in-plane shear strength and damage fragility functions for

Engineering Structures 242 (2021) 112569

Available online 5 June 20210141-0296/© 2021 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

In-plane shear strength and damage fragility functions for partially-grouted reinforced masonry walls with bond-beam reinforcement

Zhiming Zhang a,*, Juan Murcia-Delso b,*, Cristian Sandoval c, Gerardo Araya-Letelier d, Fenglai Wang a,e,f

a School of Civil Engineering, Harbin Institute of Technology. Harbin 150090, China b Department of Civil and Environmental Engineering, Polytechnic University of Catalonia, C. Jordi Girona 1-3 (Building C1), 08034 Barcelona, Spain c Department of Structural and Geotechnical Engineering and School of Architecture, Pontificia Universidad Catolica de Chile, Casilla 306, Correo 22, Santiago, Chile d School of Civil Construction, Pontificia Universidad Catolica de Chile, Casilla 306, Correo 22, Santiago, Chile e Key Lab of Structures Dynamic Behavior and Control of the Ministry of Education, Harbin Institute of Technology, Harbin 150090, China f Key Lab Smart Prevention and Mitigation of Civil Engineering Disasters of the Ministry of Industry and Information Technology, Harbin Institute of Technology, Harbin 150090, China

A R T I C L E I N F O

Keywords: Reinforced masonry Partially-grouted walls In-plane shear strength Fragility functions Seismic performance

A B S T R A C T

This paper presents a study on the in-plane shear response of partially-grouted reinforced masonry walls with bond-beam reinforcement. A database of 95 tests on partially-grouted walls made of concrete hollow blocks was compiled from experimental studies reported in the literature to characterize the capacity and damageability of walls subjected to in-plane lateral loading. The database has been used to evaluate the accuracy of existing design shear strength equations for partially-grouted walls. It is concluded that the shear strength expressions in the Masonry Standards Joint Committee (MSJC) code and Canadian standard are unconservative for partially- grouted walls. A modified equation based on the MSJC expression is proposed which better estimates the shear strength of this type of walls. Seismic fragility functions are also derived based on the experimental database to calculate the probability of experiencing moderate and severe damage in a partially-grouted wall for a given story-drift ratio deformation or normalized shear force demand. The resulting fragility functions show that the normalized shear demand is better correlated with the level of damage than the story-drift ratio.

1. Introduction

Reinforced masonry is widely used in low- and medium-rise con-struction, and it can be generally classified as either fully-grouted reinforced masonry (FG-RM) or partially-grouted reinforced masonry (PG-RM). In FG-RM walls, all block cells are grouted, while in PG-RM walls only the hollow cores containing vertical reinforcement are fil-led with grout. Two different methods are typically used to place hori-zontal shear reinforcement in PG-RM walls. One consists of placing the reinforcement in horizontal beams, known as bond beams, comprising special hollow units, and filling these units with grout, as shown in Fig. 1 (a). The other method consists of placing horizontal reinforcement in the bed-joint mortar (bed-joint reinforcement), as shown in Fig. 1(b). Bond- beam reinforcement is commonly used in countries such as the United States, Canada and New Zealand, while bed-joint reinforcement is mostly used in Latin America [1].

In the last decades, a number of studies have been conducted to assess the in-plane lateral response of PG-RM walls built with concrete masonry units [2–21] and clay bricks [21–23]. Experimental in-vestigations have indicated that the in-plane lateral response of PG-RM walls is controlled by a number of parameters including the properties of constituent materials, wall aspect ratio, level of axial stress, and the ratio and distribution of vertical and horizontal reinforcement. Most of the PG-RM walls tested in these investigations presented a shear-dominated behavior which is characterized by the drastic drop of strength and stiffness after major diagonal cracking occurs. Previous studies have also shown that the lateral load resisting mechanism of PG-RM walls may be quite different than that of FG-RM walls. Schultz [6] observed that vertical cracks resulting from stress concentration between ungrouted and grouted masonry appear to dominate the behavior of PG-RM walls. Minaie et al. [10] concluded that PG-RM walls respond similar to infilled frames and provide little coupling between vertical reinforcement. In terms of horizontal reinforcement schemes, some studies [19–20] have

* Corresponding authors. E-mail addresses: [email protected] (Z. Zhang), [email protected] (J. Murcia-Delso).

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Engineering Structures

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https://doi.org/10.1016/j.engstruct.2021.112569 Received 22 July 2020; Received in revised form 1 May 2021; Accepted 16 May 2021

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indicated that bed-joint PG-RM walls may present better lateral-load carrying capacity than bond-beam PG-RM walls. However, while some tests [12] have indicated that joint reinforcement provides better crack control than bond beam reinforcement, others [15] have shown that the crack pattern of walls with bond beams was more scattered than that of walls with joint reinforcement.

While many shear strength equations have been proposed for RM walls [8,24–31], most of them were derived from tests on FG-RM walls. The accuracy of these expressions for PG-RM walls has been questioned based on the results of several research studies [8,10,11,13,32,33]. Bolhassani et al. [32] proposed a new approach to calculate the shear strength of PG-RM walls based on an infilled-frame mechanism model. The shear strength predicted by the proposed model was on average in good agreement with experimental data obtained from 42 wall tests reported in the literature. Hassanli et al. [33] employed regression analysis to investigate the effect of different parameters on the accuracy of the four shear strength equations in international design codes. Their study showed in general poor correlation between code predictions and experimental results from 89 wall tests in the literature. Minaie et al. [10] established a database of 60 wall tests and concluded that the shear strength expression for RM walls provided by the 2008 version of the Masonry Standards Joint Committee (MSJC) code [28] overestimated the actual shear strength of PG-RM walls. The authors studied four different modifications to the shear strength formula in this code, which increased the level of conservatism of the predictions but presented similar or higher dispersion of the predictions. Based on the conclusions of this study and previous studies, the MSJC code equation was revised

in 2011 to include a uniform reduction factor equal to 0.75 for PG-RM walls. Recently, Elmapruk et al. [11] evaluated the accuracy of different code shear strength equations using data from 33 wall speci-mens satisfying general code requirements. Code equations provided conservative estimates for walls with small grout spacing but tended to overestimate strength when the grout spacing was large. The authors proposed applying a simple grout spacing reduction factor to the revised MSJC equation to improve the accuracy of the shear strength pre-dictions. This reduction factor was adjusted empirically and decreases nonlinearly as the grout spacing increases.

To date, few contributions have been made to predict the seismic response of PG-RM walls from a probabilistic standpoint as required in new performance-based assessment methodologies. For instance, the FEMA P-58 methodology for seismic performance assessment of build-ings [34] uses fragility functions to evaluate the damageability of structural and nonstructural components. Damage fragility functions define the conditional probability of reaching or exceeding a specific damage state for a given demand parameter (such as the story drift) obtained from structural analysis. In recent years, a number of re-searchers have developed damage fragility functions for masonry walls [1,35–42], but the work on PG-RM walls has been very limited. Murcia- Delso and Shing [39] developed fragility curves for bond-beam PG-RM walls from only 15 tests on walls subjected to either in monotonic or cyclic in-plane lateral loading. All walls were made of hollow concrete blocks. Two different sets of fragilities were developed, with one using story-drift ratio as demand parameter and the other using a normalized shear force as demand parameter. While the story-drift ratio is a simpler

Notation

An net shear area Afs face shell area Ah cross-sectional area of shear reinforcement bw thickness of wall be effective web width C1,C2,C3 shear strength coefficients in NZS 4230:2004 d distance from extreme compression fiber to centroid of

longitudinal tension reinforcement dv actual depth of a member in direction of shear considered f’

m compressive strength of masonry fyh yield strength of horizontal reinforcing steel He effective height of the wall Lw length of the wall

M maximum moment at section under consideration Pu axial load sh spacing of horizontal shear reinforcement V maximum shear force at section under consideration α angle formed between centers of load application and

reaction ρh horizontal reinforcement ratio ρv vertical reinforcement ratio σ axial stress γg1 grouted shear wall factor in MSJC (2016) γg2 factor to account for partially grouted, not greater than 0.5

in CSA S304.1–14 vbm basic type-dependent shear strength of masonry

Fig. 1. PG-RM construction: (a) wall with bond-beam, (b) wall with bed-joint reinforcement (reprinted from Araya-Letelier et al. [1]).

Z. Zhang et al.


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demand metric, the normalized shear force parameter provided better correlation with the degree of damage observed in the walls. However, due to the limited data available, only one fragility function corre-sponding to severe shear damage was derived from experimental data. Araya-Letelier et al. [1] developed fragility functions for bed-joint PG- RM walls using data from 44 and 32 tests on clay brick walls and hollow concrete block walls, respectively, subjected to in-plane cyclic lateral loading. Their fragility functions were developed using the same damage state definitions and walls demand parameters as in Murcia-Delso and Shing [39]. Their results illustrated that hollow concrete block walls present a higher deformation capacity as compared to clay brick walls. The comparison with the fragility functions derived by Murcia-Delso and Shing [39] also seemed to confirm that walls with bed-joint rein-forcement may present higher deformation capacity than those with bond-beam reinforcement as implied by previous studies [19–20], but the limited data on bond-beam walls was not sufficient to arrive to a general conclusion. It can be inferred from the studies by Murcia-Delso and Shing [39] and Araya-Letelier et al. [1] that further work is needed towards developing fragility functions for PG-RM walls, and in partic-ular for walls with bond-beam reinforcement.

This paper presents a study on the in-plane shear strength and seismic damage fragility of PG-RM walls with bond-beam reinforce-ment. A database containing test results from 95 tests on PG-RM walls made of hollow concrete blocks is compiled based on data reported in the literature. The experimental database is used to assess the accuracy of three different shear strength expressions used in design codes. Experimental data is also used to study two possible modifications to the current shear strength equation provided by the MSJC (2016) code [31]. Damage fragility functions are derived from the experimental database to assess the damageability of PG-RM following the FEMA P-58 performance-based seismic assessment framework. The derived fragility functions are compared with existing fragilities for FG-RM walls and PG- RM walls with bed-joint reinforcement.

2. Experimental database

In this study, experimental results from tests on PG-RM walls with bond-beam reinforcement have been compiled from the literature to develop a database of the in-plane shear strength and damageability of this type of walls. Specifically, the database comprises tests from studies conducted by Chen et al. [2], Thurston and Hutchison [3], Ghanem et al. [4,5], Schultz [6], Ingham et al. [7], Voon and Ingham [8], Maleki [9], Minaie et al. [10], Elmapruk et al. [11], Baenziger and Porter [12], Nolph and Elgawady [13], Hamedzadeh [14], Hoque [15], Bolhassani [16], Rizaee [17] and Calderon et al. [18]. All data is from single-story PG-RM walls tested as cantilevers (single curvature bending) or under fixed–fixed boundary conditions (double curvature bending). Wall specimens were tested under either monotonic or cyclic lateral loading histories, and they all presented a shear-dominated behavior. All the walls in the database were constructed with hollow concrete blocks and bond-beam horizontal reinforcement. For completeness, walls with no horizontal reinforcement reported in the aforementioned studies are also included to broaden the range of shear reinforcement ratios of the database. The experimental studies considered to develop the database are briefly reviewed here:

• Chen et al. [2] tested seven PG-RM walls under cyclic loading. Four of the walls were constructed with hollow concrete blocks and three with hollow clay bricks, but only the three walls with concrete blocks and vertical reinforcement have been included in the database. The test variables were the amount and distribution of horizontal and vertical reinforcement. Test results showed that a higher amount of vertical reinforcement did not modify the shear strength of walls.

• Thurston and Hutchison [3] tested eight RM walls under cyclic loading. Five of the walls are FG, and only the three PG-RM walls have been included in the database. The variables of the research

were axial stress, extent of grouting of cells and distribution of reinforcement. The authors concluded that using smaller horizontal bar diameters at a smaller distance provides better inelastic perfor-mance than using similar steel quantities in the form of fewer larger bars.

• Ghanem et al. [4,5] conducted six monotonic tests on small-scale PG- RM walls. The test variables were the axial compression level, block strength, and amount and distribution of horizontal and vertical reinforcement. Test results showed that the increase of axial compression reduced wall ductility, increased cracking strength, and changed wall behavior from flexure to shear modes. Only the four walls presenting shear failures have been included in the database. Test results also showed that increasing the amount of horizontal and vertical reinforcement increased the shear strength but did not affect the load at which diagonal cracking occurred.

• Schultz [6] tested six PG-RM walls under cyclic loading. All the walls were reinforced only in their outermost cells in the vertical direction and in a single bond beam at mid-height. The variables of the experimental study were the aspect ratio, horizontal reinforcement ratio, and axial stress. It was concluded from the tests that a higher aspect ratio increases the ultimate shear stress but it decreases the energy dissipation capacity of the walls.

• Ingham et al. [7] tested nine PG-RM walls under cyclic loading. Two of the walls are not included in the database considering their rela-tively high aspect ratio of 3.0. The test parameters were the wall aspect ratio, horizontal reinforcement ratio, and the width of the wall. The horizontal reinforcement was located at the top two courses of the walls. None of the walls had an applied axial compression load. The results showed all the walls failed in diagonal tension because of the lack of distributed horizontal shear reinforcement.

• Voon and Ingham [8] tested ten RM walls under cyclic loading. Eight of the walls are FG, and only the two PG walls are included in the database. Wall specimens had different amounts of shear reinforce-ment, axial compression load, type of grouting and wall aspect ratio. It was concluded from the test results that the shear strength equa-tion in the New Zealand code (NZS 4230:1990) was overly conser-vative in accounting for masonry shear strength.

• Maleki [9] tested five PG-RM walls under cyclic loading. The experimental program focused on the effect of reinforcement spacing and wall aspect ratio. All the walls had larger reinforcement spacing than that required by the Canadian masonry design standard (CSA S304.1 2004). From the test results, it was concluded that the global response of the test was not sensitive to the reinforcement pattern. Furthermore, good agreement was obtained between the shear pre-dictions using the CSA code and the experimental results of walls failing in shear.

• Minaie et al. [10] tested four PG-RM walls designed according to MSJC (2008), under cyclic loading. The experimental variables were the mortar formulation with different compressive strengths, level of axial stress, and boundary conditions. Experimental results indicated that the shear strength expression for RM walls provided by MSJC (2008) was unconservative for PG-RM walls. The lack of conserva-tism was attributed to the fact that the code expression was based exclusively on data from FG-RM walls.

• Elmapruk et al. [11] tested six PG-RM walls under cyclic loading. The effects of grout horizontal spacing and horizontal reinforcement ratio were investigated. Test results showed that the shear strength of the walls significantly depended on the grout horizontal spacing. Furthermore, it was concluded that beyond a certain horizontal reinforcement ratio, there was no further increase in the shear strength of the walls.

• Baenziger and Porter [12] tested eight PG-RM walls under cyclic loading. Three walls had bond-beam reinforcement and five walls had bed-joint reinforcement, but only the walls with bond beams have been included in the database. The rest of variables of the

Z. Zhang et al.


4

research were grouting ratio, aspect ratio and horizontal reinforce-ment ratio. It was concluded that the joint reinforcement provided better crack control than bond beam reinforcement. In addition, better crack control was achieved by a larger number of smaller horizontal reinforcing bars than with fewer larger bars.

• Nolph and Elgawady [13] tested five PG-RM walls under cyclic loading. The variables of the research were grout horizontal spacing and horizontal reinforcement ratio. The author concluded that the MSJC (2008) shear equation overestimated the strength of partially- grouted walls with 48 in. (1219 mm) grout horizontal spacing. This was attributed to an overestimation of the contribution of shear reinforcement. Additionally, the author proposed using the compressive strength of ungrouted prisms to calculate the masonry component of the shear resistance in the MSJC equation.

• Hamedzadeh [14] tested twenty-one PG-RM walls with widely spaced reinforcement under monotonic lateral load. The walls were constructed with half-scale concrete masonry units, and the main variables of the study were the aspect ratio and initial axial stress. A fixed–fixed boundary condition was maintained during the tests by keeping constant the displacement of the vertical actuators intro-ducing the axial load. However, the axial loads were not constant during a test. Test results showed that the shear capacity increased as the aspect ratio decreased. The initial level of axial stress did not influence the shear capacity of the walls because the actual axial load acting on the walls when the peak lateral load was achieved was relatively similar for all specimens.

• Hoque [15] tested eighteen PG-RM walls under cyclic loading. Ten walls had bond-beam reinforcement and eight walls had bed-joint reinforcement. However, only the walls with bond-beam reinforce-ment have been included in the database. The rest parameters of the research were anchorage conditions of bond-beam reinforcement, type of boundary conditions and loading protocol. No significant differences in shear capacity were observed despite the changes in

design and loading characteristics. However, it was observed that the crack patterns of walls with bond beams were more scattered as compared to walls with joint reinforcement, which presented more concentrated diagonal cracking.

• Bolhassani [15] tested seven full-scale PG-RM walls under cyclic loading. Some walls had doubly-grouted cells instead of conven-tional single-grouted cells, and some had horizontal reinforcement in both the bond beam and bed joint. However, only the three walls with bond-beam reinforcement failing in shear have been included in the database. The results showed the wall with double-reinforced cells and bond beams had a 60% increase in elastic stiffness compared to the wall with single reinforcing cells. In addition, the bed-joint reinforcement did not improve the strength and deforma-tion capacity of PG-RM walls at the ultimate limit state.

• Rizaee [17] tested fourteen PG-RM walls under cyclic loading. Two walls where the bond beam was grouted but not reinforced are not included in the database. The variables of the research are the end anchorage conditions of bond beam horizontal reinforcement and horizontal bar size. The author concluded that the end anchorage conditions employed in this study did not have a significant effect on the behavior of the walls.

• Calderon et al. [18] tested four PG-RM walls under cyclic loading. The research compared the seismic performance of walls with different horizontal reinforcement schemes. Only one wall with bond-beam horizontal reinforcement has been included in the database. The results of this study showed that different horizontal reinforcement layouts do not affect the lateral wall capacity when the horizontal reinforcement ratio and material properties are the same.

Fig. 2 summarizes the distribution of effective wall aspect ratio (effective height to section depth), horizontal reinforcement ratio, ver-tical reinforcement ratio and axial stress ratio of the wall specimens

Fig. 2. Frequency distributions of wall characteristics: (a) effective aspect ratio; (b) horizontal reinforcement ratio; (c) vertical reinforcement ratio and (d) axial stress ratio.

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comprising the database. These parameters have been computed from the data reported in the original studies using a consistent approach: reinforcing ratios are based on gross area of walls and the axial stress ratios are based on the compressive strength of hollow concrete prisms. In addition, Table A1 of the Appendix provides the design and loading characteristics of each of the 95 PG-RM specimens included in the database. The wall specimens have effective aspect ratios ranging from 0.16 to 1.50, as shown in Fig. 2(a), but 78% of them are between 0.4 and 1.0. The horizontal reinforcement ratios vary between 0 and 0.28%, and the vertical reinforcement ratios vary between 0.09% and 0.62%. The specimens were tested under constant axial compressive stress ranging between 0 and 25% of the compressive strength of hollow prisms, except in the tests by Hamedzadeh [14] in which specimens were subjected variable axial compressive stresses up to 45% of the compressive strength of hollow prisms. Finally, the ratio of the net shear area (An) to the gross area (Ag) of the wall specimens included in the database is between 40% and 65%, and the ratio of grouted cells to total number of vertical cells is between 11% and 38%.

3. Evaluation of in-plane shear strength equations

3.1. Existing shear strength equations

The database presented in the previous section is used to evaluate the accuracy of three empirically-derived shear strength equations used in masonry design codes in the United States (MSJC 2016) [31], Canada (CSA S304.1-14) [27] and New Zealand (NZS 4230:2004) [26]. These three shear strength equations are presented in Table 1. All three equations include separate terms for the strength contributions of ma-sonry and shear reinforcement, and they also take into account the effect of axial load. It is worth noting that these three expressions were initially derived for FG-RM walls [10]. The way in which these shear strength equations are used for PG-RM walls as compared to FG-RM walls varies depending on the code. MSJC (2016) uses the net shear area An of the wall (i.e., excludes ungrouted cells) and an additional reduction factor γg1 = 0.75 for PG-RM walls. The definition of net shear area for PG-RM walls is shown in Fig. 3(a). CSA S304.1-14 employs the gross shear area but modifies the contribution of masonry and axial load with a reduction factor γg2 to account for the ratio of grouting in walls which cannot be greater than 0.5. NZS 4230:2004 uses the effective face shell shear area

Table 1 Selected design shear strength equations.

Reference Shear strength equation Eq. No.

MSJC (2016) Vn1 = γg1

{

0.083[

4.0 − 1.75(

MVdv

)]

An

f ’m

√

+ 0.25Pu + 0.5fyhAh

(dv

sh

)}

Vn1 ≤ γg1(6An

f ’m

√

) whereM

Vdv≤ 0.25

Vn1 ≤ γg1(4An

f ’m

√ )where

MVdv

≥ 1.0

(1)

CSA S304.1–14 Vn2 = γg2

[

0.16(

2 −M

Vdv

)

f ’m

√

bwdv + 0.25Pu

]

+ 0.6fyhAh

(dv

sh

)

Vn2 ≤ 0.4

f ’m

√

bwdvγg2 whereHw

Lw≥ 1

Vn2 ≤ 0.4

f ’m

√

bwdvγg2

[

2 −Hw

Lw

]

whereHw

Lw< 1

(2)

NZS 4230:2004 Vn3 =

[

(C1 + C2)vbm + 0.9Pu

bedtanα + C3

Ahfyh

besh

]

bed

Vn3 ≤ 0.45

f ’m

√

bed

(3)

Note: All the variables are described in the notation list.

Fig. 3. Definition of (a) net shear area, (b) effective face shell shear area, and (c) actual face shell shear area.

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Fig. 4. Experimental to predicted shear strength ratios for (a) MSJC (2016); (b) CSA S304.1-14; (c) NZS 4230:2004; (d) Eq. (4); (e) Eq. (5).

Table 2 Summary of experimental to predicted shear strength ratios.

Statistics Vexp/Vn1(MSJC 2016) Vexp/Vn2(CSA S304.1–14) Vexp/Vn3(NZS 4230:2004) Vexp/Vn4(proposed) Vexp/Vn5(proposed)

Maximum 2.82 2.36 2.70 2.59 1.88 Minimum 0.46 0.35 0.50 0.52 0.47 Mean 1.12 0.92 1.28 1.22 1.13 Standard deviation 0.51 0.40 0.48 0.44 0.33 CV 0.46 0.43 0.37 0.36 0.29

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(Afs) to calculate the shear strength, as shown in Fig. 3(b). The shear strength of the 95 specimens in the experimental database

is calculated using the three equations presented in Table 1 based on actual material properties and nominal dimensions. For consistency, the masonry compressive strengths have been determined from reported test data on hollow masonry prisms considering the relatively low per-centage of grouted cells in the walls comprising the database, as indi-cated in a previous section. To determine the basic shear strength, vbm, in NZS 4230:2004, it is assumed that the test specimens were constructed under expert supervision (observation type A). For those specimens with only one bond beam at mid-height ([2,6,12]), the ratio between shear depth (dv in MSJC and CSA, and d in NZA) and spacing of horizontal reinforcement (sh) has been taken as 1 to properly account for the amount of steel available to resist shear. If a single bond beam was used at the top of the wall ([7,14]), the contribution from horizontal rein-forcement has been neglected due to the lack of distributed reinforce-ment to restrain shear cracks. For the specimens tested by Hamedzadeh [14], which were subjected to a variable axial load, the shear strength is calculated using the axial load reported at the peak lateral load.

Fig. 4 shows the ratios of the experimental shear strength (Vexp) to the calculated shear strength (Vn) for each one of the analytical ex-pressions. Table 2 provides the statistics of the strength ratios in terms of maximum and minimum values, mean value, standard deviation, and coefficient of variation (CV). The mean value of Vexp/Vn for MSJC (2016) is 1.12, which indicates that on average the predicted shear strength is close to the actual strength. However, there is a very large scatter in the strength ratio (CV is 0.46), which results in a significant number of cases

in which the strength is overestimated. In particular, Vexp < 0.8Vn in 34% of the cases when using the MSJC (2016) equation. Hence, this equation results in a high probability of overestimating the shear strength of PG-RM walls. The mean value of Vexp/Vn of CSA S304.1–14 is 0.92 along with a large scatter (CV is 0.43), which leads to uncon-servative estimates (Vexp < 0.8Vn in 51% of the wall tests). The mean value of Vexp/Vn of NZS 4230:2004 is 1.28 along with a smaller CV of 0.37, which results in increased conservatism with respect to the other two code equations (Vexp < 0.8Vn in 14% of the cases).

3.2. Modified MSJC equation

Two simple variations of the MSJC (2016) equation are proposed here to improve the accuracy of the shear strength predictions. The first modification is intended to better estimate the actual effective shear area of masonry for a PG wall. In particular, it is recommended to use the actual face shell shear area (Afs) of the wall, as shown in Fig. 3(c), instead of the net shear area (An), while removing the uniform reduction factor γg1. The resulting shear strength expression is shown in Eq. (4).

Vn4 = 0.083[

4.0 − 1.75(

MVdv

)]

Afs

f ’m

√

+ 0.25Pu + 0.5fyhAh

(dv

sh

)

(4)

Vn4 ≤ 6Afs

f ’m

√

whereM

Vdv≤ 0.25

Vn4 ≤ 4Afs

f ’m

√

whereM

Vdv≥ 1.0

Fig. 5. Damage states for Wall SS-1.0-20 tested by Bolhassani: (a) DS1, moderate shear damage, (b) DS2, severe shear damage, (c) Load-displacement hysteresis (adapted from [44]).

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This modification is similar to that proposed by Elmapruk et al. [11] in that the uniform reduction factor γg1 is removed and the modifications with respect to the equation for FG walls only apply to the masonry component and strength limits. However, the current proposal is directly related to a physical characteristic of the wall (Afs) while the modification suggested by Elmapruk et al. [11] relies on an empirically- derived parameter which is nonlinearly related to the vertical grout spacing.

A second modification of the MSJC formula is studied here to ac-count also for the combined effect of both wall aspect ratio and axial load level on the shear strength. A recent study conducted by Sandoval et al. [43] on PG-RM walls with bed-joint reinforcement has indicated that the influence of the axial load on the shear strength increases as the aspect ratio of the wall decreases. This effect is related to the angle of the diagonal compressive strut forming in the wall to resist shear, and it is already considered in the NZS 4230:2004 expression by multiplying the axial load term by tanα, as shown in Eq. (3). A similar approach is fol-lowed here by multiplying the current axial load term in Eq. (4) by Vdv

M , which represents the inverse of the effective aspect ratio of the wall, as shown in Eq. (5). The proposed modification is based on the simplifying assumption that the axial load term is equal to the original term (0.25Pu) for walls with an effective aspect ratio of 1, and that this contribution is inversely proportional to the effective aspect ratio. The formula result-ing from considering both the actual face shell shear and the effect of wall aspect ratio in the axial load contribution to shear resistance is provided in Eq. (5):

Vn5 = 0.083[

4.0 − 1.75(

MVdv

)]

Afs

f ’m

√

+ 0.25Pu

(Vdv

M

)

+ 0.5fyhAh

(dv

sh

)

(5)

Vn5 ≤ 6Afs

f ’m

√

whereM

Vdv≤ 0.25

Vn5 ≤ 4Afs

f ’m

√

whereM

Vdv≥ 1.0

The results of the strength predictions using the expressions Vn4 and Vn5 are presented in Fig. 4 and Table 2. As shown in Table 2, the mean value of Vexp/Vn4 is 1.22, which is slightly larger than the mean value of Vexp/

Vn1 but it is still close to 1, while the CV is reduced to 0.36. Also, the number of specimens with Vexp < 0.8Vn is reduced to 13%. For Eq. (5), the mean value of Vexp/Vn5 is 1.13, which is practically the same as for the MSJC (2016) equation. However, it has a significantly smaller scatter (CV = 0.29) and the number of specimens with Vexp < 0.8Vn is 15%. It is concluded that Vn5 provides the most accurate shear strength predictions for the walls in the database given that the strength ratio has a mean very close to 1 and the smallest dispersion among the different formulas evaluated.

4. Seismic damage fragility functions

4.1. Damage states and demand parameters

Fragility functions are developed following the discrete damage states previously defined in FEMA P-58 [34] and Murcia-Delso and Shing [39]. Each damage state is associated with different levels of repair effort. Six different damage states were proposed by Murcia-Delso and Shing [39] for RM walls based on experimental evidence. Three damage states correspond to flexure-dominated walls: slight flexural damage, moderate flexural damage, and severe flexural damage. Two damage states are defined for shear-dominated walls given their rela-tively brittle response: moderate shear damage state and severe shear damage state. For sliding shear damage, only one severe damage state is considered.

According to experimental data reported in the literature, most of

PG-RM walls tend to present a shear-dominated behavior. Hence, only shear damage states (moderate and severe) are considered here for developing fragility functions based on the assembled database, in which all specimens exhibited this type of failure. The diagonal shear behavior is relatively brittle compared to the flexural behavior. For FG- RM walls, Murcia-Delso and Shing [39] considered that the occurrence of major diagonal shear cracks triggered moderate damage in shear- dominated walls. Consistent with this definition, moderate damage (also referred to as Damage State 1 or DS1 following FEMA-P58 termi-nology) for PG-RM walls is also defined to be triggered by the first occurrence of a major diagonal shear crack which crosses the majority of the wall length or height. Fig. 5(a) shows an example of DS1 for a wall tested by Bolhassani ([16,44]). At DS1, PG-RM walls can be repaired by fully filling the cavities with grout and injecting the remaining cracks with fluid non-shrink grout [39]. If face shell spalling is observed, loose masonry needs to be removed completely and replaced by non-shrink grout. Severe diagonal shear damage (Damage State 2 or DS2) is defined as wide diagonal cracks and crushing or face shell spalling at the wall toes. Based on experimental observations, DS2 corresponds to the point at which the wall reaches its peak shear resistance. Fig. 5(b) shows DS2 for the wall tested by Bolhassani ([16,44]). For this damage state, it may not be economical to repair the component and it is likely that the entire wall component needs to be replaced [39]. Finally, Fig. 5(c) presents the lateral force-drift response of the wall tested by Bolhassani ([16,44]) with the points at which damage states DS1 and DS2 are exceeded.

Two different engineering demand parameters are used in this study to develop different sets of fragility functions. The first corresponds to the story-drift ratio (SDR), which is a simple and commonly-used mea-sure of the seismic demand on structural and non-structural compo-nents. Previous studies [35,36,39,41,42] have used SDR to develop fragility functions for unreinforced, reinforced and confined masonry walls. Murcia-Delso and Shing [39] proposed using a force-based de-mand parameter considering that SDR may present some limitations as damage indicator because it does not account for the specific design details and loading condition of a wall, which influence the lateral deformation capacity. The proposed alternative was a normalized di-agonal shear demand (NDSD) defined in Eq. (6).

NDSD =VVn

(6)

where V is the in-plane shear force acting on the wall and Vn is the shear capacity calculated in accordance with the MSJC code using expected material strengths. This demand parameter considers the variations in the wall geometry, quantity of shear reinforcement, material properties and axial loads which are accounted for when normalizing by Vn.

The wall database compiled in the present study has been used to develop statistics of the demand parameters triggering moderate and severe damage. Table B1 of the Appendix presents the values of SDR and NDSD when DS1 and DS2 are achieved for each wall specimen in the database. Since all the specimens are single-story walls, SDR is computed as wall drift divided by wall height. For NDSD,Vsimply cor-responds to the lateral force acting on the wall specimen. Two different NDSD definitions have been considered depending on the normalizing parameter: the MSJC (2016) shear strength equation (Vn1) and its modified version proposed in this study (Vn5). The values of the demand parameters at DS2 are determined based on the drift and lateral force at which the wall reaches its peak lateral strength for each of the 95 tests included in the database. Hence, the NDSD value at DS2 is equal to the ratio of the experimental to predicted shear strength. The demand parameter values for DS1 are calculated only for those tests in which the first occurrence of diagonal cracking was reported at a given drift and/or lateral force demand. The values of SDR and NDSD at DS1 were obtained from 35 and 43 wall tests, respectively.

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4.2. Development of fragility functions

Damage fragility functions are derived using the actual demand data method proposed in FEMA P-58 [34], which is briefly described here. This methodology assumes that fragility functions take the form of a lognormal distribution as shown in Eq. (7).

Fi(DP) = Φ(

ln(DP/θi)

βi

)

(7)

where Fi(DP) is the conditional probability that a component will reach or exceed a damage state i when the value of the demand parameter is DP, Φ is the standard normal (Gaussian) cumulative distribution func-tion, θi is the median value of the probability distribution at which a damage state i occurs, and βi is the corresponding dispersion. The fragility parameters in Eq. (7), θi and βi, are computed from the fragility data in Table B1 of the Appendix as follows.

For each damage state (DS1, DS2) and demand parameter (SDR, NDSD), the value of θi is calculated with Eq. (8).

θi = e

(

1N

∑M

j=1ln(DPj)

)

(8)

where DPj is the value of the demand parameter in data point (test) j at which the damage state i was exceeded, and N is the total number of data points. Two different contributions to dispersion are considered to calculate βi: random test variability and additional uncertainty. Random variability observed in experiments for a given damage state i is quan-tified with dispersion parameter βr,i, the value of which is determined using Eq. (9).

βr,i =

1N − 1

∑N

j=1

(

ln(

DPj

θi

))2√√√√ (9)

The second dispersion parameter, βu,i, accounts for the uncertainty that the tests represent the actual conditions in a real building and that available test data reflects the true random variability for damage state i. Given the relatively large size of the database and the different test configurations used in the tests, βu,i is taken equal to 0.10 based on the values recommended in FEMA P-58 [34]. The total dispersion is then computed with Eq. (10).

βi =

β2r,i + β2

u,i

√

(10)

The fragility curves for bond-beam PG-RM walls obtained from the

Table 3 Summary of derived fragility parameters for bond-beam PG-RM walls.

Damage state SDR-based fragility functions NDSD-based fragility functions using MSJC (2016) shear equation

NDSD-based fragility functions using Eq. (5)

θ βr β Lilliefors test (5%) θ βr β Lilliefors test (5%) θ βr β Lilliefors test (5%)

DS1 0.12% 0.60 0.61 Fail 0.57 0.39 0.40 Fail 0.69 0.41 0.42 Fail DS2 0.37% 0.54 0.55 Pass 1.03 0.40 0.41 Fail 1.08 0.31 0.32 Pass

Fig. 6. Fragility curves for bond-beam PG-RM walls: (a) SDR, (b) NDSD with MSJC (2016), (c) NDSD with Eq. (5).

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experimental database are presented in Fig. 6. Both empirical cumula-tive distribution curves and derived lognormal functions with random dispersion only are provided. Fig. 6(a) shows the fragility curves that use SDR as demand parameter. Fragility functions using the NDSD param-eter are shown in Fig. 6(b) and Fig. 6(c) for Vn = Vn1 and Vn = Vn5, respectively. The median, random dispersion and total dispersion of the three sets of fragility functions are presented in Table 3. The goodness- of-fit between the hypothesized lognormal distributions and empirical functions is estimated using the Lilliefors test at a 5% significance level as recommended in FEMA P-58 [34]. The lognormal distributions pass the goodness-of-fit test for two fragility functions corresponding to DS2 (see Table 3), but the rest fail the Lilliefors test at a 5% significance level by a relatively small margin, as shown in Fig. 6.

The median SDR values triggering moderate (DS1) and severe damage (DS2) are 0.12% and 0.37%, respectively. While the response of shear-dominated walls is generally considered to be brittle, these values indicate that after the walls develop their first main shear crack they can still experience a significant increase of lateral deformation before failure. The median NDSD values triggering moderate (DS1) and severe damage (DS2) are 0.57 and 1.03, respectively, when the normalization is done using the MSJC (2016) equation. The median NDSD values when normalizing with Eq. (5) become 0.69 and 1.08 for DS1 and DS2,

respectively. Hence, the first main shear crack in the wall appears, on average, when the shear demand represents approximately 60% of the theoretical shear strength.

Regarding the dispersion of the fragility functions, SDR-based fra-gilities present a higher random dispersion (βr is 0.60 for DS1 and 0.54 for DS2) than the two sets of NDSD-based fragilities (βr vary between 0.31 and 0.41). This observation is in agreement with the sets of fragility functions developed for FG-RM walls by Murcia-Delso and Shing [39] and those developed for PG-RM walls with bed-joint reinforcement by Araya-Letelier et al. [1]. The smaller dispersion of NDSD fragilities can be explained by the fact that the normalized force-based parameters are better correlated with the level of damage than SDR as they take into account design and loading characteristics that affect the capacity of the wall.

4.3. Comparison with other fragility functions for RM walls

4.3.1. Comparison with FG-RM walls The fragility functions for PG-RM walls derived in this study are

compared to those derived by Murcia-Delso and Shing [39] for FG-RM walls presenting shear damage. The fragility functions for FG-RM walls were derived based on experimental data from 36 and 27 wall

Fig. 7. Comparison of fragility functions for PG-RM and FG-RM walls with bond beam reinforcement; (a) based on SDR, and (b) based on NDSD with MSJC (2016).

Table 4 Comparison with fragility parameters derived for other types of RM walls.

Damage state Wall type SDR-based fragility functions NDSD-based fragility functions using MSJC (2016) shear equation

θ βr θ βr

DS1 PG walls with bond-beam scheme 0.12% 0.60 0.57 0.39 FG walls with bond-beam scheme 0.36% 0.54 0.86 0.22 PG walls with bed-joint scheme 0.12% 0.60 0.61 0.38

DS2 PG walls with bond-beam scheme 0.37% 0.54 1.03 0.40 FG walls with bond-beam scheme 0.59% 0.44 1.05 0.21 PG walls with bed-joint scheme 0.54% 0.55 1.08 0.41

Fig. 8. Comparison of fragility functions of bond-beam and bed-joint PG-RM walls: (a) based on SDR, and (b) based on NDSD with MSJC (2016).

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tests for moderate and severe shear damage, respectively. Note that the definitions of the damage states employed in both studies are equivalent. Fig. 7 presents the fragility functions for PG and FG walls considering random dispersion only, and their fragility parameters are also sum-marized in Table 4. As shown, the median SDR values triggering damage states DS1 and DS2 for PG walls are 67% and 37% smaller, respectively, than those for FG walls. Hence, FG walls generally present a larger deformation capacity than PG walls. The median NDSD values using the MSJC (2016) shear strength equation are 34% and 2% lower for PG walls than for FG walls at DS1 and DS2, respectively. Note that the shear strength equation for FG-RM is identical to Eq. (1) except that it does not include the γg1 = 0.75 reduction factor. Hence, the load-carrying ca-pacity of PG walls is only slightly smaller as compared to the theoretical predictions using the MSJC (2016) formula. Moderate damage caused by diagonal cracking (DS1) occurs relatively earlier for PG walls as indicated by their lower drift ratio and NDSD values. This can be explained by a faster propagation of diagonal cracking through ungrouted panels between grouted cells, as compared to walls with no ungrouted cells. Finally, the fragility functions for PG-RM walls present a significantly higher dispersion than those of FG-RM walls, as shown in Fig. 7 and Table 4. This higher dispersion can be explained by the less predictable behavior of PG-RM, and in particular for NDSD due to the fact that the normalizing term Vn1 was derived from test data from FG walls and does not have the same accuracy for PG walls.

4.3.2. Comparison with PG-RM walls with bed-joint reinforcement Fig. 8 compares the fragility functions for bond-beam PG-RM walls

with those derived by Araya-Letelier et al. [1] for PG-RM walls made of concrete blocks and bed-joint reinforcement. The fragility parameters for bed-joint PG-RM walls were determined for the same damage states as in the current study using data from 32 tests. The experimentally- derived fragility parameters are summarized in Table 4. As shown in Fig. 8(a), the SDR-based fragility functions derived for DS1 (moderate shear damage) are very similar for both bond-beam and bed-joint PG- RM walls. The latter might be explained since the distribution of the horizontal reinforcement does not have a significant impact on the generation of the first major diagonal crack (DS1). In fact, a previous study [43] has shown that the main design parameter influencing this damage state is the wall aspect ratio. Fig. 8(a) also shows that fragility functions for DS2 (severe shear damage) present important differences in terms of median (0.37% and 0.54% for bond-beam and bed-joint schemes, respectively), which may be explained due to the larger ductility capacity provided by a more distributed horizontal scheme when using bed-joint reinforcement. This observation is in agreement with previous comparisons reported in Araya-Letelier et al. [1]. In terms of NDSD-based fragility functions using the MSJC (2016) equation, Fig. 8(b) and Table 4 show that the fragility curves for both type of walls are very similar for both DS1 and DS2, but the median values for bed- joint PG-RM walls are between 5% and 7% higher than those for bond-beam PG-RM walls. Finally, both types of walls present almost identical random dispersion values for each of the sets of fragilities and damage states. Given that the size of both databases is practically the same for DS1, their similar dispersions indicate that both types of walls present a similar variability in their damage response.

5. Conclusions

This paper has presented a study of the in-plane shear strength and seismic damage fragility of partially-grouted reinforced masonry (PG- RM) walls made of concrete hollow blocks and bond beams for shear reinforcement. A database of 95 tests on PG-RM walls subjected to in- plane lateral loading was compiled from published literature. The experimental database was used to examine three existing shear strength equations in design codes of the United States (MSJC 2016), Canada (CSA S304.1-14) and New Zealand (NZS 4230:2004). The database was also used to derive seismic fragility functions for this type of walls for

two levels of damage (moderate and severe). Two different sets of fragility functions were derived, one uses the story-drift ratio (SDR) as demand parameter and the other uses a normalized diagonal shear de-mand (NDSD) parameter. The following conclusions can be drawn from this research:

(1) The average values of the experimental to predicted shear strengths (Vexp/Vn) using MSJC (2016) and NZS 4230:2004 are 1.11 and 1.28, respectively. The average value of Vexp/Vn for CSA S304.1–14 is 0.92, which indicates that the expression tends to overestimate the shear strength of PG-RM walls. All three equa-tions present a large scatter of the Vexp/Vn values; hence, they do not provide consistently safe predictions. In the case of MSJC (2016) and CSA S304.1-14, this scatter leads to a significant number of unsafe strength predictions.

(2) A modified MSJC shear strength equation has been proposed for PG-RM walls. The revised formula removes the uniform reduction factor for PG masonry, and uses face shell shear area instead of net shear area of masonry. It also accounts, in a simplified way, for the effect of wall aspect ratio in the axial load contribution to shear resistance. The modified equation presents an enhanced estimation of the shear strength of the walls in the database. Still, further research is recommended to refine shear strength pre-dictions through mechanics-based models. Aspects that deserve more in-depth investigation include the combined effect of wall aspect ratio and axial load on the shear strength or the relation between the shear resistance of masonry and the compressive strengths of ungrouted and grouted prisms.

(3) The NDSD parameter takes into account design and loading characteristics that affect the capacity of walls. The NDSD-based fragility curves present lower random dispersion than those employing SDR as demand parameter. This reduced dispersion confirms that the NDSD parameter is better correlated to the damage observed in walls.

(4) The comparison of fragility functions for PG-RM and FG-RM walls with bond beam have indicated that PG walls present in general lower deformation capacity and higher probability of damage than FG walls. The fragility functions of PG walls also present a higher dispersion than those of FG walls, hence, the damage response prediction is more uncertain for PG-RM.

(5) The comparison of fragility functions for PG-RM walls with bond- beam reinforcement and bed-joint reinforcement has indicated that while moderate shear damage is exceeded at similar defor-mation levels, walls with bed-joint reinforcement present higher deformation capacity than those using a bond-beam scheme.

CRediT authorship contribution statement

Zhiming Zhang: Conceptualization, Methodology, Formal analysis, Investigation, Data curation, Visualization, Writing - original draft. Juan Murcia-Delso: Conceptualization, Methodology, Formal analysis, Writing - review & editing, Supervision. Cristian Sandoval: Method-ology, Formal analysis, Writing - original draft. Gerardo Araya-Lete-lier: Writing - review & editing. Fenglai Wang: Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The first author Z. Zhang gratefully acknowledges the financial

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support provided by a Harbin Institute of Technology scholarship to conduct research as a visiting scholar at the University of Texas at Austin under the supervision of the second author J. Murcia-Delso. The co- authors C. Sandoval and G. Araya-Letelier are grateful for the funding provided by the Fondo de Fomento al Desarrollo Científico y Tec-nologico (FONDEF) under Grant N◦ 17I10264 and the Fondo Nacional

de Ciencia y Tecnología de Chile (FONDECYT Regular) through Grant No. 1181598.

Appendix

(See Table A1, Table B1)

Table A1 PG-RM wall database: main wall test characteristics.

Reference Wall ID Lw (mm) He(mm) He/Lw ρh ρv σ/fm’ Loading1 Boundary Conditions

Chen et al. [2] 1 1219 711.2 0.58 0.07 0.16 0.05 C Fixed-fixed 2 1219 711.2 0.58 0.00 0.42 0.05 C Fixed-fixed 3 1219 711.2 0.58 0.28 0.42 0.05 C Fixed-fixed

Thurston and Hutchison [3] 4 1600 1200 0.75 0.07 0.14 0.00 C Fixed-fixed 5 1600 1200 0.75 0.00 0.27 0.00 C Fixed-fixed 6 1600 1200 0.75 0.12 0.27 0.00 C Fixed-fixed

Ghanem et al. [4] 7 939 921 0.98 0.12 0.12 0.04 M Cantilever 8 939 921 0.98 0.12 0.12 0.04 M Cantilever

Ghanem et al. [5] 9 939 921 0.98 0.12 0.12 0.04 M Cantilever 10 939 921 0.98 0.12 0.12 0.09 M Cantilever

Schultz [6] 11 2845 711 0.25 0.05 0.20 0.06 C Fixed-fixed 12 2032 711 0.35 0.05 0.29 0.06 C Fixed-fixed 13 1422 711 0.50 0.05 0.41 0.05 C Fixed-fixed 14 2845 711 0.25 0.12 0.20 0.06 C Fixed-fixed 15 2032 711 0.35 0.12 0.29 0.05 C Fixed-fixed 16 1422 711 0.50 0.12 0.41 0.05 C Fixed-fixed

Ingham et al. [7] 17 2600 2400 0.92 0.07 0.13 0.00 C Cantilever 18 1800 2400 1.33 0.12 0.13 0.00 C Cantilever 19 2600 2400 0.92 0.07 0.12 0.00 C Cantilever 20 4200 2400 0.57 0.12 0.12 0.00 C Cantilever 21 1800 2400 1.33 0.09 0.10 0.00 C Cantilever 22 2600 2400 0.92 0.09 0.09 0.00 C Cantilever 23 4200 2400 0.57 0.09 0.09 0.00 C Cantilever

Voon and Ingham [8] 24 1800 1800 1.00 0.00 0.62 0.00 C Cantilever 25 1800 1800 1.00 0.00 0.37 0.00 C Cantilever

Maleki [9] 26 1800 1800 1.00 0.05 0.19 0.07 C Cantilever 27 1800 1800 1.00 0.05 0.18 0.06 C Cantilever 28 1800 1800 1.00 0.05 0.16 0.07 C Cantilever 29 1800 900 0.50 0.05 0.19 0.07 C Cantilever 30 1800 2700 1.50 0.04 0.19 0.07 C Cantilever

Minaie et al. [10] 31 3860 2640 0.68 0.16 0.50 0.04 C Cantilever 32 3860 2640 0.68 0.16 0.50 0.04 C Cantilever 33 3860 1320 0.34 0.16 0.50 0.00 C Fixed-fixed 34 3860 1320 0.34 0.16 0.50 0.00 C Fixed-fixed

Elmapruk et al. [11] 35 2642 1524 0.58 0.13 0.33 0.01 C Cantilever 36 2642 1524 0.58 0.13 0.33 0.01 C Cantilever 37 2642 1524 0.58 0.18 0.33 0.01 C Cantilever 38 2642 1524 0.58 0.25 0.33 0.01 C Cantilever 39 2642 1524 0.58 0.13 0.33 0.01 C Cantilever 40 2642 1524 0.58 0.13 0.33 0.01 C Cantilever

Baenziger and Porter [12] 41 2850 2640 0.93 0.05 0.24 0.00 C Cantilever 42 2850 2640 0.93 0.05 0.28 0.00 C Cantilever 43 4270 2640 0.62 0.05 0.19 0.00 C Cantilever

Nolph and Elgawady. [13] 44 2631 2337 0.89 0.09 0.46 0.02 C Cantilever 45 2631 2337 0.89 0.12 0.46 0.02 C Cantilever 46 2631 2337 0.89 0.17 0.46 0.02 C Cantilever 47 2631 2337 0.89 0.09 0.45 0.02 C Cantilever 48 2631 2337 0.89 0.09 0.46 0.02 C Cantilever

Hamedzadeh [14] 49 1233 617.5 0.50 0.09 0.18 0.24 (0.06)2 M Fixed-fixed 50 1233 617.5 0.50 0.09 0.18 0.24 (0.06)2 M Fixed-fixed 51 1233 617.5 0.50 0.09 0.18 0.18 (0.06)2 M Fixed-fixed 52 1233 617.5 0.50 0.09 0.18 0.20 (0.06)2 M Fixed-fixed 53 1233 617.5 0.50 0.09 0.18 0.22 (0.06)2 M Fixed-fixed 54 1233 617.5 0.50 0.09 0.18 0.20 (0.06)2 M Fixed-fixed 55 2372 617.5 0.26 0.09 0.14 0.27 (0.06)2 M Fixed-fixed 56 2372 617.5 0.26 0.09 0.14 0.27 (0.06)2 M Fixed-fixed 57 2372 617.5 0.26 0.09 0.14 0.28 (0.06)2 M Fixed-fixed 58 2372 617.5 0.26 0.09 0.14 0.24 (0.25)2 M Fixed-fixed 59 2372 617.5 0.26 0.09 0.14 0.28 (0.25)2 M Fixed-fixed 60 2372 617.5 0.26 0.09 0.14 0.34 (0.25)2 M Fixed-fixed 61 2372 380 0.16 0.15 0.19 0.45 (0.06)2 M Fixed-fixed 62 2372 380 0.16 0.15 0.19 0.30 (0.06)2 M Fixed-fixed 63 2372 380 0.16 0.15 0.19 0.32 (0.06)2 M Fixed-fixed 64 2372 380 0.16 0.15 0.19 0.43 (0.25)2 M Fixed-fixed

(continued on next page)

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Table B1 PG-RM wall database: demand parameter values at DS1 and DS2.

Reference Wall ID DS1 DS2

Drift (%) V/Vn1 V/Vn5 Drift (%) V/Vn1 V/Vn5

Chen et al. [2] 1 No data 0.32 1.62 1.67 2 0.36 1.57 1.48 3 0.38 1.44 1.74

Thurston and Hutchison [3] 4 No data 0.33 0.61 0.61 5 0.65 1.01 1.14 6 0.27 0.90 1.02

Ghanem et al. [4] 7 0.22 1.03 0.82 0.3 1.18 0.94 8 0.59 1.10 0.95 0.81 1.27 1.09

Ghanem et al. [5] 9 0.33 0.91 0.80 0.76 1.06 0.93 10 0.32 1.24 1.09 0.65 1.4 1.24

Schultz [6] 11 No data 0.96 0.63 0.47 12 0.76 1.10 0.91 13 0.73 0.80 0.76 14 0.73 0.74 0.61 15 0.51 0.77 0.71 16 0.34 0.79 0.91

Ingham et al. [7] 17 No data 0.20 0.73 0.63 18 No data 0.59 0.87 19 0.35 0.78 0.82 20 No data 0.77 0.79 21 No data 0.57 0.99 22 0.15 0.69 0.86 23 0.14 0.82 0.99

Voon and Ingham [8] 24 No data 0.44 1.26 1.65 25 0.44 0.99 1.07

Maleki [9] 26 No data 0.17 1.07 1.00 27 0.22 1.06 1.07 28 0.21 1.16 0.99 29 0.09 1.13 0.95 30 0.42 0.93 1.17

(continued on next page)

Table A1 (continued )

Reference Wall ID Lw (mm) He(mm) He/Lw ρh ρv σ/fm’ Loading1 Boundary Conditions

65 2372 380 0.16 0.15 0.19 0.42 (0.25)2 M Fixed-fixed 66 2372 380 0.16 0.15 0.19 0.31 (0.25)2 M Fixed-fixed 67 853 380 0.45 0.15 0.26 0.16 (0.06)2 M Fixed-fixed 68 853 380 0.45 0.15 0.26 0.13 (0.06)2 M Fixed-fixed 69 853 380 0.45 0.15 0.26 0.17 (0.06)2 M Fixed-fixed

Hoque [15] 70 1800 900 0.50 0.12 0.18 0.10 C Fixed-fixed 71 1800 900 0.50 0.12 0.18 0.10 C Fixed-fixed 72 1800 900 0.50 0.12 0.18 0.10 C Fixed-fixed 73 1800 900 0.50 0.12 0.18 0.10 C Fixed-fixed 74 1800 900 0.50 0.12 0.18 0.10 C Fixed-fixed 75 1800 900 0.50 0.12 0.18 0.10 C Fixed-fixed 76 1800 900 0.50 0.12 0.18 0.10 C Fixed-fixed 77 1800 900 0.50 0.12 0.18 0.10 C Fixed-fixed 78 1800 900 0.50 0.12 0.18 0.10 C Fixed-fixed 79 1800 900 0.50 0.12 0.18 0.10 C Fixed-fixed

Bolhassani[16] 80 3900 3900 1.00 0.11 0.11 0.01 C Cantilever 81 5700 3900 0.68 0.11 0.10 0.01 C Cantilever 82 5700 3900 0.68 0.10 0.09 0.01 C Cantilever

Rizaee [17] 83 1790 895 0.50 0.12 0.18 0.11 C Fixed-fixed 84 1790 895 0.50 0.12 0.18 0.11 C Fixed-fixed 85 1790 895 0.50 0.12 0.18 0.11 C Fixed-fixed 86 1790 895 0.50 0.12 0.18 0.11 C Fixed-fixed 87 1790 895 0.50 0.06 0.18 0.09 C Fixed-fixed 88 1790 895 0.50 0.06 0.18 0.09 C Fixed-fixed 89 1790 895 0.50 0.06 0.18 0.09 C Fixed-fixed 90 1790 895 0.50 0.06 0.18 0.09 C Fixed-fixed 91 1790 895 0.50 0.06 0.18 0.09 C Fixed-fixed 92 1790 895 0.50 0.06 0.18 0.09 C Fixed-fixed 93 1790 895 0.50 0.06 0.18 0.14 C Fixed-fixed 94 1790 895 0.50 0.06 0.18 0.14 C Fixed-fixed

Calderon et al. [18] 95 2640 2270 0.86 0.08 0.41 0.09 C Cantilever

1 M: monotonic, C: cyclic. 2 Axial load at peak shear capacity (initial axial load).

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References

[1] Araya-Letelier G, Calderon S, Sandoval C, Sanhueza M, Murcia-Delso J. Fragility functions for partially-grouted masonry shear walls with bed-joint reinforcement. Eng Struct 2019;191:206–18.

[2] Chen SJ, Hidalgo PA, Mayes RL, Clough RW, McNiven HD. Cyclic loading tests of masonry single piers, volume 2—height to width ratio of 1. EERC rep. no. 78/28.

Berkeley: Environmental Engineering Research Council. University of California; 1978.

[3] Thurston SJ, Hutchison DL. Reinforced masonry shear walls: cyclic load tests in contraflexure. Bull New Zealand Natl Soc Earthquake Eng 1982;15(1).

[4] Ghanem GM, Essawy AS, Hamid AA. Effect of steel distribution on the behavior of partially grouted reinforced masonry shear walls. Saskatoon, Canada: University of Saskatchewan; 1992.

Table B1 (continued )

Reference Wall ID DS1 DS2

Drift (%) V/Vn1 V/Vn5 Drift (%) V/Vn1 V/Vn5

Minaie et al. [10] 31 0.10 0.53 0.61 0.27 0.76 0.88 32 0.10 0.23 0.26 0.45 0.46 0.53 33 0.10 0.23 0.23 0.62 0.56 0.56 34 0.10 0.23 0.24 0.65 0.53 0.54

Elmapruk et al. [11] 35 0.06 0.54 0.63 0.18 0.79 0.91 36 0.12 0.59 0.68 0.37 0.84 0.96 37 0.09 0.6 0.72 0.20 0.79 0.94 38 0.06 0.6 0.71 0.30 0.92 1.09 39 0.08 0.68 0.84 0.32 1.06 1.31 40 0.12 0.73 0.97 0.32 1.06 1.40

Baenziger and Porter [12] 41 No data 0.46 0.81 1.02 42 0.30 0.78 1.11 43 0.32 0.80 1.07

Nolph and Elgawady [13] 44 0.33 0.49 0.56 1.30 1.11 1.26 45 0.27 0.39 0.47 1.50 1.09 1.29 46 0.33 0.47 0.56 0.65 0.97 1.15 47 0.11 0.20 0.22 1.30 1.34 1.47 48 0.22 0.38 0.41 1.50 1.57 1.67

Hamedzadeh [14] 49 No data 0.49 1.83 1.60 50 0.45 1.73 1.51 51 0.36 1.31 1.17 52 0.39 1.56 1.39 53 0.37 1.44 1.27 54 0.30 1.49 1.33 55 0.43 1.75 1.27 56 0.26 1.67 1.21 57 0.53 1.70 1.22 58 0.44 1.57 1.16 59 0.44 1.54 1.11 60 0.23 2.08 1.44 61 0.58 2.82 1.88 62 0.37 2.51 1.61 63 0.70 2.72 1.75 64 0.35 2.58 1.71 65 0.41 2.66 1.76 66 0.45 2.22 1.43 67 0.86 1.08 1.06 68 0.20 1.01 1.00 69 0.54 1.32 1.28

Hoque [15] 70 No data 0.46 0.62 0.44 0.80 1.06 71 0.55 0.73 0.36 0.79 1.05 72 0.69 0.93 0.36 0.88 1.18 73 0.61 0.81 0.36 0.81 1.08 74 0.74 0.99 0.39 0.87 1.16 75 0.68 0.91 0.42 0.85 1.14 76 0.56 0.74 0.22 0.59 0.78 77 0.82 1.10 0.27 0.82 1.10 78 0.80 1.07 0.39 0.87 1.16 79 0.65 0.86 0.25 0.68 0.90

Bolhassani [16] 80 0.30 0.57 0.51 0.43 0.59 0.52 81 0.30 No data 0.20 0.71 0.63 82 0.30 0.20 0.67 0.68

Rizaee [17] 83 0.06 0.62 0.89 0.31 0.93 1.33 84 0.08 0.63 0.90 0.17 1.12 1.59 85 0.08 0.73 1.04 0.22 1.18 1.69 86 0.08 0.72 1.03 0.45 1.14 1.63 87 0.08 0.58 0.77 0.22 0.78 1.03 88 0.08 0.57 0.75 0.11 0.72 0.96 89 0.08 0.59 0.78 0.28 0.82 1.09 90 0.08 0.56 0.74 0.25 0.75 0.99 91 0.08 0.49 0.65 0.36 0.76 1.01 92 0.08 0.50 0.67 0.28 0.71 0.95 93 0.08 0.55 0.79 0.28 0.81 1.16 94 0.08 0.56 0.81 0.25 0.80 1.15

Calderon et al. [18] 95 0.08 0.74 1.82 0.44 0.79 1.95

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