Influence of a window-type opening on the shear response of partially-grouted masonry shear walls

Engineering Structures 201 (2019) 109783

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Influence of a window-type opening on the shear response of partiallygrouted masonry shear walls

T

Sebastián Calderóna, Cristián Sandovala,b, , Ernesto Inzunzaa, Carlos Cruz-Noguezc, Amr Ba Rahimc, Laura Vargasa ⁎

a

Department of Structural and Geotechnical Engineering, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile School of Architecture, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile c Department of Civil and Environmental Engineering, University of Alberta, Edmonton, Canada b

ARTICLE INFO

ABSTRACT

Keywords: Reinforced masonry Partially grouted Shear wall Window opening Perforated wall Shear equation DIC technique

This paper focuses on assessing the effects of the presence of a window-type opening on the shear behavior of partially-grouted reinforced masonry (PG-RM) shear walls. For this purpose, this work analyzes previously reported test results of three PG-RM walls tested under in-plane cyclic lateral loading. The main variable of the tested walls is the size of a central window. Therefore, all the walls had the same external dimensions (a heightto-length aspect ratio equal to 1.02). The performance of the walls is analyzed and discussed in terms of maximum shear strength, drift, degradation of stiffness, disp1acement ductility, dissipation of energy, and equivalent viscous damping. The damage evolution in the tested walls is investigated using digital image correlation (DIC) technique. Finally, the paper compares the accuracy of the Strut-and-Tie method and five existing shear equations to estimate the maximum lateral capacity of PG-RM walls with openings. In this comparative analysis, seven numerical cases of walls with openings previously validated are also considered. Based on the results, the study has confirmed that the presence of a central window in a PG-RM shear wall reduces the shear strength and ductility of the entire wall, and increases the rate of degradation of lateral stiffness. Although a significant disparity was not seen in the shear strength of the two tested walls with openings, they exhibited differences in the damage progression, the total amount of dissipated energy and the equivalent viscous damping during the tests. In addition, the experimental evidence indicates that although damage is concentrated on the piers of the walls, it also propagates to the segments below the openings. Therefore, there is uncertainty about how much the properties of wall segments near the openings affect the global behavior of the wall. The assessment of different methods to estimate the lateral resistance of a PG-RM shear wall with openings shows that the sum of individual contributions of the piers gives accurate predictions provided the expression used to estimate the resistance of piers is accurate enough.

1. Introduction In recent years, considerable amount of experimental research on the in-plane seismic behavior of partially-grouted reinforced masonry (PGRM) shear walls has been carried out [19,46,30,28,16,32,10,33, 3,38,5,41], among others). From previous experimental studies, it has been concluded that the seismic response of PG-RM walls depends mainly on parameters such as wall geometry (aspect ratio), axial load level, vertical and horizontal reinforcement (amount and spacing), and mechanical properties of the constituent materials and their interfaces. Despite the significant advances made in understanding and predicting the seismic

response of such walls, the influence of these parameters and their potential interactions are still under study because the experimental dataset is not exhaustive enough [22]. In addition, most of the specimens tested and reported in the literature do not have openings even though the presence of doors or windows is a typical feature of real buildings. The seismic response of PG-RM shear walls with openings has proved to be different and much more complex than the response of walls without openings [14,21]. Despite this, very limited experimental research has been carried out in this matter. Elshafie et al. [17] performed an experimental program to study the lateral response of shear walls with openings using 1/3-scale reinforced concrete masonry shear

⁎ Corresponding author at: Department of Structural and Geotechnical Engineering, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile. E-mail address: [email protected] (C. Sandoval).

https://doi.org/10.1016/j.engstruct.2019.109783 Received 21 April 2019; Received in revised form 11 September 2019; Accepted 10 October 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.

Engineering Structures 201 (2019) 109783

S. Calderón, et al.

Notation

ftg fyv , fyh

The following symbols are used in shear expressions

An Ash Asv Asvt , Asht AT d' dv dsv , dsh Em' fcg fm'

ftm

hw kbv , kbh Lw M P sh t V

= Net cross section of wall = Area of a horizontal reinforcement (in this case two rods of 4.2 mm) = Area of a vertical reinforcement = Total area of vertical and horizontal reinforcement, respectively = Total cross section of wall = Cover of reinforcement of the vertical edge = Effective depth of wall = Diameter of vertical and horizontal bars, respectively = Young’s modulus of masonry = compressive strength of grout

v,

= Compressive strength of masonry = Tensile strength of mortar

n

walls. In this research, walls were designed to behave mainly in a flexural mode by forming plastic hinges at the ends of the members. This research concluded that for walls with similar overall dimensions and flexural reinforcement arrangement, the reduction in stiffness due to openings is comparable to the reduction in shear strength, regardless of the opening size and location. Voon and Ingham [46] conducted experimental tests on eight partially grouted concrete masonry walls with openings, which were subjected to cyclic lateral loading without axial compression load. The Voon and Ingham’s study also focused on validating Strut-and-Tie (ST) models to predict the maximum shear strength of walls taking into account the openings. From the results, the authors concluded that the shear capacity of the walls is highly affected by the height of the opening because if the diagonal compression strut is steep it is less efficient to transfer lateral forces. Furthermore, ST models showed a better prediction of the maximum lateral load than the New Zealand masonry code [34]. Recently, Fortes et al. [18] tested 6 half-scale, three-story, partially grouted reinforced concrete masonry walls. Three specimens were built with a central door in each level, while the other three specimens were constructed with a central window in each level. In both configurations, the openings were coupled with masonry beams. The results of this study suggest that the lateral capacity of PG-RM walls is independent of the type of opening, as long as the widths of the openings are reasonably similar, though their ductility is sensitive to the opening type. The influence of a window-type central opening in PG-RM shear walls has been also investigated by Calderón et al. [8]. In this research, three PG-RM walls, one solid and two with a central window of the same width but of different height, were tested under cyclic lateral loading. The solid wall presented higher shear strength than walls with a central window. However, both walls with opening had comparable shear strength. This is consistent with the observations made by Fortes et al. [18], but it is contrary to that was reported by Voon and Ingham [46]. Koutras and Shing [29] presented the results and findings of a shake-table test conducted on a full-scale, one-story, partially grouted, and reinforced concrete masonry building. An important conclusion of this study is that the base-shear capacity of a PG-RM structure with openings cannot be calculated as the simple sum of the shear strengths of the contributing piers because of the disparity in the stiffness and the brittle behavior of the piers. It is clear that very few experimental investigations have taken into account the influence of the openings on the shear response of PG-RM shear walls. On the other hand, the complex behavior of PG-RM walls under seismic loads has hindered the development of analytical models that accurately predict their shear strength. In general, existing expressions estimate the maximum shear strength of a PG-RM wall as the

h

= Tensile strength of grout = Yielding strength of vertical and horizontal reinforcement, respectively = Height of wall = Term depending on the anchorage conditions of bars = Length of wall = Moment applied to wall = Axial load applied to wall = Vertical spacing of horizontal reinforcement = Gross thickness of wall = Shear applied to wall = Boundary condition factor (0.60 for cantilever support condition) = Type of masonry grouting factor (0.60 for PG-RM) = Vertical and horizontal reinforcement ratio referred to gross cross-section area of walls, respectively. = Axial stress of wall referred to gross cross section

sum of the contributions of masonry, horizontal reinforcement, and axial compressive load. However, the equations used to describe each contribution vary considerably among studies, resulting in different levels of complexity and accuracy [1]. On the other hand, although openings can have a significant effect on the seismic response, none of the existing expressions include modifications or reduction factors when estimating the shear strength of a perforated wall. In fact, the shear strength of a wall with openings is typically calculated as the sum of the shear strength of its piers [14,46,7]. That is, walls with openings can be considered as consisting of a system of piers which ignores any coupling element. However, according to Voon and Ingham [46] and Koutras and Shing [29], this consideration can lead to unsatisfactory predictions of the shear capacity of a PG-RM wall with openings. In the same line, design standards (such as CSA S304-14 or TMS 402/406) do not prescribe methods or specific procedures to account for window or door openings on the PG-RM wall response. Therefore, while it is known that some characteristics of the piers (such as their aspect ratio) influence the response of the wall, there is not sufficient evidence on how much of the design process should focus on those. To overcome this drawback, some researchers [46,13,45,7,12] have suggested the use of Strut-and-Tie (ST) models as a proper tool to evaluate the shear strength of PG-RM walls with openings. Although these models can produce satisfactory results, they have several limitations due to the fact that choosing an appropriate layout of struts and ties is a process that requires multiple iterations and it is somewhat subjective [7]. In summary, improving the predictive capacity of existing methods by considering all design parameters that control the seismic response of PG-RM walls, including the effect of window or door openings, requires further research efforts. This is echoed by the recent state-of-the-art review made by El-Dakhakhni and Ashour [15]. In a previous study [8], three full-scale PG-RM square shear walls made of multi-perforated clay bricks under in-plane cyclic lateral loading were tested in order to achieve a better understanding about the influence of a window-type opening on their shear response. A comprehensive mechanical characterization was carried out on main constitutive materials and their interfaces in order to obtain the input parameters required for proper numerical simulation of the tested walls. Thus, detailed micro-models of the walls were developed and validated by comparing numerical and experimental results. Once validated, micro-models were used to numerically investigate the influence of the opening size and the horizontal reinforcement ratio of the piers on the lateral response of PG-RM shear walls. Therefore, Calderón et al.’s study [8] focused more on the numerical implementation rather than to investigate how the presence of a window-type opening can affect the seismic performance of a PG-RM shear wall. 2

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H

1320

762.5

b) Wall O2

with welded steps every 300 mm) located every two courses, and a total of four steel bars arranged as vertical reinforcement (two of 25 mm diameter and two of 22 mm diameter). It should be noted that a high flexural reinforcement ratio was provided to the three walls in order to guarantee a shear failure mode during the tests. As commented, a classic assumption in the design of a PG-RM shear wall with openings is to consider that the lateral loads are resisted by the piers formed at the sides of openings. Therefore, following that approach, the total shear capacity of walls O1 and O2 would be the sum of the shear capacities provided by their two piers highlighted as the shaded area in Fig. 2. Table 1 summarizes the characteristics of the resistant cross-sectional areas for the three walls. Note that the net cross-sectional area of walls was calculated considering the grouted cells and subtracting the hollows of the units. The test setup was the same for all walls and consisted of a cantilever system (fixed at the base and free at the top), as shown in Fig. 3a. The loading protocol was a series of displacement-controlled loading cycles with two cycles per deformation level (Fig. 3b). The magnitude of the horizontal force was recorded with a load cell incorporated in the actuator, while the horizontal displacements of each pier of walls O1 and O2 were measured by the sensors C11 and C16 located at the center of the first course immediately below the top RC beam, as illustrated in Fig. 3a. In the case of wall S1, without an opening, a single sensor (C11)

(b) Wall O1

(c) Wall O2

762.5

762.5 655

(d) Lateral view 140

762.5 2 4.2 every two courses

1320

2 4.2 every two courses

800

500

3230 2230

2 4.2 every two courses

500

655

2 2 2 2 2 2 2 2

4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2

200 300

2480

700

(e) Typical cross section 140

1 25

(f) Clay brick

1 22 620

1 22 770

1.73

Fig. 2. Identification of piers (shaded areas) in walls O1 (a) and O2 (b). Notes: Aspect ratios of piers in red text. Dimensions in mm. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Three full-scale PG-RM shear walls were fabricated and tested under cyclic lateral loading by Calderón et al. [8]. One wall was solid (S1) and the other two (walls O1 and O2) had a window-type opening. Fig. 1(a–d) shows the characteristics of the wall specimens tested. In conformity with the characteristics of partially grouted walls, only cells with the presence of vertical steel bars were filled with grout, as can be seen in the cross-section of walls (Fig. 1e). All the walls were constructed using multi-perforated clay bricks with average length, thickness, and height equal to 290, 140, and 112 mm, respectively (Fig. 1f), and were tested without external axial load. The three walls had the same external dimensions (a height-tolength (aspect) ratio of 1.02), a thickness of 140 mm, and an average thickness of mortar joints of approximately 15 mm. Wall S1, without opening, was considered as a control wall, while walls O1 and O2 had a centered opening with dimensions (length × height) of 655x800 mm2 and 655x1320 mm2, respectively. These openings represent an area of 10.7% and 17.8%, respectively, of the total masonry surface of the solid wall (S1). In order to keep the focus on the influence of the openings, all walls had the same reinforcement scheme: bed joint (shear) reinforcement (pre-fabricated ladder-type with two 4.2 mm diameter rods and

762.5

762.5 655

1.73

a) Wall O1

2.1. Test specimens and test setup

2180

1.05

762.5 655 762.5

2. Overview of Calderón et al.’s [8] experimental program

(a) Wall S1

1.05

R.C. beam 0.35H

800

R.C. beam

0.6H

In this context, this paper presents and discusses aspects that complement and extend the research previously reported by Calderón et al. [8]. Structural parameters such as shear strength, drift, degradation of stiffness, displacement ductility, dissipation of energy, and equivalent viscous damping are quantified in order to establish the influence of the presence of a central window in the cyclic behavior of PG-RM sheardominated walls. The capability of a 2D-DIC measurement system to track the displacement field and crack pattern on the surface of each tested wall is also investigated. Finally, the maximum shear strength of PG-RM walls with openings is evaluated using different approaches, among them, the Strut-and-Tie method and some shear equations available in the literature. This later analysis uses the results of the three tested walls and of other seven numerically simulated walls, all of them validated and reported in Calderón et al. [8].

1 25 112

620

290

2180

140

Fig. 1. Specimen characteristics (dimensions in mm). 3

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Table 1 Identification of resistant cross-sectional area in wall specimens. Wall

S1 O1 O2

Opening size

Net cross sectional area (mm2 )

Length (mm)

Height (mm)

Opening area ratio (%)

– 655 655

– 800 1320

0 10.7 17.8

(1)

155133 112291(2) 112291(2)

was placed laterally centered at the same height as in the other two walls. Out-of-plane displacements were prevented by means of two steel hinged rods spaced at 1500 mm, as shown in Fig. 4a. Full details on the test setup and instrumentation used in the tests can be found in Calderón et al. [8]. During the tests, all walls were also monitored with a 2D digital image correlation (DIC) measurement system (Fig. 4). This technique has proven to be a powerful tool for measuring displacements on large masonry wall specimens and provide faithful progressive crack maps [20,39,6,36]. As discussed later, the performance of DIC method was verified by tracking the movement of selected points on the painted surface of walls at the same position of LVDT sensors (but on the opposite surface of the walls) and comparing the results with the recorded ones by traditional LVDT sensors. Once validated, digital image processing is used to visualize the displacement map developed on the surface of each tested wall. In this study, digital images were taken with a CANON T5i ® digital camera (maximum resolution of 18 Megapixels) placed at 3.5 m far from the wall face at a height of 1.45 m mounted on a standard tripod, as shown in Fig. 4a. The images were taken at an approximate rate of 1 photo every 5 s, capturing more than 500 pictures per test. The speckle pattern required by the DIC technique was applied on the front side of the specimens by using a multi-perforated plate (Fig. 4b). The correlation process was done by employing the Ncorr 2DDIC Toolbox [4] on MATLAB ® [31]. This toolbox constructs a grid on the region of images of interest in order to define the units of analysis that are compared and tracked with subsequent pictures [35].

Steel bars

Steel bars

(3)

(%)

0.071 0.074 0.075

Reinforcement of piers;

(4)

(3) (3)

2 25 + 2 22 2 25 + 2 22 2 25 + 2 22

v

(%)

1.12 1.51 1.51

(4) (4)

Reinforcement ratio considers the

3. Test results 3.1. Hysteretic response The horizontal load – drift hysteretic curves of the specimens are shown in Fig. 5. For comparison purposes, the curve of solid wall S1 is also included in the graphs of walls O1 and O2. As can be observed, all walls exhibited a fairly symmetrical behavior, although the maximum horizontal force in the forward cycle was 10%, 17%, and 13% higher than in the backward cycle for walls S1, O1, and O2, respectively. This occurs because when a reverse cycle is applied, the wall is already damaged by the previous deformation cycle in the forward direction, as has been also reported elsewhere [23,38]. From the graphs (Fig. 5), it

Typical materials used in Chilean PG-RM constructions were used to build the three walls which were investigated in this study. In agreement with NCh167.Of2001 [26], multi-perforated clay bricks presented an average compressive strength (on the basis of the gross area) of 15.3 MPa with a coefficient of variation (CV) of 7.4% and an average percentage of voids equal to 53.3% (CV = 0.8%). Regarding the mortar

(b)

C11

h

used, an average compressive strength of 6.8 MPa (CV = 1.0%) and an average Young’s modulus of 8295 MPa (CV = 37.2%) were determined according to NCh1037.Of77 [25]. Meanwhile, the grout used for filling the cells containing vertical steel bars presented an average compressive strength of 8.6 MPa (CV = 2.6%) and an average Young’s modulus of 14,392 MPa (CV = 18.1%), both also obtained according to NCh1037.Of77 [25]. Steel reinforcement was tested in tension according to NCh200.Of72 [24]. For horizontal reinforcement, composed of ϕ4.2 mm AT560-500H steel bars, an average yielding stress of 578 MPa (CV = 2.4%) and an average Young’s modulus of 205 GPa (CV = 1.3%) were obtained. Whilst for vertical reinforcement, where A630-420H steel was specified for ϕ22 and ϕ25 mm bars, an average Young’s modulus of 233 GPa was determined. In order to characterize the compressive properties of masonry as a composite material, three prisms were constructed and tested under compression according to NCh1928.Of2003 [27], obtaining an average compressive strength of 7.77 MPa (CV = 10.5%) and an average Young’s modulus of 8830 MPa (CV = 38.6%). A detailed mechanical characterization of the materials used in the construction of walls, mainly aimed at the generation of numerical models, can be found in Calderón et al. [8] and Sandoval and Arnau [40].

2.2. Material properties

300 kN Horizontal actuator

Vertical reinforcement

8 − 2 4.2 3 − 2 4.2(3) 5 − 2 4.2(3)

Notes: (1) Ratio of opening area to full area of wall S1; (2) Net area that considers both piers; contribution of the two piers located at both sides of the opening.

(a)

Horizontal reinforcement

C16

Lateral stopper

Fig. 3. Typical test setup (a) and displacement protocol imposed (b). 4

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(a)

(b)

Two hinged steel rods

Surface of patterned specimen 3.5m

Digital camera 1.45m

Surface of specimen instrumented with sensors Halogen lights Fig. 4. Implemented 2D-DIC measurement system.

can be observed that all walls had a similar initial stiffness and an approximately linear elastic behavior until they reach lateral drifts of 0.05–0.058%. After that, the curves of walls exhibited nonlinear behavior which is related to the appearing of small stepped diagonal cracks. With the continuous increment of lateral displacement and the progression of the damage, gradual degradation of stiffness occurred, the area enclosed by hysteresis cycles began to increase indicating an increment in the dissipated energy. On the other hand, the degradation of lateral force turned out to be quite similar for both walls with opening, although a greater lateral displacement was reached by the wall O2 due to the higher aspect ratio of its piers. It should be noted that the maximum lateral force of walls S1, O1, and O2 was reached after 16, 13, and 14 displacement cycles, respectively. As expected, the presence of a window-type opening had a significant effect on the recorded maximum horizontal force, as well as on 250

200

Horizontal Force (kN)

a) Wall S1

150

250

=210.3 kN

100 50 0 -50 -100 -150 -200 -250

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

150 100

Wall S1 Wall O1

50 0 -50 -100 -150 -250

0.8

=131.1 kN

b) Wall O1

-200

Wall S1

=191.4 kN

-0.8

250 200

-0.6

-0.4

-0.2

=133.5 kN

c) Wall O2

150 100

Wall S1 Wall O2

50 0 -50 -100 -150 -200 -250

Wall O2

=118.1 kN -0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Drift (%) Fig. 5. Hysteresis loops of walls tested: (a) S1, (b) O1, and (c) O2. 5

Wall O1

=111.4 kN

Drift (%)

Horizontal Force (kN)

Horizontal Force (kN)

200

its lateral deformation capacity and on damage patterns. Thus, while the solid wall S1 reached an average maximum lateral load of 200.8 kN, the wall O1 reached an average maximum lateral load of 121.2 kN, and the wall O2 reached an average maximum lateral load of 125.8 kN, all in the forward loading direction. This means a decrement of the maximum lateral load of 39% and 37%, respectively. The estimation of the lateral displacement by means of the DIC method for the wall S1 is shown in Fig. 6 together with the measurements recorded by a traditional displacement transducer (sensor C11). Although displacements are slightly over-estimated by the DIC-method, DIC-measured and LVDT-measured displacements are very similar through the test, presenting a relative error of 15% and 14% for the displacement of maximum lateral load in the forward and backward loading directions, respectively. In addition, the hysteresis envelopes are calculated by relating the DIC-displacements with the corresponding

0

0.2

Drift (%)

0.4

0.6

0.8

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Lateral displacement (mm)

20

analysis units in the incorrectly painted regions. Another aspect is that as displacements are computed in terms of pixels, using a wrong scale factor to transform them into a conventional measurement unit can alter the magnitude of the calculated displacements. This could have a great influence owing that displacements are very small in relation to the dimensions of the specimens, especially in the early stages of the tests. Likely, scale factor does not affect the direction of the displacements, easing the identification of the problem in the stage of analysis of results. Even though these aspects are detrimental to the performance of the method, in this case, DIC provides comparable measurements. Therefore, DIC measurements are then used to generate displacement maps in the entire surface of the tested walls. This information is essential given that provides empirical evidence of the progression of damage and of the failure modes of the walls under study. In this sense, a good description of the progression of damage in this type of walls would facilitate the development of seismic fragility curves for different damage states, as reported in a recent study by Araya-Letelier et al. [2] for the case of partially-grouted masonry shear walls with bed-joint reinforcement.

15 10 5 0 -5 -10

DIC Sensor C11

-15 -20

0

1000

2000

3000

4000

5000

6000

Time (s) Fig. 6. DIC-measured horizontal displacements for wall S1.

3.2. Damage evolution and displacement maps Fig. 8 presents the DIC-measured horizontal displacement maps at the maximum displacement in the forward loading directions of selected cycles for the wall S1. An elastic deformation profile can be seen for small deformations, as shown in Fig. 8a. In this stage, narrow flexural cracks were observed in both side edges. Subsequently, with the successive increments on the lateral displacement, these cracks turned progressively into shear stair-stepped cracks along with unitsmortar interfaces. Then, diagonal cracks began growing from the edges of the wall and propagated inwards throughout the height of the wall with an approximate angle of 45°. The formation and progression of

(a) Wall S1

Horizontal Force (kN)

250 200 150 100 50 0 -50 -100 -150 -200 -250

Experimental DIC -18 -15 -12 -9 -6 -3 0

3

6

9 12 15 18

250 200 150 100 50 0 -50 -100 -150 -200 -250

Lateral displacement (mm)

Horizontal Force (kN)

Horizontal Force (kN)

measured forces (Fig. 7). Although there are small differences, DIC-based envelopes have a good correspondence with LVDT-based ones, exhibiting similar shapes and stiffness degradation. The differences can be explained by factors inherent to the DIC technique, such as the variation in the illumination which can generate random errors and affect the determination of displacements since the algorithm compares the intensity of pixels in successive images. In fact, in this research, each wall specimen was constantly lit up with halogen lights throughout the test, but the existence of some constructive irregularities on the surface of the walls caused shadows that certainly could have influenced DIC calculations. According to Ghorbani et al. [20], another relevant factor that can affect is the speckle pattern implemented that might not be the most optimal for large walls like these. This hinders the creation of adequate

250 200 150 100 50 0 -50 -100 -150 -200 -250

(b) Wall O1

Experimental DIC -18 -15 -12 -9 -6 -3 0

3

6

9 12 15 18

Lateral displacement (mm)

(c) Wall O2

Experimental DIC -18 -15 -12 -9 -6 -3 0

3

6

9 12 15 18

Lateral displacement (mm)

Fig. 7. DIC results for walls: (a) S1, (b) O1, and (c) O2. 6

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Fig. 8. DIC-measured displacement maps of Wall S1 in the forward loading direction.

these cracks can be observed as differential displacements in Fig. 8b–f. When reaching the maximum lateral capacity, a cracked diagonal band can be easily identified (Fig. 8e). By the final of the test, the reinforcement kept the wall as a whole because diagonal cracks split the wall into two parts (Fig. 8f). Fig. 9 shows the main damages observed in wall S1 at the end of the cyclic loading test. As can be observed in Fig. 9a, the presence of the grouted cells in the central part of the wall influenced the progression of the major diagonal cracks. In fact, diagonal cracks started at the upper and bottom corners of the wall and they stopped at the position of the

two central vertical reinforcement because of the increment of crosssection provided by the grouted cores. Then, horizontal cracks that connected the two outer parts were developed at the mortar bed joints since these zones were the weakest failure plane. It should be noted that this effect on the crack pattern, caused by the presence of grouted cells in the interior of a PG-RM wall, was also reported by Dhanasekar & Haider [10]. Additionally, spalling of the faces of bricks and penetration of cracks into head beams at the toes of the wall were also observed in that stage of the test as is shown in Fig. 9b and c, respectively. The final state of damage of wall S1 in both loading directions is shown in

Grouted cells

(b)

(c)

(a)

Fig. 9. Main damage patterns in wall S1 at the end of the test: (a) effect of grouted cells, (b) face spalling of bricks, and (c) penetration of cracks into head beams. 7

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appeared and grew on piers of Wall O2 according to lateral deformation increased. These cracks, which can be identified as differential displacements in Fig. 12b–d, formed at unit-mortar interfaces but they were less steepened than those ones observed on wall O1. Some cracks went through bricks and mortar joints with the increment of the lateral deformation. Similarly to the wall O1, the piers of the wall O2 had a different displacement profile when the wall’s maximum resistance was reached (Fig. 12e). In the final stage of the test, notorious differential displacements can be identified (Fig. 12f). Moreover, diagonal cracks formed throughout the entire height of the piers, with cracks penetrating on the cap beam, and spalling at the brick face occurred. However, unlike the wall O1, the damage in the wall O2 was mostly concentrated in the upper half of each pier, as can be seen in Fig. 10c and Fig. 12f. Sandoval et al. [41] also reported a similar damage pattern in solid walls with an aspect ratio equal to 1.95. In addition, it is important to remark that damage is not limited to piers because some diagonal cracks extended to the fifth course under the window opening.

Fig. 10a, where it can be observed that main cracks were accompanied by a large number of small cracks. In the case of wall O1, small stepped cracks were detected before the test at edges of the wall below the sixth course, which were caused by the tensioning of the anchor bolts when connecting them to the slab of the laboratory. However, it is believed that they did not affect the response of the wall because pre-test cracks were far away from the zones where damage produced by the lateral loading was concentrated. This can be verified by observing Fig. 11, which shows the DIC-based displacement maps for different stages of the test in the forward loading direction. It can be seen that for a small lateral deformation (Fig. 11a), both piers deform uniformly in shear but there are some differential displacements below the left pier. Then (Fig. 11b–e), it can be seen that a diagonal crack appears at the left pier and then at the right pier, cracks that propagated along unit-mortar interfaces. Afterward, these diagonal cracks grew including fracture of bricks and mortar as deformation increased. These displacement maps demonstrate that despite both piers worked jointly in the elastic range, a different situation could occur in the nonlinear range. In fact, displacement differences between piers at maximum resistance of wall (Fig. 11e) suggest that both piers did not reach their maximum resistance at the same instant. These differences remain up to the final of the test, as shown in Fig. 11f. In relation to the cyclic response, the right pier concentrated the damage at the final stage of the test with wider x-shaped diagonal cracks than the left pier, as can be seen in Fig. 10b. In addition, spalling of the face of brick and penetration of cracks into the head beam at the corners of the wall were also observed. It is worth noticing that only a few cracks were developed below the piers, as depicted in Fig. 10b. Analyzing the piers of wall O1 as individual elements, it could be noticed that their damage progression is quite similar to that observed in wall S1 because both structural elements had a similar aspect ratio (close to 1.0). However, in the case of piers of wall O1, the damage was mainly concentrated in some main cracks due to the lack of interior grouted columns, unlike wall S1. Fig. 12 shows the progression of horizontal displacement maps developed on the surface of wall O2 according to the forward loading direction. In this wall specimen where the window height is greater than the wall O1, narrow horizontal cracks were observed at the firstbed joint mortar of both piers in early stages of the test. This implies that the higher aspect ratio of the piers of this wall generated an inplane response with a higher flexural component than that of wall S1 and the piers of wall O1. This could be corroborated by observing Fig. 12a that shows the DIC-based displacement map of wall O2 for a small lateral deformation. Nevertheless, the high vertical reinforcement ratio shifted the failure mode from flexure to shear. This observation was confirmed because these first horizontal cracks opened and closed depending on the loading direction, but its length and width did not increase during the test. As can be seen in Fig. 12b–d, diagonal cracks

a) Wall S1

4. Quantification of seismic performance parameters In this section, seismic performance parameters such as shear strength, stiffness degradation, displacement ductility, energy dissipation, and equivalent viscous damping are quantified in order to establish the effects of the presence of a central window in the cyclic response of PG-RM shear walls. 4.1. Maximum shear strength Table 2 summarizes the results for tested walls where the terms max and max correspond to the lateral displacement and drift associated with the maximum lateral load (Vmax ) recorded in each loading direction. Table 2 also shows the maximum shear strength based on the gross cross-sectional area ( gross ) and based on the net cross-sectional area ( net ). As commented in Section 2.1, the net cross-sectional area of walls was calculated considering the grouted cells and subtracting the hollows of the units. As mentioned before, the presence of a window in walls O1 and O2 reduced the maximum lateral load in comparison with the solid wall (S1) in 40% and 37% respectively. It should be noted that Voon and Ingham [46] also reported a quite similar reduction of lateral resistance for walls with similar opening sizes. Since the walls O1 and O2 had the same horizontal reinforcement amount than wall S1, the reduction in the shear strength must depend on the presence and size of the opening. Even though walls O1 and O2 had different opening sizes, they resisted similar shear strength, indicating that the aspect ratio of the piers around the opening would not have an influence on this parameter. However, as mentioned before, the wall O1 had pre-existing diagonal cracking before the tests, so the maximum lateral load of the wall O1

b) Wall O1

c) Wall O2

Fig. 10. Crack patterns at the end of tests: (a) S1, (b) O1, and (c) O2. 8

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Fig. 11. DIC-measured displacement maps of Wall O1 in the forward loading direction.

would have been higher if it had been intact. Therefore, more experimental tests are necessary to conclude on the influence of the aspect ratio of piers and the aspect ratio of an opening on the lateral capacity of a PG-RM wall.

Regarding the displacement capacity, the wall O1 reached its peak strength at a lower average drift than the wall O2 (0.36% compared with 0.45%). Thus, the drift at peak strength of both walls with openings presented a decrease of about 31% and 15% with respect to the

Fig. 12. DIC-measured displacement maps of Wall O2 in the forward loading direction. 9

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Table 2 Summary of experimental capacities of tested walls. Wall

Loading direction

Experimental elastic limit SL

(mm)

VSL (kN)

Idealized elastic limit E

(mm)

Maximum lateral load

VE (kN)

max

(mm)

max

Ductility

Vmax (kN)

(%)

gross (MPa)

net

(MPa)

µd

µ¯d

S1

Forward Reverse

1.2 1.1

41 55

2.35 2.6

113.3 120.4

12.5 10.8

0.56 0.48

210.3 191.4

0.69 0.63

1.36 1.23

5.3 4.2

4.7

O1

Forward Reverse

1.2 1.3

49 49

2.15 2.15

87.7 81.0

9.1 6.9

0.39 0.31

131.1 111.4

0.61 0.52

1.17 1.00

4.2 3.2

3.7

O2

Forward Reverse

1.3 1.4

55.5 49

2.12 2.5

90.5 87.5

9.8 10.4

0.44 0.47

133.5 118.1

0.63 0.55

1.19 1.01

4.6 4.2

4.4

solid wall (S1). This shows a clear effect of the presence of an opening in the drift levels of a PG-RM wall. Fig. 13 shows the envelopes of the hysteresis curves for the three walls tested. In these curves, shear stresses were calculated on the net area of each wall (reported in Table 1, column 5). As can be observed, the solid wall (S1) developed lower shear stresses than walls with openings up to a drift about 0.3%. After this drift level, the solid wall exhibited higher shear stresses than the other two walls because walls with openings concentrated damage in a smaller zone and lost lateral stiffness. It is important to remark that despite walls O1 and O2 withstood lower lateral loads than wall S1, the three walls exhibited quite similar shear stress-drift envelopes when the resistant cross-sectional area of each wall is considered.

of the first cycles of each displacement level is presented in Fig. 15a. As can be seen, the three walls dissipated almost the same energy up to a drift of 0.2%. For a drift of 0.4%, which is when walls with openings had reached their maximum resistance, wall O1 dissipated more energy than the other two walls. By the end of the tests, wall S1 dissipated more energy than walls with openings, because it had the largest crosssectional area among the three tested walls and it developed a greater number of loading cycles before reaching its maximum resistance. The wall O2 had the lowest dissipation capacity due to its behavior has a higher flexural component, as has been commented previously, but their piers cannot develop the required deformations for dissipating a great amount of energy via this mechanism. In this regard, walls O1 and S1 were able to dissipate more energy than wall O2 because the shear component was fully activated when achieving their maximum resistance. Therefore, it is clear that the presence of an opening is detrimental for the energy dissipation capacity, and that the aspect ratio of the resistant zones can influence it as well.

4.2. Ductility Displacement ductility of each wall tested was calculated by idealizing their hysteresis envelope with a bilinear curve, following the methodology shown in Fig. 14a. This consists of identifying (1) the initial secant stiffness (KE ), which was calculated as the secant stiffness up to the point where there is a notable variation on the tangential lateral stiffness of the envelope ( SL, VSL ), (2) the idealized elastic limit ( E , VE ) is determined by making the energy of the experimental envelope equal to the energy of the bilinear up to the maximum lateral resistance of the experimental envelope ( MR, VMR ), and (3) the displacement ductility ( µd ) is calculated as the ratio between the displacement of the maximum resistance and the displacement of the elastic limit ( µd = MR / E ). The parameters that define the idealized envelopes of the tested walls, which are shown in Fig. 14b, and their corresponding ductility values are summarized in Table 2. The solid wall S1 showed an average displacement ductility (µ¯d ) equal to 4.7, a value that is quite similar to those reported in some previous studies for similar walls [41]. Walls O1 and O2 presented a decrement of 21% and 6%, respectively, on the average displacement ductility in relation with the wall S1, indicating that the presence of an opening reduces the displacement ductility. Probably, this is due to the opening produces damage localization at the piers. Although wall O2 presented higher displacement ductility than wall O1, their idealized envelopes are quite similar up to the failure of wall O1, as can be seen in Fig. 14b. The difference between these two walls is because piers of wall O2 had a higher aspect ratio, and as a consequence, their behavior is more influenced by a flexural component. This leads to the wall O2 reaching a greater displacement than the wall O1 at the time of maximum lateral load and, consequently, exhibiting a higher ductility. It should be noted that the same conclusions can be drawn using the values of maximum displacement ductility instead of the average displacement ones (µ¯d ). This behavior has been also reported in other investigations of PG-RM solid walls [42,38,41].

4.4. Stiffness degradation As is known PG-RM walls suffer lateral stiffness degradation proportionally to the level of damage. The evolution of the stiffness degradation can be assessed by testing walls under incremental cyclic deformations. In this regard, the secant stiffness for a cyclei was calculated according to the Eq. (1), where Vmax , i and Vmin, i are the maximum and minimum lateral load of the cyclei , respectively; and max, i and min, i are their corresponding displacements. Initial stiffness (K 0 ) corresponds to the stiffness of the first load cycle.

K s, i =

Vmax, i

Vmin, i

max , i

min, i

(1)

The progression of the degradation is usually analyzed in terms of the ratioKs, i/ K 0 , which is shown in Fig. 15b. As can be observed, the solid wall exhibited very little stiffness degradation up to a drift of

1.5

Shear stress (N/mm2)

1.2 0.9 0.6 0.3 0 -0.3 -0.6

The enclosed area by one force–displacement diagram cycle defines its corresponding dissipated energy. The accumulated dissipated energy

Wall O1

-1.2 -1.5

4.3. Energy dissipation

Wall S1

-0.9

Wall O2 -0.8

-0.6

-0.4

-0.2

0

Drift (%)

0.2

0.4

0.6

0.8

Fig. 13. Shear stress – drift envelopes referred to net cross-sectional area. 10

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Horizontal force ( )

(

,

(

,

250

)

) =

Experimental envelope Bilinear idealization

= 5.3

(b)

200 150

Horizontal Force (kN)

Delimitate equal areas

(a)

100

= 4.2

50 0 -50 -100 = 4.2

-150 -200 -250

0

= 4.6

= 3.2

= 4.2

-18 -15 -12 -9 -6 -3

Wall S1 Wall O1 Wall O2 0

3

6

9

12 15 18

Lateral displacement (mm)

Lateral displacement ( )

Fig. 14. (a) Displacement ductility calculation, (b) bilinear idealization of tested wall envelopes.

100

100 )

(a)

80

⋅

60

40

,

( ⋅

/

60

40

,

Wall S1 Wall O1 Wall O2

20 0

Wall S1 Wall O1 Wall O2

(%)

80

0

0.1

0.2

0.3

0.4

0.5

20 0

0.6

(b) 0

0.1

Drift (%) 16

0.2

0.3 0.4 Drift (%)

0.5

0.6

Wall S1 Wall O1 Wall O2

14 12

(%)

10

8 6 4 2 0

(c) 0

0.1

0.2

0.3 0.4 Drift (%)

0.5

0.6

Fig. 15. Energy dissipation (a), lateral stiffness degradation (b), and equivalent viscous damping (c).

0.05%, then it showed a more gradual degradation than the perforated walls. This difference is probably due to wall S1 had a larger shearresistant cross-section than the other walls. Hence, the presence of an opening increases the rate of stiffness degradation. On the other hand, no difference could be discerned between walls O1 and O2, so the effect of the aspect ratio of the piers beside the opening cannot be assessed.

Eq. (2). eq, i

=

Ei 2· · Ks, i·

2 max , i

(2)

where Ei is the dissipated energy (area of the load – displacement diagram enclosed by the cycle) and Ks, i is the secant stiffness. As can be seen in Fig. 15c, the wall O2 had higher equivalent viscous damping than the other two walls for drift values up to 0.1%. This can be explained by the fact that piers of wall O2 were relatively slender and they developed more flexural cracks at early stages of testing than wall S1 and piers of wall O1. Afterward, all walls exhibited similar equivalent viscous damping, between 5% and 7%, at the middle stage of the test. At a deformation corresponding to the maximum resistance, the wall O1 had a higher equivalent viscous damping than the other walls because it showed a larger cracked portion of its resistant area. Walls O1 and O2 showed their maximum equivalent viscous damping at different drift levels, so the aspect ratio of piers had a

4.5. Equivalent viscous damping The equivalent viscous damping is used as an indicator of the real damping of a structure. In this regard, cyclic loading tests provides valuable information because allow studying how the equivalent viscous damping evolves according to damage progresses. For each complete cycle i with maximum deformation max, i , the specimen is represented by a linear equivalent Kelvin’s system that dissipates the same amount of energy than the specimen for max, i . Then, the equivalent viscous damping of the cycle i ( eq, i ) is calculated with the 11

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notable effect on this seismic parameter. On the other hand, wall S1 exhibited an almost constant equivalent viscous damping during the test.

5.1. Numerical cases

(b) O1M

2 4.2 every 2 4.2 every 3 courses 2 courses

(a) S1M

655mm

655mm

(d) O1-UR

(e) O2-UR

655mm

655mm

(f) O1-OR

(g) O2-OR

2 4.2 every 2 courses

Fig. 16. Walls simulated via detailed micro-modelling in Calderón et al. [8]. 12

2 4.2 every 2 courses

970mm

(c) O2M

2 4.2 every 2 courses

2 4.2 at each joint

970mm

2 4.2 every 2 4.2 every 3 courses 2 courses

2 4.2 every 2 courses

In this section, the maximum shear strength of PG-RM walls with openings is predicted by means of the strut-and-tie method and shear expressions from the literature, including some international masonry codes. For this purpose, a set of ten PG-RM walls is considered. This set of walls is composed of the three tested walls (S1, O1, and O2) and seven other walls that were numerically simulated by Calderón et al. [8]. It should be noted that those seven walls were not incorporated in the quantification of seismic performance parameters because they come from pushover-type monotonic analyzes.

2 4.2 at each joint

5. Prediction of shear strength

2 4.2 every 2 courses

Fig. 16 shows the seven walls numerically simulated by Calderón et al. [8]. These seven walls had the same external dimensions than the tested walls, but different opening sizes and horizontal reinforcement layouts in the piers were investigated. The effect of the opening size was studied by considering walls O1M and O2M, which had a window opening with the same height as those of walls O1 and O2, respectively, but 970 mm wide. This modification forced to move inner vertical reinforcement 56 mm outward and, therefore, wall S1M represents the non-perforated version of the modified layout of vertical reinforcement. On the other hand, the effect of the horizontal reinforcement ratio was analyzed by means of the walls O1-UR, O1-OR, O2-UR, O2-OR. Walls O1-UR and O2-UR had reinforcement every 3 mortar joints (instead 2 as in the walls O1 and O2), for this reason, these walls are considered as under-reinforced (UR) walls. While walls O1-OR and O2-OR represent variants of walls O1 and O2 with higher reinforcement ratio given that

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each horizontal mortar joint had reinforcement and consequently these specimens are called over-reinforced (OR) walls. Regarding vertical reinforcement, all seven walls had two 25 mm diameter bars on the outer edges and two 22 mm diameter bars in the interior of the panel. Table 3 summarizes the geometrical characteristics of these seven numerical cases, the amount of provided reinforcement, and the predicted lateral resistance via detailed micro-modelling.

factors to account for the effect of openings in the shear strength of a wall. For this reason, the shear strength of a PG-RM shear wall with openings is typically calculated as the sum of the shear strength of its piers. Although it is known that characteristics of the piers (e.g. their aspect ratio) influence the response of a wall, there is no certainty that the properties of piers are sufficiently determinant to justify that the wall’s design process is focused on them. The accuracy of expressions is evaluated in terms of the Vexp/ Vn ratio, where Vexp is the experimental lateral resistance of a wall and Vn the predicted resistance. Then, a value of Vexp/ Vn close to 1.0 corresponds to a satisfactory prediction, a value of Vexp/ Vn higher than 1.0 correspond to an underestimation and a value of Vexp/ Vn lower than 1.0 correspond to an overestimation. In Sandoval et al. [41], a statistical analysis of the Vexp/ Vn ratio for different shear prediction expressions was presented, obtaining mean values of Vexp/ Vn ratio close to 1.0 for the expressions presented by Shing et al. [43], Psilla and Tassios [37], Aguilar et al. [1] and by the Canadian masonry design standard CSA-S304 [9]. Therefore, these expressions were used here to evaluate the in-plane shear resistance of the tested and the simulated walls. Additionally, the expression presented in the U.S. masonry code TMS 402/602 [44] is also considered in the analysis. These formulas are presented in Table 5. In the Psilla & Tassios’s equations, the term kbh is taken as 3.0 and kbv is taken as 2.0 according to anchorage conditions used in the tested walls. As in Sandoval et al. [41], the equation of Chilean reinforced masonry code [27] is not included in the analysis because it is based on the allowable stress philosophy.

5.2. Strut-and-tie method Strut-and-Tie models (STM) were used to predict the strength of the walls accounting for the presence of the openings. STM layouts were determined based on the recommendations by Voon [47]. The results from Dillon [13] were used to determine the strut widths, strut angles, strut strengths, and nodal strengths. The effective compressive strength of struts was taken as fcs = 0.8 s fm' , where s is a strut efficiency factor, assumed as 1.0 for prismatic shaped struts, and is the strut inclination factor, which takes into account how masonry compressive strength varies according to the angle between the line of action of the resultant force and the principal axis (perpendicular to bed-joints) of the masonry composite ( = 1.0 for load normal to the bed joints or = 0, and = 2/3 for > 37.5°, with interpolated for intermediate values). The effective compressive strength of nodal zones was taken as fn = 0.8 n fm' where n is the node efficiency factor, equal to 1.0 for nodal zones anchoring one tie only. The capacity of the ties was calculated as Asv fyv . The average ungrouted compressive strength, which was measured from the prism tests, was used as the masonry compressive strength fm' . Fig. 17 presents the strut-and-tie schemes used for the walls investigated, while Table 4 summarizes the predicted shear forces for the different cases under study. As seen in Fig. 17b and c, walls with the same opening configuration but different pier’s horizontal reinforcement ratio (e.g. walls O1, O1-UR and O1-OR) have the same STM layout. This is because the method neglects the contribution of horizontal reinforcement to the strutcompressive strength. On the other hand, Table 4 shows that the actual strength of the walls was greater than that predicted when taking into consideration the strut inclination angle, indicating a conservative prediction when this value is used. Neglecting the effect of the strut inclination angle is still conservative, by exception of walls S1, S1M, O1, and O1-OR. Possible reasons for the difference may include the fact that the STM recommendations used here were developed originally for concrete blocks, and that more detailed STM layouts may be more appropriate for these walls, especially for the wall with larger openings or that with piers that have an aspect ratio greater than 1.0.

5.4. Assessment The accuracy of the selected methods is assessed following the methodology presented in Aguilar et al. [1]. The results for different walls are presented in Fig. 18 and the statistical analysis of them are presented in Table 6. In this analysis, the results of STM are also included. Contrarily to the Vexp/ Vn ratios reported by Sandoval et al. [41] for solid PG-RM multi-perforated clay brick masonry walls, results obtained with the Psilla & Tassios’s equation have the lowest accuracy. In fact, this expression is the most conservative and presents the Vexp/ Vn ratios with the highest variability among the studied expressions. This is because the term associated to the contribution of the vertical reinforcement (Eq. (6)) is zero for all perforated walls, and as a consequence, the effect of vertical reinforcement on the shear resistance is neglected. The TMS 402/602’s [44] and the CSA-S304’s [9] equations are also conservative. It is worth noting that CSA S304′s equations give the same resistance values for walls O1 and O2 because they had aspect ratios (M /(Vd v ) ) higher than 1.0, and as results, the influence of the aspect ratio of piers is disregarded. Despite TMS 402′s equation considers the effect of the aspect ratio of piers, it gives an average Vexp/ Vn ratio of 1.8 and has the second greater variability, probably due to it was calibrated for solid RM walls of concrete blocks. It should be noted that a recent study by Dillon and Fonseca [11] recommended that new

5.3. Existing shear equations As discussed by Aguilar et al. [1] and Sandoval et al. [41], there are different shear equations to predict the in-plane shear strength of PGRM walls. However, none of them include modifications or reduction Table 3 Characteristics of numerical wall specimens. Wall

S1M O1M O2M O1-UR O1-OR O2-UR O2-OR

Piers’ dimensions Length (mm)

Height (mm)

– 605 605 762.5 762.5 762.5 762.5

– 800 1320 800 800 1320 1320

Opening area ratio (%) (1)

0 16.0 26.3 10.7 10.7 17.8 17.8

Net cross sectional area (mm2) 155133 91654 (2) 91654 (2) 112291 (2) 112291 (2) 112291 (2) 112291 (2)

Horizontal reinforcement

Vertical reinforcement

Steel bars

Steel bars

8-2 3-2 5-2 2-2 5-2 3-2 9-2

4.2 4.2 4.2 4.2 4.2 4.2 4.2

Notes: (1) Ratio of opening area to full area of wall S1; (2) Net area that considers both piers; contribution of the two piers located at both sides of the opening. 13

(3) (3) (3) (3) (3) (3)

(3)

h

(%)

0.071 0.074 0.075 0.049 0.124 0.045 0.135

2 2 2 2 2 2 2

25 + 2 25 + 2 25 + 2 25 + 2 25 + 2 25 + 2 25 + 2

Reinforcement of piers;

v

22 22 22 22 22 22 22 (4)

(%)

1.12 2.02 2.02 1.55 1.55 1.55 1.55

(4) (4) (4) (4) (4) (4)

Micro-model Vmax (kN) 207.3 123.9 108.4 151.5 166.8 124.9 176.4

Reinforcement ratio considers the

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Fig. 17. STM layouts. (Forces in kN. Values in parenthesis exclude

shear strength models should be developed to replace the TMS and CSA equations due to, among other reasons, the original expression was not implemented for their use in partially grouted walls. On the other hand, the accuracy and the variability of the Vexp/ Vn ratios of the expressions proposed by Shing et al. [43], Aguilar et al. [1], and the STM are appropriated. Though the three groups of estimations over-predict the resistance of some walls, the average values of Vexp/ Vn ratio are slightly above 1.0 and their standard deviation are acceptable. The most accurate resistance estimations in average terms were obtained by means of the equation by Aguilar et al. [1], which was calibrated over a large database of tests of PG-RM shear walls with bedjoint reinforcement. When using the STM the resistance of a wall with openings is not computed as the sum of individual resistances of the piers, as it was done with the expressions presented in Table 5, but considering the flow of the forces within the wall. The STM has the advantage of accounting the geometry of a wall explicitly in the calculations, helping designers with the analysis of masonry regions near openings and piers. The STM results were comparable to the other methods. However, it is

in calculations).

noted that phenomena such as the contribution of distributed horizontal reinforcement, reinforcement anchorage, size/strength of the nodal zones, and the effect of the strut orientation on the strength of masonry assemblies are still open areas for research in current STMs for PG-RM shear walls. 6. Conclusions This paper analyzed previously reported test results of three fullscale partially grouted reinforced masonry (PG-RM) walls that were tested under cyclic in-plane lateral loads without external axial load. The three walls had the same external dimensions (aspect ratio of 1.02), but one wall was solid and the other had a centered perforation with different aspect ratios (1.22 and 2.02). The performance of the walls was analyzed in terms of their peak strength, ductility, energy dissipation, stiffness degradation, and equivalent viscous damping. Furthermore, the DIC technique was validated and used for tracking the damage evolution through tests. Moreover, a Strut-and-Tie Model (STM), which is based on the works of Voon [47] and Dillon [13], and

Table 4 Predicted shear forces by means of Strut-and-Tie models. Wall S1

O1

O2

Reference shear force (kN)

Experimental Numerical

200.8

121.3

125.8

STM shear force (kN)

Including

169.0

97.2

73.8

Excluding

253.3

145.65

110.6

S1M

O1M

O2M

O1-UR

O2-UR

O1-OR

O2-OR

207.3

123.9

108.4

151.5

166.8

124.9

176.4

178.0

68.0

50.0

97.2

73.8

97.2

265.0

14

102.0

74.0

145.65

110.6

145.65

73.8

110.6

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Table 5 Shear expressions selected in this study. Reference

Shear expression

Shing et al. [43]

Eq.

Vn1 = (0.166 + 0.0217 v f yv ) An fm' + 0.0217 n An fm' +

Psilla and Tassios [37] 1 M Vdv

Vn2 =

Aguilar et al. [1]

h

=1

v

=1

(0.03 fm' L w t + 0.3P ) + (

0.6 f yh dsh kbh ftm Lw 0.6 f yv dsv kbv ftg hw

(

Vn3 = 1.061AT

2 h Asht f yh 3

Lw

2d'

(3)

1 Ash f yh

sh

(4)

+ 0.2 v Asvt f yv ) (5)

0

(6)

0

(7)

)

0.5568

+ 0.9309

1+e

(8)

= 1 M ' fm Vdv 7.624

8.0997

6.9163

h f yh 0.72

v fcg f yv

6.1045

1.222

' h f yh fm 2.838

Vn4 = ( m

m tdv

+ 0.25P )

(

M Vdv

= 0.16 2

0.25 An AT

TMS 402/602 [44]

Vn5 =

)

An AT

+ 0.6Asht f yh

fm'

0.4 fm' tdv

Vn4,max =

0.4 fm' tdv

M Vdv

dv sh

(10)

0.4 fm' An An AT

An (2 AT

hw Lw

, if hw ) lw

, if

hw Lw

(11)

>1

1; with

hw Lw

0.5 (12) (13)

g ·(Vnm

g ·Vn5, max , g

+ Vns )

Asht sh

1.75M /(Vdv )) An

(14)

= 0.75 for PG-RM fm'

(15)

+ 0.25P

(16)

fyh dv 0.5An fm' , n

Vn5, max = (0.56

, if

0.23 M /(Vdv )) An fm' , n if 0.33An

fm' , n

M /(Vdv )

1.0

Min Max Mean Std. Desv.

TMS 402/602 [44]

Shing et al. [43]

Psilla and Tassios [37]

Aguilar et al. [1]

CSA S304 [9]

STM

1.19 2.98 1.80 0.54

0.85 1.61 1.15 0.23

1.34 3.48 2.05 0.68

0.82 1.32 1.06 0.18

1.40 1.97 1.57 0.17

0.78 1.59 1.13 0.26

Psilla and Tassios (2009) S&T Method

3.0 2.5 2.0 1.5 1.0

Fig. 18. Accuracy of selected methods. 15

O2-OR

O2-UR

O1-OR

O1-UR

O2M

O1M

S1M

O2

O1

0.5

S1

(17)

0.25

Table 6 Statistical values for selected expressions and ST models.

– The presence of a central opening in a PG-RM wall reduces the shear strength and the displacement ductility and increments the stiffness degradation rate in comparison with a solid wall. – The damage in perforated walls is not restricted only to piers. In

Shing et al. (1990) CSA-S304 (2014)

M /(Vdv )

0.25 < M /(Vdv ) < 1.0 if

five existent expressions [43,37,1,9,44] were used to estimate the maximum lateral resistance of the tested walls and of seven additional walls in order to verify the accuracy of their predictions. The following conclusions have been obtained from this research:

TMS 402 (2016) Aguilar et al. (2016)

(9)

Vn4, max

0.5 for PG walls

Vns = 0.5

ratio

n 1.49

1.0

Vnm = 0.083·(4

0.0

1.1012

+ 9.0745

CSA-S304 [9]

3.5

+ 0.4315

Engineering Structures 201 (2019) 109783

S. Calderón, et al.

–

–

–

–

–

spite of the properties of the piers (such as aspect ratio, horizontal reinforcement ratio) control the lateral response of the wall, attention must be given to the design of the masonry zones nearby piers and perforations. The aspect ratio of piers has a significant influence on the behavior of a wall with an opening. Although a significant disparity was not seen in the shear strength of the two tested walls with openings, they exhibited differences in the damage progression, the total amount of dissipated energy and the equivalent viscous damping during the tests. The validation of the DIC method confirms its potentiality as a powerful experimental measurement system. It is remarked the usefulness of the DIC method for identifying the formation of cracks (as differential displacements) in early testing stages when the eye inspection is not able to detect the incipient damage. STM accurately predicts the resistance of walls by explicitly including the geometry of the wall. It is noted that the method must be improved for properly including the effect of distributed shear reinforcement on the masonry shear strength. Estimating the resistance of a PG-RM shear wall with openings as the sum of individual contributions of the piers gives accurate predictions provided the expression used to estimate the resistance of piers is accurate enough. Among the evaluated expressions, the proposed by Aguilar et al. [1] was the most accurate. Finally, it is clear that more experimental and numerical work is needed in order to quantify the influence of the openings in the shear response of PG-RM shear walls, in particular when wall segments between openings (piers) have different aspect ratios. Similarly, greater efforts are also needed in order to develop more accurate methods to achieve a better prediction of the lateral capacity of this type of walls with openings.

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Declaration of Competing Interest The authors declare that there is no conflict of interest. Acknowledgements The authors are grateful for the funding provided by the Fondo de Fomento al Desarrollo Científico y Tecnológico (FONDEF-Chile) under Grant N°17I10264 and the Fondo Nacional de Ciencia y Tecnología de Chile (FONDECYT Regular) through Grant N°1181598. Sebastián Calderón thanks the support provided by CONICYT by its program CONICYT-PCHA/Doctorado Nacional/2017-21170992. Laura Vargas also thanks the support provided by CONICYT by its program CONICYT-PCHA/Doctorado Nacional/2019-21191181. Finally, the second author also thanks the financial support given by the “Vicerrectoría de Investigación de la Pontificia Universidad Católica de Chile, mediante el Concurso Pasantías y Estadías Breves de Investigación Convocatoria 2019”. References [1] Aguilar V, Sandoval C, Adam JM, Garzón-Roca J, Valdebenito G. Prediction of the shear strength of reinforced masonry walls using a large experimental database and artificial neural networks. Struct Infrastruct Eng 2016;12(12):1661–74. [2] Araya-Letelier G, Calderón 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. [3] Arnau O, Sandoval C, Murià-Vila D. Determination and validation of input parameters for detailed micro-modelling of partially grouted reinforced masonry walls. In 10th Pacific conference on earthquake engineering. 6–8 November 2015. Sydney, Australia; 2015. [4] Blaber J, Adair B, Antoniou A. Ncorr: open-source 2D digital image correlation Matlab software. Exp Mech 2015;55(6):1105–22. [5] Bolhassani M, Hamid AA, Moon FL. Enhancement of lateral in-plane capacity of partially grouted concrete masonry shear walls. Eng Struct 2016;108:59–76. [6] Bolhassani M, Hamid AA, Rajaram S, Vanniamparambil PA, Bartoli I, Kontsos A. Failure analysis and damage detection of partially grouted masonry walls by

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