STRUCTURAL ANALYSIS IN EARTHQUAKE ENGINEERING – A BREAKTHROUGH OF SIMPLIFIED NON-LINEAR METHODS fajfar

Published by Elsevier Elsevier Science Ltd. All rights reserved 12th European Conference on Earthquake Engineering Paper Reference 843

STRUCTURAL ANALYSIS IN EARTHQUAKE ENGINEERING – A BREAKTHROUGH OF SIMPLIFIED NON-LINEAR METHODS P. Fajfar University of Ljubljana, Faculty of Civil and Geodetic Engineering Ljubljana, Slovenia

ABSTRACT Structural response to strong earthquake ground motion cannot be accurately predicted due to large uncertainties and the randomness of structural properties and ground motion parameters. Consequently, excessive sophistication in structural analysis is not warranted. For the time  being, the most rational analysis and performance evaluation methods for practical applications seem to be simplified non-linear procedures, which combine the non-linear static (pushover) analysis of a relatively simple mathematical model and the response spectrum approach. In recent years, a breakthrough of these procedures has been observed. They have  been implemented into the modern guidelines and codes. The paper discusses such  procedures. After a brief overview of the methods, the major attention is focused on the N2 method, which has been implemented into the recent draft of the Eurocode 8 standard. The theoretical background of the extended version of the method, which can be applied for asymmetric structures, is presented. The similarities and differences between different methods, the determination of target displacement in the capacity spectrum method, the  problems related to the application of simplified methods to analysis of 3D models, the approximations and limitations of the simplified inelastic methods, and the direct displacement-based design are discussed. Although different methods may yield in many cases similar results, they differ in respect to simplicity, transparency and clarity of theoretical  background. The most important difference is related to the determination of displacement demand. The use is inelastic spectra is considered to be more appropriate than the use of highly damped equivalent elastic spectra.

 Keywords: non-linear analysis; pushover analysis; simplified non-linear methods; displacement-based design; inelastic spectra; torsion; asymmetric structures; Eurocode 8 INTRODUCTION The structural analysis in earthquake engineering is a complex task because (a) the problem is dynamic and usually non-linear, (b) the structural system is usually complex, and (c) input data (structural properties and ground motions) are random and uncertain. In principle, the non-linear time-history analysis is the correct approach. However, such an approach, for the time being, is not practical for everyday design use. It requires additional input data (timehistories of ground motions and detailed hysteretic behaviour of structural members) which cannot be reliably predicted. Non-linear dynamic analysis is, at present, appropriate for

research and for design of important structures. It represents a long-term trend. On the other hand, the methods applied in the great majority of existing building codes are based on the assumption of linear elastic structural behaviour and do not provide information about real strength, ductility and energy dissipation. They also fail to predict expected damage in quantitative terms. For the time being, the most rational analysis and performance evaluation methods for practical applications seem to be simplified inelastic procedures, which combine the non-linear static (pushover) analysis of a relatively simple mathematical model and the response spectrum approach. The paper discusses such procedures. The major attention is focused on the N2 method, which has been implemented into the recent draft of the Eurocode 8 standard. The theoretical background of the extended version of the method, which can be applied for asymmetric structures, is presented. The similarities and differences between different methods, the determination of target displacement in the capacity spectrum method, the problems related to the application of simplified methods to analysis of 3D models, the approximations and limitations of the simplified inelastic methods, and the direct displacement-based design are discussed. BRIEF OVERVIEW OF SIMPLIFIED NON-LINEAR METHODS In 1975, Freeman at al [1] developed a rapid evaluation method, which can be considered as a forerunner of the today’s “Capacity spectrum method”. In 1981, Saiidi and Sozen [2]  proposed to perform non-linear dynamic analyses on an equivalent SDOF system. Based on this idea, Fajfar and Fischinger [3,4] developed in mid-1980s the first version of the N2 method (N stands for Non-linear and 2 for two mathematical models – a SDOF and a MDOF model). However, the earthquake engineering community has not paid much attention to simplified non-linear approaches until mid-1990s, when a breakthrough of this approaches occurred. Present examples of simplified non-linear methods are the so-called Capacity spectrum method, which is in different variants applied in ATC 40 [5], in Trisevices’ manual (see [6]), and in Japanese Building Standard Law (see [7]), the non-linear static procedure, applied in FEMA 273 [8] and further developed in FEMA 356 [9], the N2 method [10], which has been implemented in the draft Eurocode 8 [11], and the Modal pushover analysis [12]. A method, similar to the non-linear static procedure in FEMA 273, has been included in the most recent NEHRP provisions [13]. Several simplified non-linear methods are implemented in the SEAOC Blue Book [14]. All methods combine the pushover analysis of a multi-degreeof-freedom (MDOF) model with the response sp ectrum analysis of an equivalent singledegree-of-freedom (SDOF) system. Inelastic spectra or elastic spectra with equivalent dam ping and period are applied. As an alterna tive representation of inelastic spectrum the Yield point spectrum has been developed [15].

With the exception of the methods implem ented in FEMA documents, these methods are formulated in the acceleration - displacement (AD) format. In this format, the capacity of a structure is directly compared with the demands of earthquake ground motion on the structure. The graphical presentation makes possible a visual interpretation of the procedure and of the relations between the basic quantities controlling the seismic response. The capacity of the structure is represented by a force - displacement curve, obtained by non-linear static (pushover) analysis. The base shear forces and roof displacements are converted to the spectral accelerations and spectral displacements of an equivalent single-degree-of-freedom (SDOF) system, respectively. These spectral values define the capacity diagram. The definition of seismic demand spectrum represents the main difference between different methods. In all cases, the intersection of the capacity curve and the demand spectrum provides an estimate of the inelastic acceleration (strength) and displacement demand.

Some other promising simplified procedures aimed for deformation-controlled design (called also displacement-based design, see a special chapter in this paper) have been developed, e.g. the approaches developed by Panagiotakos and Fardis [16] and by Priestley and co-authors [17-19]. They do not use non-linear analysis explicitly and they will be not discussed in this  paper. All methods are basically limited to planar structures. Recently, attempts have been m ade to extend the applicability of the methods to asymmetric structures, which require a 3D analysis. In this paper, the extended version of the N2 method is presented. It is based on 3D pushover analysis. A 3D pushover analysis has been used also by Ayala et al [20]. Another approach, which combines a 3D elastic dynamic (modal) analysis of the model representing the whole structure with 2D pushover analyses of most critical substructures (e.g. planar fram es or walls) was proposed by Moghadam and Tso [21]. Discussion on analysis of 3D m odels is  provided in chapter Asymmetric structures. DESCRIPTION OF THE N2 METHOD In this chapter, the steps of the simple version of the N2 method, extended to asym metric structures, are described. A simple version of the spectrum for the reduction factor is applied. It should be noted, however, that the suggested procedures used in particular steps of the method can be easily replaced by other available procedures. Additional inform ation on the (planar version) of the N2 method can be found in papers [10,22,23].

Step 1: Data A 3-D model of the building structure is used. The floor diaphragms are assumed to be rigid in the horizontal plane. The number of degrees of freedom is three times the number of storeys N. The degrees of freedom are grouped in three sub-vectors, representing displacements at the storey levels in the horizontal directions x and y, and torsional rotations UT = [UxT, UyT, UzT]

In addition to the data needed for the usual elastic analysis, the non-linear force - deformation relationships for structural elements under monotonic loading are also required. The most common element model is the beam element with concentrated plasticity at both ends. A  bilinear or trilinear moment - rotation relationship is usually used. Seismic demand is traditionally defined in the form of an elastic (pseudo)-acceleration spectrum S ae (“pseudo” will be omitted in the following text), in which spectral accelerations are given as a function of the natural period of the structure T . In principle, any spectrum can  be used. However, the most convenient is a spectrum of the Newmark-Hall type. The specified damping coefficient is taken into account in the spectrum. Step 2: Seismic Demand in AD Format Starting from the usual acceleration spectrum (acceleration versus period), inelastic spectra in acceleration – displacement (AD) format can be determined. For an elastic SDOF system, the following relation applies S de

=

T 2 4 π2

S ae  

(1)

where S ae and S de  are the values in the elastic acceleration and displacement spectrum, respectively, corresponding to the period T  and a fixed viscous damping ratio.

For an inelastic SDOF system with a bilinear fo rce - deformation relationship, the acceleration spectrum (S a) and the displacement spectrum (S d)  can be determined as [24] S  S a = ae   (2)  Rµ S d 

=

µ  Rµ

S de

=

µ

T 2

 Rµ 4 π 2

S ae

=µ

T 2

S a  

4 π2

(3)

where µ is the ductility factor defined as the ratio between the m aximum displacement and the yield displacement, and  Rµ is the reduction f actor due to ductility, i.e., due to the hysteretic energy dissipation of ductile structures. Note that Rµ is not equivalent to the reduction factor R used in seismic codes. The code reduction factor  R , which is in Eurocode 8 called behaviour factor q, takes into account both energy dissipation and the so-called overstrength  Rs. It can be defined as R = Rµ Rs. Several proposals have been made for the reduction factor Rµ. In the simple version of the N2 method, we will make use of a bilinear spectrum for the reduction factor Rµ  Rµ

= (µ − 1)

 Rµ

=µ

T  T C 

+1

T  < T C  T  ≥ T C   

 

(4) (5)

where T C  is the characteristic period of the ground motion. It is typically defined as the transition period where the constant acceleration segment of the response spectrum (the short period range) passes to the constant velocity segment of the spectrum (the medium-period range). Eqs 3 and 5 suggest that, in the medium- and long-period ranges, the equal displacement rule applies, i.e., the displacement of the inelastic system is equal to the displacement of the corresponding elastic system with the same period. Eqs 4 and 5 represent a simple version of the formulae proposed by Vidic et al [24]. Starting from the elastic design spectrum, and using Eqs 3 to 5, the dema nd spectra for the constant ductility factors µ in AD format can be obtained (Fi gure 1). They represent inelastic demand spectra. It should be noted that the construction of these spectra is in fact not needed in the com putational procedure. They just help for the visualisation of the procedure. Spectra in Figure 1 are based on EC8 [25] elastic spectrum for subsoil class B. In the draft of the standard [11] the constant displacement range of the spectrum begins at the period equal to 2 s, whereas the displacements in the very long period range (above 4 s) generally decrease with increasing period to the value of the peak ground displacement.  Note that any other inelastic spectrum can be employed in analysis. Inelastic demand spectra can be determined also by a rigorous procedure by using non-linear dynami c analysis. Step 3: Pushover Analysis A pushover analysis is performed by subjecting a structure to a m onotonically increasing  pattern of lateral forces, representing the inertial forces which would be experienced by the structure when subjected to ground shaking. Under incrementally increasing loads various structural elements yield sequentially. Consequently, at each event, the structure experiences a loss in stiffness.

Sa (g) T=0.6

3 T=0.15 2.5

µ=1 T=1

2 1.5

1.5

2

T=2

1 3

T=3

4

0.5

T=4

0 0

20

40

60

80

100

120

Sd (cm)

Figure 1: Demand spectra for constant ductilities in AD format normalized to 1.0 g peak ground acceleration, based on EC8 (1994) elastic spectrum for subsoil class B.

Using a pushover analysis, a characteristic non-linear force - displacement relationship of the MDOF system can be determined. In principle, any force and displacement can be chosen. Usually, base shear and roof (top) displacement are used as representative of force and displacement, respectively. The selection of an appropriate lateral load distribution is an important step within the  pushover analysis. A unique solution does not exist. Fortunately, the range of reasonable assumptions is usually relatively narrow and, within this range, different assumptions produce similar results. One practical possibility is to use two different displacement shapes (load  patterns) and to envelope the results. According to the draft of EC8 standard [11], a “uniform”  pattern, based on lateral forces that are proportional to mass regardless of elevation, and a “modal” pattern, consistent with the lateral force distribution determined in elastic analysis, are required. Lateral loads are applied in mass centres of different storeys. The vector of the lateral loads P, which generally consists of components in three directions (forces the x and y direction and torsional moments) is determined as P = p

 = p M

(6)

where M  is the mass matrix. The magnitude of the lateral loads is controlled by  p. The distribution of lateral loads is related to the assumed displacement shape . (Note that the displacement shape is needed only for the transformation from the MDOF to the equivalent SDOF system in Step 4). Consequently, the assumed load and displacement shapes are not mutually independent as in the majority of other pushover analysis approaches. The procedure can start either by assuming displacement shape and determining lateral load distribution according to Eq 6, or by assuming lateral load distribution  and determining displacement shape from Eq 6. Note that Eq 6 does not present any restriction regarding the distribution of lateral loads. Generally, can consist of non-zero components in three directions (two horizontal directions and of torsional rotation). In such a case (coupled displacement shape) lateral loads also consist of components in three directions. The procedure can be substantially simplified

if  lateral loads are applied in one direction only. This is a special case that requires that also the assumed displacement shape has non-zero components in one direction only, e.g.  = [

T

x

, 0T, 0T]

T

 

(7)

This special case is used in the present version of the N2 method. It should be noted, however, that even in this special case of uncoupled assumed displacement shape, the resulting displacements, determined by a pushover analysis of an asymme tric structure, will be coupled, i.e. they will have components in three directions. From Eqs 6 and 7 it follows that the lateral force in the x-direction at the i-th level is  proportional to the com ponent Φx,i  of the assumed displacement shape x, weighted by the storey mass mi  P  x,i = p mi Φx,i

 (8)

Such a relation has a physical background: if the assumed displacem ent shape was equal to the mode shape and constant during ground shaking, i.e. if the structural behaviour was elastic, then the distribution of lateral forces would be equal to the distribution of effective earthquake forces and Eq 6 was “exact”. In inelastic range, the displacement shape changes with time and Eq 6 represents an approximation. Nevertheless, by assuming related lateral forces and displacements according to Eq 6, the transformation from the MDOF to the equivalent SDOF system and vice-versa (Steps 4 and 6) follows from simple mathematics not only in elastic but also in inelastic range. No additional approximations are required, as in the case of some other simplified procedures. Step 4: Equivalent SDOF Model and Capacity Curve In the N2 method, seismic demand is determined by using response spectra. Inelastic  behaviour is taken into account explicitly. Consequently, the structure should, in principle, be modelled as a SDOF system. Different procedures have been used to determine the characteristics of an equivalent SDOF system. One of them, used in the current version of the  N2 method, is discussed below. The starting point is the equation of motion of a 3D structural model (with 3N degrees of freedom) representing a multi-storey building

&& + R  = −M s a   MU

(9)

U  is a vector representing displacements and rotations, R   is a vector representing internal forces, a  is the ground acceleration as a function of time, and s  is a vector defining the direction of ground motion. In the case of uni-directional ground motion, e.g. in the direction x, the vector s consists of one unit sub-vector and of two sub-vectors equal to 0. sT = [1T, 0T, 0T ] 

(10)

In the N2 method, ground motion is applied independently in two horizontal directions. Consequently, two separate analyses have to be performed with two different s vectors (vector (10) and a similar vector that corresponds to the ground excitation in the y-direction). For simplicity, damping is not included in the derivation. Its influence will be included in the design spectrum. It will be assumed that the displacement shape is constant, i.e. that it does not change during the structural response to ground motion. This is the basic and the most critical assumption within the procedure. The displacement vector U is defined as U=

 Dt 

(11)

where Dt  is the time-dependent top displacement. way that the component at the top is equal to 1.

is, for convenience, normalized in such a

From statics it follows P = R  

(12)

i.e., the internal forces R  are equal to the statically applied external loads P. By introducing Eqs 6, 11, and 12 into Eq 9, and by multiplying from the left side with obtain T 

&& +  D t 

M

T 

M

 p

=−

T 

Msa  

After multiplying and dividing the left hand side with equivalent SDOF system can be written as

T

, we (13)

T 

M s, the equation of motion of the

&& * + F * = − m *a   m * D

(14)

where m* is the equivalent mass of the SDOF system m* =

Φ T  M s  

(15)

It depends on the direction of ground motion. For ground motion in the x-direction, Eq 15 can  be written as m x * =

∑m Φ i

 x , i

 

(16)

 D* and F * are the displacement and force of the equivalent SDOF system  D* =

 Dt 

 F * =

,

Γ

V 

Γ

(17),(18)

V  is defined as V  =

T 

M s p

=  pm *  

(19)

It is the base shear of the MDOF model in the direction of ground motion. For ground motion in the x-direction, the following relations apply V  x

= ∑ pmi Φ x,i = ∑ P  x,i  

(20)

The constant Γ  controls the transformation from the MDOF to the SDOF model and viceversa. It is defined as

Γ=

T  T 

M s

M

=

m*  L *

 

(21)

 Note that m* depends on the direction of ground motion. Consequently, Γ, D*, and  F * also depend on the direction of ground motion. In the case of ground motion in one direction (Eq 10) and assuming a simple uncoupled displacement shape (Eq 7), Eq 21 can be written as

Γ=

∑ m ⋅ φ  ∑ m ⋅ φ  i

 x ,i

i

2  x ,i

 

(22)

which is the same equation as in the case of planar structures. Consequently, the transformation from the MDOF to the SDOF system and vice versa is exactly the same as in the case of a planar structure.

Γ is usually called the modal participation factor. Note that the assumed displacement shape is normalized – the value at the top is equal to 1. Note also that any reasonable shape can be used for  . As a special case, the elastic first mode shape can be assumed. The same constant Γ applies for the transformation of both displacem ents and forces (Eqs 17 and 18). As a consequence, the force - displacement relationship determ ined for the MDOF system (the V  -  Dt   diagram) applies also to the equivalent SDOF system (the  F * -  D* diagram), provided that both force and displacement are divided by Γ. This can be visualized  by changing the scale on both axes of the force – displacement diagram. The initial stiffness of the equivalent SDOF system remains the same as that defined by the base shear – top displacement diagram of the MDOF system. In order to determine a simplified (elastic - perfectly plastic) force – displacement relationship for the equivalent SDOF system, engineering judgement has to be used. In regulatory documents some guidelines may be given. In Annex B of the draft standard EC8 [11] the  bilinear idealization is based on the equal energy principle. Note that the displacement demand depends on the equivalent stiffness which, in the case of the equal energy approach, depends on the target displacement. In principle, an iterative approach would be needed. However, it is questionable if it is warranted. In the draft of EC8, the displacem ent at the formation of plastic mechanism is used for the determination of equivalent stiffness based on equal energy. This approach yields generally a conservative estimate of displacement demand. If displacement demand is expected to be much lower than that at the plastic mechanism, it is reasonable to base the equal energy rule on a smaller displacement which leads to a higher equivalent stiffness. The graphical procedure, used in the simple N2 method, requires that the post-yield stiffness is equal to zero. This is because the reduction factor  Rµ is defined as the ratio of the required elastic strength to the yield strength. The influence of moderate strain hardening is incorporated in the demand spectra. It should be emphasized that moderate strain hardening does not have a significant influence on displacement demand, and that the proposed spectra approximately apply for systems with zero or small strain-hardening (see chapter Approximations and limitations). The elastic period of the idealized bilinear system T * can be determined as ∗

T 

m ∗  D y∗

=2π

 F  y∗

(23)

where  F  y∗  and  D y∗ are the yield strength and displacement, respectively. Finally, the capacity diagram in AD format is obtained by dividing the forces in the force deformation ( F * - D*) diagram by the equivalent mass m* S a

=

 F ∗ m*

 

(24)

Step 5: Seismic Demand for the Equivalent SDOF System The seismic demand for the equivalent SDOF system can be determined by using the graphical procedure illustrated in Figure 2 (for medium- and long-period structures; for short period structures see e.g. Fajfar, 2000). Both the demand spectra and the capacity diagram have been plotted in the same graph. The intersection of the radial line corresponding to the

elastic period of the idealized bilinear system T * with the elastic demand spectrum S ae defines the acceleration demand (strength) required for elastic behavior and the corresponding elastic displacement demand. The yield acceleration S ay represents both the acceleration demand and the capacity of the inelastic system. The reduction factor  Rµ  can be determined as the ratio  between the accelerations corresponding to the elastic and inelastic systems  Rµ

=

( )

S ae T ∗

(25)

S ay

 Note that Rµ is not the same as the reduction (behaviour, response modification) factor R used in seismic codes. The code reduction factor R takes into account both energy dissipation and the so-called overstrength. The design acceleration S ad   is typically smaller than the yield acceleration S ay. If the elastic period T * is larger than or equal to T C, the inelastic displacement demand S d  is equal to the elastic displacement demand S de (see Eqs 3 and 5, and Figure 2). From triangles in Figure 2 it follows that the ductility demand, defined as

µ = S d / D y∗ , is equal to Rµ

T * ≥ T C 

S d = S de (T * )

(26)

µ = Rµ

(27) Sa *

T Sae

µ = 1 (elastic) µ

Say Sad *

Dd Dy*

Sd = Sde

Sd

Figure 2: Elastic and inelastic demand spectra versus capacity curve.

If the elastic period of the system is smaller than T C, the ductility demand can be calculated from the rearranged Eq 4

µ = ( Rµ − 1)

T C  T ∗

+1

T * < T C  

(28)

The displacement demand can be determined either from the definition of ductility or from Eqs 3 and 28 as S d 

 1 + ( R − 1) T C     µ  Rµ   T ∗  cases ( T ∗ < T C    and T ∗ ≥ T C )

= µ D y∗ =

S de

(29)

In both the inelastic demand in terms of accelerations and displacements corresponds to the intersection point of the capacity diagram with the demand spectrum corresponding to the ductility demand µ. At this point, the ductility factor determined from the capacity diagram and the ductility factor associated with the intersecting demand spectrum are equal.

All steps in the procedure can be performed numerically without using the graph. However, visualization of the procedure may help in better understanding the relations between the  basic quantities. At this stage, the displacement demand can be modified if necessary, e.g. for taking into account larger displacements observed for systems with narrow hysteresis loops or negative  post-yield stiffness. Steps 6 And 7: Global and Local Seismic Demand for the MDOF Model The displacement demand for the SDOF model S d   is transformed into the maximum top displacement  Dt   of the MDOF system (target displacement) by using Eq 14. Under monotonically increasing lateral loads with a fixed pattern (as in Step 3), the structure is  pushed to Dt . It is assumed that the distribution of deformations throughout the structure in the static (pushover) analysis approximately corresponds to that which would be obtained in the dynamic analyses. Separate 3D pushover analyses are performed in two horizontal directions. Relevant results (displacements, storey drifts, joint rotations, and forces in brittle elements which should remain in elastic region), obtained by two independent analyses, are combined  by the SRSS rule.  Note that  Dt   represents a mean value for the applied earthquake loading, and that there is a considerable scatter about the mean. Consequently, it is appropriate to investigate likely  building performance under extreme load conditions that exceed the design values. It is appropriate to carry out the analysis to at least 150% of the calculated top displacement. Step 8: Performance Evaluation (Damage Analysis) Expected performance can be assessed by comparing the seismic demands, determined in Step 7, with the capacities for the relevant performance level. The determination of seismic capacity is not discussed in this paper. Global performance can be visualized by comparing displacement capacity and demand. Step 9: Graphical Presentation If the procedure is formulated in the acceleration - displacement (AD) format, the visual interpretation of the procedure and of the relations between the basic quantities controlling the seismic response is possible. Some approaches (e.g. ATC-40) use graphical procedures for the determination of the “performance point”, which represents seismic demand. The graphical  presentation of the basic parameters (of the equivalent SDOF system) in the N2 method (simple variant, as implemented in EC8) for medium- and long-period structures, for which the “equal displacement rule” applies, is given in Figure 2. The intersection of the radial line corresponding to the elastic period T * of the idealized bilinear system with the elastic demand spectrum defines the acceleration demand (strength), required for elastic behaviour S ae, and the corresponding elastic displacement demand S de. The yield acceleration S ay represents both the acceleration demand and the capacity of the inelastic system. The reduction factor  Rµ represents the ratio between the accelerations corresponding to the elastic and inelastic systems. If the elastic period T * is larger than or equal to T C ( i.e., if the period of the structure is in the medium- or long-period range), the inelastic displacement demand S d  is equal to the elastic displacement demand S de  (“equal displacement rule”). From triangles in Figure 2 it follows that the ductility demand is equal to the ratio between S ae and S ay, i.e. to Rµ. S ad represents a typical design strength, i.e. strength required by codes for ductile structures, and  Dd * is the corresponding displacement obtained by linear analysis. The inelastic demand in terms of accelerations and displacements corresponds to the intersection point of the capacity diagram with the demand spectrum corresponding to the ductility demand µ. At this point, the

ductility factor determined from the capacity diagram and the ductility factor associated with the intersecting demand spectrum are equal. Note that all steps in the procedure can be  performed numerically without using the graph. However, visualization of the procedure may help in better understanding the relations between the basic quantities.

SEISMIC DEMAND IN THE “CAPACITY SPECTRUM METHOD” The acceleration – displacement (AD) format was proposed in conjunction with the “capacity spectrum method” [1,6]. In this method, and in other methods based on equivalent linearization, inelastic seismic demand is estimated from a highly damped elastic spectrum. An equivalent linear system is determined with lower stiffness and higher damping than those of the actual system. The equivalent (effective) elastic period of the system corresponds to the secant stiffness at the maximum displacement. Different procedures have been proposed for determination of equivalent viscous damping, which is intended to take into account the hysteretic energy dissipation. The equivalent damping is usually a function of the ductility demand µ. Very different ductility – damping relations have been proposed. A partial compilation was made by Freeman [6]. Large ranges of equivalent damping values correspond to a specific ductility value. For the ductility demand equal to 3, for example, values from 16 to 39% have been suggested.

In AD format, seismic demand is defined by the intersection of the capacity curve and the demand spectrum. If an equivalent elastic spectrum is used, displacement demand depends on equivalent stiffness and equivalent damping. On the other hand, both quantities depend on the ductility demand, which is related to displacement demand. Consequently, all quantities are interrelated and, in principle, iteration is needed. The procedure is as follows. A value of the target displacement has to be assumed. Based on this assumed value, the equivalent elastic  period and equivalent viscous damping of the SDOF system are determined. The new value for target displacement is determined from the elastic spectrum for the equivalent damping as a function of the equivalent period. The iterative procedure is supposed to converge to the final value of the target displacement. However, it may not converge as demonstrated by Chopra and Goel [26]. The details of the procedures for the determination of target displacement based on equivalent elastic system will not be discussed here. The reader is referred to a recent overview paper by Miranda and Ruiz-Garcia [27].

COMPARISON OF SIMPLIFIED METHODS A study performed on a simple planar structure [28] indicates that the procedures, based on  pushover analysis, generally yield results of adequate accura cy. The global quantities (like top displacement) are generally more accurate than the local ones (like rotations at member ends). However, different approaches differ in regard to simplicity, transparency, and the clarity of the theoretical background.

The essential difference is related to the determination of the displacement demand (target displacement). If an equivalent elastic spectrum is used, an iteration is needed, as discussed in the previous chapter. (Note that iteration is not needed if a procedure based on equivalent elastic spectrum is used for direct displacement based design in which the target displacement is fixed. Often, iteration is not performed if the procedure is used for checking if displacement demand is smaller than displacement capacity.) The use of highly damped elastic spectra for the determination of seismic demand is a controversial part of the capacity spectrum method

and other methods based on equivalent linearization. The quantitative values of equivalent damping, suggested by different authors, differ considerably. It is interesting to note that Freeman, the author of the “Capacity spectrum method”, derived the equivalent damping (employed in Triservices’ manual procedure) by equating the peak deformation of the equivalent linear system, determined from the elastic spectrum, to the peak deformation of the inelastic system, determined from the inelastic (Newmark-Hall) spectrum. The question arises, why this detour? The inelastic design spectra can be used directly and they do not require iteration. Inelastic spectra can be used not only for analysis and performance evaluation, but also for direct displacement-based design as indicated in [22] and demonstrated by Chopra and Goel [29]. The above arguments suggest the superiority of the  procedures based on inelastic spectra to those based on equivalent elast ic spectra. Different procedures differ also in the assumed lateral load pattern, used in pushover analysis, and in the displacement shape, used for the transformation from the MDOF to the SDOF system (and vice versa). Only if the two vectors are related (Eq 6), the transformation from the MDOF to the SDOF system is based on a mathematical derivation. This feature leads to a transparent transformation. If the two vectors are independent, additional approximations are implicitly introduced, and the clarity of the theoretical background and the transparency are lost, although the accuracy of results may be adequate. The use of lateral load pattern, which is related to the assumed displacement shape, does not present any restriction, because any shape can be used. Bilinear idealization of the pushover curve is required for the methods using inelastic spectra. The procedures employing equivalent elastic spectra do not need initial stiffness. However, the equivalent damping is usually based on the ductility, therefore a bilinear idealization is needed also for these procedures (partial exception is the approach in the Japanese code). If the bilinear idealization depends on the displacement demand, than the computational  procedure becomes iterative even in the case when inelastic spectrum is used. It is questionable if this complication is warranted. For practical applications and for educational purposes a graphical representation of the  procedure is extremely important. A breakthrough of the simplified methods was possible when the acceleration – displacement format was implemented, which allows a visualization of important demand and capacity parameters (even if all results can be obtained numerically). It seems that the use of this format is essential for the appreciation of the  procedure in practice. ASYMMETRIC STRUCTURES Originally, all simplified inelastic methods have been limited to planar structural models. Consequently, in principle, they were not applicable to asymmetric structures, where translational and rotational vibrations are coupled and a 3D model is needed. Only very recently, some attempts have been made to extend the applicability of the simplified methods to 3D structural models.

An approach based on elastic dynamic analysis and 2D pushover analysis was proposed by Moghadam and Tso [21]. This approach extends the widely used equal displacement principle to 3D structures. It is assumed that the displacements at the top of the building are essentially the same, whether the building is responding in its elastic state or is excited into the inelastic range. First, an elastic dynamic analysis of the 3D structural model is performed and multiple target displacements are determined, one for each resisting element in which the damage

assessment is sought. Then, the damage pattern of an asymmetric building is evaluated by  performing a number of 2D pushover analyses on resisting elements of interest. This approach avoids the need to perform a 3D pushover analysis and is applicable to both torsionally flexible and stiff buildings (see discussion in the further text). However, it is not directly applicable for short period structures, for which the equal displacement rule does not apply. So far, a reasonable accuracy of the approach has been demonstrated for very simple frame  buildings regular in elevation. The question is whether the approach can be extended to more general structures, which consist of load-bearing elements of different types (for example frames and walls) and to structures with considerable changes in stiffness and strength along the height of the building. Some results, obtained for a frame-wall structure, are presented in [30]. In the N2 method, presented in this paper, the torsional effects are incorporated by using  pushover analyses of a 3D structural model. (3D pushover analysis has been discussed, e.g., in [31,32].) The lateral loads are applied in mass centres in one direction only. Based on the obtained base shear – roof (top) displacement relationship, an equivalent SDOF system is determined. Top displacement corresponds to the mass centre. Two independent analyses are  performed, one for each horizontal direction. The transformations from the MDOF to the SDOF system and vice versa, and the procedure for determining the target displacement of the equivalent SDOF system are exactly the same as in the case of planar structures. Seismic demands are determined by pushing the structure to the target displacement defined at the mass centre. Relevant results (displacements, storey drifts, joint rotations, and forces in brittle elements which should remain in elastic region), obtained by two independent analyses, are combined by the SRSS rule. The applicability of the approach is limited to torsionally stiff structures as discussed in further text. A similar procedure is used also by Ayala and co-workers. Recently they proposed an advanced approach [20], in which the vector of the lateral loads in pushover analysis includes also torsional moments, i.e. lateral loads are applied eccentrically in respect to mass centres. The extension of the N2 method to asymmetric structures has been based on research on the elastic and inelastic seismic response of asymmetric structures, which contributed to a better understanding of torsional effects. It has been found that there is a substantial difference  between the seismic response of torsionally stiff and torsionally flexible structures, as discussed below. The most important structural parameters, which control the torsional behaviour in elastic range, proved to be two frequency ratios, which are defined as uncoupled torsional frequency divided by uncoupled translational frequencies for two horizontal directions. Depending on the value of these ratios, the structures can be classified as torsionally stiff (ratios larger than 1, first two modes are translational) or torsionally flexible (ratios smaller than 1, first mode is torsional). In the case of a torsionally stiff structure, a single (predominantly translational) mode controls the displacements in one direction. Typical dynamic behavior of such structures is qualitatively similar to the static response, i.e., displacements are larger at the flexible edge and generally smaller at the stiff edge. The seismic response of torsionally flexible buildings is qualitatively different from that observed in torsionally stiff buildings and from that obtained in the case of static loading in mass centre. Both predominatly translational and predominantly torsional mode are important. As a result, not only displacements at the flexible edge increase due to torsion, like in the case of torsionally stiff structures, but also displacements at the stiff edge. So, displacements at both flexible and stiff side are generally

larger than those of the symmetric structure. In many cases, torsional amplification at all edges may be very large. The above conclusions have been mainly drawn from results of elastic analyses obtained by applying uni-directional ground motion. In the case of bi-directional ground motion, which requires a structural model with load-bearing elements in two directions, additional considerations are needed. Structures can be torsionally stiff in one direction, and torsionally flexible in the other one. In the inelastic range, the stiffnesses of structural elements and, consequently, the translational and torsional stiffnesses are changing with time. Torsional  plastic mechanisms are usually formed for short periods of time. Discussion of these effects, which may considerably influence the structural response, is beyond the scope of this paper. In the case of torsionally flexible buildings, where more than one vibration mode significantly contributes to the response in one direction, pushover analysis, in principle, cannot provide adequate results.  For torsionally stiff structures, where the structure vibrates mainly in  predominantly translational modes, pushover analysis yields qualitatively correct displacements, which are larger at the flexible edge than at the stiff edge. Torsional influences are usually somewhat underestimated at flexible edges. At stiff edges, the results of a  pushover analysis may be considerably different from those obtained by dynamic analysis. Until more research results are available, a conservative approach is suggested, which does not consider any favourable effects due to torsion: for all load-bearing elements located  between the mass centre and the stiff edge assume a constant value of displacement, equal to the displacement at the mass centre. The results of pushover analysis can be improved by using effective eccentricities. Effective eccentricities are intended to take into account dynamic effects and to produce the same peak response as dynamic analysis. Such an approach is usual in seismic codes which are based on elastic analysis. However, no consensus has been reached on the amplitudes of effective eccentricity and different values are used in different codes. In principle, two analyses are required, one with increased eccentricity intended to simulate the dynamic amplification at the flexible edge, and an other with reduced eccentricity intended to simulate the dynamic influences at the stiff edge. The difference between static and dynamic results at the flexible edge can be reduced by increasing the eccentricity of the static lateral loading, i.e. by moving the point of application from the mass centre toward the flexible edge. A different effective eccentricity is appropriate for matching the results at the stiff edge. For elastic analysis, several effective eccentricities are used in different codes. Note the similarity with the 2D  pushover analysis, where two different patterns of lateral loading are used, one of them is intended to simulate the higher mode effect. The option of using effective eccentricities has not been included in the N2 method yet. In the theoretical background, presented in chapter Description of the N2 method, it is assumed that the lateral loads are applied in mass centres. The problem of effective eccentricity, discussed above, can be stated in an other way: at which point to apply lateral forces? In addition to this problem, another problem arises in the case of a 3D structure: how to combine the influence of the two horizontal excitations? Independent uni-directional static load applications in two directions without the combination of results usually yield unconservative results. On the other hand, simultaneous load application to full target displacement in two horizontal directions is exceedingly conservative. It always leads to a torsional plastic mechanism and, as a consequence, to reduction of strength compared to the corresponding symmetric structure. Inelastic dynamic time-history analyses of torsionally stiff structures generally do not support such a behavior. During a time-history, torsional plastic mechanisms usually occur. However, they last only a

short time and usually (but not always) do not influence detrimentally the overall structural  behavior. In general, improved estimates can be obtained if results of two independent unidirectional analyses with ground motion applied separately in two horizontal directions are combined using the SRSS rule. This rule has been widely applied in elastic range and it can  be used, subjected to more dispersion, also in inelastic range [33,34]. APPROXIMATIONS AND LIMITATIONS The described methods are, like any approximate method, subject to several limitations. Applications of the majority of the method are, for the time being, restricted to analysis of  planar structures. In planar analysis, there are two main sources of approximations and corresponding limitations: pushover analysis and determination of target displacement.

 Non-linear static (pushover) analysis can provide an insight into the structural aspects which control performance during severe earthquakes. The analysis provides data on the strength and ductility of the structure, which cannot be obtained by elastic analysis. Furthermore, it exposes design weaknesses that may remain hidden in an elastic analysis. On the other hand, the limitations of the approach should be recognized. Pushover analysis is based on a very restrictive assumption, i.e. a time-independent displacement shape. Thus, it is in principle inaccurate for structures where higher mode effects are significant, and it may not detect the structural weaknesses, which may be generated when the structure’s dynamic characteristics change after the formation of the first local plastic mechanism. A detailed discussion of  pushover analysis can be found in papers by Krawinkler and Seneviratna [35] and Elnashai [36]. Additional discussion on the relationship between MDOF and SDOF systems is  presented in [37]. One practical possibility to partly overcome the limitations imposed by pushover analysis is to assume two different displacement shapes (load patterns), and to envelope the results. Another more complex possibility is to use lateral load distribution which changes in each step of analysis, i.e. the Adaptive pushover analysis [36]. The other important source of inaccuracy is the determination of target displacement (displacement demand) for the equivalent SDOF system. If inelastic spectra are used, the displacement demand depends on the initial period of the system and on the spectrum. Initial  period of the equivalent SDOF system is not uniquely defined, it depends on the bilinear idealization of the actual base shear – top displacement curve and is, to some extent, based on engineering judgement. Inelastic spectra are based on statistical analyses of structural models and may not apply for structures, whose inelastic behaviour is basically different from that assumed in statistical analyses. The simplest possibility for inelastic spectra is to apply, in the medium- and long-period range, the “equal displacement rule.” The equal displacement rule has been used quite successfully for almost 40 years. Many statistical studies (see, e.g., the discussion in [10]) have confirmed that the equal displacement rule is a viable approach for structures on firm sites with the fundamental period in the medium- or long-period range, with relatively stable and full hysteretic loops. In many cases a conservative estimate of the mean value of the inelastic displacement may be obtained [38]. The equal displacement rule, however, may yield too small inelastic displacements in the case of near-fault ground motions, hysteretic loops with significant pinching or significant stiffness and/or strength deterioration, and for systems with low strength (i.e., with a yield strength to required elastic strength ratio of less than 0.2). Moreover, the equal displacement rule may be not satisfactory for soft soil conditions. In these cases, modified inelastic spectra should be used. Alternatively, correction factors for displacement demand (if available) may be applied.

In the case of short-period structures, inelastic displacements are larger than the elastic ones. The transition period, below which the inelastic to elastic displacement ratio begins to increase, is roughly equal to the characteristic period of the ground motion T C . In the short period range, the sensitivity of inelastic displacements to changes of structural parameters is greater than in the medium- and long period ranges. Consequently, estimates of inelastic displacement are less accurate in the short-period range. However, the absolute values of displacements in the short-period region are small and, typically, they do not control the design.  Note that for the methods, based on inelastic spectra, any realistic elastic and corresponding (compatible) inelastic spectrum can be applied. For example, for a specific acceleration timehistory, the elastic acceleration spectrum as well as the inelastic spectra, which take into account specific hysteretic behavior, can be computed and used as demand spectra. Moreover, any reasonable  Rµ  spectrum, compatible with the elastic spectrum, can be used. (Note that elastic spectra for specific accelerograms and smooth  Rµ  spectra are not compatible.) Examples are presented in [26, 39]) . In the case of equivalent elastic spectra, the major source of errors is the determination of equivalent viscous damping. In the short-period range, the equivalent elastic spectra differ a lot from realistic inelastic spectra. The results of the proposed method are intended to represent mean values for the applied earthquake loading. There is a considerable scatter about the mean. Consequently, it is appropriate to investigate likely building performance under extreme load conditions that exceed the design values. This can be achieved by increasing the value of the target displacement. For methods, which can be applied for analysis of 3D structural models, all limitations discussed above remain. As an additional limitation, the N2 method is not appropriate for torsionally flexible structures. The pushover approach is based on the assumption that the structure vibrates predominantly in the fundamental mode. For torsionally flexible buildings, more than one mode contributes significantly to the overall responses in the direction of excitation. Thus the basic assumption of the pushover approach is violated and the difference  between static and dynamic results may be unacceptably large. However, with an adequate choice of two effective eccentricities, the envelope of the results of pushover analyses might approximate the results of dynamic analysis. The larger one is intended to control the flexible edge and the smaller one the stiff edge. The eccentricities given in codes could be used until more research results are available. A conservative approach is not to consider any computed favourable effect of torsion. The approach, proposed by Moghadam and Tso, which is based on equal displacement  principle, cannot be directly applied to short-period structures. However, there are no limitations regarding torsionally flexible structures. The simplified methods usually do not take into account the effect of cumulative damage. This effect can easily be taken into account by using the so-called equivalent ductility factor (e.g. [40,41]). The idea behind the equivalent ductility factor is to reduce the monotonic deformation capacity of an element and/or structure as a consequence of cumulative damage due to the dissipation of hysteretic energy [23]. Alternatively, the influence of cumulative damage can be taken into account by increasing seismic demand (e.g. [42,43]).

DIRECT DISPLACEMENT-BASED DESIGN “The most suitable approach for seismic design practice appears to be Deformation Controlled Design. In this approach deformation targets defined by performance based engineering criteria are used. Two methods can be employed: • The traditional force based design approach combined with required deformation target verification; • The direct deformation based design approach in which the design starts from the target deformations.” (see Conclusions in [44])

Direct displacement-based design is often used as another name for Direct deformation-based design. In principle, direct displacement-based design can be applied within the framework of the simplified non-linear methods formulated in the AD format. Four quantities define structural behaviour: strength, displacement, ductility and stiffness. Design and/or  performance evaluation begins by fixing one or two of them. The others are determined by calculations. Different approaches differ in the quantities that are chosen at the beginning of the design or evaluation. For example, for a seismic performance evaluation of a newly designed or existing structure, stiffness and strength have to be known, whereas a direct displacement-based design approach starts from a predetermined target displacement. A great majority of presently available direct displacement-based design procedures use equivalent linear systems. So, a wrong impression has been obtained that the equivalent linear system is an essential part of a direct displacement-based procedure. However, such an impression is false. The use of an equivalent elastic system with high damping and secant stiffness, corresponding to the maximum response, is only one of possible approaches for determination of seismic demand. Inelastic spectra can be used as well, as indicated in [22] and demonstrated by Chopra and Goel [29]. It should be noted, however, that in the case of a direct displacement-based procedure, in which the assumed target displacement is the starting  point, there is no need for iteration, even if an approach based on equivalent linearization is used. Examples of recent direct-displacement based approaches are methods developed by Priestley [17] and Panagiotakos and Fardis [16]. The former does not require structural analysis and is  based on equivalent linearization. The later is based on elastic analysis and equal displacement rule. CONCLUSIONS Structural response to strong earthquake ground motion cannot be accurately predicted due to large uncertainties and the randomness of structural properties and ground motion parameters. Consequently, excessive sophistication in structural analysis is not warranted. The simplified non-linear methods, which are, like any approximate method, subject to several limitations,  provide a tool for a rational yet practical evaluation procedure for building structures for multiple performance objectives. The formulation of the methods in the acceleration – displacement format enables the visual interpretation of the procedure and of the relations  between the basic quantities controlling the seismic response. This feature is attractive to designers. In general, the results obtained using the simplified methods are reasonably accurate, provided that the structure oscillates predominantly in the first mode. The applicability of the methods can be easily extended to torsionally stiff asymmetric structures, where the response is dominated by translational vibration modes. The influence of higher

modes in elevation can be approximately covered by using at least two different distribution of lateral loads. The use of two effective eccentricities in plan represents a similar idea, which may cover the influence of torsional modes of vibration. Although different methods may yield in many cases similar results, they differ in respect to simplicity, transparency and clarity of theoretical background. The most important difference is related to the determination of displacement demand. The use of inelastic spectra is considered to be more appropriate than the use of highly damped equivalent elastic spectra. Inelastic spectra are based on a more solid theoretical background and they do not require iteration. ACKNOWLEDGEMENTS The results presented in this paper are based on work continuously supported by the Ministry of Education, Science and Sport of the Republic of Slovenia. This support is gratefully acknowledged. The author is indebted to Professor M. Fischinger for important contributions at the initial stage of development of the N2 method, to the past and present Ph.D. and M.Sc. students M. Dolšek, D. Drobni č, P. Gašperšič, V.Kilar, D.Marušić, I.Peruš, and T. Vidic, and to visiting researchers G.Magliulo (Univ.of Naples) and D.Zamfirescu (Technical Univ. of Civ.Eng., Bucharest). The results of their dedicated work are included in this paper. The  benefits of the continuous cooperation with Professor H.Krawinkler from Stanford University are invaluable. REFERENCES 1. Freeman SA, Nicoletti JP, Tyrell JV. Evaluations of existing buildings for seismic risk  – A case study of Puget Sound Naval Shipyard, Bremerton, Washington. In: Proceedings of the 1 st U.S. National Conference on Earthquake Engineering, Berkeley: EERI, 1975:113-122. 2. Saiidi M, Sozen MA. Simple nonlinear seismic analysis of R/C structures. Journal of Structural Engineering, ASCE 1981; 107: 937-52. 3. Fajfar P, Fischinger M. Non-linear seismic analysis of RC buildings: Implications of a case study. European Earthquake Engineering 1987; 1(1): 31-43. 4. Fajfar P, Fischinger M. N2 – A method for non-linear seismic analysis of regular  buildings. In: Proceedings of the 9th  World Conference on Earthquake Engineering, Tokyo, Kyoto 1988; Tokyo: Maruzen, 1989: Vol.5:111-16. 5. ATC. Seismic evaluation and retrofit of concrete buildings. Vol. 1, ATC 40, Redwood City: Applied Technology Council, 1996. 6. Freeman SA. Development and use of capacity spectrum method. In: Proceedings of the 6th U.S. National Conference on Earthquake Engineering, Seattle, CD-ROM, Oakland: EERI, 1998. 7. Otani S, Hiraishi H, Midorikawa M, Teshigawara M. New seismic design provisions in Japan. In: Proceedings 2000 Fall ACI Convention, Toronto, 2000. 8. FEMA. NEHRP guidelines for the seismic rehabilitation of buildings, FEMA 273, and  NEHRP Commentary on the guidelines for the seismic rehabilitation of buildings, FEMA 274. Washington, D.C.: Federal Emergency Management Agency, 1997. 9. FEMA. Prestandard and Commentary for the Seismic Rehabilitation of Buildings, FEMA 356. Washington, D.C.: Federal Emergency Management Agency, 2000. 10. Fajfar P. A nonlinear analysis method for performance-based seismic design. Earthquake Spectra 2000, 16(3): 573-92.

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