Numerical modeling of reinforced masonry walls under lateral loading at the component level response as opposed to system level response

Ain Shams Engineering Journal 10 (2019) 435–451

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Civil Engineering

Numerical modeling of reinforced masonry walls under lateral loading at the component level response as opposed to system level response Ahmed Abdellatif, Marwan Shedid ⇑, Hussien Okail, Amr Abdelrahman Structural Engineering Department, Ain Shams University, Cairo, Egypt

a r t i c l e

i n f o

Article history: Received 31 May 2017 Revised 14 August 2018 Accepted 18 December 2018 Available online 26 February 2019 Keywords: Finite element Nonlinear models Reinforced masonry Shear wall Numerical analysis

a b s t r a c t Many experimental and analytical investigations were carried out on fully-grouted reinforced masonry shear walls types (rectangular, flanged or end-confined) to investigate their behavior under lateral loads. These studies mainly focused on evaluating the seismic response parameters for reinforced masonry shear walls (RMSW) such as ductility capacity, energy dissipation, stiffness degradation and strength. Yet, most of the research was conducted on studying each wall individually (component level response) and quite few investigations were carried out considering the system level response when different wall types are combined in a single building/system. In this paper, a simple numerical macro finite element model for walls is verified and used to simulate the in-plane response of a structure composed of ten RMSW having different ductility capacities but designed to have the same ultimate strength. The model was initially verified against available experimental data in the literature, then a parametric study was introduced to represent the effect of reinforcement ratio and axial compression on wall behavior prior to modeling structures composed of several walls. The current investigation intended to introduce how the structure ductility is affected when walls, having different ductility capacity, are interacting within one lateral load resisting system in the structure. Three methods were proposed to measure the yield displacement of the entire system to determine the value of displacement ductility for the structure and compared it to that of the individual walls within the system. Finally, a set of fragility curves were represented to illustrate the enhancement of seismic performance of masonry structures through adding end confined and flanged walls inside the structure. The results of this study showed that the displacement ductility of a structure could be significantly improved when flanged and end-confined boundary walls are included in the system as opposed to that constructed using only rectangular walls. The effect of adding end-confined masonry walls in improving structure displacement ductility is found to be more significant compared to adding flanged ones. Using fragility curves, the effect of end confined and flanged wall in the enhancement of the structure performance and delaying the damage state appears clearly in third and fourth damage states but did not have any contribution in enhancement of the first and second damage states. Ó 2019 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Ain Shams University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-ncnd/4.0/).

1. Introduction

⇑ Corresponding author. E-mail addresses: [email protected] (A. Abdellatif), marwan.shedid@ eng.asu.edu.eg (M. Shedid), [email protected] (H. Okail), amr@aace-eg. com (A. Abdelrahman). Peer review under responsibility of Ain Shams University.

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Fully grouted reinforced masonry shear walls have been used in many countries especially in North America as the main lateral load resisting systems in low and medium rise buildings because of their inherently large lateral stiffness and strength. The behavior of these walls depends on the behavior of several constitutive materials that have different characteristics such as blocks, mortar, vertical and horizontal reinforcement and grout which makes the nonlinear analysis of RMSW challenging. Consequently, there is a need to develop a numerical model for such walls that is both accurate and simple to be used in modeling the response of struc-

https://doi.org/10.1016/j.asej.2018.12.003 2090-4479/Ó 2019 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Ain Shams University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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tures composed of many walls under lateral loading as opposed to individual wall elements. A simplified numerical model using primarily fiber-based beam column elements capable of capturing flexure response is used to simulate the behavior of three different wall types: rectangular, flanged and end confined reinforced masonry shear walls under cyclic loading. A detailed description of the model followed by description of the experimental program used to validate the proposed modeling technique is presented in this paper. In this regard, the experimental results of cyclic response, strength, displacement ductility, energy dissipation and effective stiffness is compared to model results in order to evaluate the effectiveness of the modeling approach of an individual wall (component level). Afterwards, a structure (system level) will be modeled to investigate the seismic performance of a structure using the verified individual wall modeling technique. It worth mentioning that a few and recently experimental investigations were carried out to evaluate the building response. In 2011, Heerema et al. [1] tested a two-story third scale building with RM walls that have no coupling through the concrete diaphragm. That is done through reducing the slab thickness between the walls. Also, Ashour et al. [2] tested the same building configuration that was tested by Heerema, but the coupling effect was investigated through keeping the slab thickness constant. Finally, Ezzeldin et al. [3] tested the same two-story building tested by Ashour but changed the main walls from rectangular to confined walls with boundary element. Although many numerical models were used to simulate response of individual elements, a simple numerical model is needed to simulate the building performance with accurate representation. The nonlinear finite element program SeismoStruct v6.5 [4] is used in this study to conduct static pushover and cyclic analysis for previously experimentally tested reinforced masonry shear walls. The program is capable to consider both geometric nonlinearity due to P-delta effects and material nonlinearity through introducing material constitutive rules. The model was validated against the results of six reinforced concrete block structural walls having rectangular, flanged and end-confined cross sections, and subjected to in-plane cyclic lateral loading presented by Shedid et al. [5]. In this study, the validated simple macro models captured to a good extent the cyclic load-displacement relationships, energy dissipation, strength and stiffness degradation, and displacement ductility. These models were then used in a parametric study to investigate the effect of axial stress ratio and reinforcement ratio on yielding drift, ultimate drift and displacement ductility. Moreover, a 3D-structure is modeled and investigated regarding strength and displacement ductility using a set of walls having different ductility capacity which is difficult to be performed experimentally due to laboratory limitations and costs.

2. Nonlinear analytical modeling In the proposed modeling technique each wall is represented by a beam-column element that fully accounts for both geometric nonlinearity (p-delta effect) and material nonlinearity (plasticity). Geometric nonlinearity is considered through the employment of a total co-rotational formulation developed by Correia and Virtuoso [6], while material nonlinearity is considered in the model by using distributed plasticity element. Such element considers spread of plasticity along wall height and has many advantages in earthquake engineering compared to simpler lumpedplasticity models [7]. The latter has the disadvantage of considering inelastic deformation in local zones (localization) and therefore Reinforced Concrete (RC) and Reinforced Masonry (RM) walls having ductile behavior with plasticity extending over part of the wall height will not be accurately represented by lumped-plasticity

models. The type of distributed plasticity model implemented in this study is displacement-based formulation (DB) element. This type of formulation assumes a displacement field with a linear curvature variation along the wall element and hence the nonlinear behavior at wall base (Inelastic curvature) will not be captured correctly and numerical instability may occur. Therefore, subdivision of wall element at the base into small segments must be conducted to overcome the assumption of a linear curvature field. A subdivision of wall into six elements per storey yields a close match with the experimentally recorded response as will be discussed later. This subdivision provides a better representation for the plastic hinge length Lp which is determined based on Eq. (1) as reported by Bohl and Adebar [8] and previous test results of RMSW [9]. This equation was based on nonlinear finite element analysis results of 22 isolated walls and found to give the best estimate of the plastic hinge length for the three walls. The formula is a function of the wall length (Lw), moment-shear ratio (Z), gross area of wall cross 0 section (Ag), concrete compressive strength (f c ) and axial compression (P).

Lp ¼ ð0:2Lw þ 0:05Z Þ 1:0  1:5

P 0 f c Ag

!

ð1Þ

The cross-section analysis and the generated behavior are based on a fiber-section approach. In such approach, the cross section is divided into small fibers and each fiber is associated with a uniaxial stress-strain relationship, the strain or stress distribution over each section is then obtained through the integration of the nonlinear uniaxial stress-strain response of the individual fibers according to number of fibers per section. The constitutive materials such as reinforcement steel, unconfined and confined masonry are represented as fibers. The cross section of the three wall types (Rectangular, Flanged and End-Confined) is discretized into fibers and as the number of fibers increases the fiber area decreases leading to a higher accuracy in solution results. The main advantages of using fiber section method are that both moment-curvature analysis of members and element hysteretic response are not required prior to analysis as they are already defined by the material constitutive models. The solution is obtained by assuming a linear strain distribution across the slender wall cross section (according to Navier-Bernoulli assumption that plane sections remain plane) and calculating the stresses in each fiber using the material constitutive models. For the flexural dominated walls considered later in this study, the assumption was found to yield accurate results up to the damage level considered (15% strength degradation). This may not be the case for higher loading levels where severe damage is expected along the length of the wall in the plastic hinge zone, however this may not be of interest in seismic analysis related to specific performance limits such as life safety/collapse prevention states [10]. The assumed strain distribution is adjusted in an iterative procedure until equating the internal axial force to the applied axial load. This adjustment is performed by changing the strain values in the first and last fibers and assuming a linear distribution of strains relaying on compatibility of strains. After obtaining the correct strain distribution in each step of the analysis, the curvature (/), the axial force (N) and the moment (M) in the section can be computed using Eqs. (2)–(4).

/¼ N¼ M¼

es þ em d X

ri Ai

X

ri Ai yi

ð2Þ ð3Þ ð4Þ

where: es is the steel strain, em is the masonry strain, d is the distance from outermost bar to the compression wall end, ri is the

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fiber axial stress, Ai is the fiber cross section area and yi is the fiber distance from neutral axis. It is worth noting that all walls modeled in this study are flexural dominated with large safety against shear failure as stipulated by major design codes. In such cases, shear effects and deformations are to be minimal; therefore, ignoring shear effect associated with fiber-section formulation is not expected to jeopardize the analytical results and findings [10]. Moreover, the assumption of linear strain profile for flexural dominated walls shown to be acceptable based on previous experimental test programs conducted on RMSW with aspect ratios larger than 1.5 [5,11], hence it is considered valid for modeling. 3. Constitutive material model In this paper, three previously experimentally tested types of RMSW (rectangular, flanged and end confined boundary wall) were modeled. Unconfined masonry models were used for Rectangular and Flanged wall types due to the lack of confinement of the grout core as a result of having a single horizontal reinforcement and single layer of vertical reinforcement in block cell. On the other hand, the end confined boundary wall type was represented with two masonry materials model types, namely confined and unconfined model. The confined model was used at the end of the wall with confining parameters due to the presence of end boundary characterized by the presence of closed ties that prevented and delayed buckling of vertical reinforcement, in addition to providing confinement to the grout core. The second masonry material model presented at middle of the wall without confining condition (unconfined masonry) similarly to rectangular wall type as shown in Fig. 1.

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3.1. Masonry material Although, Mander et al. model [12] is originally a concrete model, the stress-strain characteristics of fully grouted masonry prims are very similar to those of concrete given that proper values of the elastic modulus, compressive strength and ultimate strain are used in modeling [13,14]. Mander et al. nonlinear concrete model was modified and used to represent fully grouted masonry elements. In this model the effect of transverse reinforcement and stirrups in increasing ductility of confined core material was considered due to the effect of confinement which increased the deformability of ultimate compressive masonry strain. A confinement factor was applied to Mander et al. concrete model to account for the effect of confinement stirrups and defined as the ratio between the confined and unconfined compressive stress of masonry to scale up the stress-strain relationship (about 13%) as shown in Fig. 2a. As recommended, the tensile strength of masonry was neglected and taken equal to zero and this is due to the fact that crack opening may introduce numerical instability in the analysis [4]. This assumption is accepted when predicting the global response of an element such as top displacement (which is the focus of the paper) rather than accurately representing the local response of elements and sections such as local strains. The parameters used for masonry modeling are listed in Table 1. 3.2. Reinforcement material The steel reinforcement was modeled using Menegotto-Pinto [15] steel model that is defined in Siesmostruct [4] materials model. Menegotto-Pinto model is defined as a uniaxial steel model

Fig. 1. Cross-section details of analyzed walls.

Fig. 2. (a) Stress-Strain relationship for Mander model, (b) Menegotto-Pinto steel model.

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Table 1 Parameter used for masonry and steel modeling. Material

Parameter

Masonry Model [12]

Young’s modulus (Em )

Value 0

Maximum compressive strength (f m ) Strain at maximum compressive strength tension strength (rt ) Confinement factor Steel Reinforcement Model [15]

Young’s modulus (Es ) Yield strength (Fy) Strain hardening parameter Transition curve initial shape parameter R0 Transition curve shape calibrating coefficient–A1 Transition curve shape calibrating coefficient–A2 Isotropic hardening calibrating coefficient–A3 Isotropic hardening calibrating coefficient–A4 Fracture/buckling strain

13,700 MPa 16.40 MPa 0.002 0 1 (for Rectangular and Flanged walls) 1.13 (for End-Confined walls) 200.6 GPa 495 MPa 0.01 20 19.4 0.15 0 1 0.1

enhanced by the isotropic hardenings rules proposed by Filippou et al. [16], and the additional memory rule introduced by Fragiadakis et al. [17] for higher numerical stability and accuracy under transient seismic loading. This model is defined by Modulus of elasticity, yield strength, strain hardening parameter, fracture/ buckling strain and some coefficients representing Bauschinger effect [18], pinching of hysteretic loops and transition from elastic to plastic zone. This model considers the effect of reversal cycles (Bauschinger effect & hysteretic behavior) of tension and compression and is characterized by the stress-strain hysteretic loops shown in Fig. 2b. This formulation allows modifying the shape of the branches at each cycle depending on the plastic excursion of the previous cycle. All parameters values used for the steel model are shown in Table 1.

4. Modeling approach The displacement based formulation as opposed to force-based formulation was used as the post peak behavior in the latter will be underestimated if the number of integration points (NIP) and consequently its weighted length does not match the location of plastic hinge at the base of the wall. On the other hand, only a refined meshing is required in displacement based formulation to achieve an accurate response [19]. The choice of element length is an important aspect when displacement-based elements are used with distributed plasticity due to strain localization, in which plastic deformation in a vertical cantilever wall tends to be concentrated in the first element above the base of the wall, while the top elements remain elastic. Because of strain localization, the numerical results are very sensitive to the first element length near the base, or simply to the plastic hinge length estimation. It is worth noting that in this situation, with a displacement based formulation a refined division of the structural element is needed for accurate response. A sensitivity study was conducted to determine the number of elements to be used for wall meshing and Fig. 3 shows load displacement relationship for W1 using 3 and 6 element meshing. Based on the plotted results, it was shown that 3 element meshing resulted in overestimating strength and ductility of the wall. Based on several iterations, it was found that 6 element meshing yield reasonable results and denser meshing would not cause significant enhancement in load displacement relationships. Therefore, 6 element meshing was used for all wall modeled in this study. Also, to capture the structural response, two types of deformations are required to be considered [20]; the first is flexural deformation causing inelastic strains in reinforcing steel and masonry while the second deformation is member end rotation due to reinforce-

Fig. 3. Model sensitivity to number of subdivisions for wall element.

ment slip and strain penetration in the foundation. Experimental studies done by Kowalsky et al. [21] and Saatcioglu et al. [22] showed that the top displacement resulting from end rotation produced by strain penetration may occupy up to 35% of the total lateral deformation of flexural members which may result in underestimation of wall drift as well as overestimation of wall stiffness. Yield penetration causes increase in base rotations which are considered by reducing the bending stiffness of an elastic element at the wall base. This elastic element is modeled with small length of 10 mm as suggested in the verification manual of SeismoStruct v6.5 [4] adopted for flexural elements. To account for strain penetration deformations and bond slip, an elastic element is modeled at the base of the wall to consider the initial deformation. The elastic stiffness (EI) of this element is considered, as stated by (FEMA-356) [23], to be equal to 0.5EI of gross wall section. This value is considered when the ratio between the applied axial load (P) to the gross cross-sectional area (Ag) multiplied by 0 masonry compressive strength (f m ) is less than 0.3. Therefore, the reduced stiffness will account for the additional deformation (i.e., wall end rotation) due to bar slip. 5. Modeling verification In this paper, the generated numerical models for the different types of RMSW specimens were verified with experimentally tested walls conducted by Shedid et al. [5]. The results of the six rectangular, flanged and end confined walls (specimens W1–W6) were compared with the modeling results to validate the numerical model. A brief overview of the experimental program is provided herein. The experimented walls were tested under fully

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reversed displacement controlled quasi-static cyclic loading up to 50% degradation in strength while being subjected to 160 kN vertical axial load. The selection of the tested walls was based on a criteria to allow verifying the analytical model under various conditions including amount of vertical reinforcement (1.17– 0.55%), cross section shape (Rectangular, Flanged and End confined) and aspect ratios (1.5 and 2.2). Table 2 summarizes the different design aspects of the walls used for model calibration, showing wall type, axial load level, reinforcement, dimensions and the cross-section details of the different wall types, which are presented in Fig. 1b. A comparison between analytical and experimental results is conducted with respect to cyclic response, peak load, displacement ductility, energy dissipation and effective stiffness. As a measure for the evaluation, the error in the results will be quantified as in Eq. (5) being equal to the ratio of the difference between analytical and experimental result to the experimental value.

Error ¼

 Analytical mean v alue  Experimental mean Experimental mean v alue



v alue

 100 ð5Þ

5.1. Cyclic response All walls were loaded with target multiples of yield displacement in reversed cycles until walls lost 50% of their ultimate capacity (Qu) which is considered the failure state in the analytical study similar to the experimental investigation. Fig. 4 shows that there is a good agreement between experimental hysteresis loops and the corresponding loops from the nonlinear cyclic analysis for all wall types. The model was able to capture the most relevant characteristics of the cyclic wall response, including initial stiffness, ultimate resistance, stiffness degradation, strength deterioration and hysteretic shape. Fig. 5 shows the comparison between experimental and analytical values of ultimate strength, yield displacement, displacement at 20% strength degradation and displacement ductility at 20% strength degradation. The error calculated is found to range between 3.8% and 12%. It was observed from Fig. 5 that end confined and flanged walls have more ductile capacity compared to rectangular walls, which is mainly due to the increased thickness at wall ends (flanges or end-confined) leads to a significant decrease for the required length of compression zone, and therefore increasing curvature at ultimate load and displacement ductility. Moreover, thickened wall ends will provide out of plane

Table 2 Data used for model verification (Shedid et al. [2]). Specimen

Wall dimensions

Vertical reinforcement

Horizontal reinforcement

Number of bars and bar size

qv (%)

No. D4 @ spacing (mm)

qh (%)

W1 W2 W3

1802 mm  3990 mm Length  Height

19 M10 11 M10 11 M10

1.17 0.55 0.55

1 @95 1 @95 1 @95

0.30 0.30 0.30

1.09 0.89 0.89

W4 W5 W6

1802 mm  2660 mm Length  Height

19 M10 11 M10 11 M10

1.17 0.55 0.55

2 @95 2 @95 2 @ 95

0.60 0.60 0.60

1.05 0.88 0.88

Fig. 4. Experimental and Numerical cyclic results for Reinforced Masonry Walls.

Axial stress (MPa)

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Fig. 5. Experimental and Numerical response for Reinforced Masonry Walls, (a) Ultimate strength, (b) Yield displacement, (c) Displacement at 20% strength degradation, (d) Displacement ductility at 20% strength degradation.

stability and the closed ties at end boundary will prevent vertical reinforcement buckling and thus provide confinement for compression zone. 5.2. Energy dissipation Energy dissipation through hysteretic damping Ed, is an important aspect in seismic design because it reduces the amplitude of the seismic response. Previous work [24] showed that the envelope of the load-displacement hysteresis loops is relatively insensitive to the imposed displacement increments and to the number of cycles. Therefore, the energy dissipation Ed, is represented, as suggested by Hose and Seible [25] by the area enclosed within the load-displacement curve passing through the peak values of each loading cycle. Fig. 6 illustrates that the energy dissipation was low for the loading stages prior to significant inelastic deformation in the masonry and reinforcement took place. For higher displacement levels, the energy dissipation increased significantly compared to early stages of loading for both experimental and analytical results. The maximum difference between experimental and numerical results for each wall was 14%, 6%, 15%, 11%, 9.5% and 15% for walls 1–6 respectively and the average difference for all the six walls was equal to 11.75%. 5.3. Effective stiffness To assess the variation of stiffness with increased displacement, the effective stiffness was calculated according to ASCE/SEI 7-10 (ASCE/SEI 2010) [26] as shown in Eq. (6).

K eff ¼

jF þ j þ jF  j jDþ j þ jD j

ð6Þ

where F+ and F are positive and negative lateral resistance of shear wall at D+ and D respectively, and D+ and D are the maximum

push and pull displacement for each cycle. Fig. 7 shows that the numerical model well predicts the effective stiffness which is determined experimentally with an average difference of 9.5%, 5.3%, 5.4%, 10.4%, 9.9% and 12.7% for walls W1 to W6 respectively. It is observed from Fig. 7 that the rate of change by which the effective stiffness decreases is higher at the early stage of loading until yielding compared to the post-yield stage due to the progression of cracks and reduction of gross inertia in the early stages. It can be concluded from this study that the proposed models can to a great extent capture the cyclic response, the ultimate strength, displacement ductility, energy dissipation, strength degradation and stiffness of the experimentally tested walls throughout the entire loading history. Therefore, the numerical model can be used to conduct extensive parametric study to illustrate the effect of reinforcement ratio and axial compressive stress on the behavior of different wall cross section types and aspect ratios. 6. Parametric study A parametric study was performed to investigate the effect of vertical reinforcement ratio (qv ), axial compressive stresses and aspect ratio variation on the ultimate strength, displacement at reinforcement yielding and at 20% strength degradation and finally displacement ductility for different wall sections (rectangular, flanged and end-confined). Two phases of analysis are considered, the first phase is intended for studying the effect of vertical reinforcement ratio variation from 0.4% to 1.3% for rectangular, flanged and end confined wall sections while taking into consideration aspect ratio difference of 2.2 and 1.5 and keeping the axial compression constant and equal to 160 kN (similar to experimental walls). While, the second phase is intended for the studying the effect of axial compressive stress variation from 0.8 MPa to 0

0

3.0 MPa (from 5% f m to 18% f m ) on the three wall types and also with different aspect ratios of 2.2 and 1.5 for walls with vertical

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Fig. 6. Experimental and Numerical Energy Dissipation Results.

Fig. 7. Experimental and Numerical Effective Stiffness Results.

reinforcement ratio held constant at 0.8%. All wall heights, lengths, block dimensions, reinforcement pattern and distribution are taken the same as the six previously verified walls shown in Fig. 1b. It should be noted that increasing reinforcement ratio is done through increasing bar diameter to maintain same reinforce-

ment pattern. Finally, the effect of these parameters on lateral wall resistance, lateral displacement and displacement ductility are investigated and discussed. For all reinforced concrete masonry shear walls analyzed in this study, the material properties were taken as shown in Table 1.

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6.1. Effect of vertical reinforcement The effect of significant variation in vertical reinforcement ratios (0.4–1.3%) on wall strength, lateral displacement and displacement ductility is discussed in this section. The axial compressive load applied on walls and the horizontal reinforcement ratio 0 are held constant at about 5% fm and at qh = 0.30%, respectively. Whereas, the vertical reinforcement ratio (qv ) for all wall section types ranged from 0.4% to 1.3% for aspect ratios of 2.2 and 1.5. 6.1.1. Ultimate strength As shown in Fig. 8, for flexural dominated walls, the wall ultimate strength tends to be very sensitive to the amount of vertical reinforcement. For rectangular walls the increase in reinforcement ratio by 225% (from 0.4% to 1.3%) increases wall strength by 108% for both aspect ratios while for flanged walls and end confined walls a similar increase in reinforcement ratio lead to increase in wall strength by 120% on average for both aspect ratios. It was also observed that end-confined and flanged walls slightly gain more strength compared to their rectangular counterparts and this is mainly due to the effect of bar configuration which allows concentration of reinforcement at wall ends where maximum stresses are located. It is also shown from Fig. 8 that walls having end-confined boundary element have higher strength capacity compared to flanged and rectangular walls when the same vertical reinforcement ratio is used for the three wall section types. Similarly, flanged walls possess higher section capacity compared to rectangular wall section. Normalization for the relationship between ultimate strength (Q) and drift for the three wall types is illustrated in Fig. 8 where ultimate strength of each wall is divided by the ultimate strength (Qq = 0.4%) of walls with minimum considered reinforcement ratio qv ¼ 0:4% to represent an expression for strength variation with reinforcement ratio variation. It is clear that both aspect ratio (2.2 and 1.5) curves coincide as flexure strength is a section level property. 6.1.2. Drift at yield The top drift at first yield is affected by increasing the amount of vertical reinforcement as shown in Fig. 9a where lateral displacement at first yield (Dy) increases with increasing of the amount of vertical reinforcement. For rectangular walls the increase in reinforcement ratio by 225% increases wall yield drift by 26% for both aspect ratios while for flanged walls and end confined walls a similar increase in reinforcement ratio lead to increase in wall

yield drift by 22% for both aspect ratios. This increase is mainly due to increasing the yielding curvature as a result of increases of reinforcement ratio and consequently the displacement at yielding increases (for the same load level the curvature decreases with the increase of reinforcement ratio). The effect of aspect ratio also is presented in Fig. 9a where yield drift increases with the increase of aspect ratio and consequently walls having aspect ratio 2.2 will have higher yield drift values compared to walls having aspect ratio of 1.5. This is mainly due to yield displacement strongly depending on wall height as from Eq. (7).

Dy ¼

Øy H 2 3

ð7Þ

where H is the wall height and Øy is the yield curvature. 6.1.3. Drift at 20% strength degradation The effect of vertical reinforcement ratio on top wall drift at 20% strength degradation is presented in Fig. 9b where drift decreases with the increase of amount of vertical reinforcement. For rectangular walls, top drift at 20% strength degradation decreases by 27% on average for both aspect ratios corresponding to the increase in reinforcement ratio by 225% (from 0.4% to 1.3%) while for flanged walls and end confined walls similar increase in reinforcement ratio lead to decreases in wall drift by only 14% on average for both aspect ratios. The reason for such decrease in displacement capacity with the increase of reinforcement ratio is that as reinforcement increases the depth of the neutral axis increased and so ultimate curvature will decrease and hence decreasing the ultimate displacement and drift for flexural dominated walls. The reason that rectangular walls are more affected by the decrease of drift at ultimate stage (27%) compared to flanged and end confined (14%) is due to the higher influence of reinforcement ratio on neutral axis depth and consequently compression zone as a result of smaller thickness of cross section. On the other hand, the presence of flange or end boundary decreases the rate by which neutral axis moves leading to lesser decrease of ultimate curvature. Also, it can be observed that with the increase of aspect ratio from 1.5 to 2.2 the lateral drift at 20% strength degradation increases. 6.1.4. Displacement ductility at 20% strength degradation Fig. 9c shows the effect of vertical reinforcement ratio on displacement ductility at 20% strength degradation. Displacement ductility at 20% strength degradation is affected by the increase of amount of vertical reinforcement. For rectangular walls the

Fig. 8. Effect of vertical reinforcement on ultimate capacity.

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Fig. 9. Effect of vertical reinforcement on (a) %drift at first yield (%Dy), (b) %drift at 20% strength degradation (%D0.8u), (c) displacement ductility (m0.8).

increase of vertical reinforcement from 0.4% to 1.3% leads to a decrease in the displacement ductility at 20% strength degradation decreases by 41% on average (43% for aspect ratio 1.50 and 39% for aspect ratio 2.2), while for flanged and end-confined walls a similar increase of vertical reinforcement results in decrease in displacement ductility by 28% on average (30% for aspect ratio 1.50 and 26% for aspect ratio 2.2). The decrease of overall displacement ductility is mainly due to the increase yield drift and the decrease of drift at 20% strength degradation with the increase of reinforcement ratio as discussed earlier. The aspect ratio also affects displacement ductility, as shown in Fig. 9c where walls with smaller aspect ratio have higher displacement ductility compared to walls with higher aspect ratio and this is mainly due to smaller yield drift values for low aspect ratio walls (aspect ratio 1.5) while walls having higher aspect ratio (2.2) exhibit yielding at higher drifts. Although higher aspect ratio walls (2.2) have higher lateral drift at 20% strength degradation compared to lower aspect ratio walls (1.50) yet the variation in yield drift significantly affects the displacement ductility (refer to Fig. 9a and b). It is also observed that for the same vertical reinforcement ratio, the end-confined wall

sections have higher displacement ductility in comparison with flanged and rectangular wall section for both aspect ratios. Similarly flanged walls has higher displacement ductility capacity compared to rectangular wall section as shown in Fig. 9c for qv = 0.8%.

6.2. Effect of axial compression The effect of variation in applied axial compression on wall ele0

ment (5–18% of f m ) is discussed in this section with respect to wall strength, lateral displacement and displacement ductility. The vertical reinforcement ratio and the horizontal reinforcement ratio in this section are held constant and equal to 0.8% and 0.30%, respectively for the three wall section types. While the axial compressive stress and aspect ratio are variables. The axial loads acting on walls are 160 kN, 195 kN, 320 kN, 400 kN and 485 kN corresponding to 0

6%, 7.3%, 12%, 15%, 18.3% f m respectively for rectangular section 0

and 5%, 6%, 10%, 12.2%, 15% f m respectively for flanged and end confined section.

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6.2.1. Ultimate strength Fig. 10 shows the effect of axial compressive stress on ultimate strength of walls. As shown in the figure, the ultimate strength is less sensitive to the applied compressive stresses compared to the effect of vertical reinforcement ratio variation. The increase in axial load by 200% (from 160 kN to 485 kN) increases the ultimate strength on average by 20%, 19.2% and 16.6% for rectangular wall section, flanged and end-confined walls respectively. It is obvious from Fig. 10 that walls having end-confined boundary element have higher strength capacity when the same axial stress level is applied for the three wall section types, similarly flanged walls have higher section capacity compared to rectangular wall section. Normalization for the curves of the three wall types is illustrated where ultimate strength (Q) of each wall with a different axial compression load is divided by that subjected to the minimum applied axial load of 160 kN (Qp min). The rate by which the strength increases with the increase of axial load is the same for both aspect ratios (2.2 and 1.5) as both curves coincide on each other. 6.2.2. Drift at yield The lateral drift at first yield slightly changed with increasing the axial load for the different wall section types (rectangular, flanged and end-confined) and for both aspect ratios of 1.5 and 2.2 as shown in Fig. 11a. The increase in axial load by 200% increases the yield drift on average by 6% for rectangular wall section while a similar increase in axial load increases the yield drift on average by 7% for flanged and end-confined walls. The effect of aspect ratio also is presented in Fig. 11a where yield drift increase with the increase of aspect ratio and this is due to high dependency of yield displacement on wall height as shown in Eq. (7). Also, the trend of variation of yield drift for both aspect ratios is similar for the tested axial load range. 6.2.3. Drift at 20% strength degradation A shown in Fig. 11b, the lateral drift at 20% strength degradation decreases significantly with the increase of applied axial compressive stress. the increase in axial load by 200% decreases on average wall drift at 20% strength degradation by 62%, 63% and 52% respectively for both aspect ratios for rectangular walls, flanged walls and end confined walls. The reason for the decrease in drift and displacement at 20% strength degradation (displacement capacity) with the increase of axial stress is due to the increase of compression zone length and shifting of neutral axis towards tension side

which decreases the ultimate curvature and consequently ultimate displacement and drift. 6.2.4. Displacement ductility at 20% strength degradation Fig. 11c shows the effect of axial compression on displacement ductility at 20% strength degradation. Displacement ductility at 20% strength degradation is significantly affected by the increase of compressive stresses. For a 200% increase in axial load the displacement ductility at 20% strength degradation decreases on average by 60%, 63% and 53% for rectangular walls, flanged and endconfined walls, respectively. 7. System level ductility capacity In this part of the study an analytical investigation is carried out for a system composed of ten reinforced masonry shear walls to compare the component level response (Individual wall) to that of the system level with respect to ductility capacity. The investigation is intended to understand how system ductility varies with the variation of individual wall ductility especially, when composed of rectangular, flanged and end-confined walls having different ductility capacities. It is clear from the previous experimental and analytical investigations reported and discussed in previous sections that end-confined and flanged walls have more ductility capacity compared to rectangular wall section. It is worth noting that the purpose of the analysis presented in the following section is not to simulate real building construction as the study is only concerned with translation displacement while ignoring rotation at the system level. The purpose of the study is to test the interaction of walls with different ductility capabilities at the post-peak loading stage as opposed to single wall element. Therefore, extreme cases are considered to test the entire spectrum of wall combination. The three wall types used in this study to construct the structures are the same walls verified in the previous section (W1, W2 and W3) having the same length and height of 1800 mm and 3990 mm respectively. The vertical reinforcement ratio and axial stress level for the walls are shown in Table 2. Each wall of the ten walls forming the structure is subjected to an axial compression load of 160 kN. It is worth noting that the strength of the three wall types used in the structure was designed to be approximately the same so that the ductility capacity of each wall is the only influencing factor in

Fig. 10. Effect of axial compressive stress on ultimate capacity.

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Fig. 11. Effect of axial compression on (a) %drift at first yield (%Dy), (b) %drift at 20% strength degradation (%D0.8u), (c) displacement ductility (m0.8).

the outcome global displacement ductility of the entire structure. The arrangement of the ten walls forming the structure is shown in Fig. 12 where walls are spaced by 6 m and placed in a symmetrical manner. The structures are loaded by a displacement incremental load acting at their center of rigidity to which the top node of all walls is constrained as this study is only concerned with

translational deformation. It shall be noted that the aim of the paper is not to simulate real construction, however, the main purpose is to document the post-peak response of a group of walls having different ductility and displacement capabilities at the system level. Therefore, extreme cases are considered to test the entire spectrum of wall combination.

Fig. 12. Reinforced masonry walls arrangement: model configuration and individual wall type analytical load-displacement curve.

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The analysis conducted in this study is nonlinear static pushover analysis to study the effect of combining a ductile element (flanged or end-confined wall) with a less ductile one (rectangular wall) on the overall displacement ductility of the structures. Prototype structures are modeled to represent all the available combinations between individual wall types. The analytical pushover curves for the three individual wall types which illustrate the displacement ductility capacity of different wall types are shown in Fig. 12. It was observed a sudden drop in strength in Fig. 12 for the end-confined wall at the ultimate displacement and this is mainly due to outermost reinforcement bars reaches fracture strain. This agrees with experimental test result reported by Shedid et al. [5] during the repeated loading cycle at 101.2 mm where two vertical reinforcement bars fractured at both ends of the end confined wall. Regarding structures matrix data, three control structure prototype are considered and composed of ten identical walls and ten structures with different combinations between wall types, as presented in Table 3, are modeled to investigate their effect on overall system displacement ductility.

7.1. Displacement ductility calculation In this study, the displacement ductility (mD0.8u), defined as the ratio of the displacement associated with of 20% strength degradation to the effective yield displacement, will be used as a measure for structure performance under lateral load. The value of the displacement corresponding to 20% drop in peak strength is easily determined from the load-displacement curve of the global structure system generated from the nonlinear pushover analysis. However, the value of the displacement corresponding to yielding of the entire structure is not well defined from the loaddisplacement curve especially when the structure is composed of walls with different ductility capacities. Consequently, three approximate methods are proposed to determine the yielding displacement for the structure as illustrated below:

Dytotal = Yield displacement of structure; DyðAÞ = Yield displacement of individual wall A; DyðBÞ = Yield displacement of individual wall B; N A = Total number of wall type A; N B = Total number of wall type B. ii. Tangent Stiffness Method In this method, the yielding displacement of the structure is considered by calculating the tangent stiffness incrementally from the load-displacement curve of the structure. The increment showing a significant drop in stiffness value is then considered the yielding increment and the corresponding displacement value will be taken as the yield displacement of the whole structure. A constant displacement increment of 0.2 mm is used to calculate the tangent stiffness and resulted in an acceptable measure for the rate of change of load with respect to imposed displacement and hence a good capture for stiffness variation along loading stroke. Fig. 13 illustrates the method of calculation where tangent stiffness and lateral resistance are presented with imposed lateral displacement. As shown in the figure structure 1 was used as a demonstrative example where a significant variation of stiffness corresponding to a displacement of 9.6 mm which is considered yielding displacement of the structure. iii. Design load (Qd) Method In this method the yielding load Qy for the structure will be taken equal to the design load Qd following ASCE/SEI 7-10 (ASCE/ SEI 2010) [26] which defines the overstrength factor X as the ratio between the ultimate load Qu and designed load Qd and taking this ratio equal to 0.6. This assumption is also consistent with the definition proposed by Uang [27] indicating that Qd corresponds to the load level at which a significant deviation from the system linear elastic response starts to develop, therefore the yielding load Qy of the system will be assumed to be equal to 0.6Qu in this method.

i. Weighted Average Method In this method the yielding displacement of the structure is determined based on the yielding of its constitutive walls occurring at different displacements depending on the wall type. Thus, structure yielding displacement can be calculated by the following proposed Eq. (8).

Dytotal ¼

N A  DyðAÞ þ N B  DyðBÞ NA þ NB

ð8Þ

Where:

Table 3 Different walls combination in structures. Structure ID

Type Wall

Wall Combination

Control 1 Control2 Control3

Rectangular (R) Flanged (F) End-Confined (C)

10R 10F 10C

1 2 3

R&C

8R + 2C 6R + 4C 4R + 6C

4 5 6

R&F

8R + 2F 6R + 4F 4R + 6F

7 8 9 10

F&C

2C + 8F 4C + 6F 6C + 4F 8C + 2F

7.2. Results The load-displacement curves for the ten prototype structures are shown in Fig. 14. It is observed from the curves that the effect of adding a more ductile element enhances the overall structure displacement ductility. The detailed values of yield displacement Dy, displacement at ultimate load Du, displacement at 20% strength degradation D0.8u and displacement ductility at 20% strength degradation l0.8u and their corresponding lateral resistance Q and secant stiffness for the whole structure are illustrated in Table 4. The secant stiffness is given as a percentage of the initial stiffness to show the variation of the lateral stiffness with increased top wall displacement. From the previous table it is clear that the first and second method for calculating an approximate value for structure yield displacements are almost similar and more conservative than the third method thus, the first and second method are selected by the authors to determine structure displacement ductility in this study. Regarding structure displacements, the yield displacement did not vary significantly for all structures and ranged from 8.8 mm to 9.8 mm with an average yield drift of 0.235%. A significant variation was observed for displacement at ultimate load Du and ranged between 0.615% and 0.86% drift at ultimate load for structure 1 to structure 3 respectively (Group I), while a minor variation ranging from 0.595% to 0.675% drift at ultimate load for structure 4 to structure 6 respectively (Group II) was observed and finally a significant variation of 1.08–1.89% drift at ultimate load was observed for structure 7 to structure 10 respectively (Group III). This illustrates the major

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Fig. 13. Tangent Stiffness Method demonstration, (a) Load–displacement curve and stiffness variation, (b) Structure 1 Load and Stiffness variation with lateral displacement.

Fig. 14. Buildings Load-Displacement Curves.

effect of the presence of end confined wall in a structure (Group I and III) in increasing the drift at ultimate load. On the other the effect of flanged wall participation was smaller (Group II) compared to end confined wall. A significant variation For displacement at 20% strength degradation D0.8u was observed and ranged between 1.37% and 1.78% drift for structure 1 to structure 3 respectively (Group I), while a variation of 1.22–1.47% drift for structure 4 to structure 6 respectively (Group II) was observed and finally a significant variation of 1.82–2.27% drift was observed for structure 7 to structure 10 respectively (Group III). These results showed that increasing the participation of end confined or flanged walls in a structure consisting of rectangular walls (Control 1) did not affect the structure displacement at first yield; however, it significantly increased the attained displacements and drift level prior to any significant loss in lateral resistance of structure. It shall be noted that the NBCC (2005) [28] specifies 1% drift as a drift limit

for post-disaster structures, which are required to operate in full capacity after a seismic event with an insignificant strength degradation, while a 2.5% drift is permitted for structures of normal importance. For ASCE/SEI 7-10 (ASCE/SEI 2010) [26] it specifies a maximum of 1% drift for cantilever shear wall. Regarding structures ultimate capacity, all structures approximately have the same ultimate strength Qu as shown in Table 4 and did not vary significantly as the individual walls were designed to approximately have the same ultimate capacity. For structure secant stiffness, it was observed from Table 4 that all structures have a similar stiffness until the onset of yielding, as the yielding stiffness varies between 40.2% and 36.1% of initial stiffness from structure 1–10 respectively. However, a significant degradation in stiffness was observed at the post peak loading level especially at 20% strength degradation. This observation is clear for structure 1, 2 and 3 (Group I) and structure 7, 8, 9 and 10 (Group III) and as

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Table 4 Structure response. Global Structure Response Top displacements (mm)

Displacement ductility

Secant stiffness (kN/mm)

Dy

Du

D0.8u

mD0.8u

Initial (Kg)

9.8 9.6** 8.0*** 9.6* 9.6** 7.6*** 9.4* 9.6** 7.2***

24.6

55

333

26.4

65.6

34.4

71.2

5.6 5.7 6.8 6.8 6.8 8.6 7.5 7.4 9.8

9.8* 9.6** 7.9*** 9.6* 9.2** 7.6*** 9.4* 9.0** 7.1***

23.8

49

24.6

52

27

59

9* 8.8** 8.8*** 9* 8.8** 8.8*** 9* 8.8** 8.8*** 9* 8.9** 8.8***

43.4

73

56

75

64.8

83

75.6

91

Control 1 (10R)

10

23

Control 2 (10F)

9

Control 3 (10C)

9

GROUP I

Building 1 (8R + 2C)

*

Building 2 (6R + 4C)

Building 3 (4R + 6C)

GROUP II

Building 4 (8R + 2F)

Building 5 (6R + 4F)

Building 6 (4R + 6F)

GROUP III

Building7 (2C + 8F)

Building8 (4C + 6F)

Building 9 (6C + 4F)

Building 10 (8C + 2F)

* ** ***

Ultimate Strength (kN)

At yield (Ky) (%Kg) 134 (40.2%)

At Qu (Ku) (%Kg) 76.5 (23.0%)

At 20% Qu (K0.8u) (%Kg) 28.5 (8.6%)

Qu

350

137 (39.1%)

70.3 (20.1%)

24.3 (6.9%)

1855

367

140 (38.1%)

53 (14.4%)

20.5 (5.6%)

1820

5 5.1 6.2 5.4 5.6 6.8 6.3 6.5 8.3

334

134 (40.1%)

78.8 (23.6%)

31.3 (9.4%)

1876

352

138 (39.2%)

75.2 (21.4%)

29.2 (8.3%)

1850

369

140 (37.9%)

67.1 (18.2%)

25.3 (6.9%)

1811

8.1 8.3 8.3 8.3 8.5 8.5 9.2 9.4 9.4 10.1 10.2 10.3

400

144 (36.0%)

39.1 (9.8%)

18.5 (4.6%)

1696

399.5

144 (36.0%)

30.4 (7.6%)

18.1 (4.5%)

1700

399.5

144 (36.0%)

26.5 (6.6%)

16.5 (4.1%)

1716

399

144 (36.1%)

23 (5.8%)

15.3 (3.8%)

1737

46.5

4.65

314

129.5 (41.2%)

82.1 (26.1%)

33.3 (10.6%)

1891

42

69

7.6

401

142.8 (35.6%)

40.2 (10.0%)

22.1 (5.5%)

1685

85

96.5

10.7

397

143.5 (36.1%)

20.6 (5.1%)

17.2 (4.3%)

1750

1882

Yield displacement calculated based on weighted average method. Yield displacement calculated based on Tangent stiffness method. Yield displacement calculated based on yielding load at 0.6 Qu.

the participation of end confined increases inside the structure, the stiffness in post peak loading (At maximum load (Ku) and at 20% strength degradation (K0.8u)) decreases significantly. It was observed through structures 1, 2 and 3 (Group I) where 20%, 40% and 60% of rectangular walls were replaced by more ductile end-confined walls that the displacement ductility of the structure increased by 18%, 46% and 62% respectively compared to a structure that consists only of rectangular walls (control 1). These results clarify the effect of end confined walls in enhancement the ductility of structure composed of rectangular walls. While for structures 4, 5 and 6 (Group II) replacing 20%, 40% and 60% of rectangular walls by more ductile flanged walls lead to increase in the displacement ductility of the structures by 9%, 18% and 37% respectively compared to a structure consisting only of rectangular walls (control1). These results clarify the effect of flanged walls in enhancement the ductility of structure composed of rectangular walls but not the same influence as the case of more ductile end confined walls in Group I. For structures 7, 8, 9 and 10 (Group III) replacing 20%, 40%, 60% and 80% of flanged walls by more ductile end-confined walls lead to increase in the displacement ductility of the structure by 17%, 20%, 33% and 46% respectively

compared to a structure consisting only of flanged walls (control2). These results clarify the effect of end confined walls in enhancement the ductility of the structure composed of flanged walls. To measure the level of enhancement of structure ductility, a Structure Performance Enhancement Ratio (SPER) is considered and is defined as the ratio between the displacement ductility of a structure with components having different displacement ductility levels (i.e. Rectangular and End-Confined wall system) to that of a structure composed of the less ductile elements (i.e. Rectangular wall). Fig. 15a shows the SPER with respect to the participation percentage of End-confined and Flanged walls. As shown in Fig. 15b the effect of relying on end-confined masonry walls in improving structure displacement ductility is much more significant compared to that of flanged walls. This is due to the endconfined masonry wall having much more ductility capacity compared to flanged walls, as for example the SPER values are 1.55 and 1.25 for 50% participation percentage of end-confined and flanged walls respectively. It was also observed that the response of component wall inside a system level is different compared to the same wall response when tested individually with respect to ductility capacity. This

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449

Fig. 15. Structure enhancement variation, (a) Displacement ductility of buildings, (b) SPER using end confined and flanged walls.

behavior is clear when the system level is composed of different wall types that have different ductility levels. If the system is composed of the same wall type (one type of wall) then no difference will be observed between the component wall inside the structure and the individual wall with respect to ductility (the load–displacement curve will be scaled up according to the number of walls used in the structure). It was observed that the individual wall system i.e. Flanged or end-confined affects and improves the displacement ductility of rectangular wall when both are used in same structure (Group I and II) and this is due to delaying of displacement corresponding to 20% strength degradation D0.8u for the rectangular wall and thus increase the ductility capacity of individual rectangular walls within the system due to redistribution of loads. Also, the yield displacement Dy is not significantly affected in all groups for individual walls. 7.3. Fragility assessment Performance-based seismic assessment is a formal process for design of new structures, or seismic upgrade of existing structures, which includes a specific intent to achieve defined performance objectives in future earthquakes. Performance objectives relate to expectations regarding the amount of damage a building may experience in response to earthquake shaking, and the consequences of that damage. Performance is measured in terms of the probability of incurring casualties, repair and replacement costs, repair time, and unsafe placarding. The methodology and procedures are applicable to new or existing buildings. The concept of Performance-based seismic design was implicitly incorporated in most of the modern seismic codes through limit states. However, such design mainly was attributed to fulfillment of only one performance level aimed to preserve the life safety. In response to the problems attributed to consideration of one performance level, as stated above, several performance levels were considered in Performance-based seismic design. Performance levels were classified to four categories (FEMA 356) [23]: 1. Fully operational: Facilities continue in operation with negligible damage. 2. Immediate occupancy: Facility continues in operation with minor damage. 3. Life safe: Life safety is substantially protected, damage is moderate to extensive. 4. Collapse prevention: Life safety at risk, damage is severe.

One method of evaluating seismic performance of buildings in earthquakes is by using Fragility curves. These diagrams show the probability of exceeding a specific state of damage versus demand parameter. In this section the concept of seismic fragility was used to evaluate the seismic performance of the ten structures previously described to reveal the effect of presence of ductile wall elements (flanged or end confined) on the probability of exceeding a well-known damage state by using the structure drift as the engineering demand parameter. A previous fragility study was carried out [29] to illustrate the enhancement of seismic performance of end confined and flanged walls as individual walls compared to rectangular wall; however the fragility study at the structure level is introduced in this section. Quantification of the anticipated damage patterns and behavior of traditional reinforced masonry structural shear walls is divided into four categories according to a predefined method of repair by FEMA 306 (ATC 1998) [30]: Insignificant, Slight, Moderate and Extreme. These damage states have been identified to coincide with a certain level of remediation (i.e. epoxy injection of cracks) and may be integrated into the formulation of fragility functions as described by the ATC-58-1 (ATC 2011) [31] document. Fragility functions provide a conditional probability that a particular damage sate will occur in a component for a given demand value. The peak top drift (D) of each structure is selected as the demand parameter, and the occurrence of each damage state is assumed to be sequential in nature (i.e. Damage state 2 occurs after Damage State 1). The top drift associated with the first occurrence of each damage state has been determined for each structure using the pushover curve. The ATC-58-1 (ATC 2011) [31] recommends the use of a cumulative probability function based on a log-normal probability distribution for the generation of fragility functions. The lognormal probability distribution function is shown in Eq. (9) and requires determination of the median drift for each damage state (hi) as well as the logarithmic standard deviation (dispersion) (bi) as determined by Eqs. (10) and (11) respectively. The fragility function is represented by a smoothed curve fit to the observed occurrence of each damage state in each structure.

  ln hD i F ðDÞ ¼ Uð Þ bi 1

hi ¼ eðM

PM i¼1

ðln Di ÞÞ

ð9Þ ð10Þ

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vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u  M  u 1 X Di 2 bi ¼ t ln M  1 i¼1 hi

8. Conclusions

ð11Þ

Where:

Di = Top drift; hi = Mean drift of a damage state; bi = logarithmic Standard deviation (Dispersion); M = Total number of walls; F = Log- normal probability distribution; U = Standard normal (Gaussian) cumulative distribution. As shown in Fig. 16 the three structure group I, II and III are having the same vulnerability to the first and second damage states which are considered for initiation of flexure cracks and onset of reinforcement yielding. Consequently, the presence of end confined or flanged walls in these structures did not affect the occurrence of first and second damage states. However, for the third and fourth damage states which are considered for crushing of masonry toe or buckling or fracture of reinforcement, it was observed that the presence of end confined walls in structure group I and III delay the occurrence of both state of damage compared to when flanged walls were used. As Shown in Fig. 16 the 80% probability of occurrence of third damage state was at drift ratio of 0.67%, 0.82% and 1.8% drift for Group II, I and III respectively. While, the 80% probability of occurrence of fourth damage state was at drift ratio of 1.4%, 1.8% and 2.2% drift for Group II, I and III respectively. Consequently, the combination between flanged and end confined walls (group III) represent the best alternative to delay the third and fourth damage state (life safety performance level and collapse prevention) with significant drift level with the preceding structure group II and I, thus resulting in lesser amount of damage at the same drift level.

The aim of this paper was to introduce a reliable and simple numerical model that could be used to simulate the flexural behavior of reinforced fully grouted masonry shear walls, to be used in representing the system level behavior and the overall response of a structure with respect to displacement ductility. This study investigated the behavior of structures composed of different types of walls with different ductility capacities and the factors that influenced the structure overall ductility. Three methods for representing the structure’s yield displacement were also proposed and used in calculating the system displacement ductility. A parametric study was introduced to represent the effect of reinforcement ratio and axial compression on wall behavior. Finally, a set of fragility curves were presented to illustrate the enhancement of the seismic performance (life safety and collapse prevention (performance levels) of masonry structures through adding end confined and flanged walls inside the structure. Based on the results of this study the following conclusions may be drawn:  Yield drift increases by 25% on average with the increase of vertical reinforcement by 225% while it was almost constant corresponding to an increase in axial load by 200%.  Displacement ductility decreases with the increase of axial compression and vertical reinforcement by 57% and 30%, respectively on average for three wall types corresponding to increases of axial load and vertical reinforcement ratios of 200% and 225% respectively.  Strength increases with the increase of axial compression and vertical reinforcement by 18% and 110%, respectively on average for three wall types corresponding to increases of axial load and vertical reinforcement ratios of 200% and 225% respectively.

Fig. 16. Fragility curves for building group 1, 2 and 3.

A. Abdellatif et al. / Ain Shams Engineering Journal 10 (2019) 435–451

 Effect of reinforcement increase on the reduction of the displacement ductility is more significant in rectangular walls compared to flanged and end confined walls.  The displacement ductility of the structures investigated in this study increased by 18%, 46% and 62% when the end confined wall participation increased by 20%, 40% and 60% respectively with rectangular wall combinations, which reflects the effect of end confined wall in enhancement of the structure ductility.  The displacement ductility of the structures increased by 9%, 18% and 37% when the flanged wall participation increased by 20%, 40% and 60% respectively with rectangular wall combination, which reflects the effect of flanged wall in enhancement of the structure ductility.  The effect of end-confined masonry walls in improving structure displacement ductility is more significant than the flanged one and this is due to that the end-confined masonry wall has much more ductility capacity compared to flanged walls.  Using fragility curves, the effect of end confined and flanged wall in enhancement of the structure performance and delaying the damage state appears clearly in the third and fourth damage states. However, they did not have any contribution in enhancement of the first and second damage states.

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