Frediano Church in Italy and the Cadiz Cathedral in Spain. It is predicted that, besides their aesthetics, these curvilinear concave buttresses were used in the upper parts of these structures, as they do not obstruct the view thanks to the their form. As can be seen from Fig. 8, the last buttress form considered is trapezoidal buttress of variable thickness. With the curve of the model supported with this buttress, and for comparison, the curves of the models with normal trapezoidal buttresses (Table 5, Type 1) and rectangular buttresses are shown in Fig. 17. It is seen that, except for the last stages of their analysis, the behavior of the trapezoidal buttressed models is quite similar. The seismic capacity of both trapezoidal buttress supported models is considerably higher than that of the rectangular buttressed model. In the second section of the study it was stated that trapezoidal buttresses are one of the most widely used buttresses in historical masonry structures. It is understood from the results obtained that this situation cannot be casual.
Because the ancient builders who understood the importance of the buttress must have comprehended absolutely that the buttress form would also be important.

Fig. 16
Capacity curves of semi-cylindrical, triangular prism and truncated half-conic buttressed models, and comparison with the capacity curve of the rectangular buttressed model A collective comparison of the results from these first group analyses is presented in Table 6. The

Fig. 17
Capacity curve of the model having trapezoidal buttresses with varying thickness, and comparison with the capacity curves of the rectangular and trapezoidal (Type 1) buttressed models If the capacity curves' a general assessment is made, the first thing observed is that they are not smooth. This is quite normal, because damages occur to the models with increasing horizontal load, and the curves naturally progress in a non-smooth way as these damages spread and new damages occur. The curves of all models begin with a high slope zone. This is the stage, where there is no damage to the models yet, so the lateral stiffness is highest. After this zone, a sudden drop in stiffness takes place due to the initial cracking. The curves then exhibit a second climbing zone, the slope of which gradually decreases, and reaches a maximum. Then the curves descend in general and collapse is reached at the end of the zone. Some of them show a slight recovery in the last part of this descending zone.  (Fig. 8b) 4939.04 6.22 1.46 Trapezoidal (Fig. 8c, Type 1) 4533.40 5.71 1.34 Trapezoidal (Fig. 8c, Type 2) 4738.48 5.97 1.40 Trapezoidal (Fig. 8c, Type 3) 4233.61 5.33 1.25 Two-stepped (Fig. 8d1) 4311.21 5.43 1.27 Three-stepped (Fig. 8d2) 4320.49 5.44 1.28 First type combined (Fig. 8e1) 3927.07 4.95 1.16 Second type combined (Fig. 8e2) 4385.54 5.52 1.30 Third type combined (Fig. 8e3) 4890.22 6.16 1.45 Curvilinear concave (Fig. 8f1) 5520.29 6.95 1.63 Curvilinear convex (Fig. 8f2) 4226.03 5.32 1.25 Semi-cyl. (Fig. 8g1) 2739.54 3.45 0.81 Triang. pri. (Fig. 8g2) 3149.55 3.97 0.93 Truncated half-conic (Fig. 8g3) 3534.11 4.45 1.05 Trap. with vary. thick. (Fig. 8h) 4636.14 5.84 1.37

Second group analyses
As for the analyses in this second group, here simply the bottom depths and thicknesses of buttresses are taken equal to each other. The reason for taking the buttress dimensions in this way is to determine how the buttress form affects the lateral seismic resistance of the structure if the base area of the buttresses is limited. It should be noted that, only in semi-cylindrical and truncated half-conic buttresses, Fig. 8g1 and g3, although the depth condition can be met, the thickness condition at the bottom cannot be realized due to their geometry. The dimensions of the buttresses in this group analyses are given in Table 7. The buttresses that need to be designed (arranged) have been designed with the same considerations expressed in the first group analyses. The table also presents the volumes of the buttresses and the ratio of the volume of each buttress to the volume of the rectangular buttress.
The results of the analyses, that is, the capacity curves of the models are presented in Fig. 18.
It can be seen that the curves of the semi-cylindrical and truncated half-conic buttressed models are at  Table 8, the capacities, weights, maximum seismic coefficients and the ratios of capacities of the models to the capacity of the rectangular buttressed model are given. When the semicylindrical and truncated half-conic buttressed models are excluded (because it was stated above that the buttresses in these models cannot meet the thickness requirement due to their geometry) and a comparison is made between the other models, it is seen that the three-stepped buttressed model has the highest seismic resistance, and the triangular prism buttressed one has the lowest resistance. The curvilinear concave buttressed model, which gave the highest resistance in the first group analyses, gave the lowest seismic resistance after the triangular prism buttressed model in the analyses in this group. When evaluating the results of the trapezoidal, stepped and combined buttressed models, it should not be forgetten that their buttresses were completely arranged by ourselves, adhering only to the condition of "equal base depth". In these models when the design of the buttresses is changed, it is obvious that the results will also change to a certain extent.  (Fig. 8d1) 3334.46 698.12 4.78 0.986 Three-stepped (Fig. 8d2) 3706.03 665.40 5.57 1.096 First type comb. (Fig. 8e1) 3445.65 698.12 4.94 1.019 Second type comb. (Fig. 8e2) 3549.40 649.04 5.47 1.049 Third type comb. (Fig. 8e3) 3224.28 600.94 5.37 0.953 Curv. concave (Fig. 8f1) 3011.37 536.73 5.61 0.890 Curv. convex (Fig. 8f2) 3492.15 665.40 5.25 1.032 Semi-cyl. (Fig. 8g1) 4468.32 1394.54 3.20 1.321 Triang. pri. (Fig. 8g2) 2843.30 600.94 4.73 0.840 Trunc. half-conic (Fig. 8g3) 4139.76 983.23 4.21 1.224 Trap. with vary. thick. (Fig. 8h) 3324.09 633.51 5.25 0.983 To sum up, the calculations show that the lateral seismic resistance of structures supported by buttresses of different forms with equal base depths are at different levels, as expected. However, when the results of the two groups of analysis are compared, Table 6 and Table 8, it is seen that the effect of the buttress form on seismic resistance in the case of this second group is not as high as in the first group. In addition, it should be noted that although the semi-circular buttress provided the highest seismic resistance to the structure in the second group calculations, in this case its volume and therefore the amount of material used is considerably higher than the other buttresses.

Fig. 18
Capacity curves of the models in second group analyses

Conclusions and future works
In this study, the effect of buttress form on transverse seismic resistance of high masonry walls was investigated. For this purpose, from a real historical masonry structure, a calculation model and a basic model were created. As analysis method, nonlinear static analysis method was used and calculations were carried out with the Abaqus program. Firstly, the effect of buttress depth on transverse seismic resistance of the structure was investigated by changing the buttress depth in the calculation model. Then, two groups of analyses were performed on the models obtained by mounting different buttress forms to the basic model. For buttresses, a rich variety of forms was considered. The salient results can be summarised as follows. a) Buttress depth is an important parameter on the transverse seismic resistance of high masonry walls, and thus building-type masonry structures. In rectangular buttressed structures, the maximum seismic capacity of the structure is almost linearly related to the buttress depth.
b) In the case of buttresses of equal volume (first group analyses), the form has a strong influence on the seismic resistance of the structure. Among the buttresses considered, the curvilinear concave buttress provides the highest seismic resistance to the structure, while the semicircular buttress gives the lowest support. The slope of trapezoidal buttress has a noticeable effect on the seismic capacity of the structure. Provided that depths and heights of its parts are arranged harmoniously, the behavior of a stepped buttress approaches that of a triangular buttress, as the number of steps increases.
Arrangement of a combined buttress affects the level of support that this buttress will provide to the structure.
c) In the case of buttresses with equal base depth and thickness (second group analyses), the form again has an effect on the capacity. However, the effect of the form in this case is not as striking as in the case of equal-volume buttresses.
d) The general conclusion of the study is that, no matter what form of buttress used by the ancient builders, their high awareness of the structural significance of the buttress is sufficient for us to remember them with respect and admiration. If they not had this insight, there would have been no, for example, magnificent Ottoman mosques and grandiose Gothic cathedrals that fascinate us today.
As mentioned earlier, the effect of buttress form on seismic behavior and resistance in squat historic masonry structures is investigated in an ongoing study. Apart from this, buttresses in the form of a special mathematical curve, tower-like buttresses (like those with box cross-section and ringsection), and buttresses stepped in two directions can also be studied. In addition, albeit they act like tension ropes, by taking inspiration from buttress root shapes of some trees, studies can be made on the optimum buttress forms. These are left to future studies.