Investigation on the behavior factor of leaning masonry wall structures

In force-based seismic design, the behavior factor q (or response modification factor R) is devised to prevent overdesign by reducing the design force. Different values of the behavior factor q are available for various structure types. However, masonry fence walls are not categorized in the current seismic design codes. These walls are of poor quality due to the absence of proper quality assurance and maintenance, and most of those structures are inclined. In this study, the behavior factor q of leaning masonry fence walls is experimentally estimated. We fabricate three masonry wall structures that are inclined with three different wall angles of 1.5∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^\circ$$\end{document}, 3.0∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^\circ$$\end{document}, and 6.0∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^\circ$$\end{document}, respectively. Monotonically increasing biaxial lateral loads are applied to the walls using a shaking table. The value q of each leaning wall is calculated by analyzing the dynamic load–displacement relationship. A relationship between the factor q and wall angle is discussed, and a new empirical formula representing q as a function of a wall angle is proposed. The proposed empirical formula would be useful in seismic design for severely leaning masonry walls.


Introduction
In force-based seismic design, design response spectra are reduced by introducing the behavior factor q, also known as the response modification factor R (ATC 1978). This design concept assumes that structures can develop lateral strength beyond their designed strength and sustain large inelastic deformation without collapse (Whittaker et al. 1999;Kim and Choi 2005). Inelastic deformations can be used to absorb quantifiable energy levels, which can help decrease the design force acting on the structure (Borzi and Elnashai 2000). This reduction in the design force can be attributed to q that considers the effects of overdesign, ductility, and energy dissipation on the actual structures (Hamzeh et al. 2021).
Seismic design codes and extant research focusing on q can be classified into masonry use for structural (1) framing systems and (2) members. For the structural framing systems, masonry is used as the structural frame for low-rise buildings, which withstand external forces with their own frame weight. Eurocode 8 (2004) stipulates different q for different masonry structural systems: reinforced masonry (q = 2.5 − 3.0); confined masonry (q = 2.0 − 3.0); and unreinforced masonry (q = 1.5 − 2.5). Several researchers have experimentally investigated q. For example, Tomaževič et al. 1994Tomaževič et al. , 1996Tomaževič et al. , 1997aTomaževič et al. , b, 1997Tomaževič et al. , 2006Tomaževič et al. , 2007 investigated q of reduced-scale masonry structures using a simple uniaxial shaking table and obtained q values of 2.84, 2.69, and 3.74 for unreinforced, confined, and reinforced masonry framing systems, respectively. Benedetti et al. (1998) determined q on real-scale unreinforced and retrofitted structures as 1.4 and 2.0, respectively. Zonta et al. (2001) performed shaking table tests in a reduced-scale building to determine the q of advanced reinforced masonry construction techniques, and the obtained lowest q was 3.6. For structural members, masonry plays the role of a wall, such as an infilled wall or a load-bearing wall, in the structural system. These walls complement the seismic performance of the system or divide the inner space. Relevant studies investigated the q or R varying with boundary conditions of the wall. For example, Da Porto et al. 2009 investigated the q of masonry walls with various bonding arrangements. The difference in q was not significant, and the obtained q matched well with that proposed in Eurocode 8. Shedid et al. 2011 demonstrated the R for rectangular structural masonry walls was close to 5.0, and they (Shedid et al. 2010) further investigated R for flanged and end-confined structural masonry walls that are not categorized by seismic design codes. The obtained R was at least 3.0 exceeding the maximum R (=2.0) stipulated in the seismic code. This means that the substantially improved performance of the flanged and end-confined walls was achieved.
In the above-mentioned studies and seismic codes, behavior factors (or response modification factors) are well-documented for the framing system and members. However, studies on individual frame walls that do not belong to either of these categories remain limited. The individual frame wall is a structural system that withstands external forces using only the weight of the wall without any structural members and roofs; this is a simple fence-type structure wherein several walls are connected at the wall ends. Further, the wall widths, heights, and joint types are diverse because an individual frame wall is built without following any regulations or manuals. Most of these walls are of poor quality due to the absence of proper quality assurance. In South Korea, many walls are inclined in the out-of-plane direction to an extent where they can be easily recognized by the naked eye; thus, there is always a risk of collapse for such walls. Recently, Gyeongju ( M W 5.5) and Pohang ( M W 5.4) earthquakes that occurred in South Korea severely damaged many individual walls.
Given this context, q of individual masonry walls uncategorized by the current seismic design codes are investigated in this study. Furthermore, the effect of the wall angle on q is discussed. To this end, three U-shaped masonry walls were made to lean at angles of 1.5 • , 3.0 • , and 6.0 • , respectively. Dynamic shaking table tests were performed by applying incremental biaxial horizontal ground motions to the walls. The resistance curve (load-displacement relationship) of the wall was obtained, and q was calculated from the resistance curve. The obtained q was compared with q suggested in the seismic design code, and the q values in the code could not be used for leaning masonry walls inclined by higher than 6 • . For these severely leaning walls, we proposed a new empirical formula that represents q as a function of a wall angle.
The rest of this manuscript is organized as follows. The relevant theories regarding the damage limit states and the behavior factor q are explained in Sects. 2 and 3, respectively. Section 4 provides the specimen fabrication method; Sect. 5 introduces the experimental program, and Sect. 6 discusses the experimental results. Finally, the summary and conclusions are provided in Sect. 7.

Seismic resistance and limit states
The seismic behavior of a structural system is estimated based on the relationship between the lateral resistance and the displacements. The resistance is represented by the base shear coefficient BSC, which is the ratio of the base shear to the weight of the structure. The displacement is expressed in terms of the drift ratio which is the ratio of the top drift to the height of the structure. Figure 1 shows the relationship between the BSC and that describes the global response of a structural system subjected to monotonically increasing displacements.
Typical limit states associated with the severity of damage to the structure are marked on the resistance curve shown in Fig. 1; these limit states are explained below (Tomaževič 2007 • Crack limit state: Initial cracks occur in the structure, changing the global stiffness of the structural system. The crack limit state on the curve is defined by the resistance R cr and the corresponding drift ratio cr . • Maximum resistance state: The structural system experiences its maximum resistance capacity. In the resistance curve, the maximum resistance is defined by the resistance R max and the corresponding drift ratio R max . • Design ultimate limit state: The resistance is reduced to 80% of the maximum resistance. The resistance decreases gradually after reaching the maximum value. Once the resistance is below 80% of the maximum resistance, the structural system is no longer considered for design because it only provides information regarding the additional ductility and energy dissipation capacity. The design ultimate limit on the curve is defined by the resistance 0.8R max and corresponding drift ratio 0.8R max . • Limit of collapse: The collapse of the structural system is a serious concern, and the structure may collapse if subjected to aftershocks, i.e., near or total collapse. The limit of collapse on the curve is defined by the resistance R collapse and the corresponding drift ratio collapse . The actual resistance curve is approximated as an elastoplastic curve (experimentally idealized) to simply calculate the resistance and drift ratio values at each limit state. The secant stiffness at the initial crack occurrence is defined as the initial slope of the experimental idealized curve, which is called the effective stiffness K ei and expressed as where R cr and cr represent the resistance and drift ratio at the crack limit state, respectively. The idealized maximum resistance R max,i is calculated considering equal energy dissipation capacities of the actual and idealized curves. Both areas enclosed by the actual and idealized curves are equal, and R max,i is expressed as where 0.8R max represents the drift ratio at the design ultimate state, and A env represents the area enclosed by the actual resistance curve. The idealized maximum resistance R max,i can be calculated more easily using Tomaževič et al.'s 1985 equation. The test results of more than 60 walls indicate that the R max,i ∕R max ratio of the shear failure walls is 0.9, which is expressed as In practice, Eq. (3) would be preferred over Eq.
(2) because of the equation's simplicity and acceptable accuracy. Now that R max,i and K ei are known, it is natural to calculate max,i as The ultimate drift ratio of the idealized curve that satisfies the collapse-prevention performance is defined as 0.8R max . However, the ultimate drift ratio should be determined by additionally considering the damage limit state (serviceability) that is classified according to the damage level of the structure (Tomaževič and Weiss 2010). According to EMS-98 (Grünthal 1998), the damage grade of the masonry wall where shear failure prevails is classified into Grades 2-5 by correlating them with the serviceability limit states as explained below (Tomaževič 2007): • Grade 2: Slight structural damage. First structural damage can cause a noticeable reduction in the first-mode natural frequency of the structural system. • Grade 3: Moderate damage. Numerous types of cracks are distributed on the wall. This grade is detected immediately after the maximum resistance of the wall is measured, and when the repair is possible. • Grade 4: Heavy structural damage. Individual masonry units are crushed and dislocated, and residual deformation is visible. The wall can still be repaired, but the repair may not be cost-effective. • Grade 5: Near-collapse. The crack width is greater than 10 mm in addition to widespread damage such as crushed and dislocated masonry units. The wall is completely unrepairable, and an imminent collapse is expected with the aftershock.
An acceptable level of damage is a state wherein a wall system remains safe and usable despite being damaged; this is a damaged state that does not exceed Grade 3. Figure 1 shows that the range of the acceptable damage level spans up to the drift ratio at the maximum resistance or slightly above. Subsequently, the wall system remains safe but is no longer usable, i.e., the drift ratio ranges from the maximum resistance to the ultimate limit state, and the damage state corresponds to Grade 3 or 4. The damage state at Grade 3 is occasionally observed immediately after the maximum resistance of the wall is measured. Most damage occurs at approximately three times the drift ratio of the initial cracks in the wall, i.e., at 3 cr . The damage state at Grade 4 is observed near the drift ratio, defined as the no-collapse requirement, where the resistance of the structure degrades to 80% of the maximum 0.8R max . Although the damage state at Grade 4 satisfies the no-collapse requirement, it is often unrepairable or not cost-effective to repair. Thus, the damage limitation requirement (serviceability) should be considered to determine the level of the design ultimate state. The smaller value of either 3 cr or 0.8R max should be selected as the limit of the design ultimate state u,i on the idealized resistance curve to satisfy both nocollapse requirement and damage limitation requirement (serviceability): Resistance degradation represents the no-collapse requirement, and the damage limitation requirement (serviceability) is represented by the limited drift. If a smaller value is considered for evaluating the design parameters, such as the behavior factor q, both requirements will be satisfied simultaneously.

Behavior factor
The design response spectrum in the force-based seismic design is reduced by introducing the behavior factor q because of the nonlinear behavior of the structure system, see Fig. 2. q is devised to prevent overdesign caused by exploiting benefits offered by the energy dissipation capacity of structures.
Different equations for the behavior factor q (or response modification factor R) have been proposed in various research and design codes (Kim and Choi 2005). Earlier research has suggested that q depends on the over-strength of individual members, structural redundancy, and ductility of the structure (Uang 1991;Ahmad et al. 2018;ATC 1982;Freeman 1990;ATC 1995a, b): The current North American seismic design codes (US code ASCE 7-16 (2016) and Canadian code NBCC-15 (2015)) define the response modification factor R considering two force-based seismic design constituents: overstrength factor R S and ductility factor R , under the assumption that the redundancy factor R R is equal to 1.0. Eurocode 8 (2004) considers only the ductility factor R to calculate q. Thus, substantially larger behavior factors are proposed in North American codes than those in Europe (Borzi and Elnashai 2000). In this study, the conservative European approach is selected to calculate q. Previous studies (Tomaževič 2007;Tomaževič and Weiss 2010;Ahmad et al. 2018;Lourenç et al. 2013;Chourasia et al. 2021;Magenes 2006) have already demonstrated that q or R considering only the ductility factor R is reasonable. Newmark and Hall 1982 reported that R can be defined based on either an equal displacement or equal energy principle depending on the natural period of vibration of the structure. Figure 3 shows schemes of each principle in terms of the resistance-drift ratio relationship.
The equal displacement approach is more suitable for structural systems that have a long vibration period ( T n ≥ 0.5 s) with frequencies lower than 2 Hz, while the equal-energy approach is more appropriate for structures with short vibration periods ( T n < 0.5 s) and frequencies higher than 2 Hz (Filiatrault 2013). As shown in Fig. 3(a), the equal-displacement approach, both the elastic and elastoplastic curves have the same initial stiffness and ultimate displacement ( u = e ). As shown in Fig. 3(b), in the equal-energy approach, the elastic load-displacement response is assumed to have the same initial effective stiffness as the elastoplastic response, wherein the total areas under both load-displacement curves are equal ( A E = A EP ).
The equal-energy approach is adopted in this study because the first-mode natural frequencies of the unreinforced leaning masonry walls are higher than 2 Hz. Based on the concept of the equal area between the elastic and elastoplastic load-displacement curves, R is estimated by correlating it with the ductility of the structural system as where the system ductility factor is defined as the ratio of u,i to max,i , expressed as

Fabrication of specimens
The stiffness and quality of the specimen walls were similar to those of actual masonry fences. The mortar was made with quality and strength comparable to that of the actual masonry fence by complying with the on-site water-cement mixing ratio. In addition, the thickness of the specimen walls was determined to be double-wythe brick to have a similar out-of-plane stiffness to the actual masonry fences. On the other hand, the geometric similarity was somewhat limited due to the size of the shaking table. The variety of widths and heights of actual masonry fences made it more challenging to determine the geometry of the specimen walls. Given these limitations, the width was set to the longest length that could be placed on the shaking table, and the height was limited so that even at the maximum inclination of 6 • , sufficient dynamic responses could be obtained without abrupt collapse.
A fence-type wall is a simple structure wherein several individual blind walls are connected at the wall ends. The degree of vulnerability to the inclination is different depending on the joint conditions at both ends of the wall. On the one hand, if both ends of the wall are fixed, the wall rarely tilts. On the other hand, the wall is relatively vulnerable to being inclined if either end is in a free condition. In fact, most leaning walls have at least one free end.
Schemes of a equal-displacement and b equal-energy principles A U-shaped wall structure was devised to reproduce similarity to actual leaning walls; in this structure, two transverse walls facing each other were made to lean at the same angle in the out-of-plane direction, as shown in Fig. 4.
One end of the transverse wall was a fixed condition because it intersected with the longitudinal wall, and the other end was close to a free condition by the flange. Hereafter, the longitudinal wall (width = 2700 mm) is referred to as the north wall, and the two transverse walls (width = 1990 mm) are referred to as the west and east walls. Further, the two subsidiary walls (width = 590 mm) are referred to as the southwest and southeast walls. The height of all walls was 1470 mm, without any headers. The wall thickness was 190 mm, and the wall was constructed of double-wythe brick masonry. The mortar was plastered with a thickness of 10 mm on the bed and head joints for bonding between the bricks. Interlocking bonded types were selected for the four corners. The specimen had a 160-mm-thick, 3650-mm-wide, and 3550-mm-long reinforced concrete foundation. The total mass of the specimen was 9.00 ton, of which the masses of the foundation and masonry walls were 4.70 and 4.30 tons, respectively.
The wall angles of the test walls were determined by measuring the inclined angles of structures in the three deteriorating areas (Mokpo, Seosan, and Daegu) in South Korea. In these three areas, various structural types exist, including not only masonry but also reinforced concrete and wooden structures. Figure 5 shows wall-angle maps of all structures located in the three areas, and Table 1 shows their statistical distribution of wall angles.
Most structures are inclined to within 6 • . Specifically, the structures inclined within 6 • account for 98.4%, 95.7%, and 99.4% in Mokpo, Daegu, and Seosan, respectively. From this field survey results, the maximum wall angle of the specimen was determined as 6 • , and later 3 • and 1.5 • were additionally considered.
The two transverse walls (the west and east walls) facing each other were intentionally inclined in their out-of-plane direction for calculating their behavior factors. The inclined angle is defined as , and the specimens with three different (1.5 • , 3.0 • , and 6.0 • ) were fabricated. Figure 6 shows the front view of the three manufactured specimens and their appearances. The north, southwest, and southeast walls were inclined with the same angle in the inplane direction because only the west and east walls were inclined with an angle in the out-of-plane direction. The wall was composed of a total of 21 brick layers in the height direction; therefore, careful handling was required to stack the bricks from the bottom to achieve the desired inclined angle.

Setup and instrumentation
Tests were conducted by using a shaking table (MTS systems corporation) at the Seismic Research and Test Center (SESTEC), South Korea. Figure 7 shows the plan view of the shaking table (area of 5.0 m × 5.0 m, maximum acceleration of 3.0 g, and operable frequency range of 0.1-60 Hz).
The three different wall angles of the leaning masonry specimens were tested separately using the shaking table. Two horizontal artificial ground motions were applied to the specimen simultaneously in both longitudinal and transverse directions. Further, the specimen was surrounded by a steel fence to prevent damage to the facility caused by the wall collapse.   Figure 8 shows six accelerometers and six linear variable differential transformers (LVDTs) attached to the masonry walls, foundation, and shaking table to measure the dynamic responses.
The accelerometers (PCB, 356A16) recorded the acceleration time histories in the two horizontal directions in real time. The accelerometers placed on the foundation and shaking table helped confirm if the desired input ground motion was accurately generated. In addition, the two accelerometers were installed at different positions on the two transverse walls: at the middle of the width, only the heights were different; one at half the wall height and the other at the top. Four LVDTs (TML, DP-1000E) were attached at the same positions as the accelerometers, to record the out-of-plane displacement in real time. The remaining two LVDTs were installed on the west and north sides of the foundation to estimate the relative displacement of the walls to the foundation.

Test procedure
An artificial ground motion was applied to the specimen, instead of the natural earthquakes, because this study did not cover any specific sites (Jung et al. 2017(Jung et al. , 2022. The artificial ground motion was generated from the design response spectrum (DRS) determined in compliance with the Korean design code (EESK 2019). The seismic grade of an unreinforced masonry wall was assumed to be "seismic level 2" (the grade of a facility that has little social impact even if it loses its functionality). This facility should be able to withstand an earthquake with a 2400-year return period to satisfy the "collapseprevention performance." Further, several factors needed to be determined to generate  the DRS: We assumed that the ground type S3, earthquake zone 1, and 5% damping ratio. Considering the above factors, the DRS was obtained as shown in Fig. 9(a). For the input ground motions to the shaking table, the acceleration time histories in both the longitudinal and transverse directions were derived from the DRS; their accelerograms are presented in Fig. 9(b) and (c), respectively. The total duration is 30 s, of which the peak ground accelerations (PGAs) are 0.29 g and 0.26 g for the longitudinal and the transverse directions, respectively.
The intensity of the ground motion was varied by gradually increasing and decreasing the amplitude scale factor of the DRS to obtain extensive data. The DRS was defined as EQ 100% and scaled to generate ground motions from EQ 25% to EQ 400%. For each EQ run, the artificial ground motions in the two horizontal directions were generated from the scaled DRS. Table 2 summarizes the EQ runs along with PGAs measured on the shaking table in both horizontal directions for the three specimens.
The first inputs of the three specimens were equal to EQ 25%. However, the last inputs were different because the higher the wall angle , the earlier the test ended.
As specified by the American Society of Civil Engineers (ASCE), the two-horizontal acceleration time histories should be statistically independent to ensure that the twohorizontal artificial ground motions are similar to real earthquake events: their correlation coefficient should not exceed 0.3 (ASCE 2017). A strict criterion was applied in this study, wherein the correlation coefficient should be less than 0.16 that is the value measured from actual strong earthquakes (Chen 1975;USNRC 2007). When two horizontal acceleration time histories are defined as X and Y, respectively, the MATLAB function xcorr(X,Y,'coeff') is used to calculate the correlation coefficient between the two acceleration time histories. Using this function, the time histories of the correlation coefficient were obtained for all input acceleration time histories, and their maximum values of the correlation coefficient were found not to exceed 0.16. During the shaking table tests, the correlation coefficient between the two horizontal acceleration time histories measured at the shaking table was also calculated for all EQ runs. As a result, the average of the maximum correlation coefficients were 0.11, 0.12, and 0.13 for = 1.5 • , = 3.0 • , and = 6.0 • , respectively. The performance of the shaking table was verified by comparing the target response spectra and response spectra obtained from the actual table motion to confirm that the desired artificial ground motion was input accurately. Figure 10 shows examples for EQs 50% and 125%, and they indicate that the performance of the shaking table is acceptable because the table response spectra measured in the longitudinal and transverse directions follow the trend of the target response spectra.

Damage evolution
The structural damage and cracks on the specimens were investigated through visual inspection after every EQ run ended. Figure 11 shows the planar figures for the three different specimens wherein the longitudinal wall (the north wall) is fixed in the center, and the two transverse walls (the west and the east walls) and the two subsidiary walls (the southwest and the southeast walls) spread out on both ends.
For all three specimens, the horizontal crack first developed along the bottom mortar bed joints of the southwest and southeast walls. As the intensity of the EQ run increased, this crack propagated upward along the mortar joint, causing distributed damage to the lower part of the southwest and southeast walls. Meanwhile, the crack damage also propagated to the west and  Fig. 11 Damage evolution for = 1.5 • , 3.0 • , and 6.0 • east walls and grew diagonally upward, which divided these walls into upper and lower parts. Owing to the severe flexural behavior of the west and east walls, the upper part was imminent to fall off, and this state was near collapse. For the wall with = 3.0 • and = 6.0 • , the cracks even propagated to the north wall. Table 3 summarizes the EQ runs when the first crack occurred and when the near-collapse state was reached.
The first cracks and near-collapse state were observed at an earlier EQ run with an increase in . At = 1.5 • , the west wall did not experience the near-collapse state even at the strongest intensity (EQ 400%).

Hysteresis analysis
The hysteresis loop is represented by the relationship between the base-shear coefficient BSC and drift ratio using the measured data from the accelerometers and LVDTs. The base shear is defined as the estimated maximum lateral force exerted on the base of the structure attributed to a lateral seismic load (Khan 2013); the BSC normalized by its own weight indicates the structural resistance against the lateral load. The BSCs of the west and east walls are denoted by BSC W and BSC E , respectively. Their base shears are calculated from the summation of the multiplication of each acceleration and the tributary mass; they are then normalized by the wall weight where the accelerations a i (i=1,2) are measured at half the height and at the top of the wall, respectively. The base shears are calculated by multiplying each acceleration with half the mass of the wall. We assume that the masses of the southwest and southeast walls are allocated to adjacent west and east walls, respectively. Accordingly, the total masses of the west wall m W and the east wall m E are 1.41 ton; their tributary masses m i,W and m i,E are 0.705 ton, which is half the total mass of the wall.
is defined as the displacement at the top of the wall normalized by the full height. It is a crucial damage parameter that defines the collapse probability of a structure subjected to a lateral seismic load (Yang et al. 2010). The drift ratios of the west wall ( W ) and east wall ( E ) are expressed as where h = 1,470 mm represents the height of the walls, and W and E represent the out-ofplane displacement at the top of the west and east walls, respectively. Figure 12 shows the hysteresis loop of the east wall of = 6 • for all EQ runs. The red straight line indicates the out-of-plane average stiffness during the EQ run. The red straight line is the major axis of an ellipse approximating the hysteresis loop, as shown in Fig. 13.   Fig. 12 Hysteresis loops of the east wall of = 6 • for EQ runs The ellipse was obtained by principal component analysis (PCA), which is a method of extracting representative features of data using a covariance matrix. An appropriate major axis is obtained when the method is applied to a two-dimensional elliptical distribution.
Until EQ 200%, where the first cracks occurred, the BSC increases without any significant change in the stiffness. Then, the stiffness degrades rapidly with an increase in to the last EQ 250% where the near-collapse state was observed. A similar hysteresis loop change is observed in other leaning walls with different wall angles ; however, the EQ run intensities of the first crack occurrence and the near-collapse states are different. For every EQ run, the maximum BSC and corresponding at the same instant of time are marked by white dots, and the maximum and corresponding BSC at the same instant of time are marked by black dots in Fig. 12.
The hysteresis envelope is the resistance curve against lateral loads (Uang 1986;Whittaker et al. 1987) drawn by plotting the maximum BSC (white dots) of each EQ run in the new BSC-domain and by connecting them with a straight line. However, every white dot cannot be used in the envelope because the maximum values of BSC sometimes deviate further than expected. Therefore, we draw the hysteresis envelopes by selecting valid white dots. If the wall experiences a near-collapse state, the ultimate point of the curve is the maximum (black dot) of the last EQ run. Otherwise, the maximum BSC (white dot) of the last EQ run is the ultimate point. Figure 14 shows an example of the hysteresis envelope of the east wall of =6 • .
The hysteresis envelope is drawn in quadrants 1 and 3; however, in this study, we approach conservatively and select the one with the lower ductility (lower energy dissipation capacity) of the two quadrants to calculate the behavior factor q.

Behavior factors for leaning masonry walls
The hysteresis envelopes of all leaning walls are presented in Fig. 15. The hysteresis envelope should be idealized as a bilinear curve to calculate the behavior factor q. Table 4 summarizes the values required to obtain a bilinear idealized curve. The required values are calculated following the methodology described in Sect. 2.
K ei represents the secant stiffness at the initial crack occurrence on the hysteresis envelope. The maximum BSC of the idealized curve BSC max,i is 90% of the BSC max ; its corresponding drift ratio max,i is the ratio of BSC max,i to K ei . The ultimate point of the idealized curve is defined as u,i , which is the minimum of 0.8BSC max and 3 cr . In most walls, 3 cr prevails. The system ductility is defined as the ratio of u,i to max,i by Eq. (8). q is finally obtained by substituting into Eq. (7). For each , the two behavior factors were obtained for the east and west walls. However, the lower q was selected as a conservative approach, see Table 4. The required values of the west wall with = 1.5 • could not be obtained because the west wall did not reach the near-collapse state, which indicates a linear behavior even under the maximum intensity load.
The relationship between q and is presented in Fig. 16. The selected q of =1.5 • , 3.0 • , and 6.0 • are 1.917, 1.554, and 1.469, respectively. The q decreases with an increase in : A more conservative seismic design is required for higher by weakening the amount of reduction in the spectral acceleration of design response spectra. For comparison, the ranges of q stipulated in Eurocode 8 (2004) are presented for different masonry systems: reinforced masonry (q = 2.5 − 3.0); confined masonry (q = 2.0 − 3.0); and unreinforced masonry (q = 1.5 − 2.5). The behavior factors of = 1.5 • and 3.0 • are included within the given range of the unreinforced masonry in Eurocode 8; the q for =6.0 • is slightly lower than the lower bound of the unreinforced masonry range.
For conservative seismic design, it is recommended to use the lower bound of the behavior factor given in Eurocode 8, but for structures whose behavior factor is lower than the lower bound, it indicates that even with conservative design, the earthquake resistance of the structure cannot be guaranteed. Earthquake resistance is closely related to the ductility of a structural system. Strong earthquakes cause the structure to behave in an inelastic region, where the energy is dissipated by structural deformation. In this situation, ductility is a quantitative indicator that shows the extent to which a structure can undergo large deformation without losing its serviceability. The square root of system ductility  is proportional to the behavior factor q, as shown by Eqs. (6) and (7). Therefore, based on the observation that the wall inclined more than 6 • has q lower than the lower bound in Eurocode 8, it can be inferred that wall angles larger than 6 • may provide a smaller ductility capability than that assumed by Eurocode 8. In order to achieve the desired seismic performance of masonry fence walls, strict management and regulation are required to ensure that the wall angle does not exceed 6 • . If is higher than 6 • , it may be difficult to use the q suggested in Eurocode 8. In this case, regression analysis is performed to predict q. Under the assumption that the q decreases linearly with an increase in , their relationship can be expressed as The linear regression is also shown in Fig. 16. The factor q for ≥ 6 • may be lower than the lower bound (=1.5) which is the recommended value in the code. In other words, the seismic performance of severely leaning walls (over 6 • ) cannot be guaranteed using the current seismic design code. If the severe wall-angle categories are included in the code, lower q values shall be assigned. For heavily leaning walls used for special or design purposes, the proposed empirical formula can be used to calculate the value of q varying with the design wall angle.

Summary and conclusions
In this study, the behavior factors q for leaning unreinforced masonry walls that are not categorized in current seismic design codes were experimentally estimated. The testing walls were inclined artificially by 1.5 • , 3.0 • , and 6.0 • in the out-of-plane direction. Monotonically increasing biaxial lateral loads were applied to the wall foundation using the shaking (11) q = −0.09 + 1.96, table. The resistance curve of the wall was obtained from the measured acceleration and drift ratio, and the behavior factor was calculated from the resistance curve. The behavior factors for the walls inclined at 1.5 • and 3.0 • matched well with the range of behavior factors stipulated in Eurocode 8; the behavior factor for the wall inclined at 6.0 • was slightly lower than the lower bound of the range in the code. The behavior factors stipulated in Eurocode 8 can still be used for those walls inclined by lower than 6 • . However, it was predicted that the behavior factor of the wall angle exceeding 6.0 • would be lower than the behavior factor specified in Eurocode 8. In order to achieve the desired seismic performance of masonry fence walls, strict management and regulation are required so that the wall angle does not exceed 6 • .
The relationship between the behavior factor and the wall angle was investigated; The behavior factor decreased with an increase in wall angle. Thus, a severely leaning masonry wall may require a new behavior factor. We proposed an empirical formula that represents the behavior factor as a function of the wall angle to estimate the behavior factors of walls inclined over 6 • . For heavily tilted walls used for design or special purposes, the empirical formula can be used to calculate the behavior factor varying with the design wall angle.
Although the amount of experimental data seems to be insufficient to generalize the relationship between the behavior factor and the wall angle, this study is noteworthy because, to the best of the author's knowledge, it is the first attempt at investigating the behavior factor of masonry fence walls and the effect of wall angle on the behavior factor. A further study will complement the experimental limits by developing a numerical model to generalize the results.

Declarations
Conflict of interest All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.