SPRING-TYPE ELEMENTS MODEL FOR NON-LINEAR STATIC ANALYSIS OF MASONRY BUILDINGS

Juan C. JIMÉNEZ1, Ramón G. DRIGO2, Luis G. PUJADES3, Alex H. BARBAT4

ABSTRACT The use of analytical models based on a relatively small number of mechanical and geometrical parameters is very useful to provide a preliminary evaluation of the expected seismic performance and determine whether a more refined approach is necessary. In this context, the aim of this work is to develop a simplified model for nonlinear static analysis of unreinforced brick masonry (URM) walls. The proposed model fits within the equivalent frame approach with one-dimensional members and poses an alternative assembly of macro-elements describing the behavior of piers and spandrels. Such macro-elements are composed by springs: two end springs controlling the flexural behavior and a central spring responding to shear forces. As far as the non-linear modeling is concerned, original force-displacement relationships (FDRs) were implemented in terms of multi-linear curves, in order to incorporate recent advances in the knowledge of strength degradation of the URM members. The proposed model was implemented in the Ruaumoko code and its reliability was tested by means of pushover analyses performed on three walls that have been widely used as benchmarks. The obtained pushover curves were compared with those reported by several authors, showing a good agreement in terms of elastic stiffness, base shear capacity and displacement capacity. In addition, in relation to one of the walls, the precision in the capture of the ultimate damage was discussed. The model implemented by using the Ruaumoko code, resulted simple and effective, and it has still room for improvement, so that this work poses possibilities of refinement, from the identification and discussion of their limitations. Keywords:  unreinforced brick masonry; equivalent frame approach; spring-type macro-element; pushover analysis

1. INTRODUCTION Nonlinear analysis of URM buildings by means of finite elements (micro-scale and meso-scale) demands an important computational effort for most practical problems of seismic assessment. As a result, in the macro-scale level, the equivalent frame (EF) approach has been developing since the late 70’s to simulate the seismic behavior of URM buildings governed by in-plane failure mechanisms. ANDILWall/SAM II (Calliari et al. 2010), RAN code (Augenti et al. 2010), Tremuri (Lagomarsino et al. 2013) and 3DMacro (Caliò et al. 2013) are specialized programs of seismic analysis in URM buildings, which, to a large extent, have incorporated the progress achieved in the EF approach. On the other hand, some researchers have implemented their EF models in general purpose programs such as SAP 2000 (Pasticier et al. 2008; Knox and Ingham 2012) or in software frameworks as OpenSees (Akhaveissy and Abbassi 2014; Raka et al. 2015). On the other hand, in european cities, the regular URM buildings (regularity in terms of distribution of walls in floor plan and regarding the pattern of openings in the walls) are common. In the seismic assessment of this type of buildings, above all if such assessment is of global or preliminary character, both the advanced and the simplified models show, very likely, close outcomes (Calderini et al. 2009b; 1Director of Red Sísmica del Austro, University of Cuenca, Cuenca, Ecuador, juan.jimenez@ucuenca.edu.ec 2Professor, Department of Structural Engineering, UPC, Barcelona, Spain, jose.ramon.gonzalez@upc. 3 Professor, Department of Civil and Environmental Engineering, UPC, Barcelona, Spain, lluis.pujades@upc.edu 4 Professor, Department of Civil and Environmental Engineering, UPC, Barcelona, Spain, alex.barbat@upc.edu

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Pantò et al. 2015). In this context, the aim of this work (part of a project of URM buildings modeling) is the implementation of a one-dimensional model for nonlinear static analysis of URM walls by using the Ruaumoko code (Carr 2007a). Regarding one-dimensional models, the most common are those with frame-type members (Magenes and Della Fontana 1998; Galasco et al. 2006; Pasticier et al. 2008; Knox and Hingham 2012). An alternative to these models are those with spring-type members (Chen et al. 2008; Amadio et al. 2011). In this work, from the review of EF models existing in literature and the study of the types of members available in Ruaumoko (Carr 2007b), the spring-type one-dimensional model proposed by Amadio et al. (2011) was adopted in its basic characteristics, namely: conformation of the members and kinematic model. For strength criteria of the piers, the formulation proposed by Magenes and Calvi (1997) was assumed. This formulation has been adopted by various authors and by the NZSEE code (2006), largely because it considers the boundary conditions of the piers. Regarding strength criteria of the spandrels, due to interlocking phenomena, these members can rely on the contribution of an equivalent tensile strength that influences in their flexural strength and singularize their behavior with respect to the piers. The assumption of most codes to neglect such contribution is too conservative. This is recognized by several authors (Cattari and Lagomarsino 2008; Beyer 2012) and by FEMA 306 (1998). On the other hand, FDRs for piers and spandrels have been typically represented as bilinear curves, assuming an elasto-plastic brittle behavior (Tomaževič 1996; Magenes and Calvi 1997). Recent experimental campaigns on piers and spandrels (Gattesco et al. 2008; Galasco et al. 2010; Beyer and Dazio 2012; Bosiljkov and Kržan 2012) have clarified the strength degradation exhibited by these members, showing that the deformability limits prescribed by several codes are conservative, and the bi-linear FDR, a rough idealization. From the experimental evidence and in the frame of european project PERPETUATE (Lagomarsino and Cattari 2015), significant refinements to these bi-linear curves were proposed (Cattari et al. 2012). Such improvements motivated in this work the definition of the FDRs in terms of multi-linear curves. In order to verify the ability of the PM in analyzing URM structures, pushover analyses on three walls, commonly used as a benchmark, were performed: the well known Door-Wall (DW) and Window-Wall (WW) of the prototype building of the Pavia test (Calvi and Magenes 1994) and the so-called wall D of the building in via Verdi, Catania (Liberatore et al. 2000). The pushover curves obtained herein showed a good agreement with those obtained by other authors. Thus, the PM implemented in Ruaumoko code resulted simple and effective. 2. THE PROPOSED MODEL 2.1 Discretization and kinematic model The EF approach applied in the analysis of URM buildings implies replacing the corresponding walls with equivalent frames (Fig. 1a). This approach was adopted herein using the code Ruaumoko as modeling and seismic analysis tool. Within this framework, a model based on spring-type members was assumed instead of a conventional frame-type model. The frame-type member in Ruaumoko is equipped with a very rigid hysteresis rule to describe the shear behavior of the URM members (Carr 2007c). Conversely, a solution with spring-type members entails the availability of a wide catalog of hysteresis and allows a better description of the nonlinear response to shear forces (Carr 2007c). The spring-type member of Ruaumoko 2D is, actually, a multi-spring member consisting in a package of three types of springs: longitudinal/axial, transverse/shear and rotational/flexural. This multi-spring is a four-node one-dimensional member that comprises two rigid links at the ends and a deformable central part, delimited by two central nodes (Carr, 2007b). Regarding the discretization of the frames in piers and spandrels, obeyed the known criterion of Dolce (1991). The conformation of the members as macro-elements and the kinematic model of the EF approach of Amadio et al. (2011) has been adopted in this work. The macro-element proposed by these authors is the same for both piers and spandrels. As illustrated in Figure 1b, the macro-element includes three elements: two elements at the ends responding to bending, and the central element responding to shear forces. The model proposed by Amadio et al. (2011) was validated comparing obtained results with experimental results at both member level and wall level. This numerical validation was enriched with more examples in Rinaldin et al. (2016).

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Figure 1. Proposed model for unreinforced masonry: (a) Proposed global model, based in Amadio et al. (2011) and (b) Detail of the equivalent frame model with non linear spring macro-elements

2.2 Non-linear modeling of piers and spandrels Failure modes considered for the piers and spandrels of the PM are summarized in Table 1 together with the formulas used to evaluate the corresponding strength capacities. Regarding piers, the formulas proposed in Magenes and Calvi (1997) were adopted herein. With regard to that associated to the diagonal cracking, Calderini et al. (2009a) concluded that the strength formulas based in the model of Mann and Muller, as those posed by Magenes and Calvi (1997), are proper for piers of masonry with important degree of anisotropy. Regarding spandrels, the formula proposed by FEMA 306 (1998) for the strength capacity against bending failure and those proposed by Magenes (2000) and Magenes et al. (2000) for the strength capacities against shear failure modes were adopted. The shear strength capacity of the members (piers and spandrels) was evaluated as the lowest strength capacity associated to the failure modes considered in each case. In the case of the piers: Vup = min.{Vd, Vd,b, Vss}, and in the case of the spandrels: Vusp = min.{Vsp1, Vsp2}, where Vd, Vd,b, Vss, Vsp1 and Vsp2 are defined in Table 1. The axial load interactions were not considered. Thus, the axial load used in the formulas of strength capacity for the piers and the spandrels were the static axial load and the axial load equal to zero, respectively. In the framework of the European project PERPETUATE (Lagomarsino and Cattari 2015), Cattari et al. (2012) proposed multi-linear FDRs able to describe the nonlinear shear and flexural response of piers and spandrels until severe damage levels. These FDRs considered five damage states, characterized by a drift δ and a percentage of degradation β (stepped degradation). Recommended use ranges for these parameters are given in Cattari et al. (2012). Based on these guidelines, and assuming average values for δ and β, referential FDRs were delineated (multi-linear curves with dotted line in Figs. 4 and 5). The FDRs of the PM were established from these referential FDRs, combining bi-linear and tri-linear hysteretic rules with the bi-linear strength degradation law of Ruaumoko (Carr 2007c). The shear FDRs of the PM were built so that the areas under their degrading portions equaled the areas under the degrading portions of referential shear FDRs (multi-linear curves with continuous line in Figures 2a and 3a). In contrast, the flexural FDRs were defined more simply, so that they exhibit only one degradation phase until a residual strength is reached (multi-linear curves with continuous line in Figures 2b and 3b). The tri-linear curves were adopted to take into account the first cracking, assuming a post-cracking stiffness equal to 35% of the elastic stiffness. Regarding FDRs of the spandrels, particular attention has been paid to the experimental work of Gattesco et al. (2008), as their results refers to spandrels that share basic features with those considered in this study: old brick masonry and timber-lintel. Thus, based on that work, a residual strength of 0.4Vmax was adopted for the referential FDRs. In the case of the flexural FDR of the PM (Figure 3b), a linear degradation was adopted as it fits better than the stepped one with the experimental curve obtained by Gattesco et al. (2008).

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Table 1. Considered failure modes and assumed strength capacities

Element Failure modes Strength capacity

Pier (1)

rocking 2

12up

u

D t p pM

f

diagonal-stepped cracking

'; ; ;

1d

d VV

Hc pV Dt c c

D

diagonal-straight cracking , 1

2.3 1bt

d bV bt

f pV D t

f

shear sliding 1.5

; ;1 3ss

V

c pV D t c c

c p

Spandrel (2)

vertical cracking (bending)

1sp eff bj sp eff sj eff bM d f t b f b h NB

(0.75 ) ; 0.75bj sp pilar sjf c f c

2 3 ; 2 ; 2 ; 1 2eff sp eff b sp bd h b l NR NR h h

diagonal-stepped cracking 1sp sp spV h t c

diagonal-straight cracking

'

2 '1 ; ; 0

2.3 (1 ) 2sp spbt

sp sp sp V spv bt sp

p lfV h t p

f h

(1) For piers. D: pier length; t: pier thickness; p = P/(Dt): normal stress; fu: compressive stress of the masonry; λ=0.85: coefficient of transformation to rectangular pattern of stresses; c: cohesion; κ = [1+ μ

(2Δy/Δx)]-1: correction factor proposed by Mann and Müller (1982); Δx, Δy: length and height of brick unit; μ: coefficient of friction; fbt : tensile strength of brick unit; αv: shear ratio; Hd: deformable height (from discretization); ψ’: boundary conditions factor (0.5: constrained rotations at both ends; 1: cantilever). (2) For spandrels. (γsp σpilar) normal stress in horizontal joints at the ends of the spandrel (estimated as a percentage γsp of σpilar); σpilar: average compressive stress of the adjacent piers to the spandrel; deff: effective arm; fbj: shear strength in horizontal joints; fsj: strength from cohesion of the mortar layer between simple walls; tsp: wall thickness; beff: effective width of the brick unit; hsp: height of the spandrel; lb: length of the brick unit; NR: number of horizontal joints of mortar; hb: height of the brick unit; NB: number of simple walls; (NB-1): number of interfaces of mortar between simple walls; η: factor to estimate the average stress on the non-cracked spandrel.

Figure 2. Proposed normalized force-displacement relationships for piers and the curves obtained by Cattari et al. (2012): a) Shear; b) Flexural moment

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Figure 3. Proposed normalized force-displacement relationships for spandrels and the curves obtained by Cattari et al. (2012). a) Shear; b) Flexural moment

3. APPLICATIONS WITH THE PROPOSED MODEL 3.1 Characteristics of the walls and of the performed pushover analysis

The results of tests performed at the University of Pavia by Calvi and Magenes (1994) on a prototype building and the simulations carried out, as part of the Catania project, on a selected wall of the building in via Verdi (Magenes et al. 2000; Brencich et al. 2000; Liberatore and Spera 2000) were used in this study to validate the PM. Thus, pushover analyses on three walls were performed: the Pavia walls (the door-wall: DW, and the window-wall: WW) and the so-called wall D of the building in via Verdi; Table 2 shows their mechanical properties.

Table 2. Mechanical properties corresponding to Pavia walls (Calvi and Magenes 1994) and the wall D of the building in via Verdi, Catania (Magenes et al. 2000)

Pavia-walls Wall “D” – Catania Masonry Young modulus (E) [MPa] 1900 1500 Shear modulus (G) [MPa] 570 150 Compressive strength (fu) [MPa] 6.2 2.4 Shear strength (fvk0) [MPa] 0.2 0.2 Specific weight [kN/m3] 19 19 Brick units Tensile strength (fbt) [MPa] 1.07 2.0 Mortar joints Cohesion [c] [MPa] 0.14 0.20 Friction coefficient () - 0.55 0.50

After adopting the boundary conditions factors ψ' (1.0 for the Pavia walls; 0.5 for the wall D of the building in via Verdi), nine pushover analyses, whose characteristics are summarized in Table 3, were performed using Ruaumoko. Since the diagonal cracking was the predominant failure in the three studied walls (Magenes et al. 1995; Magenes et al. 2000), only the shear FDRs in the piers were conveniently modified. Furthermore, since the reference curves of the walls (including the experimental curves) did not exhibit a well developed degradation, the discussion of results will focus on the drift of the piers associated with the first strength decay (δ1). Those analyses that consider δ1=0.45% (see Table 3) kept the shear FDR in the piers represented in Figure 2a. The curves PM-calib1 and PM-calib2, corresponding to the Pavia walls, were obtained with the aim of fitting the experimental curves (Calvi and Magenes 1994; Magenes et al. 1995) in terms of the displacement associated with the first strength decay. The asterisked PM-calib curves represent the best fit. The other two curves PM-calib were included to illustrate the sensitivity of the displacement of the wall with respect to the drift variation δ1.

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Table 3. Characteristics of the performed pushover analysis

analized wall curve load δ1(%) δ2(%) δ3(%)

door-wall

PM-unif. uniform 0.45 0.60 1.2

PM-1st mode first mode 0.45 0.60 1.2

PM-calib1* uniform 1.2* 1.5 2.5

PM-calib2 uniform 1.0 1.2 2.0

window-wall

PM-unif. uniform 0.45 0.60 1.2

PM-calib1 uniform 1.0 1.2 2.2

PM-calib2* uniform 0.8* 1.0 2.0

“D”-Catania PM-unif. uniform 0.45 0.60 1.2

PM-1st mode first mode 0.45 0.60 1.2

* Drifts δ1 considered in the best fit with the experimental curves of the Pavia walls

3.2 Application with the proposed model: Pavia walls Calvi and Magenes (1994) tested and analyzed a full-scale prototype of a two-story building. This prototype comprised two longitudinal walls: the DW and WW, and two transverse walls. The setup was planned so that the DW and the WW worked independently. Figures 4 and 5 shows their dimensions, global model, nodal masses and the vertical loads on the piers.

Figure 4. Door wall: a) Dimensions; b) Discretization; c) Nodal masses and vertical loads

Figure 5. Window wall: a) Dimensions; b) Discretization; c) Nodal masses and vertical loads

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The modal analysis of the DW provided T1=0.095s for the first mode of vibration. In regard to the results of the pushover analysis, the final state of the FDRs of the springs of the PM corroborated that the stepped diagonal cracking in the piers of the first floor determined the global failure mode (Calvi and Magenes 1994). In the case of the spandrels, however, it was noted that the PM underestimates to some extent the damage presented on the DW (Figures 6a and 6b). Figure 7a shows the four pushover curves obtained with the PM (PM-curves) superimposed over those obtained by other authors using different software codes. Concretely, we included in this figure the curves obtained with the SAM code, (Magenes 2000), Tremuri code (Galasco et al. 2006; Calderini et al. 2009b), and FREMA code (Sabatino and Rizzano 2010). The curve Calvi and Magenes corresponds to the envelope of the experimental cyclic curve obtained by Magenes et al. (1995). It is relevant to note that the Galasco-b curve was obtained by using the failure drifts recommended by the CEN-Eurocode 8 (2005), while the Galasco-a curve was obtained with failure drifts in the piers of 0.8%: shear, and 1%: bending (Galasco et al. 2006). The PM-unif. and the PM-1st mode curves showed low ductility in comparison with the experimental curve. With the aim of a better fit with the experimental curve in terms of displacement capacity, we performed the pushover curves PM-calib1 and PM-calib2, modifying the shear FDRs of the piers (Table 3) and by varying the DUCT parameter (where DUCT = δ/δy). For the best fitted curve, PM-calib 1, a post-cracking stiffness of the 15% of the elastic stiffness, instead of the 35%, was used.

Figure. 6 Door wall: a) Evolution of FDR’s in selected piers and spandrels in the proposed model; b) Crack

pattern at ultimate state

Figure 7. Pavia walls: Comparison of the pushover curves obtained with the proposed model and the obtained ones by other authors: a) Door wall; b) Window wall

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On the other hand, the modal analysis of the WW provided T1=0.11s for the first mode of vibration. Figure 7b shows three pushover curves performed with the PM (PM-curves) superimposed over the experimental curve obtained by Calvi and Magenes (1994), and two pushover curves obtained by Galasco et al. (2006). The curves Galasco-a and Galasco-b were obtained by using elasto-plastic fragile FDRs, considering failure drifts in the piers of 0.8% associated to shear and 1% associated to bending. The PM-unif. curve showed a lower ductility than the corresponding to the experimental curve in both DW and WW walls (Figure 7). With the aim of improving the fit in terms of displacement capacity, we performed the PM-calib 1 and PM-calib 2 (see Table 3). 3.3 Application with the proposed model: wall D – Building at Via Verdi, Catania In the context of the Catania project (Liberatore et al. 2000), the seismic performance of two URM buildings were assessed. The wall considered herein, identified with “D” in Liberatore et al. (2000), belongs to the building located in Via Verdi. Figure 8a shows its dimensions; in Jiménez-Pacheco (2016) can be found the discretization, nodal masses and the vertical loads on the piers. The modal analysis reported T=0.33s for the first mode of vibration. Figure 8b shows the pushover curves obtained herein (PM and PM-1st. mode) and those obtained by Amadio et al. (2011) (Amadio et al.-unif. and Amadio et al.-1st mode), Magenes et al. (2000) (Magenes et al.-GE and Magenes et al.-PZ), Pasticier et al. (2008) (Pasticier et al.-triang.), Brencich et al. (2000) (Brencich et al.-GE) and Liberatore and Spera (2000) (Liberatore and Spera-PZ). The curves in Magenes et al. (2000) were obtained using two lateral load distributions Fi/Ft: GE = {0.220, 0.403, 0.377; Ft = 1110kN }, and PZ =

{0.205, 0.338, 0.457; Ft = 960kN }, proposed by research groups of the universities of Genoa and Basilicata, respectively. Liberatore and Spera (2000) used the PZ-distribution and Pasticier et al. (2008), a triangular distribution.

Figure 8. Wall D – Catania Project: a) Dimensions; b) Comparison of the pushover curves obtained with the proposed model and the obtained ones by other authors

4. DISCUSSION OF RESULTS 4.1 Pavia walls It is pertinent to note two aspects of the Pavia test with a view to establishing conclusions. First: the values of the global drift reported by the Pavia test were unexpectedly high (Magenes et al. 1995). Second: the WW, being weaker than the DW, it was the first to fail (Calvi and Magenes 1994). From this last consideration, it should be expected that the WW, in comparison with the DW, shows a more genuine behavior of independent wall, and its pushover curve, a better fit to the experimental curve obtained in the Pavia test. In this sense, the calibration of the PM in terms of displacement capacity evidenced the most appropriate simulation represented by the pushover analysis of the WW: the fit of the pushover curve with the experimental curve was achieved adopting in this study, as done in

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Galasco et al. (2006), a drift δ1 of 0.8% in the shear FDR of the piers (PM-calib2 and Galasco-b curves in Fig. 7b). Likewise, in relation to the displacement capacity, it is relevant the good approximation that exhibited the PM-unif curves with the Galasco-b (in the DW) and with the Galasco-a (in the WW) in the onset of their degradation (Figs. 7a and 7b). A shear drift of 0.40% in the piers was used to obtain the referred Galasco-curves (Galasco et al. 2006), practically similar to that used to obtain the PM-unif curves (δ1=0.45%). Considering that the Galasco-curves were obtained using the Tremuri program (Galasco et al. 2006), this good agreement reveals the consistency of the PM with respect to the assessment of the displacement capacity. On the other hand, regarding the stiffness and the capacity base shear of the pushover curves, the PM-curves and the reference curves (including the experimental curves) showed acceptable agreement, chiefly in relation to the WW (Figures 7a and 7b). Thus, the curves obtained by Galasco et al. (2006) and the PM-curves exhibit practically the same elastic stiffness in the case of the WW (Figure 7b). In the same way, concerning the base shear capacities, while the best fit of the PM-curves with respect to the experimental curve is that obtained in the case of the WW, in the case of the DW, a small discrepancy of approximately 5% was evidenced (Figures 7a and 7b). With regard to the ability to capture the damage, as mentioned above, the PM underestimated to some extent the damage level in the spandrels. Here, it is important to indicate that in Rinaldin et al. (2016), deepening of the work of Amadio et al. (2011), basic reference for the PM, this problem with the spandrels of the Pavia walls was evidenced also. While it is true that the consideration of the interaction with the axial force would improve the damage control in the spandrels, the authors believe that such underestimation has more to do with an aspect of the setup of the Pavia test, where two steel beams were used to introduce the horizontal forces and where it seems these beams worked as tie-rods. The axial restraint provided by this structural element would entail a better coupling between piers and spandrels and a more uniformly distributed damage among them. This situation was confirmed in Cattari and Beyer (2015), where different modelling assumptions for spandrels in EF-models and their effect on the global response of a URM wall were discussed, adopting the DW of the Pavia test as object of study. 4.2 Wall D – building at via Verdi - Catania project Regarding the pushover curves of the wall D (Catania project), it must be highlighted their sensitivity with respect to the lateral load pattern. The significant variations of the elastic stiffness in the curves obtained by Magenes et al. (2000) and of the base shear capacity in the curves obtained by Amadio et al. (2011) under different lateral load patterns illustrate this fact (Figure 8b). In this context, the most appropriate curves as reference for the PM-curves were those obtained by Amadio et al. (2011) since they were obtained with the same lateral load patterns. Even more, the same formulas for the diagonal cracking strength and almost the same shear failure drift for the piers δ1 (0.40% in Amadio et al., and 0.45% in the PM) were used for these curves. While the comparison exhibited a discrepancy of around of 15% in the elastic stiffness, the base shear capacity and the ductility of the PM-curves and the corresponding curves of Amadio et al. (2011) showed a very good agreement (Figure 8b). In particular, for the curves associated to the lateral load proportional to the first mode of vibration, the base shear capacity is around 470 kN and the displacement capacity (associated to the onset of degradation) is around 24mm, with differences lower than 5% (Figure 8b).

5. CONCLUSIONS AND RECOMMENDATIONS This work explored the ability of the proposed EF model (based on spring-type macro-elements) to develop nonlinear static analysis of URM walls including openings. The pushover curves obtained on the three walls used as benchmark showed good agreement with several pushover curves reported by other authors (included experimental curves). The attainment of pushover curves from multi-linear curves considering the first cracking and the strength degradation according to a bi-linear law would enable a better identification of the damage states and a more reliable seismic assessment of the URM buildings. The consideration, in an integrated manner, of the interactions with the axial loading and the review of the flexural strength of the spandrels are improvements that would equate it with advanced models.

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Another implementation that could be considered, trying to capture more realistically the coupling between piers and spandrels, is the insertion between them of an interaction element instead of the rigid link. Nonetheless, this simplified model gives acceptable results in terms of pushover curves and damage scenarios and Ruaumoko constitutes an effective and reliable tool for the seismic assessment of walls with a regular pattern of openings. 6. ACKNOWLEDGEMENTS AND FUNDING This research has been partially funded by the Ministry of Economy and Competitiveness (Ministerio de Economía y Competitividad-MINECO) of the Spanish Government and by the European Regional Development Fund (Fondo Europeo de Desarrollo Regional-FEDER) of the European Union (UE) through projects referenced as: CGL2011-23621 and CGL2015-65913-P (MINECO/FEDER, UE). Mr. Juan Jiménez Pacheco, sponsored by the University of Cuenca, obtained a scholarship from the Ministry of Higher Education, Science and Technology, SENESCYT, Government of Ecuador, for doctoral studies in the Earthquake Engineering program of the UPC-Barcelona Tech. 7. REFERENCES Akhaveissy AH, Abbassi M (2014). Pushover analysis of unreinforced masonry structures by fiber finite element method. Research in civil and Environmental Engineering 2(3): 96-119.

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