Displacement-based seismic performance evaluation and vulnerability assessment of buildings: The N2 method revisited

Structures 24 (2020) 41–49

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Displacement-based seismic performance evaluation and vulnerability assessment of buildings: The N2 method revisited

T

Gonçalo Correia Lopesa, , Romeu Vicentea, Tiago Miguel Ferreirab, Miguel Azenhab, João Estêvãoc ⁎

a

RISCO – Aveiro Research Centre of RIsks and Sustainability in COnstruction, University of Aveiro, Civil Engineering Department, Campus Universitário de Santiago, Aveiro 3810-193, Portugal b ISISE – Institute for Sustainability and Innovation in Structural Engineering, University of Minho, School of Engineering, Civil Engineering Department, Azurém Campus, Guimarães 4800-058, Portugal c Department of Civil Engineering, University of Algarve, Faro 8005-139, Portugal

ARTICLE INFO

ABSTRACT

Keywords: Nonlinear static procedures N2 method Pushover analysis Inelastic spectra Displacement-based design Deformation-controlled design

The so-called N2 method is a widely used nonlinear static method for the seismic design and/or assessment of different building typologies. In this paper, the idea of reversing the so-called N2 method currently adopted in the Annex B of Eurocode 8 (the code on which the present paper is based) in a direct displacement-based procedure is presented in a simple alternative format. In this way, it is possible to obtain the percentage of the seismic action associated to any performance displacement. By doing so, it is also possible to determine the target displacement without the need for an iterative procedure. The proposed method presents several advantages for professional engineering practice in respect to the seismic vulnerability assessment of new and existing buildings. The comparison of the original N2 method with the proposed one is illustrated by means of a test structure example.

1. Introduction The rational seismic design of new buildings, as well as the structural performance evaluation of existing structures subjected to earthquake loading, may be carried out using different analysis methods (linear or nonlinear, static or dynamic, force or displacement-based). Between the linear static methods and non-linear dynamic time history analysis, the Nonlinear Static Procedures (NSP) based on pushover analysis prove to be a good trade-off between the computational burden and the accuracy and objective of such approach [1,2], whose reliability and effectiveness has been analysed by several authors [3–8]. In this context, Folić and Ćosić [9] present a review and a systematisation of the NSP for structural performance analysis developed in the last twenty years. Amongst the most widely known NSP are the Capacity Spectrum Method (CSM) [10] (defined in ATC-40 [11] and its improved version provided in FEMA 440 [12]), the Displacement Coefficient Method (DCM) (provided in FEMA 273 [13] and its improved versions in FEMA 356 [14]), the secant method [15], the Modal Pushover Analysis (MPA) [16], the Simple Lateral Mechanism Analysis (SLaMA) [17,18], and the N2 method [19] (a variant of the CSM based

on inelastic design spectra) which is currently adopted in Eurocode 8 (EC8) [20] – the code on which the present paper is based. The so-called N2 method proposed by Fajfar in the 1980s [19] owes its designation to the fact that it requires the application of a nonlinear method – “N” – and the construction of two – “2” – mathematical models: a Multi-Degree-Of-Freedom (MDOF) model for the non-linear static analysis and an equivalent Single-Degree-Of-Freedom (SDOF) model. This method is driven by structural and ground motion data, which are used to define the capacity of a structure and the seismic demand, respectively. The inelastic response spectra need to be defined to account for the dissipation of energy (as an alternative to the nonlinear dynamic analysis), using for that purpose an equivalent nonlinear SDOF system based on the bilinear force-deformation relationship. The method yields an adequate estimate of demand in terms of structural stiffness, strength, ductility and energy dissipation, considering the uncertainties related with the input data [21]. As pointed out by Fajfar [22], like any approximation method, the N2 method is subjected to several limitations, which can be divided into two main sources of uncertainty: the pushover analysis and the inelastic

Corresponding author. E-mail addresses: [email protected] (G. Correia Lopes), [email protected] (R. Vicente), [email protected] (T.M. Ferreira), [email protected] (M. Azenha), [email protected] (J. Estêvão). ⁎

https://doi.org/10.1016/j.istruc.2019.12.028 Received 16 September 2019; Received in revised form 17 December 2019; Accepted 30 December 2019 2352-0124/ © 2020 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.

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G. Correia Lopes, et al.

spectra. Apart from the approximation of the pushover curve – “the Achilles’ heel of the N2 method” according to Cerovečki et al. [5] – amongst the most referred limitations are: the influence of higher modes, problems of shear demand and pure flexural systems, the discrepancy between the results of dynamic and push-over analysis, the influence of different load patterns, the sensitivity of inelastic displacements to changes of structural parameters in the short-period range, the soil-structure interaction effects, etc. [5,21,22]. In this context, some of authors have even proposed possible ways of overcoming them [22–28]. Even so, the N2 method has many advantages, whereby it is widely used either in scientific research or in everyday engineering practice for the seismic structural design of new buildings and seismic assessment of the existing buildings. Nevertheless, it has still not been fully explored in the case of structures without rigid diaphragms (e.g., timber and masonry buildings) [5]. One recent example of the applicability of this method is the relatively new interest in the scope of the seismic assessment of existing masonry buildings [28–33], which represents the largest share of the world’s building’s stock [34]. Since the target displacement is not known beforehand, most current codes specify iterative procedures to determine the performance point. However, in addition to their inefficiency, sometimes they lead to the lack of convergence and accuracy [35]. In order to overcome this issue, there are several proposals of non-iterative methods (e.g., [2,35,36]). However, these methods do not lead specifically with the actual N2 method followed in the Annex B of EC8 which provides an unclear manner of determining the idealised elasto-perfectly plastic force – displacement relationship (also known as simplified/equivalent/bilinear capacity curve/diagram), particularly for structural systems that exhibit degrading pushover curves. In EC8, its determination starts at the point where the pushover curve reaches the maximum base shear force, at the abscissa dm (see Fig. 3 (c)). Then, the first approximation of the value of the target displacement dt can be determined from the elastic acceleration response spectrum. However, since the displacements dm and dt may differ considerably, an iterative graphical procedure should be used in order to converge the value of the target displacement, required to plot the capacity diagram. Although the iterative procedure is indicated as optional in EC8, it is important to highlight that, despite being a slower procedure, it is recommended as it leads to more accurate results, which can differ significatively from a non-iterative to an iterative procedure [37]. This procedure is schematically illustrated in Fig. 1 (a), where the convergence is reasonably attained at the third iteration (black line). As stated by several authors [1,2,38], displacement-based approaches (e.g., [39–42]) have proven themselves more advantageous than the traditional force-based approaches, whereby newer generations of seismic design codes and rehabilitation procedures are driving greater attention to deformation controlled design, both by enhancing force-based design through verification of deformation targets and by the development of direct deformation based design approaches.

Based on this premise, in contrast with the original N2 method, the method proposed in this paper offers an alternative approach for the graphical representation of the seismic performance of buildings, by reversing the original procedure in a direct displacement-based design. By doing so, it is also possible to determine the target displacement without the need for an iterative procedure. Unlike other procedures that pursue solely one performance point of a determined seismic intensity, the proposed methodology draws the curve of the percentage (hereafter referred to as “scale factor”) of the design ground acceleration (i.e., the design ground acceleration on rock, ag , or the design acceleration on the ground, ag ·S ) as function of the control node displacement, over its entire domain, without iterations. Thus, the proposed displacement-based method avoids the need to plot both the demand spectrum and the capacity curve, since these are implicitly considered in its formulation. Moreover, once plotted the scale factor ( ) for the design ground acceleration as function of the control node displacement (d ), the target displacement is easily determined by linear interpolation at the point where the scale factor is d = dt ), as graphically illustrated in in equal to one ( (d ) = 1 Fig. 1 (b). Furthermore, since the obtained curve is an injective function in the interval [0; du ], it is possible to quantify the reserve of capacity of the structure beyond the target displacement through the value of the scale factor for the design ground acceleration at the ultimate displacement du (which shall be duly limited depending on the type of structure, in accordance with the specifications provided in EC8). In the illustrated case given in Fig. 1 (b), it is possible to observe that the ultimate displacement would have been reached for a seismic demand (i.e., design ground acceleration) 50% higher than the one considered. At the same time, this procedure proves to be also advantageous for the seismic vulnerability assessment of buildings. Since it is a direct displacement-based design approach, starting from a predetermined target displacement, corresponding to a damage state threshold, it gives the percentage of the ground acceleration associated to that displacement limit threshold. 2. Displacement-based N2 method 2.1. Representation of the scale factor for the design ground acceleration as function of the control node displacement In this method, the Multi-Degree-Of-Freedom (MDOF) pushover curve, which represents the relation between base shear force and control node displacement, should be determined by nonlinear static (pushover) analysis in terms of roof displacement (control node displacement). For the numerical and/or graphical determination of the target displacement and/or the percentage of the ground acceleration associated to a given displacement, the following steps shall be taken. The boxed formulas are meant to represent an array of values as function of

Fig. 1. Graphical comparison of the procedures for the determination of the target displacement (example of a structure in the short period range, T < TC ). 42

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Fig. 2. Calculation procedure to be implemented in a spreadsheet.

[0; du ], and shall correspond to an inthe independent variable d dividual column of values if this method is to be implemented in a spreadsheet (see Fig. 2). Step 1. Transformation to an equivalent Single Degree of Freedom (SDOF) system The mass of the equivalent SDOF system m is determined as: m =

mi

Firstly, the area Em under the pushover curve corresponding to the initial limit point (the pair (dm; Fm) ) is determined, and the coordinates (d y 0; F y 0) of the correspondent yield displacement limit point of the idealised SDOF system are determined (first column of Table 1) so that there is a balance between the energy-dissipation capacity of the actual and idealised structures. Then, an analogous process is carried out for each value of the [0; du ], in order to obtain the final cocontrol node displacement d ordinates of the yield displacement limit point as function of d (d y (d ); F y (d )) (last two columns of Table 1). For this purpose, two different approaches should be used, depending on the relationship between the values of the displacement corresponding to the maximum base shear force (dm ), and any given value for the control node displacement (d ):

(1)

i

and the transformation factor (also called modal participation factor) is given by:

=

m mi

(2)

2 i

The force F and displacement d of the equivalent SDOF system are computed as:

d =

dn

, F (d ) =

Fb (dn )

dm (variable initial stiffness), the ultimate strength of the a) if d idealised system equals the value of the force of the pushover curve at the value d (F y (d ) = F (d ) ); b) if d > dm (constant initial stiffness), the ultimate strength of the idealised system is determined so that the slope of the linear branch (i.e., the initial stiffness of the idealised system) corresponds to the one associated to the maximum base shear force.

(3)

where Fb and dn are, respectively, the base shear force and the control node displacement of the Multi Degree of Freedom (MDOF) system. Step 2. Determination of the idealised elasto-perfectly plastic force – displacement relationship As noted by Fajfar [38], engineering judgement has to be used in this step since different guidelines may be given in different regulatory documents. For example, the American FEMA-273 [13] and the Italian NTC [43] require that the effective lateral stiffness shall be taken as the secant stiffness calculated at a force equal to 60% of the yield strength and equal to 60% of the maximum strength, respectively. Additionally, according to EC8 [20] the yield force F y is equal to the base shear force at the formation of the plastic mechanism (ultimate strength of the idealised system) and the value of the initial stiffness is obtained so that the areas below the actual and idealised curves are equal. However, no guidance is given for the cases when the pushover curve goes beyond the peak strength (e.g., structures with degrading behaviour). The graphical comparison of these procedures is illustrated in Fig. 3. The procedure herein proposed for the determination of the yield displacement limit point coordinates (d y (d ); F y (d )) as function of the of the control node displacement d is based on the procedure for the bilinear idealisation of the pushover curve described by Estêvão [37], which can be applied independently of the shape of the pushover curve (i.e., both for buildings showing brittle or ductile behaviours). The difference is that, in the present method, the bilinear idealisation of the pushover curve is done for the entire domain of the independent variable d , and not only for each iterated value of dt .

Step 3. Determination of the horizontal inelastic spectral acceleration Once known the force F y (d ) , which represents the ultimate strength of the idealised system as function of the control node dis[0; du ], it is possible to transform this value in terms of placement d horizontal inelastic spectral acceleration by dividing it by the equivalent mass according to the following expression:

S (d ) =

F y (d ) m

(7)

Step 4. Determination of the period of the idealised equivalent SDOF system The period T (d ) of the idealised equivalent SDOF system, as function of the control node displacement d , is determined by:

T (d ) = 2

d y (d ) S (d )

(8)

Step 5. Determination of the horizontal elastic spectral acceleration The horizontal elastic acceleration response spectrum Se (T (d )) is

Fig. 3. Comparison of the procedures for the bilinear idealisation of the pushover curve. 43

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Table 1 Procedure for the determination of the yield displacement limit point coordinates. Initial values (maximum base shear force)

Final values

d

Em =

dm 0

F (d )·dd

dm

E (d ) =

F y0 = Fm

d > dm

d 0

F (d )·dd

d 0

E (d ) =

F y (d ) = F (d )

F y 0· d

d 2

F y (d ) =

d y 0 = 2 dm

Em F y0

d y (d ) = 2 d

E (d ) F y (d )

d y (d ) = F y (d )·

given by the following piecewise-defined function:

ag · S · 1 +

T (d ) ·( TB

·2.5

ag ·S · ·2.5 Se (T (d )) =

ag ·S · ·2.5 ag ·S · ·2.5

(

(

TC T (d )

)

TC·TD (T (d ))2

)

1)

, 0

(d ) =

T (d ) < TB

, TB

T (d ) < TC

, TC

T (d ) < TD

, TD

T (d ) < 4s

det 1 + (qu qu

1)

TC T

(9)

(d ) =

T (d )·det T (d )·d

(10)

Se (T (d ))· m S (T (d )) = e Fy S (d )

2

d ·4 2 Se (T (d ))·(T (d )) 2

(14.a)

(14.b)

(15)

dt = · dt

(11)

2.3. Determination of the displacement ductility demand factor The displacement ductility demand factor µ (d ) associated with the scaled design ground acceleration as function of the control node displacement shall be calculated using the following equations: a) For structures in the short-period range (T (d ) < TC ):

(12)

Additionally, according to EC8, the qu factor is given by:

qu =

S (d )·T (d )·TC S (d )·(T (d )) 2 + d ·4 Se (T (d ))· T (d )·TC

Once plotted the scale factor for the design ground acceleration as function of the control node displacement (d ) , the target displacement dt of the idealised equivalent SDOF system with period T and unlimited elastic behaviour may be easily determined by linear interpolation (without significant deviation) at the point where the scale d = dt ). factor is equal to one ( (d ) = 1 Finally, the target displacement correspondent to the control node of the MDOF system is given by:

2

TC ·det S (d )·T (d )·(T (d ) TC ) = e TC ·d d ·4 2 Se (d )·T (d )·TC

(6)

F y0

2.2. Determination of the target displacement

Rewriting expression (10) in order to qu , and replacing dt by d , and substituting det by expression (11), yields:

qu =

dy 0

dy 0

Now, it possible to plot the scale factor for the design ground acceleration as function of the control node displacement.

and the target displacement of the idealised equivalent SDOF system with period T and unlimited elastic behaviour is given by:

T det = Se (T (d )) 2

(5)

2E (d )·dy 0 F y0

b) For structures in the medium and long period range (T (d ) TC ), (as well as for structures in the short-period range with an elastic response), according to EC8, the target displacement is simply given by expression (11) (i.e., dt = det ). Then, substituting the term Se (T (d )) by (d )· Se (T (d )) , and replacing dt by d in that expression, and rewriting it in order to (d ) yields:

Step 6. Determination of the scale factor for the design ground acceleration For each given value of the control node displacement d , it is possible to calculate the corresponding scale factor (d ) of the design ground acceleration ag (implicitly considered in expression (9)), that would be necessary to attain that displacement. The deduction of the different expressions needed to plot the scale factor (d ) are indicated below. a) For structures in the short-period range (T (d ) < TC ), with a nonlinear response (S (d ) < (d )·Se (T (d )) ), according to the EC8, the target displacement is given by:

dt =

(4)

F (d )·dd

µ (d ) = 1 + (qu (d )

1)

TC T (d )

(16.a)

b) For structures in the medium and long period range (T (d ) TC ):

(13)

Equating the expressions (12) and (13), and substituting the term Se (T (d )) by (d )· Se (T (d )) , and rewriting it in order to (d ) yields:

µ (d ) = qu (d ) 44

(16.b)

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Fig. 4. Test structure [44].

Fig. 5. Demand spectra for three levels of ground motion and corresponding capacity diagrams for the test example with the representation of the iterative procedure recommended in EC8 (iterated values between brackets).

where the factor qu (d ) (i.e., the ratio between the acceleration in the structure with unlimited elastic behaviour and in the structure with limited strength) as function of the control node displacement is calculated according to the following expression:

qu (d ) =

(d )· Se (T (d )) = S (d )

(d )· Se (T (d )) (F y (d )/m )

(17)

3. Test structure example In this section, the proposed method for evaluating how the structure will perform when subjected to an earthquake ground motion is compared with the original N2 method in order to demonstrate its applicability. For that purpose, the analysed structure is a four-storey reinforced concrete frame building (Fig. 4), designed according to EC8 [20], as a high ductility structure for a design ground acceleration of

Fig. 6. Scale factor for the design ground acceleration as function of the control node displacement for three levels of ground motion for the test example.

45

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Table 2 Calculation procedure implemented in a spreadsheet (ag = 0.3g ). Step 1

Step 2

Input: MDOF pushover curve

Indep. var.

dn

d

(m) 0.000 0.004 0.008 0.013 0.020 0.028 0.037 0.046 0.057 0.065 0.076 0.087 0.099 0.112 0.127 0.141 0.155 0.170 0.184 0.199 0.214 0.228 0.241 0.250

Fb (dn) (kN) 0.000 108.062 200.686 290.738 385.935 483.705 571.184 650.943 746.141 810.463 887.650 951.973 1013.720 1057.460 1085.760 1113.608 1139.790 1160.380 1180.960 1198.970 1219.550 1237.560 1255.570 1265.870

(m) 0.000 0.003 0.006 0.010 0.015 0.021 0.027 0.034 0.043 0.049 0.057 0.065 0.074 0.084 0.095 0.105 0.116 0.127 0.137 0.148 0.160 0.170 0.180 0.187

F (d ) (kN) 0.000 80.643 149.766 216.969 288.011 360.974 426.257 485.778 556.822 604.823 662.425 710.428 756.507 789.149 810.269 831.051 850.590 865.955 881.313 894.754 910.112 923.552 936.993 944.679

E (d )

F y (d )

(kNm) 0.000 0.107 0.445 1.178 2.524 4.427 7.050 10.210 14.517 17.922 22.989 28.660 35.307 42.928 51.667 60.488 69.154 78.533 87.847 97.550 107.893 117.423 126.845 132.863

(kN) 0.000 80.643 149.766 216.969 288.011 360.974 426.257 485.778 556.822 604.823 662.425 710.428 756.507 789.149 810.269 831.051 850.590 865.955 881.313 894.754 910.112 923.552 936.993 944.679

d y (d ) (m) 0.000 0.003 0.005 0.008 0.012 0.017 0.022 0.027 0.033 0.038 0.044 0.049 0.054 0.059 0.062 0.065 0.069 0.072 0.075 0.078 0.082 0.086 0.090 0.092

Step 3

Step 4

Step 5

S (d )

T (d )

Se (T (d ))

(d )

(s) 0.000 0.532 0.548 0.573 0.605 0.636 0.662 0.687 0.714 0.731 0.751 0.767 0.784 0.797 0.808 0.820 0.832 0.843 0.855 0.866 0.880 0.892 0.905 0.913

2

(-) 0.000 0.044 0.087 0.135 0.192 0.254 0.322 0.390 0.465 0.516 0.585 0.656 0.732 0.816 0.911 1.000 1.081 1.168 1.249 1.331 1.412 1.483 1.549 1.590

2

(m/s ) 0.000 0.372 0.690 1.000 1.327 1.663 1.964 2.239 2.566 2.787 3.053 3.274 3.486 3.637 3.734 3.830 3.920 3.991 4.061 4.123 4.194 4.256 4.318 4.353

(m/s ) 0.000 8.459 8.459 8.459 8.383 7.978 7.663 7.392 7.107 6.941 6.759 6.613 6.471 6.364 6.284 6.192 6.100 6.021 5.937 5.861 5.770 5.689 5.608 5.561

Step 6

of the target displacement. Nevertheless, sometimes this process turns out to be impossible to be completed whenever the calculated target displacement is greater than the ultimate displacement (the end point of the pushover curve). This situation means that the structure has insufficient dissipation capacity to withstand such seismic demand. This procedure is graphically illustrated in Fig. 5, for the test example, where the convergence is reasonably attained with three iterations for two levels of design ground acceleration (ag equal to 0.15 g and 0.3 g), whereas for the remaining case (ag equal to 0.6 g) the structure is not capable to withstand the applied seismic demand, whereby the iteration procedure is impossible to be computed. In contrast with the original N2 method, in the proposed displacement-based method, one curve is enough to convey the value of the target displacement, as well as to graphically represent the seismic performance of a building subjected to a known seismic demand. This method is illustrated in Fig. 6 for the test example. From this figure, it is possible to obtain the values of the target displacements for two levels of design ground acceleration (dt = 0.047 m and dt = 0.105 m for ag equal to 0.15 g and 0.3 g, respectively), whereas for the remaining case (ag equal to 0.6 g) the structure is not capable to withstand the applied seismic demand since the curve is inferior to = 1.0 over its entire domain. Furthermore, it is possible to quantify the reserve of capacity of the structure beyond the target displacement through the value of the scale factor for the design ground acceleration at the ultimate displacement. From Fig. 6, it is possible to observe that the ultimate displacement corresponding to the three levels of design ground acceleration would have been reached for a design ground acceleration (d u ) × ag = 3.180 × 0.15g = 1.590 × 0.3g = 0.795 × 0.6g = 0.477g . of In other words, the structure would attain the ultimate displacement for a design ground acceleration 20.5% lesser, 59% greater and 218% greater, than the ones considered, respectively for ag equal to 0.6 g, 0.3 g and 0.15 g. In order to illustrate the calculation procedure when implementing this method in a spreadsheet, Table 2 presents all the values used to plot the scale factor for the design ground acceleration ( ) as function of the

Fig. 7. Scale factor for the design ground acceleration as function of the control node displacement obtained from three different pushover curves.

0.3 g, for a ground type C and type 1 spectrum (TC = 0.6 s) and 5% damping. This test example-building is the same as the one used by Fajfar [22] (originally from [44]) and it is a reproduction of the fullscale building tested pseudo-dynamically in the European Laboratory for Structural assessment (ELSA) of the Joint Research Centre of the European Commission in Ispra (Italy). The building was subjected to three levels of ground motion, corresponding to a design ground acceleration ag equal to 0.15 g, 0.3 g (the design value), and 0.6 g, with the intention of assessing different performance objectives. The storey masses from the bottom to the top amounted to 87, 86, 86, and 83 tons, the normalised displacements (assuming a linear displacement shape) are respectively 0.28, 0.52, 0.76 and 1.00, and the resulting base shear coefficient amounted to 0.15. By transforming the MDOF system into an equivalent SDOF system, the equivalent mass amounts to m = 217 tons and the transformation constant is = 1.34 . More detailed description of the structure and mathematical modelling can be found in [44]. Following the original N2 method, to plot the capacity diagram it is required an iterative graphical procedure in order to converge the value 46

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Fig. 8. Flowchart of the proposed displacement-based procedure for the seismic performance evaluation and vulnerability assessment of buildings.

control node displacement (d ) for the test example correspondent to a design ground acceleration ag equal to 0.3 g (the design value). In this table, the values listed in bold correspond to the target displacement d = dt ). point ( (d ) = 1 In addition to the example above – where for the same pushover

curve it was possible to compare the scale factor for the design ground acceleration for three levels of ground motion –, another interesting example would be the comparison of the scale factor for the design ground acceleration for three different pushover curves for the same level of the design ground acceleration. This comparison is illustrated in 47

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Fig. 7, where the blue curve is the same as in the example illustrated in Fig. 6 (i.e., obtained from the same pushover curve for a design ground acceleration ag equal to 0.3 g), and the green and red curves correspond, respectively, to an increase of 50% of the force Fb and to an increase of 50% of the displacement dn of the MDOF pushover curve used in the test example. From this figure it is possible to observe that: (a) an increase of 50% of the force Fb (green curve) has led to a decrease of 20.3% of the value of the target displacement and to an increase of 22.5% of the value of the scale factor for the design ground acceleration at the ultimate displacement; (b) an increase of 50% of the displacement dn (red curve) (which also corresponds to an increase of 50% of the ultimate displacement du ) has led to an increase of 19.2% of the value of the target displacement, and to an increase of 22.5% of the value of the scale factor at the ultimate displacement. Finally, Fig. 8 summarises the proposed displacement-based procedure for the seismic performance evaluation and vulnerability assessment of buildings.

visualise both the ultimate displacements (last point) and the performance point in the same curve: the target displacements are located at the intersection of the scale factor for the design ground acceleration – control node displacement curve with the horizontal line with ordinate of 1; c) The need for the iterative procedure recommended in EC8 for the determination of the target displacement is avoided, since it can be obtained by linear interpolation of the curve of the scale factor for the design ground acceleration as function of the control node displacement (d ) at the point where the scale factor is equal to one d = dt ). ( (d ) = 1 Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments

4. Conclusions

The first author would like to acknowledge the PhD grant (PD/BD/ 135201/2017) provided by Foundation for Science and Technology (FCT) (Portugal), within the scope of the Doctoral Programme InfraRisk – (Analysis and Mitigation of Risks in Infrastructures). Thanks are also due to University of Aveiro (Portugal), FCT/MEC for the financial support to the research Unit RISCO – Aveiro Research Centre of Risks and Sustainability in Construction – (FCT/UID/ECI/04450/2013) and to the Research Unit ISISE – Institute for Sustainability and Innovation in Structural Engineering (Portugal) – (UID/ECI/04029/2019).

Notwithstanding the limitations associated with any approximate method, the N2 method based on pushover analysis is a simple and fast nonlinear approach for the seismic design and performance assessment of buildings which has been in the focus of interest of many researchers due to its simplicity and ability to provide reasonably accurate results when compared to more advanced and time-consuming methods (e.g.: the nonlinear dynamic analysis) [22]. Although the idea of reversing the N2 procedure in a direct displacement-based design is not new [38], an iteration process is still needed since the target displacement is not known beforehand. In order to tackle this limitation, the procedure proposed in this paper offers a straightforward alternative for its computation, as well as for evaluating graphically of how a structure will perform when subjected to an earthquake ground motion. In this context, the proposed method presents several advantages:

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a) In the traditional bilinear format, for the comparison of the capacity of the structure with the demand of an earthquake ground motion it is necessary to represent both the idealised elasto-perfectly plastic force – displacement relationship and the inelastic demand spectrum, whose intersection defines the performance point. In the proposed method, the inelastic demand spectrum is implicitly considered in the scale factor for the design ground acceleration – control node displacement curve ( (d ) ). Therefore, with just one curve, it is possible to compare directly the capacity of the structure with the demand of an earthquake ground motion on the structure, not only for the target displacement, but for all range of control node displacements [0; du ]. In this way, the seismic performance can also be evaluated by means of the value of the scale factor for the design ground acceleration at the ultimate displacement point (end of the curve): if (d u ) 1, the in-plane capacity of the structural capacity is proven satisfactory for the direction and magnitude of the seismic action, since the target displacement is attained before the ultimate displacement; if (d u ) < 1, the structure is not able to withstand the prescribed seismic demand and the collapse will occur before the attainment of the target displacement. For this reason, it helps on the seismic vulnerability assessment of buildings, since for a given value of damage state threshold it is possible to easily know the correspondent percentage of the seismic action to attain that displacement; b) This procedure normalises the values of the spectral accelerations, thus permitting to assess the seismic performance of buildings independently of the values of the acceleration at the yield point. For this reason, it is particularly useful when comparing the seismic performance of two or more buildings, since it is possible to 48

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