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Engineering Failure Analysis 41 (2014) 48–64

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Engineering Failure Analysis

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Failure analysis of RC shear walls with staggered openingsunder seismic loads

1350-6307/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engfailanal.2013.07.037

⇑ Tel./fax: +40 256226277.E-mail addresses: [email protected], [email protected]

Marius Mosoarca ⇑‘‘Politehnica’’ University Timisoara, 2A Traian Lalescu Street, Timisoara, Romania

a r t i c l e i n f o a b s t r a c t

Article history:Available online 23 August 2013

Keywords:Failure mechanismEarthquakesShear wallsStaggered openingsReinforcement

Reinforced concrete shear walls are used to design buildings located in seismic areas,because of their rigidity, bearing capacity and high ductility. Until now many theoreticaland experimental tests on shear walls with or without openings have been made, thereforetheir failure modes have been analysed and are rather very well-known; the researchresults being confirmed by real failure modes of RC walls after earthquakes.

Design codes and standards based on the knowledge of the failure modes of the rein-forced concrete walls were developed in order to obtain the ductile failure mechanisms.

A special case is the failure mode of the reinforced concrete shear walls with verticalstaggered openings. If at coupled walls the elements must be designed so that the plastichinges appear at the ends of the coupled beams and then in the pier, this thing is more dif-ficult at shear walls with staggered openings.

Theoretical and experimental studies on structural walls with staggered openings, lamel-lar walls and walls with bulbs at the end have been made recently. There have also beenstudied the followings: the degradation of the stiffness, the ductility function to the inten-sity of the seismic force, the presence of the vertical forces, the position and the size of theopenings and the reinforcing ways.

The article presents the results of the theoretical and experimental tests on failure modesof three types of reinforced concrete shear walls with staggered openings which are com-pared to those obtained from walls with vertical ordered openings as far as the seismicresponse is concerned.

The failure modes of the structural walls under seismic stress have been identified usingcalculus programs and cyclic alternated experimental tests.

The theoretical research on the failure modes was the basis for the elaboration of a sim-plified methodology for the calculus of the maximum theoretical seismic force that pro-duces the concrete crushing in the ultimate limit stage. The results theoreticallyobtained with the help of the calculus programs have been confirmed experimentally.The analysis of the failure modes, obtained with the computing methodology proposed,contributed to the completion of the seismic design codes for shear walls with staggeredopenings.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

After earthquakes, the structural elements of the buildings record damages and some of them even fail in various modes.The reports based on the field inspections and the field recordings made present the causes and the failure modes of thebearing elements of the buildings.

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M. Mosoarca / Engineering Failure Analysis 41 (2014) 48–64 49

It is extraordinary to find out that in almost 60 years of research in the field of failure modes of shear walls, there are stilltypes of walls which have not developed failure mechanisms, but only recorded some cracks in certain zones.

Therefore, we think that research has to be conducted on these kinds of structural walls that even after strong repeatedearthquakes have not collapsed, as it is important to predict their potential failure modes.

We think that the structural walls whose failure modes must be predicted by theoretical and experimental research arethe walls with staggered openings.

The reports made after the inspection of 13 buildings with reinforced concrete shear walls, after the earthquake fromChile from 1985, showed that the walls with staggered openings did not collapse, but only recorded minor cracks [1]. Whilethe walls with staggered openings of the ‘‘Torre del Amendral’’ building from Valparaiso, built in 1972, recorded only minorshear compression cracks, the walls with ordered openings of the ‘‘Hanga-Roa’’ building recorded brittle failures, but only atthe level of the coupling beams. In the zone with staggered openings only cracks were recorded in the wall.

We can conclude that these two failure modes indicate the bearing capacity and high rigidity given by the disposition ofthe openings in the reinforced concrete shear walls.

Although these walls recorded minor damages after the earthquake in Chile in 1985, they did not collapse nor even afterthe earthquake in Chile in February 2010 (with a magnitude of 8.8).

Therefore we think that it is necessary to develop design codes for these types of shear walls, which have a very goodseismic behaviour given by the staggered disposition of the openings. It is also important to research all the failure modeswhich these walls can develop, function to various parameters, such as aspect ratio, disposition of the openings andreinforcing.

2. Theoretical and experimental research conducted in the failure domain of the reinforced concrete shear walls withstaggered openings

2.1. Research made from 1985 until 1994 on the failure mode of the reinforced shear walls with staggered openings

The theoretical and experimental research made by Aejaz and Wight [2] on shear walls with reduced percentages of rein-forcement, staggered openings and bulbs at the ends were made function to the values of the a angle. The a angle is definedby the line which connects two consecutive vertical openings and a horizontal line, and has 3 values, i.e.: 32�; 45� and 62�.The experimental studies showed that these walls recorded a ductile failure by reaching the yielding limit in the verticalreinforcement for a total drift of 0.75%, followed by the crushing of the concrete, at the base of the small pier, for valuesof the total drift between 1.25% and 1.5%.

The research made by Yanez et al. [3], Yanez [4] were performed only on 3 shear walls with staggered openings subjectedjust to horizontal forces. The differences consisted in the dimensions and the disposition of the openings. The experimentalmodes recorded a ductile failure by reaching the yielding limit of the vertical reinforcement, followed by the crushing of theconcrete from the base of the small pier.

Thus, the walls with staggered openings develop a ductile failure by reaching the yielding limit of the vertical reinforce-ment, followed by the crushing of the concrete in the small compressed pier. For all the studied walls, after the crushing ofthe concrete at the base of the small pier, the vertical reinforcement buckled very fast.

2.2. Theoretical and experimental research made from 2003 until 2011 on the failure modes of the shear walls with staggeredopenings subjected to seismic loads

Due to the development of a new design and new reinforcing ways for the reinforced concrete shear walls in seismiczones, new failure modes have been recorded. Since 2003, at the Constructions Faculty of Timisoara, the Civil EngineeringDepartment, a new theoretical and experimental research program was started to study these new failure modes. The re-search program was made on shear walls with staggered openings [5], precast [6,7] and composite walls [8–10]. Withinthe program, the behaviour of these walls to seismic actions was studied, after the consolidation with FRP [11,12], until theyreached the failure stage. Since not even after the Chile earthquake of 2010 the walls with staggered openings did not fail, theresearch was continued in order to simulate and study the seismic behaviour [13], the rigidity degradation [14] and also toanalyse the seismic energy dissipation [15]. The research on the failure modes of the buildings in seismic areas was made byGioncu and Mazzolani [16], Anastasiadis [17], and Mosoarca [18] and was supported by the Faculty of Architecture.

2.2.1. Description of the walls studiedThe research was conducted on four types of shear walls with openings: three staggered shear walls: SW23, SW45, SW67,

SW8 (with vertically ordered openings) and one without openings SW1, [5] [18].The walls with openings studied were differentiated by the values of the a angle, as it can be seen in Table 1 and Fig. 1a.

The wall dimensions used as models are shown in Table 2 and the physical and mechanical characteristics of the concreteand of the reinforcements are shown in Table 3. In Fig. 1b, there are shown the percentages of the reinforcements of the wallmodels. Due to the limitations of the test stand, the models were reduced to a 1:4 scale, and we studied the failure modes of4 storey rectangular walls.

Table 1Notation of models.

Seismic load direction Shear walls with openings Shear wall without opening

a = 90� a = 45� a = 32� a = 18�

Left seismic bad (WEST) SW8 SW2 SW4 SW6 SW1Risht seismic load (EAST) SW8 SW3 SW5 SW7 SW1

Fig. 1. (a) Notations of dimensions; (b) zones with different vertical reinforcement percentage and (c) zones with different horizontal reinforcementpercentage.

Table 2Dimensions of the structural wall and of the experimental model.

Dimension Notation Shear wall (mm) Experimental model (mm)

Wall height hw 10,400 2600Wall width lw 5000 1250Wall thickness bw 250 80Storey height hs 2600 650Doors height hd 2000 500Doors width Id 1000 250

Table 3Physical–mechanical characteristics of the r.c. structural shear walls.

Type Rcbar diameter Yield strength Ultimate strength Modulus of elasticity

Rebars 6 mm fy = 0.386 kN/mm2 fu = 0.551 kN/mm2 Ea = 210 kN/mm2

Average tensile strength fctm Average compressive strength fcm Compressive strain Modulus of elasticityConcrete 0.003 kN/mm2 0.005 kN/mm2 3.5‰ Eb = 34 kN/mm2

50 M. Mosoarca / Engineering Failure Analysis 41 (2014) 48–64

On the wall a constant vertical force of 50 kN acted and produced a mean compressive stress of 0.5 N/mm2. A largervertical force was not used in order to avoid loss of stability and to avoid the modification of the failure mechanism.(see Table 4)

The experimental program consists in studying the failure modes of 5 types of walls by a theoretical numerical pushoveranalysis Fig. 2 shows the dimensions and the reinforcement layouts of the experimental models [13–15] [18].

The experimental models were obtained by casting, in a steel formwork, the concrete in the walls and in the foundation,in a horizontal position. After reaching the maximum compressive strength in the concrete, the walls were placed in a ver-tical position on the test stand. The steel formwork was strong enough and hindered the models from cracking while beingplaced on the test stand. It was studied the failure modes of the walls with the same amount of reinforcement, function tothe position of the openings. The experimental models were reinforced with steel wire mesh of 6 mm diameter, disposed on

Table 4Reinforcement percentage of experimental walls.

Wall Vertical reinforcement Horizontal reinforcement

pv1 pv2 pv3 po1 po2 po3 po4

(%) (%) (%) (%) (%) (%) (%)

SW1 0.85 – – 1.01 – –SW2SW3 1.56 2.15 0.31 0.79 1.61 0.31 0.79SW4SW5 1.57 1.57 0.31 0.79 1.61 0.31 0.79SW6SW7 2.15 1.4 0.31 0.79 1.61 0.31 0.79SW8 1.07 0.31 - 0.79 1.61 – –


both sides of the wall. The maximum distance between the vertical bars in the centre of the wall was of 140 mm and be-tween the horizontal bars of 130 mm. The concrete cover layer had a thickness of 15 mm. Supplementary reinforcementmade of 4 bars of 6 mm diameter with closed 90� stirrups was provided at the wall extremities and near the openings.Fig. 2 presents the reinforcement layout during the mounting of the reinforcement in laboratory. All vertical rebars startfrom the foundation block which has the following dimensions: 350 mm � 400 mm � 175 mm. The foundation blocks werereinforced with steel rebars and were provided with pipes through which anchor bolts connected the models to the teststand. Steel profiles were provided at the upper part of the models in order to avoid the local crushing of the concrete atthe application point of the horizontal and vertical loads.

2.2.2. Description of the seismic analysis methods2.2.2.1. Theoretical methodology. The analysis of the behaviour of the reinforced concrete structural walls in the post-elasticdomain was made with the aid of BIOGRAF 02 software [19]. The software performs a 2D nonlinear analysis based on incre-menting the loads introduced by the user. The surface elements used in the nonlinear analysis were of triangular type andwere loaded with forces in their plan. The finite elements were of anisotropic type in plane tension state. The analysis helpedus determine the stress and strain state of the concrete and the reinforcement, as well as the physical state of the concrete(un-cracked, cracked, plasticized, crushed) for each loading step. The dimensions of the triangular finite elements wereestablished function to the position of the reinforcements in the concrete walls.

2.2.2.2. Testing methodology. To understand the failure mechanism of the walls subjected to seismic action all the 5 wall spec-imens erected at scale 1:4 were subjected to quasi-static reversed cyclic horizontal loads. The horizontal forces acted at thesuperior part of the wall specimens. The values of the horizontal displacements imposed on the upper part of all the exper-

Fig. 2. Reinforcement arrangement of experimental models.

Fig. 2 (continued)



imental models led to different values of the horizontal forces applied. These displacements were imposed in an alternatecycle, until the strength of the specimens decreased to 85% of the peak horizontal load [18]. The value of this horizontal forceis noted with P85%. The loading methodology was identical to the experimental research made by Aejaz and Wight [2]. Theseismic behaviour of each wall was studied for 7 values of the horizontal displacement, as it can be seen in Fig. 3. The teststand is presented in Fig. 4.

The measurements were made for each value required by the horizontal displacement at the upper part of the wall givenby two East–West cyclic loads [5]. At every loading cycle, there were recorded all the new cracks, the horizontal and the ver-tical displacements in 10 points (Fig. 4) as well as the stress and the strain state in the concrete and in the reinforcement. Thebehaviour of the experimental specimens was monitored by pressure transducers, displacement transducers, strain gaugesand by optical laser measurements. The position of the strain gauges on the concrete and in the reinforcement is presented inFig. 5. Each cycle was followed by a few minutes stop to record the crack development in the specimens [18].

2.2.3. Types of failure for the experimental modelsUntil the failure stage, all experimental models subjected to an alternating cycle load or the theoretical models subjected

to a pushover force, passed the following stages:

Fig. 3. Number of cycles and values of horizontal imposed displacements.

Fig. 4. The test stand.

Fig. 5. Positioning of the strain gauges on the concrete and in the reinforcements (a) staggered openings specimen and (b) regular openings specimen SW8.

Table 5Values of the horizontal forces and of top horizontal displacement at which concrete cracks in the models.

Model Experimental analysis Theoretical analysis

P Total drift P Dx Total drift(kN) (%) (kN) (mm) (%)

SW23 40 0.05 25.12 0.40 0.17SW45 36 0.05 25.13 0.40 0.17SW67 27 0.05 25.15 0.50 0.21SW8 22 0.05 17.70 0.40 0.17


(a) Elastic behaviour stage – This stage contains all the phases covered by the models from the moment of applying theforce until the development of the first cracks.

The theoretical and experimental analysis indicates that cracks appear in all the walls at similar values for the total driftand for the horizontal forces. In the theoretical analysis, as it can be seen in Table 5, the walls with staggered openings de-velop cracks for a total drift of 0.02%, while the experimental models crack at a value of the total drift of 0.05%. The differenceis given by the type of the loading of the experimental models. The exact data was provided by the software BIOGRAF. Thecomparative values, at which the first cracks appear, are presented in Table 5. The models record the first cracks in threedifferent ways:

(i) Model SW8 records cracks from bending in the coupling beams from the first and second storey.(ii) Models SW23 and SW45 record from bending the first cracks at the base.

(iii) Model SW67 develops the first cracks from bending on the full height of the small pier from the ground and first floor.

For a total drift of 0.15–0.25% the bending cracks appear in model SW8 at the base of the piers, and inclined shear cracksappear between the openings, in the walls with staggered openings. Inclined shear cracks are also recorded at the base and inthe piers of the SW8 model for a drift between 0.25% and 0.35%. In the walls with staggered openings, in the failure stage,horizontal cracks from tension and vertical cracks from compression appear at the base of the small pier for a drift between0.75% and 1%. In SW8 model, after a total drift of 0.35–0.5% no further cracks develop, but the concrete from the couplingbeams is crushed in the direction of the inclined cracks. Subsequently, vertical compression cracks appear at the base ofthe pier [18].

(a) Nonlinear behaviour stage – This stage contains all the phases since the cracking of the concrete until the occurrence ofthe first plastic hinge in the models.

(i) Nonlinear behaviour of the reinforcement

As far as the order of the yielding of the reinforcement is concerned, it was observed that in walls with staggered open-ings, the first rebars to reach yielding were the vertical ones in point 1, followed by the horizontal ones, as shown in Fig. 6.This order was not respected by the coupled wall SW8, where the order of the yielding of the reinforcement was reversedand the wall experienced a brittle mode of failure, due to the shear forces which could have not been overtaken by the rein-forcement in the coupling beams [18]. The values of the horizontal displacements and of the seismic forces at which the rein-forcement of the structural walls yields are presented in Table 6.

Fig. 6. (a and b) Theoretical and experimental failure modes of shear walls SW2-3 and SW4-5 (c and d) Theoretical and experimental failure modes of shearwalls SW6-7 and SW8.


As it can be observed in Table 6, in specimen SW8, the first reinforcements to yield were the horizontal ones, at theextremities of the coupling beam, for a total drift of 0.07–0.08% and were followed by the vertical reinforcement, at the baseof the piers, for a total drift between 0.15% and 0.25%. In the walls with staggered openings, the first yielding was reached by

Fig. 6 (continued)


the vertical reinforcement, at the base of the small piers, for a total drift between 0.10% and 0.23%, followed by the horizontalreinforcement, at the first level, for a total drift between 0.17% and 0.30%. The horizontal reinforcement did not reach yield-ing in walls SW4 and SW5, but the vertical reinforcement of these walls yielded at a very small value of the horizontal force.

Table 6Values of the horizontal forces and of the displacements for which the reinforcement reaches the yielding limit.

Model Theoretical results Experimental results

P Dx Total drift P Dx Total drift(kN) (mm) (%) (kN) (mm) (%)

Vertical rebars yieldingSW23 701.2 5.60 2.33 500 2.40 1.00SW45 661.5 2.30 0.96 410 3.00 1.25SW67 766.5 2.50 1.04 3800 3.00 1.25SW8 572.0 3.70 1.54 600 6.00 2.50Horizontal rebars yieldingSW23 726.2 6.00 2.50 600 5.00 2.08SW45 0.0 0.00 0.00 700 6.00 2.50SW67 781.5 6.30 2.63 710 7.00 2.92SW8 377.0 1.60 0.67 400 2.00 0.83


(ii) Occurrence of plastic hinges.

The experimental models with staggered openings developed plastic hinges in a different way than model SW8. The wallswith staggered openings formed plastic hinges at the base of the small pier, while the model SW8 formed plastic hinges inthe coupling beams and then at the base of the piers. The values of the horizontal displacements and the total drift at whichplastic hinges formed in the concrete section, for constant values of the horizontal forces, are presented in Table 7.

The walls with staggered openings developed plastic hinges at horizontal forces larger than those of the model SW8. InTable 7, it can be observed that the walls with staggered openings develop plastic hinges for a total drift of 0.38% (SW6) to0.57% (SW3), i.e.: for a close disposition of the openings. The reinforcing requirements are not so special in comparison withthe disposition of the openings closer to the edge, as in the case of wall SW6. At the model with ordered openings, SW8, theexperimental tests recorded plastic hinges and the local crushing of the concrete at the ends of the coupling beams for a totaldrift of 0.35% and further development of plastic hinges, at the base of the small pier, for a drift of 0.64%.

(c) Failure stage – This stage corresponds to the crushing of the concrete in the zones with maximum compression andenhanced development of the deformations under constant horizontal loads. For a better understanding of the failure modesdeveloped by these analysed experimental models, there were defined two limit stages [18]:

(i) Limit stage – corresponding to the maximum horizontal force at which it is produced the crushing of the concrete byshear in the coupling beams (SW8) and the crushing of the concrete at the base of the small pier in the models withstaggered openings;

(ii) Failure stage – corresponding to 85% of the maximum horizontal force noted with P85%. At this maximum seismic valuethe concrete is crushed in point 1 and vertical reinforcement buckles in models with staggered openings. Model SW8records concrete crushing in points 1 and 2.

The values of the forces and displacements at which the crushing of the concrete occurs, obtained by theoretical andexperimental analysis, are presented in Table 8.

The experimental results were confirmed by the theoretical research. As it can be seen in Table 8 and Fig. 6, there werenot recorded large differences between the results. The research indicated different failure modes, although the models wereanalysed for the same concrete class and the same amount of reinforcement. Thus, in the limit stage, model SW8 with or-dered openings recorded a brittle failure, by the crushing of the concrete from the coupling beams, before the yielding of the

Table 7Values of the horizontal forces and of the top horizontal displacements for which the concrete reachedplasticisation.

Model Plasticized concrete

P Dx Total drift(kN) (mm) (%)

SW1 113.63 14.00 5.83SW2 95.62 10.00 4.17SW3 100.12 13.70 5.71SW4 88.13 10.90 4.54SW5 88.63 13.00 5.42SW6 88.40 9.00 3.75SW7 94.45 118.80 49.50SW8 69.70 15.30 6.38

Table 8Values of the horizontal forces and of the top horizontal displacements for which the concrete crushes.

Model Theoretical analysis Experimental analysis

Limit stage Failure stage

P Dx Total drift P Dx Total drift P85% Dx Total dnft(kN) (mm) (%) (kN) (mm) (%) (kN) (mm) (%)

SW1 114.43 27.80 11.58 115.00 19.50 8.13 95.00 27.8 11.58SW23 102.62 25.00 10.42 97.50 19.70 8.21 82.00 25.00 10.42SW45 92.03 23.20 9.67 87.50 13.00 5.42 78.00 24.00 10.00SW67 95.60 17.10 7.13 84.00 11.00 4.58 70.00 23.00 9.58SW8 73.80 20.40 8.50 42.50 15.00 6.25 71.00 7.00 2.92


horizontal rebars from these beams. The models with staggered openings failed by the crushing of the concrete, at the base ofthe small pier, by shear-compression (SW23, SW45), and by tension–compression (SW67). The theoretical research indicatedthat the concrete at the base of the small pier at models with staggered openings is crushed for values of the total drift of0.70–1.05%, while the experimental tests proved a total drift of 0.95–1.05% and the crushing of the concrete is produced forsmaller values of the total drift, of 0.6–0.85%. After the concrete from the models with staggered openings is crushed at thebase in the failure stage, the vertical reinforcements buckled. The crushing of the concrete in the coupling beams is reachedat a value of the total drift of 0.25–0.30%.

The maximum seismic force which produced the crushing of the concrete had similar values for both the experimentaland the theoretical models. The concrete in the models with staggered openings crushed at larger values of the seismic loadsthan the concrete of the walls with ordered openings (SW8). Out of all the models with staggered openings, SW6 records thecrushing of the concrete at the smallest value of the seismic force.

Fig. 7. Stress–strain diagram for concrete in point number 1 from Fig. 4 (a) SW23, (b) SW45 (c) SW67 (d) SW8.


2.2.4. Comparative studiesThe failure modes of the shear walls subjected to seismic action were explained based on the recordings made with strain

gauges fixed on the concrete and in the reinforcement. The strains obtained by testing were compared with theoretical onesobtained by the pushover analysis (Figs. 7 and 8). Although the seismic loading was different, there are no great differencesof the values in the failure stage [13–15,18].

2.2.4.1. Strain analysis.(a) Concrete strainSpecial care was given to the study of the concrete strains at the bottom of the experimental modes in points 1 and 2. It

can be seen that by reducing the value of the angle a, the compressive strains decrease in point 2, but the values for the ten-sile strains increase, whereas in point 1 (Fig. 4) an inverse phenomena occurs. This strain state for the concrete in point 2 isconfirmed by the cracks recorded in the failure stage. Models SW23 and SW45 along with the vertical compression cracksdevelop inclined shear cracks and horizontal cracks in the failure stage, while model SW67 develops only vertical compres-sion cracks and horizontal cracks. This type of cracks and strains show the necessity to impose higher percentages for thevertical and horizontal confining rebars for the model SW67 than for the models SW23 and SW45, on the full height ofthe small pier from the base of the walls. By decreasing the value of the angle a, the values of the horizontal forces whichproduce the crushing of the concrete also decrease in point 2, but in point 1 they do not vary that much. The strain gaugesfixed at the ends of the coupling beams for the model SW8, recorded the crushing of the concrete from shear forces, at smal-ler values of the forces and of the horizontal displacements than those which produced the crushing of the concrete at thebase of the piers, in points 1 and 2, at all the other models. In Fig. 7 there is presented in comparison the stress–strain dia-gram for concrete in point number 1.

(a) Strain of the steel rebars

The strains recorded for the vertical rebars in points 1 and 2, shown in Fig. 4 confirm the strain state recorded by thestrain gauges fixed on the concrete. Thus, in point 2 at the greatest value of the seismic force, by decreasing the value of

Fig. 8. Stress–strain diagram for steel vertical rebars in point 1 in Fig. 4 (a) SW23, (b) SW45, (c) SW67, and (d) SW8.


the angle a, the tensile strains increase, recording the following values: esteel max = 2‰ in SW23, esteel max = 3‰ in SW45,esteel max = 4‰ in SW67. In points 1 and 2 of the model SW8, the tensile strain reaches its maximum value esteel max = 2‰ afterreaching the yielding limit of the horizontal rebars in the coupling beams. These experimentally recorded values are pre-sented in Fig. 8.

The maximum strains on the horizontal rebars, at the level of the first floor, recorded experimentally, were confirmed bythe theoretical studies. Research shows the fact that by decreasing the value of angle a, the values of the tensile strains of thehorizontal rebars decreases, recording the following values: esteel max = 3‰ in SW23, esteel max = 2‰ in SW45, esteel max = 1‰ in

Fig. 9. Shear force vs. displacement P–D comparative curves from theoretical analysis and experimental results (a) SW1, (b) SW23, (c) SW45, (d) SW67 and(e) SW8.


SW67. These values of the strains show a higher requirement for horizontal rebars in these zones for 45� > a > 32� and lessrequirement for 32� > a > 18�. At all the models, the greatest value of the tensile strain was recorded in the horizontal rebarsfrom the coupling beams of model SW8, esteel max > 4‰. These strains from the coupling beams were recorded for a smallervalue of the seismic load than those recorded in the models with staggered openings. This confirms the theoretical results ofFig. 6, the occurrence of the plastic hinges at the ends of the coupling beams, developed in the experimental models, andexplains their brittle failure in this zone.

2.2.4.2. Force–displacement analysis. The seismic response of the models was studied by marking the P–D diagrams for eachdirection of seismic action [5,13–15,18]. In Fig. 9 it can be observed that although the theoretical curves were plotted for apushover analysis, and the experimental curves were plotted based on the cyclic alternating loads, there are no major dif-ferences between the recorded values. The numerous cracks developed by the experimental models, have severely reducedthe rigidity of the walls and have reached the failure stage at smaller values of the forces and of the displacements than theones resulted from theoretical analysis. This difference can be observed in the failure stage only for models SW3, SW5 andSW7 when only compressive stresses are recorded at the base of the large pier, in point 1. Similar values of the seismic loadsand displacements in the failure stage were also recorded for models SW2, SW4 and SW6, but only when the compressivestresses were recorded in point 2.

From all the experimental models, the largest degradation of the rigidity in the failure stage was recorded by the modelwith ordered vertical openings SW8, due to the crushing of the concrete in the coupling beams and due to severe cracking ofthe beams and piers. Lacking special reinforcing provisions for the extremities of the coupling beams for overtaking the shearforces, in the failure stage the walls record a horizontal displacement which slightly decreases in time, while the seismic loadcorresponding to this displacement decreases a lot from one cycle to another. Models with staggered openings did not recordsudden rigidity degradations in the failure stage, due to the horizontal rebars from the levels of the floors and due to thesufficient confinement of the compressed concrete section at the bottom of the small pier, in point 2. The best compliancebetween the experimental and theoretical results was recorded by the solid model SW1, without any openings; fact whichconfirms the credibility of the results obtained by both methods.

3. Simplified computational method to determine the maximum seismic load which produces the crushing of concrete

The computation method determines the maximum top horizontal force Pu that produces the crushing of the extremecompressed fibre at the base of the small pier (point 2) for the much reduced unconfined concrete sections. The results werecompared with the values of the ultimate horizontal force resulting from a nonlinear static analysis of biographical type, per-formed with the BIOGRAF software and the results obtained by experimental models [5,13].

Simplified computational hypotheses:

i. Seismic horizontal force is applied at the top of the wall and the resultant gravitational force was applied at the sym-metry axis of the upper wall (Fig. 10).

ii. Computational method disregards the effects of shear forces in the piers.iii. The bending moment at the base of the small pier is very small and is disregarded in computation due to its low

stiffness.

Fig. 10. Scheme for computing the ultimate strength Pu.


iv. The entire section of the concrete is cracked at the base piers on the entire height even since the first steps of the appli-cation of seismic forces. It was considered that the entire section is subject to eccentric tension with lower eccentricityand the sectional strength is provided entirely by the vertical reinforcement.

v. The contribution of the compressed concrete at the bottom of the small pier is neglected because of its small section.vi. The concrete section at the bottom does not require supplementary confinement of the concrete zone.

vii. The tensile strength of the concrete was neglected at computations.viii. The simplified method introduced the contribution of the vertical reinforcing bars to the tension efforts in these areas

to determine the ultimate strength.

The computations were based on the calculus recommendations and on the formulas contained in the design code of rein-forced concrete Romanian STAS 10107/0-90 [20].

Calculation steps:

Step 1: Calculation of the axial force of first piers with the next formula:

N1 ¼Xn

i¼1

AaiRai ð1Þ

From the equation of the vertical projection of forces:

� #X

Fv ¼ 0 : V þ N1 � N2 ¼ 0 ð2Þ

Step 2: Calculation of the axial force N2:

N2 ¼ V þ N1 ð3Þ

Step 3: Determination of Mcap1 corresponding to N1 with the following formula:

Mcap1 ¼Xn

i¼1

AiRih0i � N1h1

2� a0

� �ð4Þ

where hi – distance from the resultant tensile force support on the concrete section to extreme fibre of concrete section(Fig. 10); h0i – the distance from the edge of the support until the axis of each bar, see Fig. 10; Pu – maximum value ofthe top horizontal force.

Step 4: calculating formula for the bending moment capacity in the symmetry axis of the wall:

Mcap1 ¼ PuH � N1a1 � N2a2 ð5Þ

Step 5: Determination of Pu.

Pu ¼1HðMcap1 þ N1a1 þ N2a2Þ ð6Þ

where a1,2 = distances from the symmetry axis of the wall to the point of the application of the base piers sectionalefforts. Example:

Model SW6 : a ¼ 18�M2 ffi 0

N1 ¼ 26� 28:274 mm2 � 0:386kN

mm2 ¼ 28:37 kN ð7Þ

N2 ¼ 50 kNþ 283:75 kN ¼ 333:75 kN ð8Þ

Xn

i¼1

AiRih0i ¼ 2� 28:274 mm2 � 0:386kN

mm2 � 54750 mm ð9Þ

Xn

i¼1

AiRih0i ¼ 119;506 kN mm ð10Þ

N1h1

2� a0

� �¼ 283:75 kNð425 mm� 18 mmÞ ¼ 115;486 kN mm ð11Þ

Mcap1 ¼ 4019 kN mm ð12Þ


Pu simplified method ¼1

2600 mm� ð4019:5 kN mmþ 283:75 kN� 200 mmþ 333:75 kN� 550 mmÞ ¼ 94 kN ð13Þ

Pu pushover:analysis ¼ 95:9 kN ð14Þ

Pu eperimental test ¼ 84 kN ð15Þ

It can be observed, that between the value of horizontal ultimate force Pu, determined with the proposed simplified com-putational method, and the force resulting from the nonlinear biographical analyses and the experimental test, there is avery small difference [13].

4. Conclusions

This article presents a comparison between the failure mode of a wall with regular vertical openings, three walls withstaggered openings and a solid wall, all subjected to seismic loads. Their behaviour is presented in the elastic and post-elasticstage until they reach the failure stage. All these walls have been reinforced with the same amount of rebars, having the samedimensions and the same concrete class. The failure modes determined theoretically by a biographic pushover analysis, werecompared to the experimental failure modes laboratory obtained. The strain gauges provided information about the failuremode of the concrete and of the rebars, and indicated the zones in which the failure occurred.

By comparing the results obtained theoretically and experimentally, the following conclusions about the failure modescan be drawn:

1. The walls with staggered openings are more rigid and have a higher bearing capacity in comparison with the walls withordered openings, with the same amount of reinforcement.

2. The sequence of the occurrence and the distribution of the cracks until the failure stage are different. The first bendingcracks were recorded at the base of the walls, while the walls with regular openings developed cracks at the ends ofthe coupling beams and at the base of the wall. In the failure stage, there are recorded vertical compression cracksand inclined shear cracks at the base of the small pier. Function to the distance between the openings, the small pierat the base of the wall fails by shear – compression at a > 18� and at a < 18� by tension – compression. Despite the lackof some special reinforcing provisions of the coupling beams, for the ultimate limit state, shear cracks appear in thebeams and cracks from compression and bending at the base of the piers.

3. The walls with staggered openings fail in a different way than those with ordered openings. The walls with regular open-ings had a brittle failure by crushing the concrete in the coupling beams due to shear force, followed by yielding of thehorizontal reinforcement in these beams. The walls with staggered openings had a ductile failure by the yielding of thevertical reinforcement at the base of the piers followed by the crushing of the concrete in that zone. In the failure stage,after the crushing of the concrete at the base of the small pier, the vertical compressed rebars buckled right away.

4. Two behaviour modes are observed for the walls with staggered openings. For a ductile failure at high seismic forces, it isrecommended that a should vary between 32� and 45�; in this case, the wall behaves similarly to a truss. For a between18� and 32�, the central pier becomes very rigid and acts like a cantilever. The marginal piers are subjected to tension or tocompression only.

5. By displacing the openings towards the extremities of the walls, the tensile strains in the horizontal rebars decrease, atthe level of the floors, but the tensile strains increase in the vertical rebars and also the compression strains of the con-crete near the edges of the walls increase at the level of the built-in support. We recommend values for angle a of >32�and a denser reinforcing with horizontal and inclined bars in the zones between two consecutive openings. For values ofthe angle a < 32�, it is required a strong reinforcing with vertical rebars in order to: increase the bearing capacity of theconcrete sections subjected to bending, to decrease the distance between the stirrups, to avoid the buckling of the verticalrebars in order to enhance the compressive capacity by confinement.

6. The walls with staggered openings fail at levels of seismic forces and at horizontal displacements higher than the forcesand the horizontal displacements recorded in the failure mode of the walls with regular openings.

7. The proposed computational model is important as the engineers will be able to establish very fast, together with thearchitects, the most advantageous location of staggered openings even in the preliminary design stage. The proposed sim-plified computational model simulates with accuracy the seismic behaviour of the staggered openings shear walls and isable to identify easily the maximum seismic force which produces the crushing of concrete at the bottom of the wall, inthe small piers.

The results of the theoretical analysis described in this article were confirmed by the test results. The failure mode of thewalls with staggered openings from Chile is not well known, that is why we think that further research must be made inorder to create proper design codes for these structural walls.


Acknowledgements

The author would like to thank Prof. Valeriu Stoian, PhD, for his expertise, recommendations, and permanent assistanceprovided during this research. The author would also want to thank Prof. Victor Gioncu, PhD, for the guidance towards thecorrect understanding of the failure modes of the buildings. For the financial support and help in writing the article, theauthor thanks H.I. STRUCT and MAISON STYLE design office.

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