pasticier_nonlinear_seismic_analysis of a masonry building in sap2000

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2008; 37:467–485Published online 9 November 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.770

Non-linear seismic analysis and vulnerability evaluation ofa masonry building by means of the SAP2000 V.10 code

Laurent Pasticier1,§ , Claudio Amadio2,‡ and Massimo Fragiacomo3,∗,†,‡

1Via Schiaparelli 8, 34143 Trieste, Italy2Department of Civil & Environmental Engineering, University of Trieste, Piazzale Europa 1, 34121 Trieste, Italy

3Department of Architecture & Planning, University of Sassari, Piazza Duomo 6, 07041 Alghero, Italy

SUMMARY

The aim of the paper is to explore the possibilities offered by SAP2000® v.10, a software package withuser-friendly interface widely used by practising engineers, for seismic analyses of masonry buildings.The reliability of the code was first investigated by carrying out static push-over (SPO) analyses of twowalls, already analysed by other researchers using advanced programs. The equivalent frame modellingwas employed in all analyses carried out. The code was then used to investigate the seismic performanceof an existing two-storey building typical of the north-east of Italy, with the walls being made of roughlysquared stones. An SPO analysis was performed first on the most significant wall, followed by a numberof time-history analyses aimed to evaluate the dynamic push-over curves. Finally, the seismic fragilitycurves were derived, considering the seismic input as a random variable. Copyright q 2007 John Wiley& Sons, Ltd.

Received 14 November 2006; Revised 12 July 2007; Accepted 5 October 2007

KEY WORDS: equivalent frame; fragility curves; incremental dynamic analysis; masonry building;push-over analysis; SAP2000

1. INTRODUCTION

The assessment of the seismic vulnerability of existing buildings is one of the priorities of manyEuropean countries. A large number of old masonry buildings, often characterized by degradation,and in some cases with significant historical value, are in fact located in earthquake-prone areaswith different levels of seismic hazard. A proper evaluation of the seismic risk in existing buildings

∗Correspondence to: Massimo Fragiacomo, Department of Architecture & Planning, University of Sassari, PiazzaDuomo 6, 07041 Alghero, Italy.

†E-mail: [email protected]‡Associate Professor.§Consulting Engineer.

Copyright q 2007 John Wiley & Sons, Ltd.

468 L. PASTICIER, C. AMADIO AND M. FRAGIACOMO

Figure 1. Failure mechanisms of a masonry pier: (a) rocking; (b) sliding shear;and (c) diagonal shear cracking.

is a necessary step in order to recognize the most critical areas and assess the priorities of theretrofit work. In order to achieve this result, a proper modelling of the masonry structure is needed.

Several models, with different theoretical approaches [1], have been developed to date. The finiteelement models [2, 3], based on proper constitutive laws for the masonry components [4], allow anaccurate determination of the critical points in the structure including the failure mechanisms, butare time consuming and require the use of expensive and complex software. Other simpler modelsare based on ‘macromodel’ modelling, where the masonry building is divided into a number ofone- or two-dimensional ‘macroelements’ [5–14]. In most of the models based on two-dimensionalelements, the hypothesis of material with no tensile strength is assumed [15], which usually resultsin a quite complex iterative process. Among the models using one-dimensional elements, thePOR method is well known and extensively used. Such a method assumes, in its original versionimproved later [16, 17], that the structural collapse occurs because of a storey mechanism. Thefailure is assumed to take place only in the piers, and no allowance for the possible damage of thespandrel beams is made.

An improvement of the POR method is provided in the so-called ‘equivalent frame’ method,which allows the user to carry out a global analysis of the building. In such a method, a highernumber of possible failure mechanisms occurring inside each macroelement, such as shear withdiagonal cracking, shear with sliding, and rocking (Figure 1), can be considered. In accordancewith the use of this approach, which was used for example in the SAM code [10–12] and incomparative analyses between macromodels and finite element models [18], the spandrels andthe piers are regarded as elastic, their intersections are modelled as fully rigid, and the possiblemechanical non-linearity is concentrated in some well-defined cross-sections inside the elasticparts. The use of this approach is allowed by the FEMA 356 [19], the new Italian Seismic Code[20], and the latest draft of the European Code (Eurocode 8) [21]. Both Italian and EuropeanCodes encourage the use of non-linear static push-over (SPO) analysis and require a control of thespandrels, which is not possible using the POR method. The use of suitable programs is thereforeneeded for the design of masonry buildings according to those regulations.

The aim of this paper is to explore the possibilities offered by a widespread, user-friendly andwell-known package for structural analysis such as SAP2000® v.10 [22] for seismic design ofmasonry buildings. This code was already used for seismic analysis of masonry buildings usingboth one- and two-dimensional elements [18, 23]. The reliability and limitations of the code werefirst investigated by performing an SPO analysis of two multi-storey walls, already analysed byother authors using advanced programs. Some non-linear static and dynamic analyses were then

Copyright q 2007 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2008; 37:467–485DOI: 10.1002/eqe

NON-LINEAR SEISMIC ANALYSIS AND VULNERABILITY EVALUATION 469

carried out on a facade wall of a typical Italian masonry building using the ‘equivalent frame’method. The fragility curves were then drawn by assuming the seismic input as a random variable.

2. THE PROPOSED MODELLING

The proposed modelling of the masonry building is based on the use of the equivalent framemethod. The SAP2000® v.10 package allows the user to account for the non-linear mechanicalbehaviour of the material by introducing the following elements with lumped plasticity in theequivalent frame:

• plastic hinges;• non-linear links.

The plastic hinges were used in SPO analyses since they allow the user to accurately follow thestructural performance beyond the elastic limit at each step of the incremental analysis. The non-linear links were instead used in time-history analyses since they allow the user to accurately definethe cyclic behaviour of the elements including a proper degradation rule [24]. The mechanicalproperties of these non-linear elements were defined based on the possible failure mechanisms ofmasonry macroelements shown in Figure 1 [25–28]. The adopted modelling is described in thefollowing sections.

2.1. Modelling of the non-linear behaviour for the SPO analysis

The standard force–displacement curve that can be implemented in the SAP2000 plastic hinges isdepicted in Figure 2(a) [24]. The masonry piers were modelled as elastoplastic (as also suggestedin [20]) with final brittle failure (Figure 2(b)) [11] by introducing two ‘rocking hinges’ at the endof the deformable parts and one ‘shear hinge’ at mid-height (Figure 3(a)). A rigid-perfectly plasticbehaviour with final brittle failure was assumed for all these plastic hinges (Figure 2(c)).

The strength in terms of ultimate moment Mu is defined by Equation (1). As far as the shearstrength is concerned, according to the experimental test outcomes [28], it was decided to considertwo strength criteria. The first criterion (Equation (2)) is recommended in [20] for existing buildings.This criterion, which refers to shear failure with diagonal cracking, was originally proposed byTurnsek and Cacovic [29] and later modified by Turnsek and Sheppard [30]. The second criterion(Equation (3)) refers to shear failure with sliding and is recommended in [20] for new buildings.Although formulated differently, such a criterion is also recommended by the Eurocode [21, 31]:

Mu = �0D2t

2

(1− �0

k fd

)(1)

V fu = 1.5 fv0dDt

�

√1+ �0

1.5 fv0d(2)

V su =

3

2fv0d+�

�0�m

1+ 3H0

D�0fv0d

Dt (3)



Figure 2. (a) Standard shape of the force vs displacement curve in SAP2000® v.10 for theplastic hinge element [24]; (b) and (c): behaviour assumed, respectively, for the entire pier andthe correspondent plastic hinge; (d) and (e): behaviour assumed, respectively, for the entire

spandrel beam and the correspondent plastic hinge.

"shearhinge" forthe piers

"rockinghinge" forthe piers

"shear hinge"for thespandrels

elastic part fully rigid part

Spandrel

Joint

Pier

"shear hinge" for the piers

elastic part fully rigid part

1mGround Floor 0 2m

Analysedwall

(a) (b) (c)

Figure 3. Analysed building: (a) plastic hinges’ location in the equivalent frame model of the wall facadeused for static analyses; (b) non-linear links’ location in the equivalent frame model of the wall facadeused for dynamic analyses; and (c) ground floor plan with the analysed facade wall (measures in metres).

where �0 is the mean vertical stress, D the pier width, t the pier thickness, k the coefficient takinginto account the vertical stress distribution at the compressed toe (a common assumption is anequivalent rectangular stress block with k=0.85), fd the design compression strength, fv0d thedesign shear strength with no axial force; � (friction coefficient)=0.4, � the coefficient relatedto the pier geometrical ratio, H0 the effective pier height (distance of the cross-section in whichthe strength criterion is applied from the point of zero bending moment), and �m the safety factor(assumed to be equal to 2). For the rocking hinges the strength is given by Equation (1), andthe ultimate rotation �u corresponds to an ultimate lateral deflection �′

u equal to 0.8% of thedeformable height of the pier, minus the elastic lateral deflection, as recommended in [20]. Forthe shear hinge, the strength is given by the minimum value resulting from Equations (2) and (3).The ultimate shear displacement �u was assumed to be equal to 0.4% of the deformable height



of the pier, minus the elastic lateral deflection, as recommended in [20]. The failure was assumedto have occurred in the pier when the first between the ultimate rotation �u in the plastic hingeand the ultimate shear displacement �u in the shear hinge was attained. Since in SAP2000 it isnot possible to automatically control the total deflection of an entire macroelement if more thanone of its plastic hinges exceed the elastic limit, such a quantity was manually checked on everymacroelement at the end of each load step.

As far as the modelling of the spandrel beams is concerned, assuming the presence of a lintelproperly restrained at both supports, only one ‘shear hinge’ was introduced at mid-span (Figure3(a)), with the shear strength Vu given by

Vu=ht fv0d (4)

where h is the spandrel depth, t the spandrel thickness, and fv0d the design shear strength withno axial force. A brittle–elastic behaviour with residual strength after cracking equal to 1

4 th of themaximum strength was assumed for the entire element, with no limit in deflection (Figure 2(d)and (e)) [11].

2.2. Modelling of the non-linear behaviour for the time-history analyses

The ‘multilinear-plastic pivot’ non-linear link was used in the time-history analyses. This linkallows the user to reproduce the cyclic behaviour of the entire macroelement by defining the shapeof the hysteresis loop and the degradation of both strength and stiffness (Figure 4(a)) through aproper choice of the mechanical parameters.

Owing to the complexity of the non-linear time-history analyses, in order to reduce the compu-tational burden, only one ‘shear link’ was introduced at mid-height of the piers (Figure 3(b)). Thischoice was suggested both by the outcomes of the SPO analyses presented in the next paragraph,where the dominant failure mechanism was found to be shear in the pier, and by the geomet-rical ratio (and therefore strength) of the spandrels relative to the piers. For the sake of safety, ahysteretic behaviour characterized by shear failure with diagonal cracking was assumed. This typeof failure is, in fact, more fragile and, therefore, critical in terms of displacement demand thansliding shear.

2Fy2

2Fy22Fy2

2Fy2

Fy1

1Fy1

AB C

A'B'C'

P1P2

P4 P3

V=30%

V=30%

Fy1=Fy2

1Fy1= 2Fy2

=0.45Fy1

2Fy2= 1Fy1

=0.45Fy2

Fy2=Fy1

A

B

A'

B' v.10SAP2000et al.Magenes

-8 -6 -4 -2 0 2 4 6 8top displacement [mm]

-100-80-60-40-20

020406080

100

ba

sesh

ea

r[k

N]

(c)(b)(a)

Figure 4. (a) Standard hysteresis loop for the ‘multilinear-plastic pivot’ non-linear link in SAP2000® v10[24]; (b) reference curve assumed for the non-linear link with the corresponding values of the parametersfor stiffness and strength degradation control; and (c) comparison between the experimental curve detectedwith the quasi-static cyclic test performed on a pier by Anthoine et al. [26] and the numerical curvedetected with SAP2000® v.10 for the same pier using the multilinear-plastic pivot non-linear link with

the parameters indicated in (b).



The link parameters �1, �2, �1, and �2, which control the stiffness degradation during theunloading procedure, were chosen so as to reproduce the experimental results obtained by Magenesand Coworkers [26] on a single brick wall (Figure 4(b) and (c)). The parameters were all assumed tobe equal to 0.45. The same values were also adopted for the other stone masonry walls investigatedin this paper. The maximum shear strength and plastic displacements were assumed to be the sameas those used for the SPO analyses.

3. VALIDATION OF THE MODEL FOR SPO ANALYSES

In order to verify the reliability of the proposed modelling, two walls (designated as A and Bin Figure 5) of stone masonry buildings previously analysed in the ‘Catania Project’ [32] weremodelled with SAP2000 v.10. The ‘Catania Project’ was an extensive nationwide research projectfocused on seismic performance of existing masonry buildings. In such a project, some laboratoryand in-situ tests were performed to characterize the mechanical properties of the masonry. Inaddition, numerical modelling of the structural response was undertaken by a number of Italianuniversities, each of them using a different advanced software package. The University of Paviaused the SAM code, which is considered as an important reference for this work. Such a code, whichis based on the equivalent frame modelling, was previously validated on a number of experimentaltests providing satisfactory results [10, 25]. The pier walls are modelled using Equations (1) and(3), whereas the shear strength with diagonal cracking is evaluated using a more suitable criterionfor regular brick masonry walls [28], which is different from Equation (2). For the spandrelbeams, Equations (1) and (4) are adopted. The Basilicata research group used a no tensile strengthmacroelement model with crushing and shear failures [6, 7], while the Genoa research group useda finite elements model with layer failures [3].

The mechanical properties used in the analyses were E (Young’s modulus)=1500N/mm2,G (shear modulus)=250N/mm2, �(unit weight)=1900kg/m3, fd (design compression strength)=2.4N/mm2, fv0d (design shear strength with no vertical stress)=0.2N/mm2, and � (frictioncoefficient)=0.5. The lateral loads representing the seismic action were applied by assuming theinverted triangular distribution. The weights of each floor and the ratios between the seismic forceon the floor and the base seismic shear are reported in Table I.

There is an important difference between the SAM and the SAP2000 v.10 programs that hasto be highlighted. The SAM program updates the strengths Mu, V

fu and V s

u of the plastic hingesduring the non-linear analysis if the quantities H0 and �0 (Equations (1)–(3)) change due to theeffect of the lateral loads. While the quantity H0 almost remains constant, the quantity �0 canmarkedly change during the analysis when the lateral load rises. The SAP2000 v.10 code, however,does not allow for the automatic update of the strengths during the analysis. In order to assess thesignificance of this limitation, the two walls A and B were analysed by considering two different�0 distributions (No. 1 and No. 2) for the evaluation of the strengths of the plastic hinges. Inthe distribution No. 1, the hinge strengths were calculated using, for �0, the values read on thestructure at the step 0 of the analysis, considering only gravity loads and no lateral loads. Inthe distribution No. 2, the hinge strengths were calculated using, for �0, the values obtained byapplying the gravity load and increasing the lateral loads up to the attainment of the elastic limit ofthe frame. Four independent SPO analyses were then carried out: two on the wall A with the twostress distributions No. 1 and No. 2, denoted as SPO 1 and SPO 2, and the corresponding two onthe wall B, also denoted as SPO 1 and SPO 2. Details on the axial force, shear force, and bending



Figure 5. Validation: elevation of wall A (a) and wall B (b), with the equivalent frame modelling used inSAM and the failure mechanisms detected by the same code [5, 32] (measures in metres).

Table I. Validation: seismic weights and distribution of the lateral forces at the different floors [3].Wall A Wall B

FloorSeismic weight

Wi (kN)Seismic force/baseshear ratio Fi/�Fi

Seismic weightWi (kN)

Seismic force/baseshear ratio Fi/�Fi

2nd 548.2 0.38 290.0 0.411st 1096.3 0.40 480.0 0.37Ground 1277.0 0.22 626.0 0.22

moment distributions in the frames due to gravity only (No. 1) and both gravity and lateral load(No. 2) are reported in Pasticier [33].

3.1. Numerical comparisons

The outcomes of the numerical comparisons are displayed in Figure 6(a) for wall A and Figure6(b) for wall B. The failure mechanisms as detected by SAP 2000 in the SPO 1 and SPO 2analyses are displayed in Figure 7 for wall A. Similar to the SAM method, both SPO 1 and SPO2 analyses detected a storey mechanism at the second floor of wall A with the same value ofultimate strength. Such an ultimate strength was higher than the strength obtained by the SAMcode, but lower than those obtained by the Basilicata and Genoa research groups. The same topdisplacement was obtained in both SPO 1 and SPO 2 analyses. Such a value was close to thatdetected by the Genoa research group and fairly different from those detected by the SAM andBasilicata research group.



0 10 20 30 40top displacement [mm] top displacement [mm]

0

300

600

900

1200

1500

1800

ba

se s

he

ar

[kN

]

X

SPO 1

SPO 2SAM

Genoa R.G.

Basilicata R.G.

0 10 20 30 400

200

400

600

800

ba

se s

he

ar

[kN

]

SPO 1

SPO 2

SAMGenoa R.G.Basilicata R.G.

(a) (b)

Figure 6. Validation: comparison between the push-over curve obtained with SAP2000® v.10 and thoseobtained with the other codes for the walls A (a) and B (b).

Figure 7. Validation: deformed shape of the equivalent frame at the attainment of the ultimate deformationin the first plastic hinge in the SPO 1 (a) and SPO 2 (b) analyses for wall A, with RO, rocking; SL,sliding shear; DC, diagonal cracking shear, and underlining, attainment of the ultimate deformation.

The results of both SPO 1 and SPO 2 analyses were almost the same as those of SAM also forwall B, with a storey mechanism occurring at the second floor. The top displacement was almostthe same as that detected by the Genoa research group, but different from that detected by theBasilicata research group. The ultimate shear strength observed was different in both cases.

3.2. Discussion of the results

The proposed model in SAP 2000 led to strength evaluation close to that given by the SAM programfor both walls A and B. The ultimate displacement was similar for wall B but more different forwall A. Such a difference is due to the change from compression to tension taking place during theanalysis in some piers (the circled ones in Figure 7). Such behaviour is not automatically detectedby SAP2000 v.10, which assumes pier strength independent of the axial force. Differently, the



SAM method updates the pier strength at each step of analysis. Since no tensile strength wasassumed for the masonry piers, the analysis with the SAM code ended well before the SAP 2000model, leading to a far lower ultimate displacement capacity. The top displacement attained in theSAP 2000 model when the axial force in the piers turned from compression to tension correspondsto point X in Figure 6(a). This new value matches quite well with the one obtained as a result ofthe analysis carried out with the SAM program.

The results of SPO 1 and SPO 2 analyses in terms of ultimate strength and displacement werealmost the same for both walls. In the cases of shear walls of usual geometry, in fact, the variationof vertical stresses in the walls due to overturning moment caused by lateral load is fairly low. Themain limit of the SAP2000 v.10 model, which is the impossibility to update the strengths of thepiers based on the variation of axial force, seems therefore not to be so crucial in the SPO analysisof ordinary masonry buildings. However, in terms of failure mechanisms, only the outcomes fromthe SPO 2 analysis are close to the results obtained using the SAM program (see Figures 5 and 7).

4. THE ANALYSED BUILDING

The plan of the ground floor is displayed in Figure 3(c) for the analysed building. This is astone masonry house typical of the north-east of Italy. The vertical structure is a single layer non-retrofitted masonry made of roughly squared sandstone, while the floors are composites of concreteslabs and timber beams. In order to reduce the computational burden of the dynamic analysesneeded for the vulnerability assessment, only the facade wall was analysed using the proposedSAP2000 v.10 model. The design values assumed for the mechanical properties are based on themean values measured in situ on a number of similar buildings located in the same area as theanalysed building: fd=0.8N/mm2, fv0d=0.032N/mm2, E=1600N/mm2, and G=640N/mm2.Only the in-plane seismic performance of the wall was investigated, assuming that the wall waseffectively connected to the floors.

5. SPO ANALYSIS

The SPO curves were obtained for the wall using the same procedure described in Section 3.Such curves were needed to identify the limit states considered in the evaluation of the fragilitycurves. As recommended by recent codes of practice and regulations [20], the horizontal seismicloads were applied adopting two different distributions (Table II): (i) proportional to the productof the masses by the floor heights (inverted triangular distribution) and (ii) proportional to thefloor masses (uniform distribution). Two SPO analyses, denoted as ‘SPO 1’ and ‘SPO 2’ andcorresponding to the two different stress distributions in the wall described in Section 3, werecarried out on the two models for each load distribution, so that in total four analyses were carriedout. Both piers and spandrels were modelled as described in Section 2.1.


In the SPO 1 analysis with the inverted triangular distribution, the collapse occurred after a storeymechanism was initiated at the first floor, while with the uniform distribution the mechanismoccurred at the ground floor (Figures 8(a), 9(a), and (b)). In the first case, all the piers of the weak



Table II. Analysed building—seismic weights and distribution of the lateral forces at the different floors.

Seismic force/base shear ratio at each floor Fi/�Fi

Floor Seismic weight Wi (kN) Inverted triangular distribution Uniform distribution

1st 278.7 0.67 0.47Ground 281.5 0.33 0.53

0 2 4 6 8 10top displacement [mm] storey displacement [mm]

0

30

60

90

120

150

180

base

she

ar [k

N]

1

2

3

B

A

0 2 4 6 8 100

30

60

90

120

150

180

stor

ey s

hear

[kN

]

3'

1'

2'

SPO 1 triangular distribution


SPO 1 uniform distribution


(a) (b)

Figure 8. Analysed building: (a) SPO global curves obtained using SAP 2000 v.10, including thepoints corresponding to the attainment of the global limit states (1,2,3) on the most conservativecurve (SPO 1 with inverted triangular distribution); (b) SPO curve of the weakest floor with thepoints corresponding to the attainment of the local limit states (1′,2′,3′), with: 1′, limited damage;

2′, significant damage; 3′, near collapse.

storey collapsed due to a sliding shear mechanism. In the second case, only one pier failed dueto diagonal cracking, leading to a strength loss higher than 20% of the maximum strength, whichdefined the ultimate global displacement. Also in the SPO 2 analyses, the structure collapsed dueto a storey mechanism characterized by a type of failure similar to that occurred in the SPO 1analyses.

By comparing the four SPO curves, it is clear that the minimum base shear strength was alwaysobtained when the triangular distribution of seismic forces is applied (Figure 8(a)). By using thistype of distribution, the ultimate strengths obtained in the two corresponding analyses were almostthe same. In terms of displacement capacity, all the analyses led to nearly the same value of8.5mm. This good agreement among the different analyses is very important since the damageindex representing the limit states will be defined in terms of displacement (interstorey drift, ISD).

5.2. Choice of the damage index and limit states

The assumed damage index was the ISD. The Draft No. 7 of the Eurocode 8, Part 3 [21] defines thelimit states corresponding to the achievement of the global displacement capacity of the structureon the push-over curve. Three points were located on such a curve in terms of top displacement:(i) yield point (Limit State of Limited Damage), (ii) 3

4 th of the ultimate top displacement capacity



Figure 9. Analysed building: deformed shape of the equivalent frame at the attainment of the ultimatedeformation in the first plastic hinge in SPO 1 analysis with inverted triangular distribution ((a)—pointA in Figure 8(a)) and with uniform distribution ((b)—point B in Figure 8(a)), with: RO, rocking; SL,sliding shear; DC, diagonal cracking shear, and underlining, attainment of the ultimate deformation.

(Limit State of Significant Damage), and (iii) point corresponding to at least 20% reduction in thepeak strength (Limit State of Near Collapse).

Among the four SPO analyses carried out, the one characterized by the smallest ISD at collapsewas selected (SPO 1 with inverted triangular distribution). The three limit states were then locatedon the global push-over curve as discussed above (points 1, 2, and 3 in Figure 8(a)). From thesevalues of total displacement, the corresponding limits in terms of ISD were evaluated, and thelocal push-over curve for the weakest floor (the first floor) was drawn (points 1′, 2′, and 3′ inFigure 8(b)). For the building under study, the obtained ISD/storey height ratios were

Limit State of Limited Damage: ISD/h=0.007%;Limit State of Significant Damage: ISD/h=0.2%;Limit State of Near Collapse: ISD/h=0.3%.

Since the Limit State of Limited Damage was conservatively defined by the yielding of the veryfirst hinge in the most unfavourable of the four analyses, the corresponding displacement value isquite low. Also the other two values are fairly low, mainly due to the inherent brittle behaviour ofthe analysed stone masonry wall.

6. INCREMENTAL DYNAMIC ANALYSIS

An incremental dynamic analysis (IDA) consists of a series of non-linear time-history analyses,each one carried out using the same seismic record but a different scale factor for the seismicintensity [34, 35]. The peak ground acceleration (PGA) is properly scaled in each analysis, in orderto cover the entire range of the structural response, from the yield point to the collapse. A seismicevent can differ from another one in the frequency content, the energy content, the duration, thenumber of passages through zero of the acceleration, etc., causing therefore different effects onthe same structure [35]. Fourteen different recorded earthquake ground motions were used in the



Table III. Characteristics of the selected earthquake ground motions.

Station of PGA IA PDH td DurationEarthquake Country detection Date Component (g) (cm/s) (cm s) (s) (s)

Imperial-Valley U.S.A. El Centro 15/05/1940 SOOE 0.348 148 2.2 13 20Mexico City Mexico Sct 27 19/09/1985 N–S 1.79 244 121.5 32 180Bucharest Romania Incerc 04/03/1977 N–S 1.930 75 6.0 14 45Friuli Italy Tolmezzo 06/05/1976 E–W 0.315 117 1.5 5 35Friuli Italy Buia 15/09/1976 N–S 0.109 15 0.7 8 26Tabas Iran Boshroych 16/09/1978 N79E 1.004 28 0.3 20 35Campano Lucano Italy Irpinia, Calitri 23/11/1980 E–W 0.175 134 7.3 47 86Umbro-Marchigiano

Italy Assisi 26/09/1997 E–W 1.083 22 1.3 32 55

Kocaeli Turkey Yesilkoy 17/08/1999 N–S 0.089 19 1.4 37 106Caldiran Turkey Maku 24/11/1976 S–E 0.956 9 0.1 19 28Gazli Uzbekistan Gazli 17/05/1976 E–W 0.720 495 1.3 7 13Montenegro Montenegro Bar-S.O. 15/04/1979 E–W 0.363 303 4.4 19 48Umbro-Marchigiano

Italy Colfiorito 26/09/1997 E–W 2.968 54 0.6 5 44

Thessalonika Greece Thessaloniki-City Hotel

20/06/1978 E–W 1.431 6 0.0 22 28

0 0.5 1 1.5 2 2.5 3 3.5 4natural period [s]

0

20

40

60

80

100

120

140

Sd

[cm

]

El Centro

Mexico City

Bucharest

Friuli (Tolmezzo)

Friuli (Buia)

Tabas

IrpiniaUmbro-Marchigiano (Assisi)

Kocaeli

Caldiran

Gazli

Montenegro

Umbro-Marchigiano (Colfiorito)

Thessalonika

Figure 10. Displacement spectra of the selected earthquake ground motions.

analyses. The record properties such as the PGA, arias intensity (IA), destructive potential (PDH)

[36], time of significant damage (td), and duration are displayed in Table III, and the correspondingelastic displacement spectra are shown in Figure 10.



0 2 4 6 8 10 12top displacement [mm]

0

40

80

120

160

200

240

El Centro

Mexico City

Bucharest

Friuli (Tolmezzo)

Friuli (Buia)

Tabas

Irpinia

Kocaeli

Caldiran

Gazli

Montenegro

Thessalonika

SPO 1 Triangular distribution

Umbro-Marchigiano (Assisi)

Umbro-Marchigiano (Colfiorito)

SPO 1 Uniform distribution

base

she

ar [k

N]

Figure 11. Analysed building: comparison among the higher SPO curve, the lower SPO curve, and thecurves obtained with the IDAs.

As previously discussed in Section 2.2, the hysteresis loop used for all the links in the IDAswas chosen by assuming the only failure mechanism of shear with diagonal cracking, using thesame strength values assumed for the ‘shear hinges’ in the SPO 1.

6.1. The followed procedure

For each earthquake ground motion, the proper PGA scale factors corresponding to the achievementin the equivalent frame of the three limit states of limited damage, significant damage, and nearcollapse were determined with the bisection method. A number of other PGA scale factors wereassumed in order to draw the whole incremental dynamic push-over curve. Such a curve joinstogether the ‘base shear-top floor displacement’ points characterized by different PGA scale factorsfor the same earthquake ground motion. The maximum base shear and the maximum top floordisplacement values are reported in the curve even if they do not occur simultaneously. The 14obtained IDA curves are displayed in Figure 11 together with the two SPO curves characterizedby the largest and smallest strength values (SPO 1 with uniform distribution and SPO 1 withtriangular distribution, respectively).


Figure 11 clearly shows the strong dependence of the structural response on the seismic record usedas input. Figure 12(a) and (c) depicts the IDA curves for two of the 14 earthquake ground motions(Kocaeli and Colfiorito, respectively) together with the three points 1′,2′, and 3′ corresponding tothe three limit states, and more points corresponding to different values of the PGA scale factor. Itcan be observed that the same PGA of 0.30g would cause a very different top displacement and,



Kocaeli ground motion

0 2 4 6 8 100

40

80

120

160

200

Kocaeli



0.18g0.22g

0.24g

0.26g

0.28g0.30g 0.305g

1'

2' 3'

Colfiorito ground motion

0 2 4 6 8 100

40

80

120

160

200

Colfiorito



0.20g0.30g

0.32g

0.33g 0.35g0.37g 0.38g

0.39g

1'

2'

3'

base

she

ar [k

N]

base

she

ar [k

N]

top displacement [mm]

top displacement [mm]

(a)

(c) (d)

(b)

Figure 12. Analysed building: (a) and (c): comparison between the IDA curves and the limit SPOcurves for the ground motions of Kocaeli and Colfiorito, respectively; (b) and (d): SAP2000 v.10base shear vs top floor displacement curves for a PGA scale factor of 0.30 for the ground motion of

Kocaeli and Colfiorito, respectively.

hence, damage level in the two earthquake ground motions. The wall, in fact, would experiencehigh plastic deformations that would lead to the attainment of the limit state of significant damagewith the Kocaeli seismic record (Figure 12(b)). Conversely, the limit state of limited damage wouldjust be overcome, with little plastic deformation, with the Colfiorito record (Figure 12(d)).

Owing to the different shape of the seismic records, also the range of PGA values required tolead the structure from the elastic (limited damage) to the near collapse limit markedly changes.This is clearly shown in Figure 13(a), where it can be observed that such a range extends from0.09g to 0.54g for the Thessalonika earthquake and from 0.11g to 0.39g for the Gazly earthquake,while for the Bucharest and Mexico City ground motions it extends from 0.23g to 0.28g only.

This difference can be explained with some considerations on the shape of the accelerationspectra of the seismic events (Figure 13(b)). The reduction in stiffness due to the plasticizationcaused by the seismic actions leads to an increase in the natural period of the structure, which is



0.55g0.20g 0.25g g05.0g53.0g51.0 0.45g0.40g0.30g0.10g

acceleration

Mexico CityBucharest

Umbro (Colfiorito)Campano

Montenegro

KocaeliFriuli (Tolmezzo)

Caldiran

Friuli (Buia)Tabas

El Centro

Umbro (Assisi)Gazli

Thessalonika

Near CollapseSignificant Damage

LimitedDamage

0 1 2 3 4natural period [s]

0

5

10

15

20

Sa

[m/s

2]

Mexico CityGazli

(a) (b)

Figure 13. (a) Summary of the PGA values necessary to reach the three limit states for the different seismicevents and (b) acceleration elastic spectra of the Mexico City and Gazli earthquake ground motions.

0.1 s in elastic phase. A seismic event with a higher spectral acceleration at low natural periods,such as Gazly, can then become less destructive for higher periods, allowing the structure toresist far beyond its elastic limit. The opposite would happen for shakings with lower spectralacceleration for low natural periods, such as Mexico City. Another significant outcome is that theIDA curves give higher base shears than those obtained by the SPO analysis. Possible reasons forthis outcome are [34] (i) the influence of the higher vibration modes (particularly the 2nd mode)in the IDA curves, ignored in the SPO analysis and (ii) the use in the IDAs of recorded shakings,which are very different from the artificial motions compatible with the design spectrum that wouldgenerally lead to IDA curves inside the SPO curves.

In terms of collapse mechanisms, the results obtained with the IDAs are similar to those of theSPO analyses with uniform distribution, since both analyses detect the weak storey at the groundfloor.

7. SEISMIC FRAGILITY CURVES

The seismic fragility of a structure is defined as the probability of reaching a defined limit state incorrespondence with a specific value of the chosen seismic intensity parameter. The fragility curveswere evaluated for the wall under study by considering the seismic record as the only uncertaintyparameter and, therefore, using the results of the IDAs. No allowance for the variability of themechanical properties of the masonry wall was made, since it is believed that the uncertaintyof the seismic record is far more important than the scatter in mechanical properties. Also, noout-of-plane failure mechanisms were considered. In order to have an immediate comparison withthe PGA levels that identify the four seismic zones defined by the Italian seismic regulation [20],the PGA was considered as the seismic intensity parameter. This choice was also suggested bythe too scattered distribution of PGA values that would have been obtained for the natural periodof the analysed wall under different earthquakes if other intensity parameters such as the spectraldisplacement Sd or the spectral acceleration Sa had been used. Such a scatter of PGA values, in



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7scale factor for g scale factor for g

0

20

40

60

80

100

cum

ula

tive

pro

ba

bili

ty[%

]

IVca

tego

ry

IIIca

tego

ry

IIca

tego

ry

Icat

egor

y

Limited Damage

Significant Damage

Near Collapse

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

20

40

60

80

100

cum

ula

tive

pro

ba

bili

ty[%

]

IVca

tego

ry

III c

ateg

ory

IIca

tego

ry

I cat

egor

y

Limited Damage

Significant Damage

Near Collapse

(a) (b)

Figure 14. Analysed building: fragility curves obtained, respectively, (a) as a result of the IDAs and (b)by a linear regression of the IDA outcomes by assuming a lognormal distribution.

conjunction with the low deformation level of masonry walls, would lead to results with little orno sense. For a same value of Sd or Sa, in fact, some earthquake ground motions would lead thestructure to the attainment of the elastic limit, while some others would lead far beyond the collapselimit. Conversely, the use of the PGA as seismic intensity parameter, although characterized bysignificant variation, led to consistent results (Figure 13(a)).

The cumulative probability curves corresponding to the three limit states are depicted inFigure 14(a). They were drawn using the IDAs by evaluating, for each value of PGA, the percentageof earthquake ground motions that reached the given limit state. Two ground motions, Bucharestand Mexico City, were ignored since their peculiar features would lead to outcomes very differentfrom all the other shakings. The piecewise-linear curves in Figure 14(a) were then approximatedusing a lognormal distribution (Figure 14(b)) [35, 37]. The procedure used is based on the repre-sentation of the (PGA, normalized cumulative probability) points on a lognormal probabilisticchart. By evaluating the linear regression curve, the parameters y and �y (lognormal mean andlognormal standard deviation, respectively) were then derived and the fit cumulative distributioncurve was drawn.

From Figure 14 it can be observed that the significant damage and the near collapse curvesare very close to each other. This means that, apart from the seismic records of Bucharest andMexico City where the two limits would be almost the same, once the significant damage limitstate is reached, only small PGA increments are needed for reaching the near collapse limit state.The fragility curves allow the designer to evaluate the seismic vulnerability of the analysed wallby direct comparison with the design PGA values assumed by the Italian seismic regulation forthe four different seismic zones [20], all of those corresponding to a return period of 475 years.The analysed wall does not suffer from damage for PGA values lower than 0.1g (fourth categoryaccording to [20]), showing an elastic behaviour for all the 14 seismic records. Conversely, for aPGA of 0.35g (first category) there is a 60% probability that the wall will collapse. The analysedbuilding therefore needs proper retrofit to reduce the seismic vulnerability when located in seismic



areas characterized by a design PGA of 0.35g. For a PGA of 0.25g (second category), the samestructure is significantly damaged but still exhibits some residual strength.

8. CONCLUDING REMARKS

The first aim of this work was to test the reliability of a widespread and simple software package,such as SAP2000®, v.10 for performing SPO analyses on masonry buildings using the ‘equiv-alent frame’ simplified modelling. The proposed modelling was validated on two walls of anexisting masonry building, already analysed with different advanced programs by other researchers.A significant limitation of the SAP2000 v.10 modelling is the impossibility, during the SPO anal-ysis, to take into account the possible influence on the structural global strength of the axial forcevariation in the piers (stress �0). Based on the outcomes of some comparisons with numericalresults carried out using more advanced software, it was found that the proposed model in SAP2000 v.10 can be used for push-over analyses of masonry walls of usual and regular geometry. Inthis case the strengths of the piers can be evaluated using, for �0, the values corresponding to thestep 0 of loading (only gravity load, no horizontal loads applied yet).

The second aim of this work was to investigate the seismic performance of a typical standardmasonry building located in the north-east of Italy. An SPO analysis and an incremental dynamicanalysis were carried out on the facade wall of the building. The IDA pointed out how sensitive thestructural response is to the type of earthquake ground motion assumed as input. Same PGAs indifferent seismic records may lead to very different results in terms of displacement and strengthdemands on the same wall. The range of PGA values that led the structure from the elastic to thecollapse limit markedly changed depending on the assumed seismic event. The seismic fragilitycurves were then derived assuming the seismic event as the only uncertainty parameter. Based onthe obtained curves, there was a 60% probability for the analysed wall to reach the collapse inseismic regions characterized by PGA=0.35g, such as the first category areas according to the newItalian seismic code. A proper seismic retrofit would therefore be required to reduce the seismicvulnerability of the building under study when located in those areas. In second category areas(PGA=0.25g), the same structure was significantly damaged but still exhibited some residualstrength. Those conclusions are applicable to buildings with walls having similar geometrical andmechanical properties to the analysed one, and where adequate connection to the floors is providedso that no significant out-of-plane damage may occur.

ACKNOWLEDGEMENTS

The authors wish to thank Mr Davide Bolognini from the European Centre for Training and Researchin Earthquake Engineering (EUCENTRE) of Pavia (Italy) for the information provided on the use ofthe SAM code. Associate Professor Natalino Gattesco from the Department of Architectural and UrbanDesign, Faculty of Architecture, University of Trieste (Italy), is also acknowledged for his valuable advice.

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