Pasticier_nonlinear_seismic_analysis of a Masonry Building in SAP2000

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

Non-linear seismic analysis and vulnerability evaluation of a masonry building by means of the SAP2000 V.10 code Laurent Pasticier1, § , Claudio Amadio2, ‡ and Massimo Fragiacomo3, ∗, †, ‡ 1 Via Schiaparelli 8, 34143 Trieste, Italy of Civil & Environmental Engineering, University of Trieste, Piazzale Europa 1, 34121 Trieste, Italy 3 Department of Architecture & Planning, University of Sassari, Piazza Duomo 6, 07041 Alghero, Italy

2 Department

SUMMARY The aim of the paper is to explore the possibilities offered by SAP2000® v.10, a software package with user-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 two walls, already analysed by other researchers using advanced programs. The equivalent frame modelling was employed in all analyses carried out. The code was then used to investigate the seismic performance of an existing two-storey building typical of the north-east of Italy, with the walls being made of roughly squared stones. An SPO analysis was performed first on the most significant wall, followed by a number of time-history analyses aimed to evaluate the dynamic push-over curves. Finally, the seismic fragility curves 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 many European 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 areas with 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, Piazza Duomo 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 the retrofit 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 finite element models [2, 3], based on proper constitutive laws for the masonry components [4], allow an accurate determination of the critical points in the structure including the failure mechanisms, but are time consuming and require the use of expensive and complex software. Other simpler models are based on ‘macromodel’ modelling, where the masonry building is divided into a number of one- or two-dimensional ‘macroelements’ [5–14]. In most of the models based on two-dimensional elements, the hypothesis of material with no tensile strength is assumed [15], which usually results in a quite complex iterative process. Among the models using one-dimensional elements, the POR method is well known and extensively used. Such a method assumes, in its original version improved later [16, 17], that the structural collapse occurs because of a storey mechanism. The failure is assumed to take place only in the piers, and no allowance for the possible damage of the spandrel 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 higher number of possible failure mechanisms occurring inside each macroelement, such as shear with diagonal cracking, shear with sliding, and rocking (Figure 1), can be considered. In accordance with the use of this approach, which was used for example in the SAM code [10–12] and in comparative analyses between macromodels and finite element models [18], the spandrels and the piers are regarded as elastic, their intersections are modelled as fully rigid, and the possible mechanical non-linearity is concentrated in some well-defined cross-sections inside the elastic parts. 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 European Codes encourage the use of non-linear static push-over (SPO) analysis and require a control of the spandrels, which is not possible using the POR method. The use of suitable programs is therefore needed 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 and well-known package for structural analysis such as SAP2000® v.10 [22] for seismic design of masonry buildings. This code was already used for seismic analysis of masonry buildings using both one- and two-dimensional elements [18, 23]. The reliability and limitations of the code were first investigated by performing an SPO analysis of two multi-storey walls, already analysed by other authors using advanced programs. Some non-linear static and dynamic analyses were then Copyright q


Earthquake Engng Struct. Dyn. 2008; 37:467–485 DOI: 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 frame method. The SAP2000® v.10 package allows the user to account for the non-linear mechanical behaviour of the material by introducing the following elements with lumped plasticity in the equivalent frame: • plastic hinges; • non-linear links. The plastic hinges were used in SPO analyses since they allow the user to accurately follow the structural performance beyond the elastic limit at each step of the incremental analysis. The nonlinear links were instead used in time-history analyses since they allow the user to accurately define the cyclic behaviour of the elements including a proper degradation rule [24]. The mechanical properties of these non-linear elements were defined based on the possible failure mechanisms of masonry macroelements shown in Figure 1 [25–28]. The adopted modelling is described in the following 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 is depicted in Figure 2(a) [24]. The masonry piers were modelled as elastoplastic (as also suggested in [20]) with final brittle failure (Figure 2(b)) [11] by introducing two ‘rocking hinges’ at the end of the deformable parts and one ‘shear hinge’ at mid-height (Figure 3(a)). A rigid-perfectly plastic behaviour 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 shear strength is concerned, according to the experimental test outcomes [28], it was decided to consider two 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 by Turnˇsek and Cacovic [29] and later modified by Turnˇsek 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]: 0 0 D 2 t Mu = 1− (1) 2 k fd 1.5 f v0d Dt 0 f Vu = 1+ (2) 1.5 f v0d 3 0 f v0d + 2 m Dt Vus = 3H0 1+ f v0d D0 Copyright q


(3)


470


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

elastic part

fully rigid part

elastic part

fully rigid part

Spandrel Joint Pier Analysed wall

(a)

"shear hinge" for the piers

"rocking hinge" for the piers

"shear hinge" for the spandrels

(b)

"shear hinge" for the piers

(c)

Ground Floor

0

1m

2m

Figure 3. Analysed building: (a) plastic hinges’ location in the equivalent frame model of the wall facade used for static analyses; (b) non-linear links’ location in the equivalent frame model of the wall facade used 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 taking into account the vertical stress distribution at the compressed toe (a common assumption is an equivalent rectangular stress block with k = 0.85), f d the design compression strength, f v0d the design shear strength with no axial force; (friction coefficient) = 0.4, the coefficient related to the pier geometrical ratio, H0 the effective pier height (distance of the cross-section in which the 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), and the ultimate rotation u corresponds to an ultimate lateral deflection u equal to 0.8% of the deformable height of the pier, minus the elastic lateral deflection, as recommended in [20]. For the 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 Copyright q




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of the pier, minus the elastic lateral deflection, as recommended in [20]. The failure was assumed to have occurred in the pier when the first between the ultimate rotation u in the plastic hinge and the ultimate shear displacement u in the shear hinge was attained. Since in SAP2000 it is not possible to automatically control the total deflection of an entire macroelement if more than one of its plastic hinges exceed the elastic limit, such a quantity was manually checked on every macroelement at the end of each load step. As far as the modelling of the spandrel beams is concerned, assuming the presence of a lintel properly restrained at both supports, only one ‘shear hinge’ was introduced at mid-span (Figure 3(a)), with the shear strength Vu given by Vu = ht f v0d

(4)

where h is the spandrel depth, t the spandrel thickness, and f v0d the design shear strength with no axial force. A brittle–elastic behaviour with residual strength after cracking equal to 14 th of the maximum 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 link allows the user to reproduce the cyclic behaviour of the entire macroelement by defining the shape of the hysteresis loop and the degradation of both strength and stiffness (Figure 4(a)) through a proper choice of the mechanical parameters. Owing to the complexity of the non-linear time-history analyses, in order to reduce the computational burden, only one ‘shear link’ was introduced at mid-height of the piers (Figure 3(b)). This choice 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 geometrical ratio (and therefore strength) of the spandrels relative to the piers. For the sake of safety, a hysteretic behaviour characterized by shear failure with diagonal cracking was assumed. This type of failure is, in fact, more fragile and, therefore, critical in terms of displacement demand than sliding shear.

Fy1 1Fy1

A' C' B'

(a)

P2

P1

P3 B

Fy1=Fy2 A

C

A

1

2Fy2 2Fy2 2Fy2

Fy1= 2 Fy2 =0.45Fy1

B' V=30%

(b)

V=30% B

2Fy2= 1Fy1 =0.45Fy2

A' Fy2=Fy1

100 80 60 40 20 0 -20 -40 -60 -80 -100

base shear [kN]

P4

2 Fy2

(c)

Magenes et al. SAP2000 v.10 -8 -6 -4 -2 0 2 4 6 top displacement [mm]

8

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 parameters for stiffness and strength degradation control; and (c) comparison between the experimental curve detected with the quasi-static cyclic test performed on a pier by Anthoine et al. [26] and the numerical curve detected with SAP2000® v.10 for the same pier using the multilinear-plastic pivot non-linear link with the parameters indicated in (b). Copyright q



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The link parameters 1 , 2 , 1 , and 2 , which control the stiffness degradation during the unloading procedure, were chosen so as to reproduce the experimental results obtained by Magenes and Coworkers [26] on a single brick wall (Figure 4(b) and (c)). The parameters were all assumed to be equal to 0.45. The same values were also adopted for the other stone masonry walls investigated in this paper. The maximum shear strength and plastic displacements were assumed to be the same as 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 B in Figure 5) of stone masonry buildings previously analysed in the ‘Catania Project’ [32] were modelled with SAP2000 v.10. The ‘Catania Project’ was an extensive nationwide research project focused on seismic performance of existing masonry buildings. In such a project, some laboratory and in-situ tests were performed to characterize the mechanical properties of the masonry. In addition, numerical modelling of the structural response was undertaken by a number of Italian universities, each of them using a different advanced software package. The University of Pavia used the SAM code, which is considered as an important reference for this work. Such a code, which is based on the equivalent frame modelling, was previously validated on a number of experimental tests 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 criterion for regular brick masonry walls [28], which is different from Equation (2). For the spandrel beams, Equations (1) and (4) are adopted. The Basilicata research group used a no tensile strength macroelement model with crushing and shear failures [6, 7], while the Genoa research group used a finite elements model with layer failures [3]. The mechanical properties used in the analyses were E (Young’s modulus) = 1500 N/mm2 , G (shear modulus) = 250 N/mm2 , (unit weight) = 1900 kg/m3 , f d (design compression strength) = 2.4 N/mm2 , f v0d (design shear strength with no vertical stress) = 0.2 N/mm2 , and (friction coefficient) = 0.5. The lateral loads representing the seismic action were applied by assuming the inverted triangular distribution. The weights of each floor and the ratios between the seismic force on 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 has f to be highlighted. The SAM program updates the strengths Mu , Vu and Vus of the plastic hinges during the non-linear analysis if the quantities H0 and 0 (Equations (1)–(3)) change due to the effect of the lateral loads. While the quantity H0 almost remains constant, the quantity 0 can markedly 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 the significance 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. In the distribution No. 1, the hinge strengths were calculated using, for 0 , the values read on the structure at the step 0 of the analysis, considering only gravity loads and no lateral loads. In the distribution No. 2, the hinge strengths were calculated using, for 0 , the values obtained by applying the gravity load and increasing the lateral loads up to the attainment of the elastic limit of the frame. Four independent SPO analyses were then carried out: two on the wall A with the two stress distributions No. 1 and No. 2, denoted as SPO 1 and SPO 2, and the corresponding two on the wall B, also denoted as SPO 1 and SPO 2. Details on the axial force, shear force, and bending Copyright q



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Figure 5. Validation: elevation of wall A (a) and wall B (b), with the equivalent frame modelling used in SAM 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 Floor 2nd 1st Ground

Wall B

Seismic weight Wi (kN)

Seismic force/base shear ratio Fi / Fi


Seismic force/base shear ratio Fi / Fi

548.2 1096.3 1277.0

0.38 0.40 0.22

290.0 480.0 626.0

0.41 0.37 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 Figure 6(b) for wall B. The failure mechanisms as detected by SAP 2000 in the SPO 1 and SPO 2 analyses are displayed in Figure 7 for wall A. Similar to the SAM method, both SPO 1 and SPO 2 analyses detected a storey mechanism at the second floor of wall A with the same value of ultimate strength. Such an ultimate strength was higher than the strength obtained by the SAM code, but lower than those obtained by the Basilicata and Genoa research groups. The same top displacement was obtained in both SPO 1 and SPO 2 analyses. Such a value was close to that detected by the Genoa research group and fairly different from those detected by the SAM and Basilicata research group. Copyright q



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1800

800

base shear [kN]

base shear [kN]

X

1500 1200 900

SPO 1

600

SPO 2 SAM

300

Genoa R.G. Basilicata R.G.

0

600

400 SPO 1 SPO 2

200

SAM Genoa R.G. Basilicata R.G.

0 0

(a)

10 20 30 top displacement [mm]

40

0

(b)


40

Figure 6. Validation: comparison between the push-over curve obtained with SAP2000® v.10 and those obtained 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 deformation in 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 for wall B, with a storey mechanism occurring at the second floor. The top displacement was almost the same as that detected by the Genoa research group, but different from that detected by the Basilicata 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 program for both walls A and B. The ultimate displacement was similar for wall B but more different for wall A. Such a difference is due to the change from compression to tension taking place during the analysis in some piers (the circled ones in Figure 7). Such behaviour is not automatically detected by SAP2000 v.10, which assumes pier strength independent of the axial force. Differently, the Copyright q




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SAM method updates the pier strength at each step of analysis. Since no tensile strength was assumed for the masonry piers, the analysis with the SAM code ended well before the SAP 2000 model, leading to a far lower ultimate displacement capacity. The top displacement attained in the SAP 2000 model when the axial force in the piers turned from compression to tension corresponds to point X in Figure 6(a). This new value matches quite well with the one obtained as a result of the analysis carried out with the SAM program. The results of SPO 1 and SPO 2 analyses in terms of ultimate strength and displacement were almost the same for both walls. In the cases of shear walls of usual geometry, in fact, the variation of vertical stresses in the walls due to overturning moment caused by lateral load is fairly low. The main limit of the SAP2000 v.10 model, which is the impossibility to update the strengths of the piers based on the variation of axial force, seems therefore not to be so crucial in the SPO analysis of ordinary masonry buildings. However, in terms of failure mechanisms, only the outcomes from the 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 a stone masonry house typical of the north-east of Italy. The vertical structure is a single layer nonretrofitted masonry made of roughly squared sandstone, while the floors are composites of concrete slabs and timber beams. In order to reduce the computational burden of the dynamic analyses needed for the vulnerability assessment, only the facade wall was analysed using the proposed SAP2000 v.10 model. The design values assumed for the mechanical properties are based on the mean values measured in situ on a number of similar buildings located in the same area as the analysed building: f d = 0.8 N/mm2 , f v0d = 0.032 N/mm2 , E = 1600 N/mm2 , and G = 640 N/mm2 . Only the in-plane seismic performance of the wall was investigated, assuming that the wall was effectively 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 fragility curves. As recommended by recent codes of practice and regulations [20], the horizontal seismic loads were applied adopting two different distributions (Table II): (i) proportional to the product of the masses by the floor heights (inverted triangular distribution) and (ii) proportional to the floor masses (uniform distribution). Two SPO analyses, denoted as ‘SPO 1’ and ‘SPO 2’ and corresponding to the two different stress distributions in the wall described in Section 3, were carried out on the two models for each load distribution, so that in total four analyses were carried out. Both piers and spandrels were modelled as described in Section 2.1. 5.1. Discussion of the results In the SPO 1 analysis with the inverted triangular distribution, the collapse occurred after a storey mechanism was initiated at the first floor, while with the uniform distribution the mechanism occurred at the ground floor (Figures 8(a), 9(a), and (b)). In the first case, all the piers of the weak Copyright q



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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


Inverted triangular distribution

Uniform distribution

278.7 281.5

0.67 0.33

0.47 0.53

1st Ground

180

180 B 2

120

3

1

150 storey shear [kN]

base shear [kN]

150

A

90 SPO 1 triangular distribution

60

SPO 2 triangular distribution SPO 1 uniform distribution

30

120 2'

90

3'

1' 60 30

SPO 2 uniform distribution

0

0 0

(a)

2


10

0 (b)

2 4 6 8 storey displacement [mm]

10

Figure 8. Analysed building: (a) SPO global curves obtained using SAP 2000 v.10, including the points corresponding to the attainment of the global limit states (1, 2, 3) on the most conservative curve (SPO 1 with inverted triangular distribution); (b) SPO curve of the weakest floor with the points 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 due to diagonal cracking, leading to a strength loss higher than 20% of the maximum strength, which defined the ultimate global displacement. Also in the SPO 2 analyses, the structure collapsed due to a storey mechanism characterized by a type of failure similar to that occurred in the SPO 1 analyses. By comparing the four SPO curves, it is clear that the minimum base shear strength was always obtained when the triangular distribution of seismic forces is applied (Figure 8(a)). By using this type of distribution, the ultimate strengths obtained in the two corresponding analyses were almost the same. In terms of displacement capacity, all the analyses led to nearly the same value of 8.5 mm. This good agreement among the different analyses is very important since the damage index 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 the limit states corresponding to the achievement of the global displacement capacity of the structure on 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) 34 th of the ultimate top displacement capacity Copyright q




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Figure 9. Analysed building: deformed shape of the equivalent frame at the attainment of the ultimate deformation in the first plastic hinge in SPO 1 analysis with inverted triangular distribution ((a)—point A 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 the peak strength (Limit State of Near Collapse). Among the four SPO analyses carried out, the one characterized by the smallest ISD at collapse was selected (SPO 1 with inverted triangular distribution). The three limit states were then located on the global push-over curve as discussed above (points 1, 2, and 3 in Figure 8(a)). From these values of total displacement, the corresponding limits in terms of ISD were evaluated, and the local push-over curve for the weakest floor (the first floor) was drawn (points 1 , 2 , and 3 in Figure 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 very first hinge in the most unfavourable of the four analyses, the corresponding displacement value is quite low. Also the other two values are fairly low, mainly due to the inherent brittle behaviour of the 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 seismic intensity [34, 35]. The peak ground acceleration (PGA) is properly scaled in each analysis, in order to cover the entire range of the structural response, from the yield point to the collapse. A seismic event can differ from another one in the frequency content, the energy content, the duration, the number of passages through zero of the acceleration, etc., causing therefore different effects on the same structure [35]. Fourteen different recorded earthquake ground motions were used in the Copyright q



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Table III. Characteristics of the selected earthquake ground motions. Station of detection

Date

Component

U.S.A. Mexico Romania Italy Italy Iran Italy Italy

El Centro Sct 27 Incerc Tolmezzo Buia Boshroych Irpinia, Calitri Assisi

15/05/1940 19/09/1985 04/03/1977 06/05/1976 15/09/1976 16/09/1978 23/11/1980 26/09/1997

SOOE N–S N–S E–W N–S N79E E–W E–W

0.348 1.79 1.930 0.315 0.109 1.004 0.175 1.083

Turkey Turkey Uzbekistan Montenegro Italy

Yesilkoy Maku Gazli Bar-S.O. Colfiorito

17/08/1999 24/11/1976 17/05/1976 15/04/1979 26/09/1997

N–S S–E E–W E–W E–W

Greece

ThessalonikiCity Hotel

20/06/1978

Earthquake

Country

Imperial-Valley Mexico City Bucharest Friuli Friuli Tabas Campano Lucano UmbroMarchigiano Kocaeli Caldiran Gazli Montenegro UmbroMarchigiano Thessalonika

140

PGA IA PDH td Duration (g) (cm/s) (cm s) (s) (s) 148 244 75 117 15 28 134 22

13 32 14 5 8 20 47 32

20 180 45 35 26 35 86 55

0.089 19 0.956 9 0.720 495 0.363 303 2.968 54

1.4 37 0.1 19 1.3 7 4.4 19 0.6 5

106 28 13 48 44

E–W

1.431

0.0 22

28

2.5

3

6

2.2 121.5 6.0 1.5 0.7 0.3 7.3 1.3

El Centro Mexico City Bucharest

120

Friuli (Tolmezzo) Friuli (Buia) Tabas

Sd [cm]

100

Irpinia Umbro-Marchigiano (Assisi) Kocaeli

80

Caldiran Gazli Montenegro

60

Umbro-Marchigiano (Colfiorito) Thessalonika

40

20

0 0

0.5

1

1.5

2

3.5

4

natural period [s]

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 corresponding elastic displacement spectra are shown in Figure 10. Copyright q




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240

200

El Centro Mexico City

160

Friuli (Tolmezzo)

base shear [kN]

Bucharest Friuli (Buia) Ta bas Irpinia

120

Umbro-Marchigiano (Assisi) Kocaeli Caldiran

80

Gazli Montenegro

Umbro-Marchigiano (Colfiorito)

40

Thessalonika

SPO 1 Triangular distribution SPO 1 Uniform distribution

0 0

2

4

6 8 top displacement [mm]

10

12

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

As previously discussed in Section 2.2, the hysteresis loop used for all the links in the IDAs was chosen by assuming the only failure mechanism of shear with diagonal cracking, using the same 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 achievement in the equivalent frame of the three limit states of limited damage, significant damage, and near collapse were determined with the bisection method. A number of other PGA scale factors were assumed in order to draw the whole incremental dynamic push-over curve. Such a curve joins together the ‘base shear-top floor displacement’ points characterized by different PGA scale factors for the same earthquake ground motion. The maximum base shear and the maximum top floor displacement values are reported in the curve even if they do not occur simultaneously. The 14 obtained IDA curves are displayed in Figure 11 together with the two SPO curves characterized by the largest and smallest strength values (SPO 1 with uniform distribution and SPO 1 with triangular distribution, respectively). 6.2. Discussion of the results Figure 11 clearly shows the strong dependence of the structural response on the seismic record used as 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 to the three limit states, and more points corresponding to different values of the PGA scale factor. It can be observed that the same PGA of 0.30g would cause a very different top displacement and, Copyright q



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Kocaeli ground motion

base shear [kN]

200

3' 2' 0.30g 0.305g

0.28g

0.24g 160 0.22g 0.18g 120 1'

0.26g

80

Kocaeli SPO 1 triangular distribution SPO 1 uniform distribution

40 0 0

2

(a)


10 (b)

Colfiorito ground motion 2'

200 0.33g

base shear [kN]

160

0.35g

0.37g 0.38g

0.39g

0.30g 0.32g 0.20g 120 1' 80

3'

Colfiorito SPO 1 triangular distribution SPO 1 uniform distribution

40 0 0 (c)

2


10 (d)

Figure 12. Analysed building: (a) and (c): comparison between the IDA curves and the limit SPO curves for the ground motions of Kocaeli and Colfiorito, respectively; (b) and (d): SAP2000 v.10 base 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 experience high plastic deformations that would lead to the attainment of the limit state of significant damage with the Kocaeli seismic record (Figure 12(b)). Conversely, the limit state of limited damage would just 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 to lead 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 from 0.09g 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 acceleration spectra of the seismic events (Figure 13(b)). The reduction in stiffness due to the plasticization caused by the seismic actions leads to an increase in the natural period of the structure, which is Copyright q



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Thessalonika Gazli Umbro (Assisi) Caldiran El Centro Tabas Friuli (Buia) Friuli (Tolmezzo) Kocaeli Umbro (Colfiorito) Campano Montenegro Bucharest Mexico City

20 Mexico City Gazli

S a [m/s2 ]

15

10

5

0 0.10g

0.15g

0.20g

0.25g

(a)

0.30g

0.35g

0.40g

0.45g

0.50g

0.55g

acceleration Limited Damage

Significant Damage

0

(b)

1

2

3

4

natural period [s]

Near Collapse

Figure 13. (a) Summary of the PGA values necessary to reach the three limit states for the different seismic events 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 to resist far beyond its elastic limit. The opposite would happen for shakings with lower spectral acceleration for low natural periods, such as Mexico City. Another significant outcome is that the IDA curves give higher base shears than those obtained by the SPO analysis. Possible reasons for this 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 would generally lead to IDA curves inside the SPO curves. In terms of collapse mechanisms, the results obtained with the IDAs are similar to those of the SPO analyses with uniform distribution, since both analyses detect the weak storey at the ground floor.

7. SEISMIC FRAGILITY CURVES The seismic fragility of a structure is defined as the probability of reaching a defined limit state in correspondence with a specific value of the chosen seismic intensity parameter. The fragility curves were evaluated for the wall under study by considering the seismic record as the only uncertainty parameter and, therefore, using the results of the IDAs. No allowance for the variability of the mechanical properties of the masonry wall was made, since it is believed that the uncertainty of the seismic record is far more important than the scatter in mechanical properties. Also, no out-of-plane failure mechanisms were considered. In order to have an immediate comparison with the 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 by the too scattered distribution of PGA values that would have been obtained for the natural period of the analysed wall under different earthquakes if other intensity parameters such as the spectral displacement Sd or the spectral acceleration Sa had been used. Such a scatter of PGA values, in Copyright q



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0

(a)

0.1

0.2 0.3 0.4 0.5 scale factor for g

20

0 0.6

0.7

0

(b)

0.1

I category

I category

0

III category

20

II category

Near Collapse

Limited Damage Significant Damage Near Collapse

40 II category

Limited Damage Significant Damage

40

60

III category

60

80

IV category

80

cumulative probability [%]

100

IV category

cumulative probability [%]

100

0.2 0.3 0.4 0.5 scale factor for g

0.6

0.7

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 or no sense. For a same value of Sd or Sa , in fact, some earthquake ground motions would lead the structure to the attainment of the elastic limit, while some others would lead far beyond the collapse limit. Conversely, the use of the PGA as seismic intensity parameter, although characterized by significant variation, led to consistent results (Figure 13(a)). The cumulative probability curves corresponding to the three limit states are depicted in Figure 14(a). They were drawn using the IDAs by evaluating, for each value of PGA, the percentage of earthquake ground motions that reached the given limit state. Two ground motions, Bucharest and Mexico City, were ignored since their peculiar features would lead to outcomes very different from all the other shakings. The piecewise-linear curves in Figure 14(a) were then approximated using a lognormal distribution (Figure 14(b)) [35, 37]. The procedure used is based on the representation of the (PGA, normalized cumulative probability) points on a lognormal probabilistic chart. By evaluating the linear regression curve, the parameters y and y (lognormal mean and lognormal standard deviation, respectively) were then derived and the fit cumulative distribution curve was drawn. From Figure 14 it can be observed that the significant damage and the near collapse curves are very close to each other. This means that, apart from the seismic records of Bucharest and Mexico City where the two limits would be almost the same, once the significant damage limit state 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 wall by direct comparison with the design PGA values assumed by the Italian seismic regulation for the 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 category according to [20]), showing an elastic behaviour for all the 14 seismic records. Conversely, for a PGA of 0.35g (first category) there is a 60% probability that the wall will collapse. The analysed building therefore needs proper retrofit to reduce the seismic vulnerability when located in seismic Copyright q




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areas characterized by a design PGA of 0.35g. For a PGA of 0.25g (second category), the same structure 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 ‘equivalent frame’ simplified modelling. The proposed modelling was validated on two walls of an existing 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 analysis, to take into account the possible influence on the structural global strength of the axial force variation in the piers (stress 0 ). Based on the outcomes of some comparisons with numerical results carried out using more advanced software, it was found that the proposed model in SAP 2000 v.10 can be used for push-over analyses of masonry walls of usual and regular geometry. In this case the strengths of the piers can be evaluated using, for 0 , the values corresponding to the step 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 standard masonry building located in the north-east of Italy. An SPO analysis and an incremental dynamic analysis were carried out on the facade wall of the building. The IDA pointed out how sensitive the structural response is to the type of earthquake ground motion assumed as input. Same PGAs in different seismic records may lead to very different results in terms of displacement and strength demands on the same wall. The range of PGA values that led the structure from the elastic to the collapse limit markedly changed depending on the assumed seismic event. The seismic fragility curves were then derived assuming the seismic event as the only uncertainty parameter. Based on the obtained curves, there was a 60% probability for the analysed wall to reach the collapse in seismic regions characterized by PGA = 0.35g, such as the first category areas according to the new Italian seismic code. A proper seismic retrofit would therefore be required to reduce the seismic vulnerability 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 residual strength. Those conclusions are applicable to buildings with walls having similar geometrical and mechanical properties to the analysed one, and where adequate connection to the floors is provided so 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 Research in Earthquake Engineering (EUCENTRE) of Pavia (Italy) for the information provided on the use of the SAM code. Associate Professor Natalino Gattesco from the Department of Architectural and Urban Design, Faculty of Architecture, University of Trieste (Italy), is also acknowledged for his valuable advice. REFERENCES 1. Lourenço PB. Computations on historic masonry structures. Progress in Structural Engineering and Materials 2002; 4(3):301–319. 2. Amadio C, Fragiacomo M. Seismic analysis of a historical stone-masonry industrial building by the Abaqus code. European Earthquake Engineering 2003; 17(1):18–30. Copyright q



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