eqe2195

Seismic capacity of irregular unreinforced masonry walls withopenings

Fulvio Parisi*,† and Nicola Augenti

Department of Structural Engineering, University of Naples Federico II, Naples, Italy

SUMMARY

Masonry buildings are often characterized by geometric irregularities. In many cases, such buildings meetglobal regularity requirements provided by seismic codes, but they are composed by irregular walls withopenings. The latter are masonry walls characterized by (i) openings of different sizes, (ii) openingsmisaligned in the horizontal and/or vertical direction, or (iii) a variable number of openings per story. Anirregular layout of openings can induce not only a nonuniform distribution of gravity loads among masonrypiers but also unfavorable damage localizations resulting in a premature collapse of the wall and hence ahigher seismic vulnerability.

This paper is aimed at providing a simplified methodology to assess the effects of irregularities on the in-plane seismic capacity of unreinforced masonry (URM) walls with openings. To this end, a macroelementmethod was developed and validated through experimental results available in the literature.

The proposed methodology was based on the quantification of wall irregularities by means of geometricindices and their effects on seismic capacity of URM walls with openings through both sensitivity andregression analyses. Sensitivity analysis was based on a high number of static pushover analyses andallowed to assess variations in key seismic capacity parameters. Regression analysis let to describe eachcapacity parameter under varying irregularity index, providing empirical models for seismic assessment ofirregular URM walls with openings. The in-plane seismic capacity was found to be significantly affectedby wall irregularities, especially in the case of openings with different heights. Copyright © 2012 JohnWiley & Sons, Ltd.

Received 21 October 2011; Revised 19 March 2012; Accepted 21 March 2012

KEY WORDS: masonry buildings; structural irregularity; unreinforced masonry walls with openings;macroelement modeling; seismic capacity; irregularity indices

1. INTRODUCTION

Structural irregularities significantly affect the seismic performance of single masonry walls withopenings and thus of an entire building. Existing masonry buildings often satisfy regularity criteriaaccording to seismic codes, even though their structure is composed by irregular walls withopenings. Opposed to their regular counterparts, irregular walls have openings misaligned in thevertical and/or horizontal directions (Figure 1a–c), or different number of openings per story(Figure 1d). Such irregularities can be induced by the lack of conceptual seismic design or even bythe creation/closure of openings motivated by architectural or structural purposes (e.g., modificationof stiffness/strength distribution in plan, change of axial forces in some masonry piers). Irregularwalls can also be identified in newly designed masonry buildings because of the presence ofopenings at intermediate staircase landings. It is underlined that existing walls have often openings

*Correspondence to: Fulvio Parisi, Department of Structural Engineering, University of Naples Federico II, via Claudio 21,80125 Naples, Italy.

†E-mail: [email protected].

Copyright © 2012 John Wiley & Sons, Ltd.

EARTHQUAKE ENGINEERING & STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2013; 42:101–121Published online 16 April 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/eqe.2195

with parapets. If their thickness is significantly lower than that of adjoining piers and underlyingspandrels, and no interlocking is ensured with the surrounding masonry, parapets are to beconsidered as nonstructural elements that do not contribute to the seismic capacity of the wall. Aperforated masonry wall should then be classified as regular or irregular on the basis of the actualarrangement of openings from a structural viewpoint, that is, neglecting nonstructural parapets.

Irregularities are frequently identified in the case of inner walls, even in the presence of regularperipheral walls. An irregular layout of openings induces not only a nonuniform distribution ofgravity loads among masonry panels but also a concentration of seismic strength and drift demandsin some parts of the wall. These factors can lead to unfavorable damage concentrations increasingthe seismic vulnerability of the entire wall, as shown by past experimental tests [1–3] and on-siteinspections after strong earthquakes, such as the 2002 Molise, Italy, earthquake [4] and the 2009L’Aquila, Italy, earthquake (Figure 2).

In other words, geometric irregularities can significantly affect the seismic response of unreinforcedmasonry (URM) walls with openings because on the one hand, the damage is not affected but

Figure 1. Façade walls with openings misaligned in (a) vertical direction, (b) horizontal direction, (c) bothvertical and horizontal directions; and (d) façade wall with different number of openings per story.

Figure 2. Damage concentration at the location of significant vertical irregularity in a masonry wall withopenings (2009 L’Aquila earthquake).

102 F. PARISI AND N. AUGENTI

Copyright © 2012 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2013; 42:101–121DOI: 10.1002/eqe

potentiated by irregularities, whereas on the other hand, the damage is a feature of the seismicresponse. Reacting to this problem, the Italian Building Code (IBC) [5] states that seismic design ofmasonry buildings should be carried out by providing openings aligned along the height of eachwall (Note that no horizontal alignment is specified). Otherwise, each perforated wall should becarefully modeled and assessed, but no specific criteria are given to that aim. For the sake ofsimplicity, IBC allows to consider just continuous masonry piers from the foundation system to thetop of the building. This rule is mentioned in Eurocode 8 (EC8) [6] in the part relating to seismicdesign of simple masonry buildings. It is stressed that IBC, EC8, and also FEMA 356 [7] do notdeal with the effects of ‘horizontal irregularities’, namely the presence of openings with differentheights (i.e., doors and windows) at one or more stories, but this is clearly of concern for structuralmodeling of walls.

Seismic response of masonry buildings can be adequately simulated only if a good knowledge of thenonlinear behavior of individual walls with openings has been reached. In this way, this paper is aimedat investigating the influence of both vertical and horizontal irregularities to the seismic capacity ofURM walls with openings. To this end, a macroelement method is used and was experimentallyvalidated with reference to an irregular wall. A methodology is proposed to classify and quantifywall irregularities on the basis of a series of indices. A sensitivity analysis was carried out to capturethe main effects of irregularities on the seismic capacity of walls with openings. Finally, regressionanalysis was performed to derive empirical relationships between capacity parameters andirregularity indices. Improving knowledge about the effects of wall geometry could be helpful forengineers in both seismic assessment and retrofit of URM buildings and also for design of newmasonry structures.

2. MACROELEMENT MODELING OF UNREINFORCED MASONRY WALLS WITHOPENINGS

When dealing with standard masonry buildings, namely those composed by an assemblage of walls(with or without openings) and floor diaphragms, the first challenge is to build up a capacity modelable to reflect the effective parts of each wall that provide earthquake resistance against in-planelateral actions distributed by floors. This objective can be reached through macroelement modeling,which allows to overcome relevant difficulties of numerical modeling techniques.

2.1. Macroelement modeling and failure modes

Macroelement methods [e.g., 8–10] are widely used to assess seismic performance of masonrybuildings. They are based on an equivalent frame idealization of each wall with openings where thefollowing macroelements are identified: (i) pier panels, which are the vertical structural components;(ii) spandrel panels, which are the horizontal structural elements acting as coupling componentsbetween piers under horizontal actions; and (iii) joint panels (or cross panels), which are rigidelements connecting pier and spandrel panels. Joint panels are geometrically defined as intersectionbetween spandrels and piers, namely horizontal and vertical masonry strips between twoconsecutive series of openings, respectively (Figure 3a–c).

Figure 3. Identification of (a) piers, (b) spandrels, and (c) macroelements in a regular masonry wall with openings.

SEISMIC CAPACITY OF IRREGULAR UNREINFORCED MASONRY WALLS WITH OPENINGS 103


The failure mode of a flexible macroelement depends on its aspect ratio, the relative strengthbetween masonry components (i.e., masonry units, mortar joints, and their interface), the appliedaxial force, and the boundary conditions [11]. The failure mechanisms are usually defined as follows:

(1) Rocking failure: As the lateral force or displacement demand increases, the masonry panel suf-fers tensile cracking at the end sections (horizontal cracks in Figure 4a) and experiences rigid-body (rocking) rotation until masonry crushing is reached at the compressed toes (vertical cracksin Figure 4a).

(2) Diagonal tension cracking: Such a failure mode consists in the formation of a single diagonalcrack that typically involves both mortar joints and masonry units (Figure 4b).

(3) Shear sliding: This failure mode can develop along a single mortar bed joint (Figure 4c) or bothbed and head joints in a stepwise way (Figure 4d).

Both the in-plane lateral strength and displacement capacity of a macroelement are conditioned bythe dominant failure mode that corresponds to the minimum lateral strength Vu. In addition to theboundary conditions and the magnitude of the axial force applied to the macroelement, theaforementioned crack patterns are mainly governed by the following mechanical properties ofmasonry: (i) the tensile and compressive strengths, as well as the corresponding cracking andultimate axial strains, in the case of rocking failure; (ii) the diagonal shear strength at zero confiningstress (which can be associated with the tensile strength of masonry) in the case of diagonal tensioncracking; and (iii) the sliding shear strength at zero confining stress and the friction coefficient in thecase of shear sliding failure. The length and amplitude of cracks in any failure mode are alsoaffected by the stress–strain behavior of masonry in tension, compression, and shear, which isquantified by means of secant Young’s and shear moduli in simplified linear equivalent analyses.

2.2. Lateral strength and displacement capacity of macroelements

Let us denote as fd,m the nominal compressive strength of masonry, and l and t the length and thicknessof the URM macroelement. If an elastic–perfectly plastic stress–strain model is assumed for masonry,fd,m can be set to 0.85 times the actual peak compressive strength to account for a nonlineardistribution of axial strains over each cross section. Otherwise, if masonry is assumed to have nonlinearstress–strain behavior, fd,m can represent the actual peak compressive strength. For the sake ofsimplicity, in this study, we consider an elastic–perfectly plastic constitutive law for masonry so that theultimate axial force of an URM cross section having a depth d is defined as Nu = fd,m dt. Masonry is alsoregarded as an equivalent homogeneous, no tensile resistant material. Nevertheless, other constitutivelaws could be used to account for more real characteristics of masonry mechanics and their influenceon lateral strength.

According to the stress block criterion for flexural ultimate limit state, the nominal lateral strengthcorresponding to toe crushing (rocking failure) can be evaluated through the following equation:

Vtc ¼ d

2l01� �Ndð ÞNd (1)

Nd being the applied axial force, �Nd (=Nd /Nu) its value normalized to the ultimate axial force, and l0 thedistance between the section where flexural capacity is attained and the contraflexure point. The latterchanges with boundary conditions l0 ranging between 0.5l and l from fixed–fixed to fixed–free

Figure 4. Failure modes by (a) rocking failure, (b) diagonal tension cracking, (c) bed-joint sliding, and (d)stair-stepped sliding.



conditions. The nominal lateral strength corresponding to diagonal tension cracking is computedaccording to the failure criterion proposed by Turnšek and Čačovič [12] as follows:

Vdt ¼ bNu

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

�Nd

pb

s(2)

where b is the ratio between diagonal shear strength at zero confining stress and uniaxial compressivestrength of masonry and p is a shear stress distribution factor related to the aspect ratio of themacroelement. The nominal lateral strength corresponding to stair-stepped diagonal sliding can beevaluated via the Mohr–Coulomb friction law as follows:

Vsd ¼ 1p

gþ ma �Ndð ÞNu (3)

where g is the ratio between sliding shear strength at zero confining stress and uniaxial compressivestrength of masonry and ma is a fictitious friction coefficient of the whole masonry that is assumedequal to 0.4 by Eurocode 6 [13] and IBC [5]. The term ‘fictitious’ is motivated by the fact that stair-stepped sliding of masonry is a failure mode that locally consists in the combination of (i) tensilefailure of head joints or head joint-masonry unit interfaces and of (ii) shear failure of bed joints orbond slip over the bed joint-masonry unit interfaces. It is also underlined that ma has not the samevalue for any type of masonry and typically falls in the interval [0.3,0.8]. Finally, the nominal lateral

strength corresponding to bed-joint sliding can be computed by assuming ma ¼ 0:17=ffiffiffiffiffiffiffiffi�Nd

23p

intoEquation (3) [14]. This friction coefficient reduces as the confining pressure increases and has alocal meaning given that it is related to a mortar bed joint. Therefore, Equation (3) can be rewrittenas follows:

Vbj ¼ 1p

gþ 0:17ffiffiffiffiffiffi�Nd

3p

ÞNu

�(4)

Equal values are typically adopted for b and g because the current building codes refer to a uniqueshear strength at zero confining stress, denoted as t0, regardless of the particular failure mode (see, forinstance, [5]). Therefore, the lateral strength of the macroelement is defined as follows:

Vu ¼ min Vtc;Vdt;Vsd;Vbj� �

(5)

Limit strength domains corresponding to the aforementioned failure modes for b= g= 0.02 areplotted in Figure 5a and b for squat and slender doubly fixed macroelements, respectively.

Such domains show that diagonal tension cracking is a very limiting failure mode for both squat andslender macroelements. In particular, the squat macroelement is prone to suffer diagonal tensioncracking if 0.03⩽ �Nd ⩽ 0.85 and rocking failure for any other axial load value. The slender

Figure 5. Limit strength domains of (a) squat macroelement and (b) slender macroelement under fixed–fixedconditions.



macroelement can suffer stair-stepped sliding if 0.03⩽ �Nd ⩽ 0.18, diagonal tension cracking if 0.18⩽ �Nd ⩽ 0.65, bed-joint sliding if 0.65⩽ �Nd ⩽ 0.77, and rocking failure for other axial load values.

The lateral behavior of each macroelement is defined by force–displacement curves including bothflexural and shear contributions (see [10] for details on the algorithm). The flexural contribution iscomputed by means of a deformation-based approach where (i) sectional axial strains {ez,i(x,z)} aremonotonically increased by assuming Euler–Bernoulli hypothesis and (ii) the geometry of theeffective macroelement domain is updated by removing the cracked part subjected to tensile strainsaccording to the no tensile resistant model assumed for masonry. The effective macroelementdomain is defined as the fraction of total macroelement volume that provides lateral resistance undera given displacement, according to the constitutive model assumed for masonry.

At each maximum axial strain level efz;max, the corresponding flexural displacement df,k and lateralforce Vk are computed for the macroelement. The flexural displacement is derived by doubleintegration of axial strains over the effective macroelement domain, whereas the lateral force iscomputed by single integration of equilibrium equations at the end sections accounting for theapplied axial force. The shear contribution ds,k is then computed at the same lateral force levelthrough a stress-based, closed-form equation derived from single integration of shearing strains{gx,i(z)} over the effective macroelement domain. Ultimate displacement capacity du of themacroelement can be defined in compliance with IBC [5] or EC8 [15]; alternatively, du can bedefined as the lateral displacement corresponding to the attainment of the ultimate axial strain eu atan end section. In the latter way, one can directly include the influence of the specific masonrymechanics on the macroscopic behavior of the macroelement. Figure 6 shows the evolution in thehalf effective macroelement domain under increasing lateral drift, whereas Figure 7 gives arepresentation of the force–displacement curve of a URM macroelement failing in flexure.

Figure 6. Macroelement evolution under increasing deformation demand: (a) linear elastic state, (b)nonlinear elastic state, and (c) elastoplastic state.

Figure 7. Derivation of force–displacement behavior of a macroelement.



Such a curve consists of four branches as follows:

(1) The first branch is associated with an uncracked macroelement in elastic conditions (Note thatonly material nonlinearity at small deformations affects the flexural strength). The effectivedomain Ω has the gross macroelement geometry (Figure 6a).

(2) The second branch describes the elastic force–displacement behavior in the presence of bothmaterial and geometric nonlinearities, the latter caused by masonry cracking in tension. Theeffective macroelement domain is considered as the union of the subregions Ω1 and Ω2 relatedto the uncracked and cracked parts, respectively (Figure 6b).

(3) The last two branches simulate the inelastic force–displacement behavior in cracked conditions(Figure 6c). After ez,max reaches the cracking strain of masonry ecr at the end section, a furtherregion, denoted by Ω3, develops resulting in a hardened behavior of the macroelement (i.e.,the third branch in Figure 7). If strain softening is taken into account in the constitutivemodel of masonry, after the end section reaches a crossing inelastic axial strain, the masonrystrength degradation causes a gradual lateral strength drop and hence the softening branch inFigure 7.

3. DEFINITION OF PIER EFFECTIVE HEIGHT IN IRREGULAR WALLS WITH OPENINGS

When assessing a masonry building under earthquake actions through macroelement methods [8–10],irregularities in the arrangement of openings should be identified and included in the capacitymodel. As widely recognized in the literature, a key modeling parameter is the effective heightheff to be assumed for piers. FEMA 356 [7] allows to perform pushover analysis of perforatedURM walls under the ‘strong spandrel–weak pier’ assumption, taking into account the effect ofspandrels only in terms of boundary conditions of piers (fixed–fixed or fixed–free piers) anddefining the effective height of each pier as the height of adjacent openings. This assumption does nottypically apply to existing URM walls because they have weak spandrels without reinforcedconcrete (RC) bond beams and well-anchored lintels. For such a case, past recommendations[16] have proposed to assume heff as the interstory height for a better reproduction ofuncoupled piers. It is also stressed that the FEMA 356 approach fails in the case of coupledwalls with different opening heights even if spandrels can be considered as strong elements.Kingsley [17] recognized the need for an accurate definition of pier effective height in wallswith openings of different sizes.

A specific modeling strategy for irregular walls with openings was proposed by Augenti [18], on thebasis of visual observations of past earthquake damages to residential and school buildings [4]. Thatstrategy was also confirmed by on-site inspections on cultural heritage buildings destroyed by the2009 L’Aquila, Italy, earthquake [19]. The cyclic nature of the earthquake ground motion mayinduce nonclassical failure patterns in piers, depending on the height of adjoining openings. Figure 8

Figure 8. Shear failure of pier in unreinforced masonry wall with different opening heights (2002 Molise,Italy, earthquake).



shows large diagonal shear cracks that could be associated with two different effective heights of thepier. As a result, two different capacity models could be assumed for perforated walls with differentopening heights at the same story, depending on the orientation of seismic forces (Figure 9a–d). Inparticular, the effective height could be considered equal to that of the opening that follows the pierin the same orientation of the seismic force.

A similar modeling rule was independently proposed by Moon et al. [16] after a simulation of quasi-static lateral loading tests on a full-scale two-story URM building [2]. That rule is related to rockingfailure of piers rather than diagonal shear cracking because the effective height of a pier is definedas the height over which a compression strut is likely to develop. This definition can lead to thesame effective height according to Augenti [18] and to a pier geometry, which changes with theloading orientation. It is noted that most of piers of the model tested by Yi et al. [2] suffered stair-stepped cracks due to rocking, which propagated at an angle up to 30–45� from opening corners.Similar damage patterns were observed by Paquette and Bruneau [1] on a single-story URMbuilding with flexible diaphragm, which was subjected to pseudo-dynamic tests under increasingseismic motion intensity.

Regardless of the particular failure mode, Dolce [20] proposed a simplified formula to definethe effective height of piers, on the basis of a series of finite element analyses on 20 differentpier–spandrel subassemblages. In the case of end piers, Dolce found an upper bound slope of30� for segments simulating masonry cracks that start at the right or left corner of the adjacentopenings and propagate toward the opposite pier edges. This modeling rule was also confirmedby experimental tests in the case of piers failed in flexure rather than shear [21].

Bothara et al. [3] investigated the seismic response of a half scale, two-story brick masonry housewith timber floors by means of shaking table tests, under increasing earthquake ground motionintensity. Longitudinal walls of the test model had different types of irregularities in the arrangementof their openings. Some end piers significantly rocked and suffered oblique cracks starting fromopening corners. It is worth noting that the rule suggested by Dolce [20] could also be applied toregular walls with openings, as confirmed by damage to transverse walls of the model tested byBothara et al. [3].

On the basis of such a literature review, the macroelement modeling of a masonry wall withopenings should be carried out by taking into account the layout of openings and the orientation ofseismic forces. Regardless of the failure modes of piers, which depend on the expected axial forcelevels within them, irregular walls with openings can be modeled according to one of the modelingrules discussed previously.

Figure 9. Diagonal cracking of piers and macroelement modeling of unreinforced masonry wall withdifferent opening heights for (a,b) rightward and (c,d) leftward orientations of seismic action.



4. STATIC PUSHOVER ANALYSIS OF UNREINFORCED MASONRY WALLS WITHOPENINGS AND EXPERIMENTAL VALIDATION

Because masonry structures exhibit nonlinear behavior even under small displacement demandscorresponding to low-intensity earthquakes, linear analysis procedures are not able to simulate theirseismic response. Static pushover (SPO) analysis can then provide more reliable response predictionsfor performance-based assessment of masonry structures, ensuring a limited computational workcompared with that of nonlinear dynamic analysis. In the following sections, an SPO procedure isproposed and experimentally validated for irregular masonry walls with openings.

4.1. Static pushover procedure

On the basis of the macroelement modeling rules discussed previously [16, 18, 20], an individual URMwall with openings can be regarded as an equivalent frame composed by two-node elements with axial,flexural, and shear flexibilities (i.e., pier and spandrel panels), connected by rigid elements (i.e., jointpanels). Each flexible macroelement is affected by both mechanical and geometric nonlinearities.In the case of irregular walls, two different capacity models are developed for both orientations ofin-plane lateral actions. A force-based SPO analysis in response control can then be carried out foreach lateral load pattern separately (e.g., proportional to inertia masses or fundamental mode shapemultiplied by inertia masses), according to IBC [5] and EC8 [6]. The force–displacement curve ofeach macroelement is step-by-step updated accounting for the current axial force due to both gravityand lateral loads and the axial force–shear force interaction. The latter is simulated by means of thelimit strength domains discussed in Section 2.2.

In the procedure described in the succeeding texts, subscripts i and j indicate, respectively, thehorizontal and vertical strips at which a macroelement or opening belongs to, whereas superscripts jand k in round parentheses denote an iteration and displacement step, respectively. Horizontal andvertical strips of the URM wall are progressively numbered from the top to the base and from theleft to the right, respectively. The lateral force profile is defined by the vector F0 = (1. . .bi. . .bn)

T,where one can set the force distribution factor at the ith floor level to bi=mifi/m1 for ‘modal’ loadpattern and bi=mi/m1 for ‘uniform’ load pattern (Note that mi and fi are, respectively, the mass andmodal shape components at the ith floor level, whereas m1 is the mass at the top floor).

After inertia masses and initial stiffness of the structure have been defined, vertical loading isapplied and then SPO is carried out by keeping constant F0 and monotonically increasing the lateraldisplacement at the control point. Thus, the lateral force vector F(k) = (F1

(k). . .Fi(k). . .Fn

(k))T at the kthanalysis step is computed as follows:

F kð Þ ¼ l kð ÞF0 (6)

where l(k) is the load multiplier derived by a total updating strategy. At the jth iteration of the kthanalysis step, the displacement profile can be assumed equal to the force profile, namely

D j;kð Þ0 ¼ F0 (7)

whereD j;kð Þ0 ¼ 1 . . .o j;kð Þ

i . . .o j;kð Þn

� �Tincludes displacement distribution factors. Therefore, the lateral

displacement vector is computed as

D j;kð Þ ¼ d kð Þc D j;kð Þ

0 (8)

where

d kð Þc ¼ d k�1ð Þ

c þ ddc (9)

is the lateral displacement at the control point. The increment ddc can be kept constant during SPOanalysis, or it can be changed through a specific variation law. A series of iterations are needed to



predict the deformed shape of the URM wall corresponding at d kð Þc and F0; frame analysis is performed

for each iteration to compute horizontal forces resulting from a given trial displacement profile D j;kð Þ0 .

Force convergence is believed to be reached if the following criteria are met:

8i F j;kð Þi =F j;kð Þ

1 � bibi

��⩽tolb

j < jmax

(10)

that is, the resisting force profiles meet the load pattern defined at the beginning of SPO analysis, andthe maximum number of iterations is not attained. If Equation (10) is satisfied, the resisting base shearis computed, and a point of SPO curve can be plotted. Otherwise, the assumed displacement profile isadjusted as follows:

o j;kð Þi ¼ o j�1;kð Þ

ibi

F j�1;kð Þi =F j�1;kð Þ

1

(11)

and frame analysis is repeated until Equation (10) is satisfied. In this case, the l(k) value correspondingto the analysis step under consideration can be defined. SPO is stopped when all piers minus one havereached their ultimate displacement capacity or numerical stability is lost. This means that the SPOcurve is typically characterized by a post-peak, stair-stepped falling branch where each lateralstrength drop is associated with the collapse of a macroelement.

4.2. Experimental validation

In order to perform a simplified assessment of irregular URM walls with openings, the SPO procedurepresented in Section 4.1 was validated through experimental data provided by Paquette and Bruneau[1]. Such researchers carried out pseudo-dynamic tests on a single-story, full-scale URM buildingunder increasing seismic motion intensity. The building was approximately 4.1� 5.7m2 in plan and2.7m high and was composed by four two-wythe solid brick masonry walls with thickness of190mm. Two walls had two openings with different heights (namely a window and a door); andhence, they are considered as ‘irregular walls’ in this study. The other two walls had no openings.The walls with openings had the same geometry. All walls were realized on an RC foundation that,in turn, was anchored to the laboratory strong floor through four high-strength steel bars. Themasonry above each opening was supported by RC lintels with anchorage length of about 150mmat both ends, which is a typical length of existing masonry buildings [22]. Two 10-mm vertical gapswere left between one of the walls with openings and the others. Such a construction solution wasadopted to assess the discrepancy between response predictions from individual wall modelsconsidered by many engineers and the actual three-dimensional response. The roof was composedby 10 wood joists with 38� 286-mm2 rectangular cross section and 406-mm spacing. Such joistswere covered with diagonal and straight sheating overlays made of 19� 140-mm2 boards. The roofwas then connected to the walls by means of 19-mm diameter through-wall bolts and steel bearingplates 150� 150� 6mm3 in size. A masonry parapet was built on the wood roof along the entireperimeter of the test building, and water containers were placed on the roof to simulate a 2.4 kN/m2

live load. The two-wythe solid brick walls were laid in running and American bond with a headercourse at every six courses, tying the two wythes together. The masonry was the assemblage of90� 57� 190-mm3 bricks and mortar with cement : lime : sand in 1:2:9 proportion simulating oldmortar types. The following mean values of mechanical properties were obtained for that masonry:uniaxial compressive strength fm= 22.20MPa, tensile strength ft = 0.18MPa, bed-joint slidingstrength at zero confining stress t0,bj= 0.52MPa, diagonal shear strength at zero confining stresst0,dt= 0.42MPa, and Young’s modulus Em= 18870MPa (i.e., Em= 850 fm). It is emphasized that theaforementioned values are significantly higher than the typical values of old masonry types. Forsolid brick masonry, the IBC Commentary [23] provides fm between 2.40 and 4.00MPa and shearstrength at zero confining stress t0 between 0.060 and 0.092MPa. Such values can be amplified by



1.5 and 1.3 to account for good quality characteristics of mortar and transverse connection betweenwythes, respectively, resulting in fm,max = 7.80MPa and t0,max = 0.179MPa.

The pseudo-dynamic tests were carried out by using a synthetic acceleration time history with peakground acceleration equal to 0.453 g as input ground motion. That acceleration time history wasgenerated for La Malbaie, Canada to be representative of Eastern North America seismicity. Thetests were performed under La Malbaie time history multiplied by 0.25, 0.5, 1.5, and 2 to assess thebuilding performance at several ground motion intensities. Figure 10a shows the irregular wall ofthe building used as reference for the experimental validation. The horizontal force corresponding tothe input ground motion was applied by a hydraulic actuator at the center span of the roof level, inthe direction parallel to the walls with openings.

The irregular wall of interest for this study experienced the hysteretic force–displacement behaviorreported in Figure 10b under La Malbaie time history multiplied by 2. That seismic input inducedflexural–shear cracks with a width ranging from 2.5 to 13mm. Details on the specimen design, testsetup, instrumentation, and observed crack patterns can be found in [1]. Experimental resultsshowed (i) a negligible effect of wall continuity at the building corners on the lateral strength of theURM walls during high-intensity seismic motion, even if a different stiffness was observed duringlow-intensity seismic motion; (ii) the specimen experienced a maximum lateral drift θ ofapproximately 1% under La Malbaie time history multiplied by 2 (corresponding to a peak groundacceleration approximately equal to 0.9 g) without significant strength degradation; and (iii) thewood diaphragm remained essentially elastic at any seismic intensity level. It is emphasized that anegligible effect of wall continuity among orthogonal walls allows to assume rectangular crosssections for piers, instead of L-shaped, T-shaped, or X-shaped cross sections. Therefore, a planarcapacity model can be developed for each wall, and a spatial assemblage of walls and floors can becarried out. Especially in the case of existing buildings, wall intersections are seldom effectivebecause of the absence of both interconnections among masonry units and RC bond beams.

Two SPO analyses were carried out for the tested URM wall, one for each lateral load orientation.The modeling rule presented in [18] was used to define pier effective heights. To compare in-planeseismic capacity predictions complying with IBC [5] and EC8 [15], we considered two different setsof values for the limit state of near collapse (NC) of single macroelements. In fact, for existingURM buildings, IBC assumes that primary walls (i.e., those belonging to the lateral-force resistingsystem) reach NC in flexure if θf = 0.006 and in shear if θs = 0.004. In the same conditions, EC8establishes θf = 4/3 � 0.008 � l0/d and θs = 4/3 � 0.004 (see Section 2.2 for the meaning of l0 and d). It isunderlined that θ is defined as the ratio between the relative lateral displacement between endsections Δr and the effective length of the macroelement leff. These different definitions of NC led todifferent SPO curves for both lateral load orientations, as shown in Figure 11. It is worth noting thatthe experimental in-plane response was rather different under positive and negative displacementdemands, leading to an absolute value of lateral drift in the negative orientation about two times thatmeasured in the positive orientation. Such evidence was well captured by SPO analysis, althoughthe IBC-complying definition of limit states for single macroelements led to more conservative

Figure 10. Unreinforced masonry wall specimen: (a) elevation (dimensions in millimeter) and (b) hystereticforce–displacement response under synthetic acceleration time history multiplied by 2.



predictions of displacement capacity for the irregular wall. Furthermore, satisfactory predictions oflateral resistance were obtained for both load orientations. The collapse of a macroelement induced aresistance drop, which was more evident under negative displacement demands.

5. CHARACTERIZATION OF WALL IRREGULARITIES

To investigate the in-plane lateral behavior and seismic capacity of irregular URM walls withopenings, we grouped typical geometric irregularities in a number of classes and quantified them byindices. This classification and indexing work allowed to perform sensitivity analysis by varying oneirregularity index at a time and keeping constant the others. Key capacity parameters weremonitored to capture the main effects of irregularities. Finally, regression analysis was performed;after the best regression models were detected, empirical relationships between capacity parametersand irregularity indices were obtained and are proposed herein. The significance of seismic capacityvariations with irregularities is discussed from both statistical and structural viewpoints.

5.1. Classification of wall irregularities

In masonry walls with openings, the following basic irregularities can be identified:

(1) Horizontal irregularity: The wall has openings with different heights at the same story and equallengths along the height (Figure 12a).

(2) Vertical irregularity: The wall has openings with equal heights at the same story and differentlengths along the height (Figure 12b).

(3) Offset irregularity: The wall has horizontal and/or vertical offsets between openings with equalor different sizes (Figure 12c).

(4) Variable openings number irregularity: The wall has different number of openings per story(Figure 12d).

The first three basic irregularities are here defined for single couples of openings, whereas the lastirregularity is defined for couples of stories. The presence of more basic irregularities induces acomplex irregularity. Opening offsets are frequently identified in the vertical direction, also at thelocations of staircases in new buildings; they cause nonuniform paths of gravity loads and strangedamage patterns.

5.2. Quantification of wall irregularities

In order to assess the influence of irregularities on nonlinear response of walls with openings andvariations in their seismic capacity parameters, a series of global and partial irregularity indices aredefined herein. A global irregularity index indicates whether a perforated wall is regular (i= 0) ornot (0< i⩽ 1), given a couple of openings or stories. Given a type of irregularity, a partialirregularity index quantifies the distribution of the irregularity throughout the masonry wall.

Figure 11. Comparison between experimental and theoretical curves.



In the case of horizontal irregularity, the following global index can be defined:

iH ¼ ΔH

2Hmed¼ Hmax � Hmin

Hmax þ Hmin(12)

where Hmax and Hmin are, respectively, the maximum and minimum opening heights and ΔH is thetotal irregularity (Figure 12a) defined as follows:

ΔH ¼ Hmax � Hmin ¼ ΔþH þ Δ�

H (13)

ΔþH and Δ�

H being the distances between corresponding upper and lower opening edges.

The distribution of the horizontal irregularity with respect to the centroid of the higher opening canbe completely described by the following partial indices:

�ΔþH ¼ Δþ

H

ΔHað Þ

�Δ�H ¼ Δ�

H

ΔHbð Þ

(14)

Therefore, the amount and distribution of a horizontal irregularity can be quantified through iH andone of the partial indices given by Equation (14a,b) because their sum is equal to unity. It is noted thatsuch partial indices quantify the relative position between the lower and higher openings rather thantheir absolute position in the wall.

Similarly, in the case of vertical irregularity, the following global index can be defined:

iV ¼ ΔL

2Lmed¼ Lmax � Lmin

Lmax þ Lmin(15)

where Lmax and Lmin are, respectively, the maximum and minimum opening lengths and ΔL is the totalirregularity (Figure 12b) defined as follows:

Figure 12. Irregular walls with (a) horizontal irregularity, (b) vertical irregularity, (c) offset irregularity, and(d) variable openings number irregularity.



ΔL ¼ Lmax � Lmin ¼ ΔrL þ Δl

L (16)

ΔrL and Δl

L being the distances between corresponding right and left opening edges.

The distribution of the vertical irregularity with respect to the centroid of the longer opening can bedescribed through the following partial indices:

�ΔrL ¼ Δr

L

ΔLað Þ

�ΔlL ¼ Δl

L

ΔLbð Þ

(17)

Therefore, the amount and distribution of a vertical irregularity can be quantified through iV and oneof the partial indices given by Equation (17a,b) because their sum is equal to unity.

When quantifying an offset irregularity in the vertical direction, one should consider that thedistance ΔO between the upper edges of two openings cannot exceed ΔO,max =D� tf�H�, where Dis the interstory height, tf is the thickness of the floor slab or RC bond beam, and H� is the heightof the lower opening (Figure 12c). Furthermore, the sum of the opening heights (i.e., H+ and H�)cannot be larger than D� tf (Note that this criterion only works if floors are at constant levelthroughout the wall; otherwise, D should be considered as the maximum interstory height associatedwith the floors placed above the couple of openings under consideration). It follows that thefollowing global irregularity index can be defined:

iO ¼ ΔO

D� tf � H� (18)

A similar index can be introduced for offset irregularity between right and left edges of twoopenings, that is, in the horizontal direction of a perforated wall. In such a case, the distance ΔO

between the right and left edges of two openings cannot be larger than the right/left pier.Finally, in the case of a perforated wall with different number of openings per story (Figure 12d),

each couple of consecutive opening series can be analyzed to characterize such a wall irregularity.The following global irregularity index can be evaluated:

iN ¼ 1� Nmin

Nmax(19)

where Nmax and Nmin are the maximum and minimum numbers of openings per story, respectively. IfNmin⩽Nmax� 2, the distribution of the openings can be quantitatively described through a partialirregularity index given by the following relation:

iD ¼ 1� 2xGLw

�� (20)

where xG is the distance of the centroid G of the series of openings that correspond Nmin and Lw is theoverall length of the perforated wall. In the case of symmetric arrangement of openings (i.e., xG=Lw/2),the index iD is equal to zero. If the absolute sign in Equation (20) is not applied, then iD quantifies alsothe location of G with respect to the centerline of the perforated wall.



6. SENSITIVITY ANALYSIS

Response-controlled SPO analysis was carried out on two-story URM walls with a maximum numberof openings per story equal to 2, 3, and 4. Each parametric analysis included the case of regularperforated wall to estimate capacity variations against that case. The nonlinear behavior of URMwalls with offset irregularities and different number of openings per story does not fall in the scopeof this sensitivity study. To assess the influence of both vertical and horizontal irregularities, thebenchmark perforated wall was assumed to be composed by two stories, each of them having threepiers with length of 1.70m and effective heights according to [18], a spandrel with height of 1.00mrunning along the entire wall, and two openings with length of 1.70m and height of 2.30m(Figure 13). The total height of the wall was equal to 6.60m, whereas the thickness was assumed tobe 0.80m, which is a common value for existing URM buildings located in European countries andnot designed for earthquake resistance. The wall was supposed to be made of solid clay brickmasonry with the following mechanical properties: mean uniaxial compressive strengthfm= 4.30MPa, mean shear strength at zero confining stress t0 = 0.10MPa, mean Young’s modulusEm= 3000MPa, and mean shear modulus Gm= 1200MPa.

Each global irregularity index was changed in the interval [0,1] with a step of 0.25 considering allcouples of openings, namely several locations for the opening supposed to be the source of wallirregularity. For global irregularity indices equal to 0.25, 0.50, and 0.75, also the partial globalindices were changed in the interval [0.25,1] with a step of 0.25, resulting in 17 combinations ofglobal and partial indices for both horizontally and vertically irregular walls. SPO analysis wasperformed under both modal and uniform load patterns, as well as for both orientations in the caseof horizontal irregularity because different pier effective heights were assumed. A total amount of918 SPO analyses were carried out. Figures 14a–f and 15a–f show SPO curves corresponding tohorizontally and vertically irregular URM walls with openings, for both modal and uniform loadpatterns. In each figure, the legend shows the case of regular wall as reference and those of irregularwall. Each case of irregular wall is characterized by a code composed by six numbers: the firstnumber is the value of global irregularity index; the second number is the value of partialirregularity index; the third and fourth numbers in round parentheses defines the opening (i,j) thatinduces the irregularity type under consideration (i.e., (2,2) in Figures 14a–f and 15a–f); the fifthnumber is the number of wall stories, which is equal to 2 in this study; and the last number is themaximum number of openings per story, which was considered equal to 2, 3, and 4.

The following types of diagrams were detected in this sensitivity study:

(1) pushover curves with low nonlinearity up to the maximum resisting base shear and brittlecollapse of more macroelements;

(2) pushover curves with high nonlinearity and progressive reduction in resisting base shear due tothe collapse of macroelements at increasing drift levels; and

(3) pushover curves with no resistance drop, representing a ductile behavior of perforated walls.

Figure 13. Front and lateral views of the benchmark wall with openings (dimensions in meter).



The first type of SPO curves was typically related to a cascade of shear failures or even tensilefailure of piers, which was caused by axial forces corresponding to lateral displacement demands(i.e., axial forces induced by overturning moment at each story). Such damage localization isparticularly dangerous because it produces a sudden collapse of the wall. In this case, seismicretrofitting of the existing irregular wall should increase strength and/or ductility of macroelementsexpected to suffer brittle failure. Vertical post-tensioning could be applied to piers expected to fail intension, to avoid any mass increase. The second type of SPO curves resulted from a partial ability ofthe irregular wall to redistribute strength and drift demands among macroelements after one or morefailures. This is an intermediate nonlinear behavior of existing irregular walls where only somemacroelements suffer brittle failure and need to be retrofitted.

Finally, the third type of SPO curves was typically related to a rather uniform distribution of strength anddrift demands throughout the wall, allowing almost a full exploitation of the available ductility. In suchcases, if the existing irregular wall does not meet code requirements for the limit state of life safety orNC, seismic retrofitting should provide a uniform increase in strength and/or ductility throughout the wall.

The following parameters were computed for each pushover analysis:

(1) overstrength ratio Ω of the wall; and(2) elastic vibration period T, ultimate shear force Vu, ultimate displacement du, available

displacement ductility m, and ductility-based force reduction factor Rm of the idealized SDOF system.

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vb[

kN]

dc[mm]

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vb[

kN]

dc[mm]

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vb[

kN]

dc[mm]

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vb[

kN]

dc[mm]

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vb[

kN]

dc[mm]

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vb[

kN]

dc[mm]

(a) (b)

(c) (d)

(e) (f)

Figure 14. Pushover curves of horizontally irregular unreinforced masonry walls under leftward orientationof seismic action and horizontal irregularity at the left upper opening in the cases of (a) Nmax = 2 and modalload pattern, (b) Nmax = 2 and uniform load pattern, (c) Nmax = 3 and modal load pattern, (d) Nmax = 3 and

uniform load pattern, (e) Nmax = 4 and modal load pattern, and (f) Nmax = 4 and uniform load pattern.



The bilinear idealization of base shear (Vb) versus top displacement (dc) curves was carried out bydefining the ultimate displacement du at 20% base shear drop (limit state of life safety) and the elasticstiffness as the ratio between 0.7Vb,max and the corresponding top displacement, where Vb,max is themaximum resisting base shear [5]. The limit elastic displacement de was derived by assuming equalareas below the actual and idealized force–displacement diagrams. Hence, the availabledisplacement ductility was defined as m= du/de. The overstrength ratio was computed as Ω= au/a1,where au = 0.9amax (i.e., 0.9 times the load multiplier corresponding to Vb,max) and a1 is the loadmultiplier corresponding to the attainment of the nominal lateral strength in a pier regardless of itsfailure mode. Finally, the ductility-based force reduction factor was estimated as Rm= (2m� 1)1/2

according to the ‘equal energy rule’ typically assumed for low-period structures such as URMbuildings [24, 25]. Therefore, the total force reduction factor can be defined as R =Ω �Rm [5].

7. REGRESSION MODELS

Regression analysis was carried out for different aggregations of SPO results. The lower level ofanalysis detail was related to the variation in a given parameter c normalized to its realization forregular wall cR (i.e., c/cR), under varying global irregularity index iH (or iV). In such a case,regression analysis was performed over all SPO results related to different locations of the openings

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vb[

kN]

dc[mm]

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vb[

kN]

dc[mm]

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vb[

kN]

dc[mm]

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vb[

kN]

dc[mm]

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vb[

kN]

dc[mm]

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Vb[

kN]

dc[mm]

(a) (b)

(c) (d)

(e) (f)

Figure 15. Pushover curves of vertically irregular unreinforced masonry walls under leftward orientation ofseismic action and vertical irregularity at the left upper opening in the cases of (a) Nmax = 2 and modal loadpattern, (b) Nmax = 2 and uniform load pattern, (c) Nmax = 3 and modal load pattern, (d) Nmax = 3 and uniform

load pattern, (e) Nmax = 4 and modal load pattern, and (f) Nmax = 4 and uniform load pattern.



causing wall irregularity, partial indices, maximum numbers of openings per story, load patterns, andorientations of seismic actions. Although regression analysis without any disaggregation of data setswas less accurate than others, it allowed to derive simplified equations for seismic assessment ofURM walls with openings (Figures 16a–f and 17a–f). In particular, regression analysis was carriedout on the 5th percentile of sectional data to provide conservative estimates of capacity parameters.Each data section was represented by the vector Ψj= {Ψ1j. . .Ψij. . .ΨSj} associated with the value j ofirregularity index (where j= 0.25, 0.5, 0.75, 1; Ψij=cij/cR; and S = number of SPO results, that is,the sample size, corresponding to j) and was ordered in increasing order of magnitude. The 5thpercentile of each data section, namely the value of cj/cR not exceeded by 5% of SPO resultscorresponding to j, was estimated according to classical statistical procedures. Finally, a regressionmodel was derived for each capacity parameter under investigation by means of the ordinary leastsquare method applied to the vector Ψ5 = {1. . .Ψj. . .Ψ1}, in a way to maximize the coefficient ofdetermination R2 (Note that 1 and Ψ1 are the values of c/cR at j= 0, i.e., zero irregularity, and j= 1,respectively.). This modus operandi was motivated by the fact that in some cases, seismic capacitymay also increase under a given orientation of seismic forces, whereas a large reduction usuallyoccurs in the opposite orientation. Actually, structural engineers are interested to the worst case thatcorresponds the major reduction in seismic capacity. The scope of regression analysis was then toprovide some hints on the unfavorable effects that can be caused by the closure, modification, orrealization of openings in an existing URM wall. Regression analysis for horizontal irregularity was

T/TR = 1 - 0.14iHR² = 0.58

0

0.5

1

1.5

0 0.25 0.5 0.75 1

T/T

R

iH

0

0.5

1

1.5

0 0.25 0.5 0.75 1

iH

0

0.5

1

1.5

0

0.5

1

1.5

0 0.25 0.5 0.75 1

iH

0 0.25 0.5 0.75 1

iH

0

0.5

1

1.5

0 0.25 0.5 0.75 1

iH

0

0.5

1

1.5

0 0.25 0.5 0.75 1

iH

(a) Vu/VuR = 1 + 0.03 iHR² = 0.76

(b)

du/duR = exp(-1.98 iH)R² = 0.84

d u/d

uR

Vu/

VuR

(c)

μ/μR = exp(-1.9 iH)R² = 0.84

(d)

Rμ/RμR = exp(-1.16 iH)R² = 0.85

Rμ/

RμR

Ω/Ω

Rμ/

μ R

(e) Ω/ΩR = exp(-0.87 iH)R² = 0.64

(f)

Figure 16. Empirical models for horizontally irregular walls with openings: (a) T/TR� iH; (b) Vu/VuR� iH;(c) du/duR� iH; (d) m/mR� iH; (e) Rm/RmR� iH; and (f) Ω/ΩR� iH.



performed considering 558 SPO results at each iH value, whereas that related to vertical irregularity wasperformed accounting for 288 SPO results at each iV value. In all cases, high values of R2 indicate asatisfactory goodness of fit of the empirical models. Therefore, the equations presented in this study couldbe used to estimate the reduction in some key parameters of irregular walls with openings, such as Ω andRm, with respect to the corresponding regular wall. It is emphasized that such empirical models provideconservative predictions. In fact, it could be shown that the parameters of interest can also increase,depending for instance on the location of the opening within the wall and orientation of seismic actionsunder consideration. Furthermore, it was found that the law of variation of a parameter with a partial

irregularity index (i.e., �ΔþH or �Δr

L) may depend on the value of the corresponding global irregularity index.In general, regression analysis results indicate that seismic capacity of URM walls with openings (in

terms of ultimate displacement capacity du, displacement ductility m, and total force reduction factor R)is significantly affected by geometric irregularities, and its reduction is more significant underincreasing horizontal irregularity. Figures 16a and 17a show that the elastic period T of the idealizedSDOF system slightly reduces as a global irregularity index increases from zero to unity.

On the contrary, a little increase in Vu under increasing global irregularity index is evidenced inFigures 16b and 17b. As opposed to what appears for other capacity parameters, both T and Vu arequite insensitive to horizontal and vertical wall irregularities. In fact, SPO analyses evidenced thatsuch irregularities induce, on the one hand, little variations in the distribution of elastic demandamong macroelements and, on the other hand, significant changes in the distribution of inelastic

T/TR = 1- 0.08 iVR2 = 0.77

0

0.5

1

1.5

0 0.25 0.5 0.75 1

T/T

R

iV

0

0.5

1

1.5

0 0.25 0.5 0.75 1

iV

0

0.5

1

1.5

0 0.25 0.5 0.75 1

iV

0

0.5

1

1.5

0 0.25 0.5 0.75 1

iV

0

0.5

1

1.5

0 0.25 0.5 0.75 1

iV

0

0.5

1

1.5

0 0.25 0.5 0.75 1

iV

(a) Vu/VuR = 1 + 0.05 iVR2 = 0.60

Vu/

VuR

(b)

du/duR = exp(-1.72 iV)R2 = 0.98

d u/d

uR

(c)= exp(-1.64 iV)

R2 = 0.99

(d)

= exp(-1.02 iV)R2 = 0.99

(e) = 1- 0.25 iVR2 = 0.95

/R

/ R

R/R

R

R /R R

/R

/ R(f)

Figure 17. Empirical models for vertically irregular walls with openings: (a) T/TR� iV; (b) Vu/VuR� iV; (c)du/duR� iV; (d) m/mR� iV; (e) Rm/RmR� iV; and (f) Ω/ΩR� iV.



demand. This indicates that idealized SDOF systems of different irregular walls corresponding to agiven regular URM wall with openings have approximately the same initial stiffness and ultimateforce but different values of displacement ductility. These results could also be of interest toinvestigate how seismic demand on idealized SDOF systems changes with the type and amount ofgeometric irregularities through nonlinear dynamic analysis.

Negative exponential trends were found for all other parameters in both cases of horizontal andvertical irregularities; a particular case was that of Ω, which was found to be characterized by anegative exponential trend in the case of horizontal irregularity (Figure 16f) and a linear descendingtrend in the case of vertical irregularity (Figure 17f). In any case, a reduction in the overstrengthratio means that geometric irregularities reduce the ability of the wall to redistribute strength anddrift demands throughout the wall after the failure of a macroelement. The empirical equationsderived for T and Vu could be used to estimate inelastic displacement demand on the irregular wallunder consideration, to be compared with the ultimate displacement capacity.

8. CONCLUSIONS

A simplifiedmethodology for the assessment of the effects of geometric irregularities on the in-plane seismiccapacity of URM walls with openings has been proposed. To this end, a macroelement method wasdeveloped and validated through experimental results. The proposed methodology was based on (i) theclassification and quantification of typical geometric irregularities by means of indices and (ii) theassessment of geometric irregularity effects through both sensitivity and regression analyses. Irregularityindices allow to establish whether a wall with openings is regular or not, the total amount of irregularity,and its distribution throughout the wall. Sensitivity analysis consisted in a significant number of response-controlled SPO analyses where variations in key capacity parameters with respect to the case of regularwall were assessed. Regression analysis let to describe each capacity parameter under a varyingirregularity index, providing empirical models for seismic design/assessment of irregular URM walls withopenings. Among the geometric irregularities identified in this study, those related to different openingheights at the same story (horizontal irregularity) and lengths between two consecutive stories (verticalirregularity) were investigated because of their large presence in the worldwide building heritage.Therefore, SPO analysis was carried out on a benchmark two-story URM wall with openings byassuming several maximum numbers of openings per story, locations of the openings causing wallirregularity, and lateral load patterns. Different capacity models were considered for negative and positiveorientations of seismic actions to account for their influence on the pier effective height. SPO curves haveshown that geometric irregularities can induce damage localizations and a cascade of brittle failures,resulting in a premature collapse of the wall. Seismic design should then minimize wall irregularities thatare often induced by architectural trends; seismic retrofit of existing walls should reduce irregularities inthe arrangement of openings and mitigate their negative effects.

Regression analysis was carried out on the 5th percentile of sectional data to provide conservative estimatesof capacity parameters. Analysis results have shown that the in-plane seismic capacity of URM walls (interms of ultimate displacement capacity, displacement ductility, and force reduction factor) significantlyreduces as geometric irregularities increase. In particular, a higher sensitivity to horizontal irregularity hasbeen detected, especially in the case of global overstrength ratio. The latter decreases with an exponentialtrend in the case of horizontal irregularity and a linear trend in the case of vertical irregularity. Variations inthe ultimate displacement capacity of URM walls with openings under increasing irregularity indices havebeen also estimated for a simplified displacement-based seismic assessment.

On the basis of this study, the authors recommend to explicitly include both vertical and horizontalirregularities within capacity models of URM walls with openings. Otherwise, one can predict first thein-plane seismic capacity of the regular wall corresponding to the actual irregular wall to be designedor assessed and then potential variations caused by geometric irregularities through the proposedregression models. The latter also allow to predict whether and how seismic capacity of an existingwall may decrease after some openings are closed, modified, or realized. If seismic capacity of theas-built irregular wall is known, no structural analysis (be it linear or nonlinear) is needed to rapidlyassess potential unfavorable effects of structural modifications.



Further research is needed (i) to assess seismic capacity sensitivity to offset irregularities anddifferent number of openings, (ii) to define limits of applicability of macroelement methods to URMwalls with large irregularities (e.g., threshold irregularities derived from analytical–numericalcomparisons), and (iii) to investigate the effects of complex irregularity configurations on seismiccapacity. The effects on seismic demand could also be investigated by means of nonlinear dynamicanalyses of SDOF systems equivalent to irregular URM walls.

ACKNOWLEDGMENTS

This research was carried out in the framework of the ReLUIS-DPC 2010–2013 project (Line AT1-1.1 –‘Evaluation of the Vulnerability of Masonry Buildings, Historical Centers and Cultural Heritage’) fundedby the Italian Department of Civil Protection. The authors express their sincere thanks to Prof. MichelBruneau and Dr. Jocelyn Paquette for providing their experimental data.

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25. TomaževičM, Weiss P. Displacement capacity of masonry buildings as a basis for the assessment of behavior factor:an experimental study. Bulletin of Earthquake Engineering 2010; 8(6):1267–1294.



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