# Simplified non-linear seismic displacement demand prediction for low period structures

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## Abstract

The prediction of non-linear seismic demand using linear elastic behavior for the determination of peak non-linear response is widely used for seismic design as well as for vulnerability assessment. Existing methods use either linear response based on initial period and damping ratio, eventually corrected with factors, or linear response based on increased equivalent period and damping ratio. Improvements to the original EC8 procedure for displacement demand prediction are proposed in this study. Both propositions may be graphically approximated, which is a significant advantage for practical application. A comparison with several other methods (equal displacement rule, EC8 procedure, secant stiffness and empirical equivalent period methods) is performed. The study is based on non-linear SDOF systems subjected to recorded earthquakes, modified to match design response spectra of different ground types, and focuses on the low frequency range that is of interest for most European buildings. All results are represented in the spectral displacement/fundamental period plane that highlights the predominant effect of the fundamental period on the displacement demand. This study shows that linearized methods perform well at low strength reduction factors but may strongly underestimate the displacement demand at strength reduction factors greater than 2. This underestimation is an important issue, especially for assessment of existing buildings, which are often related with low lateral strength. In such cases, the corresponding strength reduction factors are therefore much larger than 2. The new proposals significantly improve the reliability of displacement demand prediction for values of strength reduction factors greater than 2 compared to the original EC8 procedure. As a consequence, for the seismic assessment of existing structures, such as unreinforced masonry low-rise buildings, the current procedure of EC8 should be modified in order to provide accurate predictions of the displacement demand in the domain of the response spectrum plateau.

### Keywords

Displacement-based methods Vulnerability assessment Equal displacement rule Secant stiffness Hysteretic models Seismic behavior Recorded earthquakes Displacement demand prediction Non-linear displacement demand## 1 Introduction

It is well established that structures do not remain elastic under extreme ground motion. Non-linear behavior therefore constitutes the key issue in seismic design and assessment of structures. However, to avoid the use of more elaborate analysis, structural engineering approaches are usually based on simplified static methods to determine seismic action. In these simplified methods, compared to linear behavior, seismic action is reduced according to the deformation capacity and the energy dissipation capacity of the structure as it undergoes large inelastic deformations. The majority of the building codes around the world are based on this design philosophy.

Seismic assessment using any method but non-linear time history analysis therefore requires a reliable estimation of the seismic displacement demand. Since Veletsos and Newmark (1960), it has been widely accepted that the displacements of elastic and inelastic systems are approximately the same (*Equal displacement principle*). This empirical principle was confirmed by numerous numerical and experimental investigations (e.g. Lestuzzi and Badoux 2003), except for low period structures, for which inelastic displacements are rather higher than elastic displacements. Since then, many authors tried to model inelastic displacements using linear approaches in order to allow earthquake engineers to easily perform these computations (e.g. Fajfar 1999). It should be emphasized that, until non-linear time history analysis becomes a standard procedure, there is a need for such simplified methods to estimate inelastic seismic demand.

The plateau range of the design spectra is of particular significance for seismic design and assessment. For instance, most of the buildings in Europe are lower than 5-story structures, and therefore have a natural period lower than 1 s. As a consequence, the natural period of a large part of these structures is located on the plateau of the design spectra, i.e. out of the assumed standard range of application of the equal displacement rule.

Moreover, contrary to design procedures, assessment procedures may lead to account for high strength reduction factors, due to several reasons. First, the shear strength of existing structures, especially unreinforced masonry structures, may be very low despite a creditable displacement capacity; these two parameters being not necessary linearly related. This has been shown in laboratory tests (e.g. ElGawady et al. 2005; Tomazevic and Weiss 2010). Second, some codes allow relatively high strength reduction factors, e.g. 3–4 in Germany for unreinforced masonry structures. In Switzerland, the minimum permitted compliance factor (ratio of the capacity of the structure over the demand in the current codes) is 0.25. Weak buildings therefore have to be assessed to high demands compared to their capacity, thus leading to computing the demand for high strength reduction factors (up to 8) nevertheless resulting in compliant values. Even if these computations are theoretical, they are of practical need for the engineers.

Current linearization methods can be grouped into one of two categories: (1) methods based on \(\hbox {R}-\mu -\hbox {T}\) relationships and (2) equivalent damping approaches (Lin and Miranda 2009). While the latter were developed and included in the design code in the US (Kowalsky 1994; ATC 2005), the former has been more inspiring in Europe (Fajfar 1999; CEN 2004). However, it has been shown that the simplification of the N2 method (Fajfar 1999) in EC8 led to an underestimation of the demand in some cases (Norda and Butenweg 2011). Therefore, there is a need for a simple method of estimation of the demand that should be at least conservative for structures on the plateau of the design spectra.

In this paper, different linearization methods are evaluated with respect to an inelastic model and compared to a new simple \(\hbox {R}-\mu -\hbox {T}\) relationship based on graphical assumptions.

## 2 Approaches for predicting the non-linear displacement

### 2.1 Current available approaches

#### 2.1.1 R-\(\mu \)-T relationships

Numerous studies of the so-called \(\hbox {R}-\mu -\hbox {T}\) relationships propose adjustments to this approximate prediction by giving the strength reduction factor R as a function of displacement ductility \(\mu \) and possibly period T, as reviewed by Miranda and Bertero (1994), Miranda and Ruiz-Garcia (2002) or Chopra and Chintanapakdee (2004).

#### 2.1.2 Equivalent period and damping ratios

These linearization methods are now widely used for design purposes (ATC 2005) as well as for large-scale seismic vulnerability assessment (FEMA 1999; Calvi 1999; Borzi et al. 2008; Colombi et al. 2008). This approach, slightly more complicated than the initial stiffness methods, is aimed at better describing the physics of structural damage, even if its theoretical basis remains arbitrary (Miranda 2006).

### 2.2 New proposed R-\(\mu \)-T simplified relationships

It is obvious that linear methods will never be able to reproduce the complexity of non-linear computations. Moreover, the objective of these linearization methods for the engineer is to be able to practically estimate the displacement demand that the studied structure may experience for a given hazard level. In most of the cases, the engineer prefers having (slightly) conservative values in order to be on the safe side. For more advanced applications, non-linear computations would nowadays be employed. Therefore, simple, slightly conservative and graphically obvious ways of estimating the non-linear displacement demand for low period buildings on the plateau of the design spectra are proposed here. They should only depend on the natural frequency of the structure and therefore be independent of the strength reduction factor.

**Proposition 1**

**Proposition 2**

#### 2.2.1 Graphical estimation of displacement demand prediction

Such a graphical estimation allows the displacement demand prediction to be performed with less than 5 % error compared to the exact values of the propositions.

Note that since proposition 1 constitutes the upper bound of the original EC8 displacement demand prediction, the corresponding graphical estimation (an additional one third of the spectral displacement difference) may be used for a quick check of computed displacement demand in a practical case using the current EC8 method.

## 3 Methodology

### 3.1 Ground motions

As the objective is to propose a practical but relevant method to estimate inelastic displacements, and not a physical theory aiming at explaining the phenomena, the validation is made using ground motions matching design spectra. Though these ground motions do not represent a realistic hazard, they facilitate the understanding of key parameters influencing the results.

#### 3.1.1 ESMD database of recorded ground motions

Non-linear time-history analysis may be carried out using both recorded earthquakes or artificially generated earthquakes (e.g. Schwab and Lestuzzi 2007). In the past, artificial ground motions were preferred for earthquake engineering purposes since they could be easily generated to match an elastic design spectrum and therefore be used in the frame of design codes. This approach is criticized in the literature because generated ground motions may not be realistic and do not cover the variability of actual ground motions (e.g. Lestuzzi et al. 2004). Nowadays, the amount of records of strong earthquake is exponentially increasing so that real accelerograms with given characteristics, such as their similarity with design spectra, can be selected (e.g. Iervolino et al. 2011).

Schwab and Lestuzzi (2007) selected 164 recorded ground acceleration time histories from the European Strong Motion Database (Ambraseys et al. 2002). The selection of the records in this initial database is based on structural engineering considerations rather than seismological ones. As a consequence, earthquakes with different focal mechanisms are incorporated into the dataset. The main objective is to perform a statistical study of the non-linear response of structures undergoing any earthquake record.

In order to consider earthquakes that may produce significant non-linearities in structural behavior, only records with a magnitude larger than 5 were considered for this selection. The magnitudes range from 5.0 to 7.6, the epicentral distances range from 2 to 195 km and the peak ground accelerations (PGA) range from 0.61 to \(7.85\,\hbox {m}/\hbox {s}^{2}\).

This selection was already used in other research projects in the field of seismic non-linear behavior (Lestuzzi et al. 2004 and 2007).

#### 3.1.2 Selection of sets of 12 records

Out of this preliminary selection, sets of 12 records best matching different design response spectra were extracted. For the selection of these sets, four design response spectra from EC8 are considered. These EC8 spectra are of type 1, for the usual viscous damping ratio of 5 % and for ground types A, B, C and D. They are scaled for a peak ground acceleration of 1 \(\hbox {m}/\hbox {s}^{2}\) corresponding to the design response spectra of the Swiss seismic zone 2 (SIA 2003).

Main characteristics of the 33 selected records and their distribution in the four different sets of twelve records each (as = aftershock)

Earthquake | Date | Magnitude | Distance [km] | PGA [m/s\(^2\)] | Soil A | Soil B | Soil C | Soil D |
---|---|---|---|---|---|---|---|---|

Friuli (as) | 11.09.1976 | 5.3 Mw | 8 | 1.931 | X | |||

Friuli (as) | 16.09.1977 | 5.4 Mw | 14 | 0.910 | X | |||

Volvi | 04.07.1978 | 5.12 Ms | 16 | 1.125 | X | |||

El Asnam (as) | 08.11.1980 | 5.2 Mw | 18 | 0.946 | X | |||

Friuli (as) | 15.09.1976 | 6 Mw | 11 | 1.069 | X | X | ||

Basso Tirreno | 15.04.1978 | 6 Mw | 18 | 1.585 | X | X | ||

Volvi | 20.06.1978 | 6.2 Mw | 29 | 1.430 | X | X | ||

Montenegro (as) | 24.05.1979 | 6.2 Mw | 30 | 0.754 | X | |||

Montenegro (as) | 24.05.1979 | 6.2 Mw | 17 | 2.703 | X | |||

Alkion | 25.02.1981 | 6.3 Mw | 25 | 1.176 | X | X | X | |

Aigion | 15.06.1995 | 6.5 Mw | 43 | 0.911 | X | |||

Montenegro | 15.04.1979 | 6.9 Mw | 65 | 2.509 | X | X | ||

Campano Lucano | 23.11.1980 | 6.9 Mw | 23 | 1.776 | X | |||

Campano Lucano | 23.11.1980 | 6.9 Mw | 26 | 0.903 | X | |||

Campano Lucano | 23.11.1980 | 6.9 Mw | 16 | 1.725 | X | |||

Campano Lucano | 23.11.1980 | 6.9 Mw | 33 | 0.975 | X | X | ||

Spitak | 07.12.1988 | 6.7 Mw | 36 | 1.796 | X | |||

Strofades | 18.11.1997 | 6.6 Mw | 144 | 0.907 | X | |||

Strofades | 18.11.1997 | 6.6 Mw | 32 | 1.289 | X | |||

Tabas | 16.09.1978 | 7.4 Mw | 55 | 1.003 | X | X | ||

Tabas | 16.09.1978 | 7.4 Mw | 100 | 1.002 | X | |||

Manjil | 20.06.1990 | 7.4 Mw | 81 | 0.951 | X | X | ||

Manjil | 20.06.1990 | 7.4 Mw | 131 | 1.341 | X | |||

Gulf of Akaba | 22.11.1995 | 7.1 Mw | 93 | 0.894 | X | X | ||

Izmit | 17.08.1999 | 7.6 Mw | 172 | 0.974 | X | |||

Izmit | 17.08.1999 | 7.6 Mw | 110 | 1.698 | X | |||

Izmit | 17.08.1999 | 7.6 Mw | 48 | 2.334 | X | X | ||

Izmit | 17.08.1999 | 7.6 Mw | 78 | 1.040 | X | X | X | |

Izmit | 17.08.1999 | 7.6 Mw | 96 | 1.120 | X | X | ||

Izmit | 17.08.1999 | 7.6 Mw | 10 | 2.192 | X | |||

Izmit | 17.08.1999 | 7.6 Mw | 39 | 1.266 | X | |||

Izmit | 17.08.1999 | 7.6 Mw | 34 | 3.542 | X | |||

Izmit | 17.08.1999 | 7.6 Mw | 103 | 0.871 | X | X |

#### 3.1.3 Modification of records for matching target spectra

After selection into the initial database, the 33 records are modified with the non-stationary spectral matching technique of Abrahamson (1992) in order to match individually the related design spectrum. Spectral matching for a given design spectrum is not always regarded as relevant, since design spectra are envelopes of possible earthquake spectra. The reason to perform a selection and matching of the records to the design spectra is to show the consequences of the choice of a linearization method in terms of spectra, i.e. for engineering purposes. In addition, it allows the removal of the variability due to ground motion in order to evaluate that due to the estimation of the response only. Moreover, the corner period \(\hbox {T}_{\mathrm{c}}\) is often used as a parameter to compute the reduction factors (e.g. Vidic et al. 1994). Spectral matching is therefore a way to have sets of ground motion time histories with a well-defined \(\hbox {T}_{\mathrm{c}}\).

### 3.2 Hysteretic models

The force-displacement relationship of the modified Takeda-model is specified through five parameters: the initial stiffness, the yield displacement, the post-yield stiffness, a parameter relating the stiffness degradation (\(\alpha \)) and a parameter (\(\upbeta \)) specifying the target for the reloading curve. Standard values of the parameters corresponding to the widely used “small Takeda model” (\(\alpha \,=\,0.4\) and \(\upbeta \,=\,0.0\)) are used in all analyses (Lestuzzi et al. 2007). Note that a low \(\alpha \) value improves the rate of convergence of computations. Values of 0, 5 and 10 % have been tested for the hardening coefficient r (post-yield stiffness).

### 3.3 Processing

The elastic peak displacement \(\hbox {y}_{\mathrm{el}}\) is obtained calculating the linear response of a SDOF of period T and damping ratio \(\xi \). It corresponds to the inelastic displacement according to the equal displacement rule.

- The yield displacement associated with each computation, calculated as follows:$$\begin{aligned} y_y=\frac{y_{el}}{R} \end{aligned}$$(13)
The corresponding non-linear maximum displacement value \(\hbox {y}_{\mathrm{nl}}\) using the hysteretic model.

- The displacement ductility is then computed based on the inelastic model, for a given strength reduction factor R:$$\begin{aligned} \mu =R \frac{y_{nl}}{y_{el}} \end{aligned}$$(14)
The displacement demand predictions according to the N2 method (Vidic et al. 1994), the version implemented in the EC8 (CEN 2004) and equal energy approximation are obtained following Eq. 2 and 3. It can be noticed that \(\mu \) is needed for the original Vidic et al. (1994) computation and \(\hbox {T}_{\mathrm{c}}\) is known since the response spectra is assumed.

The estimation of the linearized response as proposed by Dwairi et al. (2007) by computing the response of the SDOF for an equivalent period and damping using Eqs. 5 and 6 and the ductility.

Finally, the linearized response following Lin and Miranda (2008), which is estimated by computing the linear response of the SDOF for an equivalent period and damping following Eqs. 7 and 8.

Finally, the proposed estimates are computed using Eqs. 9 and 11.

Moreover, this representation recalls that the spectral displacement in the plateau region is a function of the square of the period, such that a small uncertainty in the period estimation leads to large errors in the displacement demand estimate. Michel et al. (2010), computed the standard deviation of period height-relationships for RC shear wall buildings in France from ambient vibration measurements and found 0.08 s, i.e. up to 50 % of the period value on the plateau. Even numerical models, based on simplified assumptions, cannot predict the period with an excellent accuracy.

The displacement demand predicted by the different approaches is compared to the statistical characteristics of the peak displacements computed according to hysteretic model for various periods and strength reduction factors. This study is focused on displacement demand prediction in the short period range corresponding to the plateau of the related design response spectrum. Therefore, the methodology detailed above has been applied for periods T between 0.1 and 1 s and strength reduction factors R of 1.5, 2, 4 and 6. The initial damping ratio \(\xi \) was set to 5 %. A constant value for strength reduction factor R is used instead of a constant displacement ductility \(\mu \) to ensure the same non-linearity level for each ground motion. The goal of the study is to determine in which cases the different methods avoid an underestimation of the peak displacement demand.

## 4 Results

### 4.1 Performance of the classical methods

For low strength reduction factors (\(\hbox {R}\le 2\)), these methods provide reasonable displacements on average in the investigated period range for all ground types, even the equal displacement rule. The non-linear phenomena are limited so that the variability of the non-linear response is not critical. EC8 is slightly conservative, especially for ground type D (not displayed). The equal energy approximation gives relevant values only below 0.5 s. For intermediate strength reduction factors (R = 4), the inelastic response deviates significantly from the elastic response. At low periods (below 0.4 s), the increased inelastic displacement can be clearly seen. The equal energy approximation provides relevant values on a narrow period range only. The EC8 approach underestimates the increase in displacement at low periods, which leads to non-conservative values of inelastic displacement for periods lower than 0.3 s, i.e. for RC shear wall buildings lower than 3-stories high according to the EC8 (CEN 2004) frequency/height relationship. The inelastic displacement even exceeds the upper bound of the EC8 (infinite R – linear trend) even for R = 4 for several soil classes. It is therefore already obvious that proposition 1 is not conservative enough for moderate to large non-linearity levels. Finally, at large strength reduction factors (R = 6), these phenomena are amplified, with the equal energy approximation being simply irrelevant and EC8 strongly underestimating the displacement at low periods.

The results are slightly affected by the used hysteretic model but same trends are obtained. It can be noticed, however, that only ductile behavior is considered here. Non-ductile behavior may change the conclusions. An important conclusion here is that reproducing the non-linear response of structures using linear SDOF systems is not an easy task, even by adding more and more parameters to the relationships. Moreover, extending the study to MDOF systems would introduce even more variability (e.g. Erduran and Kunnath 2010).

The approach of choosing intermediate periods between the initial and secant stiffness (Lin and Miranda 2008) is doubtless the most relevant from a physical point of view, but these results show that the improvements it provides are not critical. Considering the fast development of non-linear modeling, linearized models have only a future for first order design and assessment purposes. Therefore, instead of looking for more complex – but still linear - models, we propose simplified, conservative estimates for design code purposes.

### 4.2 Performance of the proposed method

Compared to EC8 displacement demands that are not on the safe side, proposition 1 leads to better results, in the engineering sense, for moderate strength reduction factors up to R = 4. Proposition 2 may be applied for higher strength reduction factors. However, for strength reduction factors higher than R = 6, displacement demand increases significantly and the equal displacement rule is no longer valid, even for intermediate and large periods.

The same trends are related to the different ground types. However, the displacement demand predictions corresponding to ground type A are less accurate than for the other ground types. The results show that for this type of design spectra, with short corner periods of the plateau, even the displacement demand predictions according to proposition 2 underestimate the results for high strength reduction factors.

## 5 Summary and conclusions

The comparison of several methods (equal displacement rule – EDR –, equal energy approximation, a secant stiffness method and two empirical equivalent period and damping methods) was performed in order to evaluate their efficiency, more specifically in the period range of the plateau of the design spectra. The study focused on the reliability of the methods’ seismic displacement demand prediction with respect to their complexity when compared to the statistical characteristics of the response of the modified Takeda hysteretic model. In order to load this non-linear model, recorded ground motions were selected and slightly modified to match design spectra of different ground types in order to study the effect of the structural response, irrespective of the ground motion variability.

As shown by previous similar studies, knowledge invested in a more complex – but still linear – method can improve the accuracy of the predictions, especially for intermediate to large periods. However, this investment may not be justified for all applications such as preliminary structural design or assessment. The results show that detailed computations lead to an incorrect understanding of accuracy, since other uncertainties, such as that of the fundamental period, limit the precision of the demand estimation in any case. For low periods, none of the examined methods perform satisfactorily in all cases. Particularly, the original procedure proposed in EC8 systematically underestimates the demand in this period range. The equal energy “rule”, still often proposed for this period range, diverges dramatically. Consequently, the linear approximation seems not to be justified for the lowest periods, corresponding to low-rise buildings.

Therefore, following the “principle of consistent crudeness” (Elms 1985) two new simple, but conservative, displacement demand estimation methods are proposed. The peak displacement predicted by the original EC8 procedure and both propositions were compared the inelastic response of to the modified Takeda hysteretic model for various periods and strength reduction factors. Both propositions may be graphically approximated which is a significant advantage for practical application. The results show that, except for low strength reduction factors up to R = 2, the propositions prevent the underestimation of the displacement demand that was observed with original EC8 procedure. The propositions may be further improved for a better prediction of displacement demand. However, such an improvement would be related with a more sophisticated expression, which may not be justified for practical applications. The reported investigations are focused on SDOF and on hysteretic models featuring ductile structural seismic behavior. These options are related to the objectives of the study. Even if some slight differences arise with MDOF (e.g. Schwab and Lestuzzi 2007), the crucial characteristics of the seismic response are captured with SDOF. Compared to ductile structural behavior, limited hysteretic energy dissipation behavior still increases the non-linear demand (Lestuzzi et al. 2007) and further investigations would be necessary in order to propose adequate displacement demand predictions for such cases.

The significance of the obtained results should be distinguished between new and existing structures. The impact is relatively limited for the design of new structures. For conventional design, current procedures are reliable because only relatively small values of the strength reduction factor are allowed. The usual construction codes practice of considering the equal displacement rule for the whole period range is even validated by the results. For ductile design, such as capacity design, restricted modifications may be involved for the low period range only. By contrast, for the seismic assessment of existing structures, such as unreinforced masonry low-rise buildings, the current procedure of EC8 should be modified in order to provide accurate predictions of the displacement demand in the domain of the response spectrum plateau. Current procedure is reliable only for small values of the strength reduction factors (\(\hbox {R}\le 2\)). For higher values of strength reduction factors, the two propositions developed in this study lead to significantly more relevant displacement demand prediction. Consequently, it is suggested to replace the current EC8 procedure by the proposition 1 for strength reduction factors between 2 and 4 and by proposition 2 for higher strength reduction factors.

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