Considerations and suggestions about the seismic assessment procedures of existing r.c. frame...

Considerations and Suggestions about the Seismic Assessment Procedures of Existing R.C. Frame Buildings According to OPCM 3431 and Eurocode 8

MPAMPATSIKOS V.1, PETRINI L.2, NASCIMBENE R.3

1 ROSE School, IUSS, University of Pavia, Pavia (Italy) 2 Structural Engineering Department, Politecnico of Milano, Milano (Italy) 3 Eucentre, European Centre for Training and Research in Earthquake Engineering, Pavia (Italy)

Email corresponding author: [email protected]

Abstract: In the European seismic countries, the majority of the building heritage was designed according to out-of-date or even non-seismic codes. The lack of considerations about the nonlinear response during the design stage leads to relevant uncertainties, regarding the potential development, location and extension of inelastic mechanisms, as well as their ductility capacity. Therefore, the reliability of the force-based assessment procedure obtained performing a linear analysis is in general questionable, due to the difficulties associated to the correct evaluation of the reduction factor of the internal forces. The direct consequence is the considerable gain in complexity of the assessment procedures proposed in the Italian seismic code and in Eurocode 8. Both codes promote the use of the nonlinear analyses and the displacement-based approach to assess the ductile response. In the light of these considerations, this work intends to furnish useful tools to perform a reliable assessment of existing r.c. frame buildings. In particular, four irregular structures, built according to out-of-date codes, were analyzed with the following aims: i) suggesting simplified approaches and improvements for the assessment procedures, concerning the evaluation of both seismic demand and capacity of the structural members; ii) checking the importance of defining properly the effective stiffness in linear analyses; iii) checking if the conditions to accept the results of linear methods proposed in the two codes are appropriate; v) giving a judgment about the reliability of the conventional 2-D pushover analysis.

Keywords: Existing r.c. frame buildings, chord rotation, shear assessment, effective stiffness, pushover analysis.

1. INTRODUCTION

In the European seismic countries, most of the existing r.c. frame buildings were designed according to out-of-date or even non-seismic codes. Therefore, the uncertainties about the nonlinear behaviour are relevant, since the presence and location of potential inelastic zones, as well as their ductility capacity are unknown. It leads to important consequences,

Environmental Semeiotics (2009) 2(1), 31-47 DOI 10.3383/es.2.1.3

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32 MPAMPATSIKOS ET AL.

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concerning the reliability of the methods of analysis and the assessment procedures. In fact, it seems opportune to limit the use of linear analyses to the buildings which show a uniform distribution of the nonlinearity. Moreover, even if these conditions are satisfied, the accuracy of the force-based assessment procedure got through linear analyses is in general questionable, due to the difficulties associated to the correct evaluation of the reduction factor of the internal forces. The present seismic codes recommend to perform nonlinear analyses and to use both force-based and displacement-based assessment procedures to control the shear and the ductility performance, respectively. The assessment process, hence, turns out to be considerably complex and lengthy.

In this work, four r.c. frame buildings built before 1980 and located in moderate-to-high seismic zones were examined according to the recommendations of OPCM 3431 and Eurocode 8 (EN 1998-3). All procedures (linear and nonlinear, static and dynamic) were critically considered, performed and compared to each other (even when their applicability conditions were not satisfied), both at the local (member) and global (structural) level. In particular, in the first part of this work, the results of sensitivity analyses will be shown, in order to suggest simplified approaches for the assessment procedure of the ductile mechanisms, concerning the evaluation of both seismic demand and capacity of the structural members. Moreover, the consistency of the brittle assessment results based on the recommendations of OPCM 3431 and EC8 will be checked, in order to test the reliability of the non-seismic approach proposed in OPCM 3431. In the second part, the attention will be shifted from the local (member) level to the global (structural) level. The results of the comparison between linear and dynamic nonlinear analyses will be shown, in order to check if the recommended conditions of applicability of linear methods are adequate, or too restrictive. Since the reliability of the results of linear analyses (particularly in terms of ductility demand) may depend strongly on a suitable modelling of the effective stiffness EIeff of the structural members, suggestions about the appropriate way to evaluate EIeff will be proposed. On the other hand, although the two codes recommend to perform the nonlinear static analysis whenever the linear analyses are not applicable, the use of the conventional pushover method may be questionable, particularly for 3-D irregular structures. Hence, the reliability of the nonlinear static analysis procedure will be tested too.

Since the four buildings are irregular in plan and elevation, with different structural configurations, geometry of the members and values of

SEISMIC ASSESSMENT PROCEDURES OF EXISTING R.C. FRAME BUILDINGS 33

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mechanical properties, the conclusions based on the obtained results can be judged as satisfactory, although further efforts will be required to get general rules, applicable to every r.c. frame building. All the results shown in the following refer to the Limit State of Significant Damage (SD LS).

2. ANALYZED BUILDINGS

The “Sede Comunale”, located in Vagli Sotto (Tuscany, Italy), designed in 1965, is a two-storey r.c. frame building with masonry infill. The shape of the building (Figure 1) is roughly rectangular (27.25m x 13.60m). The frames are mono-directional and oriented parallel to the short sides of the building. From extended in-situ inspections, fcm=8.3MPa was obtained. From the original drawings, fym=440MPa was assumed for both longitudinal and transversal steel (Feb44k).

The “Scuola Pascoli”, located in Barga (Tuscany, Italy), designed in 1978, is a two-storey r.c. frame structure with masonry infills. The shape of the building (Figure 2) is roughly square (40.90m x 35.60m). The frames are bidirectional. From extended in-situ inspections, fcm=30MPa was obtained. From the original drawings, fym=440MPa was assumed for both longitudinal and transversal steel (Feb44k).

The “Scuola Puccetti”, located in Gallicano (Tuscany, Italy), designed in 1963, is a two-storey r.c. frame building with masonry infill, a small basement floor and sloping roofs of varying height. The building is strongly asymmetric and not compact with respect to both principal directions (C-shape, Figure 3). The frames are mono-directional. From extended in-situ inspections, fcm=18MPa was obtained. From the original drawings, fym=440MPa was assumed for both longitudinal and transversal steel (Feb44k).

The “Scuola Don Bosco”, located in Rapagnano (Marche, Italy), designed in 1962, is a three-storey r.c. frame structure with masonry infills. The shape of the building (Figure 4) is roughly rectangular (23.84m x 14.44m). The frames are mono-directional and oriented parallel to the short sides of the building. From extended in-situ inspections, fcm=16.6MPa was obtained. From the original drawings, fym=215MPa was assumed for both longitudinal and transversal steel (Feb22k).

For all the buildings, a PGA of 0.25g was considered for the assessment at the SD LS.


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Figure 1: “Sede Comunale” of Vagli Sotto Figure 2: “Scuola Pascoli”

Figure 3: “Scuola Puccetti” Figure 4: Scuola Don Bosco”

3. METHODS OF ANALYSIS

Since all the structures are irregular in plan and elevation, the conditions for the applicability of the linear static analysis are not satisfied. Therefore, only the linear dynamic, nonlinear static and nonlinear dynamic analyses were performed. All analyses are based on 3-D models, as the lack of regularity did not allow to consider two separated planar models. Close results were obtained assuming fixed foundations and modelling the soil-structure interaction (dynamic springs computed according to Gazetas, 1991). Hence, fixed foundations were assumed for all the structures.

The linear dynamic analysis was carried out using the software SAP2000. The modal superposition was performed applying the CQC rule, considering enough modes to obtain a cumulative modal mass ≥ 90% of the total mass. The pseudo-acceleration elastic spectrum was determined according to Eurocode 8 (EN1998-1). Ec was evaluated according to Fib


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Bulletin 24: Ec=0.852.15104(fcm/10)1/3. This formula was used, as the expressions suggested in OPCM 3431 and EC8 are suitable for new constructions, but may be improper for the assessment of existing buildings.

The results of the assessment performed according to the linear analyses is influenced by EIeff of structural members, particularly concerning the ductile response. Usually, the stiffness degradation due to crack openings is accounted by modelling EIeff of the members as a fixed ratio (i.e. 50%) of their gross stiffness. EC8 recommends to compute EIeff as the mean value of the secant stiffness at yielding, EIsec.yield = MyLs/(3y). Assuming Ls = L/2:

4

1,,sec, 24

iiyiyyield MLEI (1)

where i refers to both positive and negative flexure at the two ends of the member. In the following sections, the values obtained by assuming EIeff equal to EIgross, 50% EIgross and EIsec.yield will be compared to those got through the nonlinear dynamic analysis, in order to test which is the most proper way to model EIeff.

The nonlinear analyses were developed through the fibre-model software SeismoStruct, which accounts for both material and geometrical nonlinearity.

The following constitutive models of the materials were assumed i) Nonlinear constant confinement concrete model (Mander, 1988) with no tensile strength; ii) Bilinear steel model, with strain hardening parameter =0.005 and εsu= 0.04 (according to OPCM 3431).

The nonlinear static analysis was based on eight conventional pushover analyses, obtained applying the “uniform” and “modal” distributions of lateral loads. With reference to the N2 method (Fajfar, 1995), the demand Di, the capacity Ci and the ratio Ri=|Di/Ci| of each i-th structural member were computed for each pushover. For each i-th member, the largest value of Ri was used to assess the response.

The nonlinear dynamic analysis was based on seven time history analyses. Hence, according to the Codes’ recommendations, the assessment of the response refers to the average of Ri,max. The seven accelerograms were artificially generated, according to the attenuation law proposed by Sabetta and Pugliese (1996), then modified to match the elastic spectrum of the SD LS.


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4. SEISMIC ASSESSMENT OF STRUCTURAL MEMBERS

In this section, the assessment of the ductile and brittle responses of structural members will be discussed. Concerning the ductile mechanisms, since the two Codes do not show significant differences, only the results based on EC8’s recommendations will be illustrated. The results of the assessment of both mechanisms will be presented through charts, which show the percentage of the mean values of |Di/Ci|, where Di and Ci are the demand and capacity of the i-th member, respectively. Although charts were developed for each analysis, in the following only the most meaningful ones (i.e. those referred to the nonlinear dynamic analysis) will be shown.

4.1 Ductile mechanisms: simplified chord rotation demand assessment

The Codes propose to assess the ductile mechanisms in terms of chord rotation, at both ends of each structural member. The chord rotation is the angle between the chord connecting the member end to the point of contraflexure and the tangent at the member end (Figure 5(a)). Hence, each structural member is formed by two cantilevers, fixed at the member ends and characterized by a length equal to the shear span Ls = M/V, where M and V are the bending moment and corresponding shear demand.

Chordrotation

Shear spanLs=M/V

(a) (b) (c)

Figure 5: (a) Def. of chord rotation; (b) chord rotation of columns and (c) of beams

Denoting as θ1 the nodal rotation of the member end and as θ2 the drift at the end of the shear span, the chord rotation demand of columns θc is obtained as θ2-θ1 (Figure 5(b)). Under seismic input, since the building is pushed laterally, θ2>>θ1. Hence, for the columns: θc = θ2-θ1 θ2.

The assessment of beams’ chord rotation demand θb is more


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complicated, due to the presence of gravity loads. Considering the beam response as the superposition of two systems (beam unloaded and end sections undergoing the nodal rotations due to the seismic input; beam fixed at both ends, loaded by gravity loads), θb may be seen as θ1+θ2 (Figure 5(c)). The nodal rotations due to seismic loads are in general much larger than the drifts due to gravity loads, while the importance of gravity loads decreases with increasing the ground motion intensity. Hence, in particular at SD and NC LS, there will be no appreciable lack of accuracy if θ2 is neglected: θb = θ1+θ2 θ1.

4.2. Ductile mechanisms: simplified chord rotation capacity assessment

The chord rotation capacity, u, depends on geometrical and mechanical properties of the member but, also, on the seismic input (Ls=M/V; N influences the assessment of the ultimate curvature capacity u). Hence, u cannot be defined as an intrinsic property of the member. The correct approach, thus, is to compute u as a function of the seismic demand. Actually, the assessment would be much simpler and faster if it were possible to eliminate the demand dependence and to replace complex theoretical calculations with simpler empirical formulas. In the following, simplified approaches will be examined and suggested, whenever they yield reliable results.

The Codes propose two formulas, one based on theoretical concepts and the other on experimental results. The empirical formula of u is:

cmywmsx ffscmu h

L

CF

f

25;01.0max

';01.0max3.0016.0

5.1

135.0225.0

(2)

where: =(NCF)/(Acfcm); sx=Asx/bwsh is the ratio of the transverse steel parallel to the loading direction; =(1-sh/2b0)(1-sh/2h0)(1-bi

2/(6h0b0)), is the confinement factor; CF is the Confidence Factor, introduced to penalize the assessment in function of the knowledge level of the structural properties (see EN 1998-3 for the meaning of the symbols).

Eq.(2) is demand-dependent. N should be obtained from the analysis under the seismic load combination. Hence, if a dynamic analysis is performed, a double assessment procedure will be required (corresponding to Nmax and Nmin). If it were possible to consider N due only to the gravity loads (which is roughly the mean N that the columns undergo during a


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seismic input), the required operations would be halved. Figure 6 shows that the seismic and gravity N yield very close values. Analogous results were found for all methods of analysis. Hence, the procedure may be simplified. For slender buildings, the seismic variation of N in perimetral columns could be substantial, possibly leading to a large reduction of their ductility capacity. Nevertheless, slender r.c. frame structures are rare in European seismic zones and, hence, the obtained results may be regarded as general.

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Ls = M/V

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Figure 6: Nonlinear static analysis, empirical

u, seismic N vs gravity N Figure 7: Nonlinear dynamic analysis,

empirical u, Ls=M/V vs Ls=L/2

Computing Ls as M/V may be complex for two different reasons: i) in dynamic analyses, M and V are given as envelope values; ii) in linear analyses, M and V grow linearly, while, when yielding takes place, Ls is likely to change with respect to its elastic value. In Figure 7 the results obtained assuming Ls=M/V are compared to those got considering Ls=L/2. Ls=L/2 and Ls=M/V yield close values (differences < 15%). Analogous results were found for all methods of analysis. Hence, the procedure may be simplified.

The theoretical expression for the ultimate chord rotation capacity of structural members is:

splplyuyu LLL 215.11 (3)

(see EN 1998-3 for the meaning of the symbols). In order to compute the ultimate curvature u, section failure was

conventionally considered to take place when the moment capacity Mc drops to 80% of its peak value, Mc,peak. If the spalling of concrete cover causes a drop > 20% Mc,peak, the curvature at spalling will be considered as u. If,


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instead, Mc computed considering only the confined core of the section is > 80% Mc,peak, u will be obtained at the failure of the confined concrete core. The confinement model for the compressive concrete advised in EC2 (EN 1992-1-1) was used. Available experimental results (Priestley, 2003) showed that y is sensitive only to h of the section and εsy of the longitudinal steel. In this work, y was computed in three different ways: i) according to the theoretical approach; ii) according to the empirical formula (4), proposed by Priestley (2003); iii) according to the empirical expression (5), suggested by Biskinis (2006).

The comparison of the results obtained applying the different ways to evaluate y is shown in Figure 8. The theoretical and the two empirical definitions of y yield close values. Analogous results were found for all methods of analysis. Hence, the procedure may be simplified.

(col.) h

beam) (T h

beam) (rect. h

syy

syy

syy

1.2;

7.1;

87.1 (4)

hsyy 75.1 (5)

The results obtained through Eq.(3) are very sensitive to Ls. In fact, if M is peculiarly small, it is likely that Ls will be shorter than Lpl. In particular, if Lpl>2Ls, u will be smaller than y or even negative. In Figure 9 the results obtained considering Ls=M/V and Ls=L/2 are compared to check the sensitivity of Eq.(3) to the way of computing Ls. Applying Ls=M/V the results are much larger than assuming Ls=L/2.

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THEOR. CURVATURE

PRIESTLEY CURVATURE

BISKINIS CURVATURE

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Ls = M/V

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Figure 8: Nonlin. dynamic anal., theoretical

u, theoretical vs empirical y Figure 9: Nonlinear dynamic analysis,

theoretical u, Ls=M/V vs Ls=L/2


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The values obtained applying the correct (seismic N, Ls=M/V, theoretical y) and the simplified (gravity N, Ls=L/2, y from Eqs.(4)-(5)) procedure of assessing u to both empirical (Eq.(2)) and theoretical (Eq.(3)) formulations are compared in Figure 10, in order to establish which approach yields more reliable results. The correct approach of Eq.(3) yields overestimated values. Since it has been shown that the way to assess both N and y does not influence the results, it is clear that Ls = M/V should not be applied to Eq.(3), as it produces inaccurate values, due to the large sensitivity to Ls. Analogous results were found for all methods of analysis.

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/COPCM 3431

EC 8

Figure 10: Nonlin. dynamic an., empirical

vs theoretical u, correct vs simplified approach

Figure 11: Nonlinear dynamic analysis, EC8 vs OPCM 3431

4.3. Brittle mechanisms: assessment of shear capacity

OPCM suggests to compute the shear resistance VR as in a non-seismic design, with the only restriction of considering a contribution of the concrete not larger than that of the same member without lateral reinforcement. In this work, the approach proposed in D.M.1996 was used. EC8, instead, accounts for the effects of both cycling loading and inelastic response in the assessment of VR (see EN 1998-3 for the meaning of the symbols):

wcccmstot

pldemccmcsR

VACFfhLCFfANLxhV

)(;5min16.01100;5.0max16.0;5min05.01)(55.0;min)2()(15.11 ,

(6)

The comparison between the results obtained applying both Codes is shown in Figure 11. It emerges that the non-seismic approach suggested in OPCM grossly overestimates VR, yielding unconservative values. Analogous results were found for all methods of analysis. Therefore, the use


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of the procedure suggested in Eurocode 8 is recommended. Since the formula proposed in OPCM is judged as inadequate, in the following charts only the results obtained according to Eurocode 8 will be shown.

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Figure 12: Nonlinear static analysis, seismic

N vs gravity N Figure 13: Nonlinear dynamic analysis,

Ls = M/V vs Ls = L/2

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CORRECT PROC.

SIMPLIFIED PROC.

Figure 14: Nonlinear dynamic analysis,

theoretical vs empirical y Figure 15: Nonlinear dynamic analysis,

correct vs simplified approach

Eq.(6) is function of N, y and Ls. In Figure 12 the comparison between the results obtained considering seismic and gravity N is shown. Seismic and gravity N yield very close values. Analogous results were found for all methods of analysis. Hence, it is possible to simplify the procedure. In Figure 13 the comparison between the results obtained considering Ls=M/V and Ls=L/2 is shown. Ls=L/2 yields a slightly (practically negligible) overestimation of the values obtained assuming Ls=M/V. Analogous results were found for all methods of analysis. Hence, the procedure may be simplified. The comparison among the results obtained considering the theoretical and the empirical formulas of y is


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shown in Figure 14. Theoretical and empirical formulas of y (Eqs.(4)-(5)) yield very close values. Analogous results were found for all methods of analysis. Hence, the procedure may be simplified. Finally, Figure 15 proves that the simplified approach may be applied without significant loss of accuracy.

5. RELIABILITY OF LINEAR AND NONLINEAR METHODS OF ANALYSIS

In this section, the reliability of the assessment based on all considered methods of analysis will be discussed. In particular, since the two Codes do not show significant differences, only the results based on EC8’s recommendations will be illustrated. The results will be presented through charts, which show the percentage of the mean values of |Di/Ci|, where Di and Ci are the demand and capacity of the i-th member, respectively.

5.1 Linear analyses

According to both Codes, the results of the linear analyses may be considered reliable only if the distribution of the ratios of the bending moment demands to the corresponding capacities is roughly constant for all structural members. In particular, considering each i-th end section of each structural member, denoting by ρi = Di/Ci the ratio of the bending moment demand Di obtained directly from the analysis to the corresponding capacity Ci, and by ρmax and ρmin the maximum and minimum values of ρi respectively, the ratio ρmax/ρmin should not exceed a maximum acceptable value (2.5 in Italy), beyond that the inelastic response of the building is considered to differ significantly from the elastic one.

Table 1: Outcome of the check on the applicability condition of linear analyses.

Applicability condition - ρmax/ρmin < 2.5 “Sede Comunale” “Scuola Pascoli” “Scuola Puccetti” “Scuola Don Bosco”

EIEFF = 100% EIGROSS ρmax/ρmin 8.03 10.14 10.58 56.61

EIEFF = 50% EIGROSS ρmax/ρmin 6.01 10.52 7.97 39.29

EIEFF = EISEC.YIELD ρmax/ρmin 5.74 5.80 4.59 24.13

In table (1) the results of the check of the applicability condition of

linear methods are shown, for the three considered way to model the


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effective stiffness of structural members (EIeff = EIgross, EIeff = 0.5 EIgross and EIeff = EIsec.yield). In no case the condition is satisfied. Despite of the Codes’ recommendations, the linear dynamic analysis was performed to test if the criterion of applicability of linear analyses is adequate.

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Figure 16: Ductile check, nonlinear dynamic

analysis vs. linear dynamic analysis with different EIeff

Figure 17: Brittle check, nonlinear dynamic analysis vs. linear dynamic analysis with different EIeff

In Figure 16 the results of the assessment of the ductile mechanisms obtained from the linear dynamic analysis performed with the three different ways of modelling the elastic stiffness are compared to those got through the nonlinear dynamic analysis, for all considered buildings. It emerges that, for each structure, there is a value of EIeff which allows to obtain, globally, reliable results, although the applicability condition is not satisfied. It means that, once modelled properly EIeff of structural members, the results furnished by the linear dynamic analysis could be useful at least to get general indications about the need of retrofitting. Figure 16 shows also that, for all the considered buildings, EIeff = EIsec.yield for each structural member leads to underestimate grossly the elastic slope of the bilinear approximation of the F- curve of the whole structure and to overestimate sensibly the chord rotation demand of the structural members. This result can be justified considering that the yielding point of the bilinear approximation of the F- curve of the structure does not correspond, in general, to the local yielding of all structural members. The majority of the members could be still in their elastic range, resulting in a higher value of EIeff with respect to that obtained assuming EIeff = EIsec.yield for each structural member. For all the considered structures, EIeff between 50% and 100% EIgross allows to obtain values close to those got through the nonlinear dynamic analysis. In particular, for the “Sede Comunale” (low value of fcm), EIeff = EIgross yields


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the most accurate results while, for the other three buildings, EIeff = 0.5 EIgross is the most reliable choice.

Figure 17 shows the results of the assessment of the brittle mechanisms. For all buildings, all the three considered ways to model EIeff yield values rather close to those got through the nonlinear dynamic analysis and, also, close to each other. It may be justified considering that the shear demands obtained directly from the linear dynamic analysis were limited (according to the Codes) to account for the development of the flexural nonlinear mechanisms. It may be concluded that, also concerning the brittle mechanisms, the linear dynamic analysis allows to obtain, globally, reliable results.

5.2 Static nonlinear analysis

The static nonlinear analysis based on 2-D conventional pushover (according to the “uniform” and “modal” distributions) applied to a 3-D model, shows several drawbacks. i) The definition of the “modal” pattern of lateral forces could be difficult, especially if the structural configuration is neither symmetric nor compact. In this case, in fact, it is probable that first mode will not be purely translational. It is in contrast with the original 2-D Fajfar’s procedure, based on unidirectional lateral forces and capacity curves. ii) According to both Codes, the “control displacement” should be that experienced at the centre of mass of the top floor. If the structure is neither compact nor symmetric, the centre of mass may rotate around a point which shifts from the centre of stiffness to the centre of resistance, besides translating towards the pushing direction. iii) If the responses of the two principal directions cannot be decoupled, the problem of how to combine the effects of two orthogonal pushovers will arise. For all these reasons, it is clear that the reliability of the static nonlinear analysis based on the conventional 2-D pushover algorithm needs to be checked, especially concerning the assessment of the response of irregular buildings.

In Figures 18 and 19 the results obtained through the nonlinear static analysis are compared to those got from the nonlinear dynamic analysis, for the ductile and brittle mechanisms respectively, for all considered buildings. In particular, for buildings characterized by small eccentricity between the centres of mass and stiffness/resistance (“Sede Comunale”, “Scuola Pascoli” and “Scuola Don Bosco”), the assessment based on the envelope results of the two distributions of lateral forces yields very conservative values. The “uniform” distribution furnished better results than the “modal” distribution,


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because of the weak-columns strong-beams mechanism which characterizes the considered structures. For strongly unsymmetrical configurations (“Scuola Puccetti”), instead, the information got through the conventional nonlinear static analysis are very poor (particularly for the ductile mechanisms), indicating the impossibility of catching correctly the effects of the torsional response through the conventional nonlinear static analysis.

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SCUOLA DONBOSCO

% D

/C

DYNAMIC NON LIN.STATIC NON LIN. - ENVELOPESTATIC NON LIN. - UNIFORMSTATIC NON LIN. - MODAL

Figure 18: Ductile check, nonlin. dynamic

analysis vs. nonlin. static analysis Figure 19: Brittle check, nonlin. dynamic

analysis vs. nonlin. static analysis

5.3. Comparison among all methods of analysis

Finally, in Figures 20 and 21 the results of all the considered analyses are compared to each other to select the “best” method to assess the response of each considered structure. They allow to draw the following conclusions. If the aim is to get general indications about the need of retrofitting, the linear dynamic analysis (once EIeff is properly modelled) may be the best solution, since it is much faster than the nonlinear analyses and allows to get an enough accurate prediction of the global response, even for strongly unsymmetrical configurations. For roughly symmetrical buildings, the nonlinear static analysis according to the “uniform” distribution may be used in alternative to the linear dynamic analysis, with the drawback of requiring larger computational efforts and the advantage of avoiding the uncertainty of the modelling of EIeff. For strongly unsymmetrical configurations, the assessment performed according to the conventional nonlinear static analysis is worse than that got through the linear dynamic analysis and, hence, it is not advised. It is clear that criteria for the applicability of the nonlinear static analysis are needed.


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DUCTILE MECHANISMS - EMPIRIC U

0

10

20

30

40

50

60

70

80

SEDECOMUNALE

SCUOLAPASCOLI

SCUOLAPUCCETTI

SCUOLA DONBOSCO

% D

/C

DYNAMIC NON LIN.DYNAMIC LIN. - 100% EI_grossDYNAMIC LIN. - 50% EI_grossSTATIC NON LIN. - UNIFORMSTATIC NON LIN. - MODAL

BRITTLE MECHANISMS

0

20

40

60

80

100

120

140

160

180

200

SEDECOMUNALE

SCUOLAPASCOLI

SCUOLAPUCCETTI

SCUOLA DONBOSCO

% D

/C

DYNAMIC NON LIN.DYNAMIC LIN. - 50% EI_grossSTATIC NON LIN. - UNIFORMSTATIC NON LIN. - MODAL

Figure 20: Ductile check, all the analyses Figure 21: Brittle check, all the analyses

6. CONCLUSIONS

The results obtained at the local (member) level can be summarized as follows. i) The assessment of the chord rotation demand can be simplified assuming, for beams, θ as the nodal rotation and, for columns, θ as the interstorey drift. ii) The theoretical and empirical formulas of the chord rotation capacity θu yield very different results. The assessment of θu according to the empirical formula can be simplified assuming N from gravity loads, Ls = L/2 and y from empirical expressions. iii) The non-seismic approach of OPCM 3431 yields an unconservative assessment of the brittle response. The EC8 formula can be simplified assuming N from gravity loads, Ls = L/2 and y from empirical expressions.

The results obtained at the global (structural) level can be summarized as follows. i) Modelling EIeff of structural members as EIsec.yield yields leads, overestimated displacements/deformations. A ratio between 50% and 100% of EIgross is, generally, suitable to predict correctly the elastic branch of the F- curve of the structure. ii) If the purpose of using linear analyses is to get general indications about the need of retrofitting, the applicability conditions proposed in OPCM 3431 and EC8 seems too restrictive. iii) For strongly unsymmetrical configurations characterized by large torsional effects, the use of the nonlinear static analysis based on the conventional 2-D pushover is not advised, as it provides poor information, in general worse than that got through the linear dynamic analysis. Hence, criteria for the applicability of the nonlinear static analysis are needed.


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References

Biskinis, D.E. (2006) Deformations of Concrete Members at Yielding and Ultimate, University of Patra, Greece.

CEN (2004) European Standard EN 1998-1-2004. Eurocode 8: Design of structures for earthquake resistance - Part 1: General rules, seismic actions and rules for buildings, Bruxelles, Belgium.

CEN (2005) European Standard EN 1998-3-2005. Eurocode 8: Design of structures for earthquake resistance - Part 3: Assessment and retrofitting of buildings, Bruxelles, Belgium.

Decreto Ministeriale (1996) Norme tecniche per il calcolo, l’esecuzione e il collaudo delle strutture in cemento armato, normale e precompresso e per le strutture metalliche, G.U. No. 29.

Fajfar, P., Gaspersic, P. (1995) The N2 Method For the Seismic Damage Analysis of RC Building, Earthquake Engineering and Structural Dynamics, 25(1), 31–46.

Fib Bulletin 24, Appendix 4-A (2003) Strength and deformation capacity of non-seismically detailed components. International Federation du Beton.

Gazetas, G. (1991) Foundation vibrations, Foundation Engineering Handbook, 2nd Ed., edited by H.Y. Fang.

Mander, J.B., Priestley, M.J.N., Park, R. (1988) Theoretical Stress-strain Model for Confined Concrete, Journal of Structural Engineering, ASCE, 114(8), 1804–1826

Mpampatsikos, V., Nascimbene, R. and Petrini, L. (2008) A Critical Review of the R.C. Frame Existing Building Assessment Procedure According to the Eurocode 8 and Italian Seismic Code, Journal of Earthquake Engineering, 12(1), 52–82.

Mpampatsikos, V., Nascimbene, R. and Petrini, L. (2008) Some Considerations About The Eurocode 8 R.C: Frame Building Assessment Procedure, 14th WCEE, Beijing, China

Ordinanza del Presidente del Consiglio dei Ministri No. 3431 (2005) Ulteriori modifiche ed integrazione all’ OPCM No. 3274, G.U. No. 105, Italy.

Panagiotakos, T.B., Fardis, M.N. (2001) Deformations of R.C. Members at Yielding and Ultimate, Structural Journal, ACI, 98(2), 135–148.

Priestley, M.J.N. (1997) Displacement-based Seismic Assessment of Reinforced Concrete Buildings, Journal of Earthquake Engineering, 1(1), 157–192.

Priestley, M.J.N. (2003) Myths and Fallacies in Earthquake Engineering, IUSS Press, Pavia, Italy. Sabetta, F., Pugliese, A. (1996) Estimation of Response Spectra and Simulation of Nonstationarity

Earthquake Ground Motions, Bull. Seism. Soc. Am., 86(2), 337–352. SAP2000 Advanced I 10.0.7, Structural Analysis Program (2006) Computer and Structures Inc. 1995,

University Ave. Berkeley, CA. SeismoSoft (2006) SeismoStruct – A computer programme for static and nonlinear dynamic analysis

of frames structures, available from URL: http://www.seismosoft.com.

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