maximum seismic displacements evaluation of steel frames from their post-earthquake residual...

Bull Earthquake Eng (2013) 11:2233–2248DOI 10.1007/s10518-013-9490-z

ORIGINAL RESEARCH PAPER

Maximum seismic displacements evaluation of steelframes from their post-earthquake residual deformation

A. A. Christidis · E. G. Dimitroudi ·G. D. Hatzigeorgiou · D. E. Beskos

Received: 16 December 2012 / Accepted: 14 July 2013 / Published online: 25 July 2013© Springer Science+Business Media Dordrecht 2013

Abstract The maximum seismic displacements of a structure can be used for the assessmentof its post-earthquake performance. In this paper, a simple and efficient procedure is proposedfor determining maximum seismic displacements of planar steel frames from their residualdeformation. More specifically, the inelastic behaviour of 36 moment resisting steel framesand 36 concentrically X-braced steel frames under one hundred strong ground motions isinvestigated. Thus, on the basis of extensive parametric studies for these structures and seismicrecords, empirical equations are constructed for simple and effective prediction of maximumseismic displacements from residual deformation, which can be measured in-situ after strongseismic events. It is found that the usage of residual deformation can be effectively utilizedto evaluate the post-earthquake performance level of steel structures.

Keywords Residual displacements · Maximum seismic displacements · Strong groundmotions · Moment resisting steel frames · Concentrically X-braced steel frames.

1 Introduction

The reliable prediction of post-earthquake structural performance is a complicated procedure,requiring by the designer to take into account numerous factors that concern the structuralresponse as well as the inherent unawareness of the structure’s dynamic characteristics andtheir variability in the presence of future earthquakes (Yun et al. 2002). Various methods

A. A. Christidis · E. G. Dimitroudi · G. D. Hatzigeorgiou (B)Department of Environmental Engineering, DemocritusUniversity of Thrace, Xanthi, Greecee-mail: [email protected]

D. E. BeskosDepartment of Civil Engineering, University of Patras, Patras, Greece

D. E. BeskosOffice of Theoretical and Applied Mechanics,Academy of Athens, Athens, Greece

123

2234 Bull Earthquake Eng (2013) 11:2233–2248

Fig. 1 Relationship betweenmaximum interstorey drift androof drift of well-designed framedstructures with 3–12 storeyssubjected to several earthquakes(from Ghobarah 2004)

and approaches have been proposed in terms of safety assessment, maintenance and repair.Thus, the post-earthquake performance level of structures provides a very important sourceof information both for rehabilitation procedures and determination of structural responseto probable incoming aftershocks (Hatzigeorgiou et al. 2011). In this respect, the dynamiccharacteristics of structures or their seismic response can be used to obtain one of the mostimportant factors of structural performance, i.e., the damage index of the entire structure (DiPasquale and Cakmak 1990; Ghobarah et al. 1999; Kim and Chun 2004).

It has been generally accepted that structural deformation can be used to gauge expecteddamage (Cosenza et al. 1993; Krawinkler et al. 2003; Erduran and Yakut 2004; Medina andKrawinkler 2005). The interstorey drift ratio (IDR), i.e., the maximum relative displacementbetween two successive stories normalized to the storey height, appears to be one of themost appropriate damage estimators, both for assessment of structural members and non-structural displacement-sensitive components. According to Ghobarah (2004), the peak roofdisplacement and the roof drift are also useful simple measures of the overall structuraldeformation. Recently, Krishnan et al. (2012) also used the roof drift as a damage estimator.For well-designed framed structures, the interstorey drift’s distribution along the height isalmost uniform and according to Ghobarah (2004), the relationship between the roof driftand the maximum interstorey drift is linear, as shown in Fig. 1. Additionally, according toFEMA P-750 (2009), the allowable storey drifts are directly connected to peak roof drift andfor design purposes, compliance with allowable story drifts is achieved by limiting the peakroof drift. It is obvious, therefore, that the evaluation of peak roof displacement is a quiteuseful parameter for the post-earthquake assessment of building structures.

Residual displacements are important parameters for the evaluation of the performancelevel of structures, as mentioned in the pertinent literature both for single (Macrae andKawashima 1997; Borzi et al. 2001; Christopoulos et al. 2003; Pampanin et al. 2003; Ruiz-Garcia and Miranda 2006a,b; Uma et al. 2006; Pettinga et al. 2007a,b; Ruiz-Garcia andMiranda 2010; Hatzigeorgiou et al. 2011) and multiple earthquakes (Hatzigeorgiou andBeskos 2009; Efraimiadou et al. 2013). For example, residual displacements are an impor-tant measure of post-earthquake functionality in bridges, and can determine whether or nota bridge remains usable following an earthquake (Lee and Billington 2010). Furthermore,Toussi and Yao (1982) and Stephens and Yao (1987) introduced a qualitative classification ofdamage, which is based on the residual inter-storey drift ratio (residual IDR) of structures. It isworth noticing that most of these research studies investigated either the quantification or thereduction of residual deformation in the framework of displacement-based seismic design.Recently, Erochko et al. (2011) found that the residual drifts of structural systems under strongearthquakes can be expressed as a function of the expected peak drifts. Simultaneously withthis study, Hatzigeorgiou et al. (2011) investigated somehow the ‘opposite problem’. Morespecifically, they examined the seismic response of single-degree-of-freedom (SDOF) struc-tures and proposed a method for evaluation of maximum displacements from their residualdisplacements. Thus, on the basis of extensive statistical analysis using a large number of

123

Bull Earthquake Eng (2013) 11:2233–2248 2235

inelastic SDOF systems under strong ground motions, they produced simple empirical rela-tions to estimate the maximum seismic displacement using the residual deformation. In anycase, the aforementioned studies of Hatzigeorgiou et al. (2011) and Erochko et al. (2011)lead to the conclusion that the residual and maximum deformations of a structure can berelated between themselves.

This paper extends the work of Hatzigeorgiou et al. (2011), which focused on SDOF sys-tems, to multi-degree-of-freedom (MDOF) systems such as multi-bay/multi-storey framedstructures. More specifically, the proposed method takes into account the fact that after astrong earthquake, residual displacements can be measured in-situ by means of various tech-niques. Characteristic examples of these techniques are the digital image correlation method(Quan et al. 2004) the global positioning system (GPS) (Lovse et al. 1995; Nickitopoulouet al. 2006) and the use of robotic theodolites (RTS) (Psimoulis and Stiros 2007), amongstothers. Furthermore, the works of Corte et al. (2005) and Chang et al. (2008) should bementioned, which examined in detail the measure of residual displacements after a strongearthquake. Additionally, residual displacements appear to be important parameters duringvisual inspection after a strong earthquake. For example, according to Field Manual for Post-earthquake Damage and Safety Assessment and Short Term Countermeasures (Baggio et al.2007) for building structures, the damage is classified as “medium–severe” if the measuredresidual drift is observable but smaller than 1 %, and “very heavy” if the measured residualdrift is in the range between 1 and 2 %. The reader can also consult the works of Yazgan(2010) and Yazgan and Dazio (2012) for more information.

Thus, in this paper, a simple method is proposed to evaluate the maximum seismic roofdisplacements of steel framed structures from their residual deformation. The procedurerelies exclusively on multi-degree of freedom models and, therefore, the potential influenceof higher mode effects has been taken into account. More specifically, the dynamic inelasticbehaviour of 36 moment resisting steel frames (MRF) and 36 concentrically X-braced steelframes (CBF) under one hundred far-field strong ground motions is investigated here byusing the Ruaumoko program (Carr 2008). Comprehensive statistical analysis of the createdresponse databank is employed in order to construct empirical equations for evaluation ofthe maximum structural displacements as functions of residual deformations. Furthermore,characteristic examples are examined to demonstrate the method and show its effectivenessand applicability. It is worth noticing that the proposed method appears to be quite simple andrequires only the knowledge of residual deformation that can be accurately measured after aseismic event in-situ using the aforementioned techniques. Therefore, the proposed methodis useful for the rapid and direct evaluation of post-earthquake performance of steel framedstructures, without requiring the knowledge of their dynamic characteristics. The method canbe applied to low-, medium- and high-rise frames (i.e., structures up to twenty storeys) andit is not intended to replace other effective and valuable methods, such as, e.g., the visualinspection, for checking the adequacy and the capacity of a structure, but rather to providean alternative way for doing so.

2 Description of structures and their modelling

2.1 Moment resisting steel frames

Thirty six planar framed structures are considered here to represent the MRF buildingsunder study. The frames have been designed by Karavasilis et al. (2007a) in accordancewith EC3 (1993) and EC8 (2005). They consist of typical beam–column steel members and

123


Fig. 2 Typical MRF

Fig. 3 Numbers of storeysversus fundamental period forMRF structures

are located in a high-seismicity region of Europe with a design/peak ground acceleration(PGA) of 0.35 g and soil class B according to EC8 (2005). They are regular and orthogonalwith storey heights and bay widths equal to 3 and 5 m, respectively. Furthermore, they havethe following characteristics: number of stories ns with values 3, 6, 9, 12, 15, and 20 andnumber of bays nb with values 3 and 6. A typical example of such frames is depicted inFig. 2.

Data of the frames, including values for ns, nb, beam and column sections, can be foundin Karavasilis et al. (2007a).

Gravity load on the beams is assumed to be equal to 27.5 KN/m (dead and live loadsof floors). The yield stress of the material is equal to 235 MPa. The relation between thefundamental period and the number of storeys of the structures appears in Fig. 3, where it isevident that the examined database covers a wide range of periods.

2.2 Concentrically X-braced steel frames

Thirty six planar structures are considered here to represent the CBF buildings under study.The frames have been designed by Karavasilis et al. (2007b) in accordance with EC3 (1993)and EC8 (2005). These frames are regular and orthogonal with storey heights and bay widthsequal to 3 and 6 m, respectively, and they have the following characteristics: number of baysnb = 3 and number of stories ns with values 3, 6, 9, 12, 15, and 20. A typical example ofsuch frames appears in Fig 4.

The characteristics of each frame such as, the number of stories, the type of columns andthe section of the X-bracing members, can be found in Karavasilis et al. (2007b).

Gravity load on the beams is assumed to be equal to 27.5 KN/m (dead and live loadsof floors). The yield stress of the material is equal to 235 MPa. The relation between thefundamental period and the number of storeys of the structures appears in Fig. 5, where it isevident that the examined database covers a wide range of periods.

123


Fig. 4 Typical CBF

Fig. 5 Numbers of storeysversus fundamental period forCBF structures

2.3 Modelling of structures

An inelastic structural Multi-Degree of Freedom (MDOF) system with linear viscous damp-ing is used to model any of the frames considered. The damping matrix is based on theRayleigh damping model and considers the current stiffness matrix at any time step as thetangent damping matrix. The inherent viscous damping ratio is assumed to be ξ = 3 % forthe first two modes. The solution of the equation of motion of the structure can be obtainedby a stepwise time integration with iterations at every time step with the aid of the Ruaumokoprogram (Carr 2008), which is an advanced finite element program for seismic analysis offramed structures.

A brief description of the modelling details is provided in the following. In this work,a two-dimensional model of each structure is created in Ruaumoko (Carr 2008) to carryout nonlinear dynamic analysis. Each beam–column finite element has two nodes and threedegrees of freedom at each node. The finite element formulation is based on the displacementmethod of structural analysis. Beam and column elements are modelled as nonlinear frameelements with lumped plasticity by defining plastic hinges at both ends of beams and columnsand assuming elastic-plastic linear hardening material behaviour, as shown in detail in Fig. 6.Bilinear hysteresis for beam and beam–column members has also been used at discrete plastichinges to capture their inelastic cyclic behaviour as in similar studies in the recent pertinentliterature, e.g. Erochko et al. (2011). Beam axial forces are assumed to be zero since all floorsare considered to be rigid in plan to account for the diaphragm action of floor slabs.

Characteristic input data for strength that are required by Ruaumoko program are thebending moment–axial force interaction diagrams for columns (Fig. 7) and bending strengthvalues for beams. More specifically, points A and E of Fig. 7 correspond to the axial com-pression and tension yield forces, respectively, and point C to the yield moment of a steelsection in the absence of axial force. Furthermore, B and D correspond to other characteristicpoints of bending moment–axial force interaction curve, which are defined in accordance

123


Fig. 6 Bilinear elastoplastichysteretic model

Fig. 7 P–M interaction diagram(Carr 2008)

with Section 5.4.8 of Eurocode 3 (1993). More specifically, the bending moment M–axialforce N interaction curve can be defined as

M

Mpl,Rd+

(N

Npl,Rd

)2

= 1 (1)

where Mpl,Rd and Npl,Rd are the bending moment and axial force plastic resistance, respec-tively. Beams and columns are connected together by rigid joints implying that panel zoneeffects are not considered. Second-order effects are accurately taken into account consideringthe large displacements formulation of the problem, while soil-structure interaction is nottaken into account, considering fixed base conditions.

Furthermore, the dynamic inelastic behaviour of brace members is simulated by theRemennikov and Walpole (1997) theory. This model has been implemented in Ruaumoko(Carr 2008) and leads to reliable results simulating accurately pertinent experimental tests,as clearly shown in Fig. 8. In order to describe the Remennikov and Walpole (1997) modelin Ruaumoko, the following parameters should be specified:

a) second moment of area and plastic section modulus, both about the minor axis, propertiesthat can be taken from Androic et al. (2000)

b) the effective length parameter, k, to determine the effective length of braces from Lef f =k · L , where L is the real member length. Taking into account that pinned-pinned braceshave been considered, this parameter is assumed to be k = 1.0.

c) both the alpha (1.0 ≤ α ≤ 1.5) and beta (1.0 ≤ β ≤ 1.5) parameters of Remennikovand Walpole (1997) theory are set equal to 1.2.

d) the initial out-of-straightness imperfection, which has units of length, is assumed to bezero.

e) finally, the type of the section of braces (RHS—rectangular hollow sections) is defined.

123


Fig. 8 The Remennikov and Walpole (1997) model: a fixed–fixed and b pinned–pinned braces

Fig. 9 PGA versus closesthorizontal distance to rupture ofrecords considered

3 Seismic input

The strong ground motion database examined in this study constitutes a representative numberof far-fault earthquakes from a variety of tectonic environments. Thus, a total of one hundredrecords were selected to cover a wide range of frequency content, duration, and magnitude.These records satisfy the following four constraints:

(a) records should not present pulse-like motions, which usually lead to different structuralresponse, and have been recorded at a small distance from the fault

(b) moment magnitude should be greater than or equal to 5.0 (Mw = 5.8–7.9 in this study)(c) PGA should be greater than or equal to 0.10 g (0.11–0.82 g in this study)(d) closest horizontal distance to the trace of the rupture that is vertically projected to the

ground surface (Joyner-Boore distance, PEER 2012) should be more than 10 km (10.8–74.5 km in this study).

The assembled database have been recorded at sites with soil type B local conditionsaccording to EC8 (2005), i.e., at soft rock or very dense soil with 360 m/s ≤ VS(30) <

750 m/s, where VS(30) is the average shear-wave velocity at a depth between 0 and 30 m.The examined 100 strong ground motions, which were downloaded from the strong motiondatabase of the Pacific Earthquake Engineering Research (PEER) Center (2012), have beenrecorded during the action of the following earthquakes: Parkfield (1966), San Fernando(1971), Imperial Valley (1979), Livermore (1980), Coalinga (1983), N. Palm Springs (1986),Chalfant Valley (1986), Whittier Narrows (1987), Loma Prieta (1989), Cape Mendocino(1992), Northridge (1994), Kobe, Japan (1995), Kocaeli, Turkey (1999), Chi-Chi, Taiwan(1999) and Duzce, Turkey (1999). Figure 9 shows the peak ground acceleration—closestdistance to rupture relation for the 100 records under consideration.

123


Fig. 10 Permanent deformation—maximum displacement diagram for the examined MRF

Fig. 11 Residual roof drift versus peak roof drift diagram for the examined MRF

From this figure, it can be observed that the records closely follow the general trendexpected from typical attenuation relationships, i.e., the maximum the horizontal distance torupture, the minimum the PGA.

4 Results

4.1 Moment resisting steel frames

The 36 MRF’s of Sect. 2.1 are subjected to the one hundred strong ground motions of Sect. 3and, therefore, an extensive databank of 3,600 seismic inelastic analyses (= 36 MRF × 100records) is created. Figure 10 shows the permanent deformation (residual roof displacement)versus maximum displacement diagrams for the examined structures.

Furthermore, Fig. 11 depicts the peak and residual roof drifts, i.e., the ratios of peak andresidual roof displacement normalized to the height of MRF buildings.

From the created databank of response results, simple empirical equations can be devel-oped to evaluate the maximum structural displacements at the top of each structure as afunction of their residual counterparts. Thus, in this work, it is assumed that the maximumtop horizontal seismic displacement, umax , can be evaluated from its residual counterpart,ures , as

|umax | = (a1 + a2ln(N ) + a3 |ures |) (1 + a4 H) (2)

where N is the number of storeys of the structures, H the post yield stiffness ratio (Fig. 6) anda1 −a4 are appropriate parameters which have been determined to provide the best data fit forEq.(2). This empirical expression was one of the simplest equations which better described thenumerical data following downward and upward concave curves, obtained by a homemadeprogram after testing hundreds of simple mathematical equations. More specifically, this

123


Fig. 12 Evaluation of maximumdisplacements for the examinedMRF

5%

95%

program fits about one thousand of built-in equations and then ranks them with respect to thecorresponding correlation factor between the ‘exact’ and ‘predicted’ results. The programhas been written in FORTRAN and the Levenberg-Marquardt algorithm (Levenberg 1944;Marquardt 1963) has been adopted to solve the least squares curve fitting problem. Thisalgorithm is based on the fundamental concepts of optimization and combines the descentmethod with the minimization of the Taylor approximation of the function. The algorithm canbe found in http://www.netlib.org/minpack/lmdif.f, and for a detailed description the readercan consult the book of Press et al. (2007). The criterion for the selection of Eq. (2) has to dowith its minimum absolute residual error using the Pearson VII limit, i.e., minimum sum ofln[

√(1 + residual2)]. Therefore, the basic criterion for this selection is the minimization of

error between the results of the empirical equation and those of the ‘exact’ dynamic inelasticanalyses. Thus, examining the whole sample of structures and records, the optimal valuesfor the parameters a1, a2 and a3 lead to the expression

|umax | = (−0.053 + 0.109ln(N ) + 1.61 · |ures |) (1 + 2.0H) (3)

The evaluation of maximum displacements using dynamic inelastic analysis (‘exactapproach’) and the proposed method (Eq. 3) is shown in Fig. 12. It is evident that the modelresults obtained from this study are in good agreement with those obtained from the ‘exact’dynamic inelastic analyses where 90 % of the databank values under consideration (5–95 %)are very close to the diagonal of the diagram of Fig. 12. The correlation coefficient for thewhole sample of MRF’s and seismic records is equal to R2 = 0.77.

Furthermore, Fig. 13 presents the time history of the top-horizontal displacement forvarious moment-resisting steel frames, assuming that H = 3 %, and shows that Eq. (3) canbe satisfactorily used to evaluate the maximum seismic displacement.

Finally, comparisons of results obtained by the proposed method with results from otherstudies appear to be useful in order to verify the efficiency and accuracy of the formerone. Thus, an 18-storey generic MRF frame that has been analyzed and designed by Ruiz-Garcia and Miranda (2005) is examined in the following. The frame has been subjected tonumerous earthquakes with various intensities. Ruiz-Garcia and Miranda (2005), in theirFigs 7.2 and 7.5, provided both the residual and maximum roof drifts. This structure isalso examined here on the basis of Eq. (3), taking into account that the storey height isequal to 3.66 m (12ft), and therefore, the total height is equal to 65.84 m (= 18 × 3.66 m).

123

http://www.netlib.org/minpack/lmdif.f


Fig. 13 Displacement time history for typical MRFs and prediction of maximum displacement

Table 1 Comparisons betweenproposed method and results ofRuiz-Garcia and Miranda (2005)

Case Residual roof driftθr,roof (%)

Maximum roof drift θroof (%)

‘Exact’ This study

1 0.03 0.45 0.45

2 0.15 0.80 0.65

3 0.43 1.17 1.12

4 0.78 1.58 1.69

Table 1 shows the maximum roof drift, θroof , and residual roof drift, θr,roof , according toRuiz-Garcia and Miranda (2005), as well as the maximum roof drifts according to Eq. (3),using exclusively the residual deformation and basic characteristics of the structure (N : 18storeys, post-yield stiffness H = 1 %, and the roof drift can be determined normalizingthe roof horizontal displacement with the total height of building). It is observed that the‘exact’ maximum roof drifts of Ruiz-Garcia and Miranda (2005), using dynamic inelas-tic analysis, can be satisfactorily evaluated using the proposed method, i.e. the empiricalEq. (3).

4.2 Concentrically X-braced steel frames

The 36 CBF’s of Sect. 2.2 are subjected to the one hundred strong ground motions ofSect. 3 and, therefore, an extensive databank of 3,600 seismic inelastic analyses (36 CBF ×100 records) is created. Figure 14 shows the permanent deformation (residual roof displace-ment) versus maximum displacement diagrams for the examined structures. Additionally,

123


Fig. 14 Permanent deformation—maximum displacement diagram for the examined CBF

Fig. 15 Residual roof drift versus peak roof drift diagram for the examined CBF

Fig. 15 depicts the peak and residual roof drifts, i.e., the ratios of peak and residual roofdisplacement normalized to the height of CBF buildings.

The form of the empirical Eq. (2) can also be used to evaluate the maximum top horizontalseismic displacements of CBF structures from their residual counterparts. Thus, examiningthe whole sample of CBF structures and seismic records, the optimal values for the parametersa1, a2 and a3 of Eq. (2) lead to the expression

|umax | = (−0.071 + 0.076ln (N ) + 0.865 · |ures |) (1 + 0.04H) (4)

The evaluation of maximum displacements using dynamic inelastic analysis (‘exactapproach’) and the proposed method (Eq. 4) is shown in Fig. 16. It is evident that the modelresults obtained from this study are in good agreement with those obtained from the ‘exact’dynamic inelastic analyses where 90 % of the databank values under consideration (5–95 %)are very close to the diagonal of the diagram of Fig. 16. In this case, the correlation coefficientfor the whole sample of CBF’s and seismic records is equal to R2 = 0.89.

Moreover, Fig. 17 presents the time history of top-horizontal displacement for variousCBF’s, assuming that H = 3 %, and shows that Eq. (4) evaluates satisfactorily the maximumseismic displacement.

4.3 Discussion about the proposed method and its limitations

As mentioned above, the main goal of this study is to propose a simple and effective methodfor the maximum seismic displacements evaluation of steel frames from their post-earthquakeresidual deformation. This method appears to be an extension of the study of Hatzigeorgiouet al. (2011), which focused on SDOF systems. Residual displacements can be reliably mea-sured after a seismic event in-situ using the techniques that have been described in studies

123


Fig. 16 Evaluation of maximumdisplacements for the examinedCBF

5%

95%

Fig. 17 Displacement time history for typical MRFs and prediction of maximum displacement

of Corte et al. (2005) and Chang et al. (2008), amongst others. Thus, the proposed methodis useful for the rapid and direct evaluation of post-earthquake performance of steel framedstructures, without requiring the knowledge of their dynamic characteristics such as the totalmass or the lateral stiffness. It should be recognized that the post-yield stiffness affects theresidual displacements of a structure (Macrae and Kawashima 1997; Hatzigeorgiou et al.2011; Erochko et al. 2011). For this reason, the influence of post-yield stiffness on residualand maximum displacements has been considered herein. Thus, the application of Eqs. (3)and (4) requires the knowledge of the hardening parameter H . It is worth noticing that the

123


Maximum top displacement H ‘Exact’ Eq. (3) error2% 21.2cm 22.1cm 4.2% 3% 20.4cm 20.6cm 0.7% 5% 19.8cm 19.3cm 2.4%

Fig. 18 Influence of hardening parameter H on maximum and residual deformation

accurate evaluation of the post-yield stiffness of a structure requires a detailed modelling ofits structural system and then the execution of nonlinear analyses under incremental static(pushover) or dynamic loading conditions. In order to keep the simple and direct characterof the proposed method, one can either assume a reliable value of the hardening parameterH , according to his previous knowledge and experience, or assume that this parameter isequal to zero, i.e., assume an elastic-perfectly plastic material behaviour. The last of theseoptions is compatible with the provisions of EC3 (1993). The influence of the hardeningparameter H on residual and maximum displacements and the effectiveness of the proposedmethod appears in Fig. 18. It is obvious that the hardening parameter H (or, equivalently,the post-yield stiffness) strongly affects the maximum displacement and especially the resid-ual displacement. Furthermore, it is obvious that the proposed empirical expressions cansatisfactorily evaluate the maximum displacement from the residual deformation.

Examining the sensitivity of Eqs. (3) and (4) concerning their post-yield stiffness, it isfound that the seismic response of moment resisting steel frames seems to be much moresensitive in comparison with the seismic response of concentrically X-braced steel frames.Thus, the parameter a4 of Eq. (2) is almost 50 times greater for the MRFs than for the CBFs,i.e., 2.0 for MRFs and 0.04 for CBFs. This has to do with their different behaviour underlateral load where MRFs carry that load by beam–column members while the members ofCBFs by axial brace members.

Additionally, it should be noted that in order to achieve rational and reliable results, thestatistical analysis that has to do with Eqs. (3) and (4) takes only into account data sets withratios between the residual top displacement over the total height of structure greater than orequal to 0.05 %, avoiding this way the cases of elastic or nearly elastic structural behaviourwith negligible residual deformation.

Moreover, as mentioned in Sect. 2.3, panel zone deformations were not considered in thiswork. It should be recognized that inelasticity in the panel zones can affect post-yield stiffnessand therefore residual and peak drifts. One can take into account the influence of panel zoneson the structural response in an indirect manner (i.e. adopting appropriate values of post-yield stiffness) since the accurate evaluation of this influence requires a detailed modelling ofthese zones. A this point it should be mentioned that other recent studies from the pertinentliterature, such as the work of Erochko et al. (2011), also ignore the deformations of panelzone.

In addition, it should be noted that the proposed method requires the knowledge of residualroof displacements, which after a seismic event also include “initial” deformations. This fact

123


can be more intense for tall structures. However, it should be recognized that “initial” defor-mations have mainly to do with vertical loads, which generally lead to vertical deformationwhile the corresponding horizontal drift is rather small. In order to quantify this problem,the tallest building that is considered here, i.e. a 20-storey structure with total height equalto 60 m, is investigated. For a post-earthquake residual roof drift equal 1 or 2 % (i.e., formedium–severe and very heavy damage levels, respectively, according to aforementionedAeDES Manual, Baggio et al. 2007), the corresponding residual horizontal roof displace-ments are equal to 60 and 120 cm, respectively (uresid. = roof − dri f tresid. × height).These values appear to be quite large in comparison with “initial” deformations or “structuraltolerances” and therefore, the influence of these tolerances on the magnitude of residual roofdisplacements appears to be immaterial.

Finally, it is worth noticing that the proposed method is not intended to substitute othereffective and valuable methods, such as, e.g., the visual inspection, for checking the adequacyand the capacity of a structure, but rather to provide an alternative way for doing so. Theauthors will examine additional parameters such as the interrelation between peak interstoreydrifts and residual interstorey drifts as well as pulse-like strong ground motions from near-fault records in an oncoming research paper.

5 Conclusions

In this paper, a new method has been developed for the prediction of maximum seismic roofdisplacement of steel structures from their residual deformation. The study focuses on 36planar moment resisting frames and 36 planar concentrically X-braced frames under 100 far-field strong ground motions. Extensive parametric and statistical studies for these structuresand seismic records lead to empirical equations for the direct prediction of maximum seismicdisplacement from its residual counterpart. On the basis of the preceding discussion andnumerical examples, the following conclusions can be drawn:

• the maximum seismic displacements can be effectively evaluated from the correspondingresidual displacements. The proposed method is successfully applied for the evaluationof maximum seismic displacements from residual deformation both for framed structuresseismically analyzed by the authors and for structures that have been analyzed by othersin the pertinent literature.

• the maximum displacements and especially the permanent deformation of a structureare strongly affected by its post-yield lateral stiffness. This is more intense for momentresisting steel frames, while the behaviour of concentrically X-braced steel frames ismildly affected by the post-yield lateral stiffness.

• the basic advantage of the proposed method has to do with the fact that the residualdeformation of a structure can be measured in situ after a strong earthquake and theevaluation of maximum seismic displacements can be achieved without requiring theknowledge of dynamic characteristics of the structure under consideration, such as itstotal mass or its lateral stiffness.

• in order to keep the simple and direct character for the application of the method, someparameters that can affect the residual and peak deformations of a structure have notconsidered, such as, the soil-structure interaction and panel zone deformations. However,the panel zone inelasticity can be indirectly considered using an appropriate value of post-yield lateral stiffness.

• the proposed method can be successfully applied to low-, medium- and high-rise frames(i.e., structures up to twenty storeys).

123


References

Androic B, Dzeba I, Dujmovic D (2000) International structural steel sections. Design tables according toEurocode 3, Ernst & Sohn, Berlin

Baggio C, Bernardini A, Colozza R, Di Pasquale G, Dolce M, Goretti A, Martinelli A, Orsini G, Papa F,Zuccaro G, Pinto AV, Taucer F (2007) Field manual for post-earthquake damage and safety assessment andshort term counter measures (AeDES), EUR 22868 EN. Joint Research Center, Ispra, Italy

Borzi B, Calvi GM, Elnashai AS, Faccioli E, Bommer J (2001) Inelastic spectra for displacement based seismicdesign. Soil Dyn Earthq Eng 21(1):47–61

Carr AJ (2008) RUAUMOKO—inelastic dynamic analysis program. Department of Civil Engineering, Uni-versity of Canterbury, New Zealand, Christchurch, New Zealand

Chang YM, Lai SH, Peng SM, Chang KT, Tu YS (2008) Data processing of 3D laser scanning on structuraldeformation. Integrating generations, FIG working week, Stockholm, Sweden

Christopoulos C, Pampanin S, Priestley MJN (2003) Performance-based seismic response of frame struc-tures including residual deformations part I: single-degree-of-freedom systems. J Earthq Eng 7:97–118

Corte G, Faggiano B, Mazzolani F, Lieto C, Melella C, Pagano L (2005) Full scale tests of a two-story RCbuilding: use of the laser scanner technology for deformation measures. In: Advances in experimentalstructural engineering, Nagaya, Japan, pp 657–666

Cosenza E, Manfredi G, Ramasco R (1993) The use of damage functionals in earthquake engineering: acomparison between different methods. Earthq Eng Struct Dyn 22:855–868

Di Pasquale E, Cakmak AS (1990) Detection of seismic structural damage using parameter-based globalindices. Probab Eng Mech 5:60–65

EN 1993-1-1 (1993) Eurocode 3. Design of steel structures. Part 1-1: general rules and rules for buildings.European Committee for Standardization (CEN), Brussels

EN 1998-1 (2005) Eurocode 8: design of structures for earthquake resistance; Part 1: general rules. Seismicactions and rules for buildings. European Committee for Standardization, Brussels

Efraimiadou S, Hatzigeorgiou GD, Beskos DE (2013) Structural pounding between adjacent buildings sub-jected to strong ground motions. Part II: the effect of multiple earthquakes. Earthquake Engng Struct Dyn(early view)

Erochko J, Christopoulos C, Tremblay R, Choi H (2011) Residual drift response of SMRFs and BRB framesin steel buildings designed according to ASCE 7–05. J Struct Eng ASCE 137:589–599

Erduran E, Yakut A (2004) Drift based damage functions for reinforced concrete columns. Comput Struct82:121–130

Federal Emergency Management Agency—FEMA (2009) NEHRP recommended seismic provisions for newbuildings and other structures (FEMA P-750). Washington, DC

Ghobarah A (2004) On drift limits associated with different damage levels. In: Proceedings of the internationalworkshop for performance-based seismic design, concepts and implementation (28 June–1 July 2004), Bled,Slovenia

Ghobarah A, Abou-Elfath H, Biddah A (1999) Response-based damage assessment of structures. Earthq EngStruct Dyn 28:79–104

Hatzigeorgiou GD, Beskos DE (2009) Inelastic displacement ratios for SDOF structures subjected to repeatedearthquakes. Eng Struct 31:2744–2755

Hatzigeorgiou GD, Papagiannopoulos GA, Beskos DE (2011) Evaluation of maximum seismic displacementsof SDOF structures from residual displacements. Eng Struct 33:3422–3431

Karavasilis TL, Bazeos N, Beskos DE (2007a) Behaviour factor for performance-based seismic design ofplane steel moment resisting frames. J Earthq Eng 11:531–559

Karavasilis TE, Bazeos N, Beskos DE (2007b) Estimation of seismic drift and ductility demands in planarregular X-braced steel. Earthq Eng Struct Dyn 36:2273–2289

Kim HS, Chun YS (2004) Structural damage assessment of building structures using dynamic experimentaldata. Struct Des Tall Spec Build 13:1–8

Krawinkler H, Medina R, Alavi B (2003) Seismic drift and ductility demands and their dependence on groundmotions. Eng Struct 25(5):637–653

Krishnan S, Casarotti E, Goltz J, Ji C, Komatitsch D, Mourhatch R, Muto M, Shaw JH, Tape C, Tromp J(2012) Rapid estimation of damage to tall buildings using near real-time earthquake and archived structuralsimulations. Bull Seism Soc Am 102(6):2646–2666

Lee WK, Billington SL (2010) Modeling residual displacements of concrete bridge columns under earthquakeloads using fiber elements. J Bridge Eng 15(3):240–249

Levenberg K (1944) A method for the solution of certain non-linear problems in least squares. Q Appl Math2:164–168

123


Lovse JW, Teskey WF, Lachapelle G, Cannon ME (1995) Dynamic deformation monitoring of tall structureusing GPS technology. J Surveying Eng ASCE 121:35–40

Macrae GA, Kawashima K (1997) Post-earthquake residual displacements of bilinear oscillators. Earthq EngStruct Dyn 26:701–716

Marquardt D (1963) An algorithm for least-squares estimation of nonlinear parameters. SIAM J Appl Math11(2):431–441

Medina RA, Krawinkler H (2005) Evaluation of drift demands for the seismic performance assessment offrames. J Struct Eng ASCE 131(7):1003–1013

Nickitopoulou A, Protopsalti K, Stiros S (2006) Monitoring dynamic and quasi-static deformations of largeflexible engineering structures with GPS: accuracy, limitations and promises. Eng Struct 28:1471–1482

Pacific Earthquake Engineering Research Center—PEER (2012) NGA database. http://peer.berkeley.edu/peer_ground_motion_database. Accessed 22 Nov 2012

Pampanin S, Christopoulos C, Priestley MJN (2003) Performance-based seismic response of frame structuresincluding residual deformations Part II: multi-degree-of-freedom systems. J Earthq Eng 7:119–147

Pettinga JD, Pampanin S, Christopoulos C, Priestley MJN (2007a) The role of inelastic torsion in the deter-mination of residual deformations. J Earthq Eng 11:133–157

Pettinga D, Christopoulos C, Pampanin S, Priestley N (2007b) Effectiveness of simple approaches in mitigatingresidual deformations in buildings. Earthq Eng Struct Dyn 36:1763–1783

Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2007) Numerical recipes in C—the art of scientificprogramming, 3rd edn. Cambridge University Press, Cambridge

Psimoulis P, Stiros S (2007) Measurement of deflections and of oscillation frequencies of engineering structuresusing robotic theodolites (RTS). Eng Struct 29:3312–3324

Quan C, Tay CJ, Huang HY (2004) 3-D deformation measurement using fringe projection and digital imagecorrelation. Optik 115:164–168

Remennikov AM, Walpole WR (1997) Analytical prediction of seismic behaviour for concentrically-bracedsteel systems. Earthq Eng Struct Dyn 26:859–874

Ruiz-Garcia J, Miranda E (2005) Performance-based assessment of existing structures accounting for resid-ual displacements. Report no. 153, John A. Blume Earthquake Engineering Center, Stanford University,Stanford

Ruiz-Garcia J, Miranda E (2006a) Residual displacement ratios for assessment of existing structures. EarthqEng Struct Dyn 35:315–336

Ruiz-Garcia J, Miranda E (2006b) Evaluation of residual drift demands in regular multi-story frames forperformance-based seismic assessment, earthquake engineering and structural dynamics. Earthq Eng StructDyn 35:1609–1629

Ruiz-Garcia J, Miranda E (2010) Probabilistic estimation of residual drift demands for seismic assessment ofmulti-story framed buildings. Eng Struct 32:11–20

Stephens JE, Yao JTP (1987) Damage assessment using response measurements. J Struct Eng ASCE 113:787–801

Toussi S, Yao JTP (1982) Hysteresis identification of existing structures. J Eng Mech ASCE 109:1189–1203Uma SR, Pampanin S, Christopoulos C (2006) A probabilistic framework for performance-based seismic

assessment of structures considering residual deformations. In: Proceedings of the 1st ECEES, Geneva,Switzerland, p 731

Yazgan U (2010) The use of post-earthquake residual displacements as a performance indicator in seismicassessment. Ph.D. thesis, Institute of Structural Engineering, ETH Zurich, Switzerland

Yazgan U, Dazio A (2012) Post-earthquake damage assessment using residual displacements. Earthq EngStruct Dyn 41:1257–1276

Yun SY, Hamburger RO, Cornell CA, Foutch DA (2002) Seismic performance evaluation for steel momentframes. J Struct Eng ASCE 128:534–545

123

http://peer.berkeley.edu/peer_ground_motion_database

http://peer.berkeley.edu/peer_ground_motion_database

maximum seismic displacements evaluation of steel frames from their post-earthquake residual...

Documents