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Reliability-based strength modification factor for seismic design spectra considering structural degradation

For earthquake resistant design, structural degradation is considered using traditional strength modification factors, which are obtained via the ratio of the nonlinear seismic response of degrading and non-degrading structural single degree of freedom (SDOF) systems. In this paper, with the aim to avoid the nonlinear seismic response to compute strength modification factors, a methodology based on probabilistic seismic hazard analyses (PSHA) is proposed in order to obtain strength 15 modification factors of design spectra which consider structural degradation through the spectral-shape intensity measure INp. PSHA using INp to account for structural degradation, and Sa(T1) which represents the spectral acceleration associated with the fundamental period and does not consider such degradation, are performed. The ratio of the uniform hazard spectra in terms of INp and Sa(T1), that represent the response of degrading and non-degrading systems, provide new strength modification factors without the need to develop nonlinear time history analysis. A mathematical expression is fitted to the ratios that 20 correspond to systems located in different soil types. The expression is validated by comparing the results with those derived from nonlinear time-history analyses of structural systems.


Introduction
Structures subjected to cyclic loading induced by intense ground motions can exhibit stiffness and/or strength degradation due to the inelastic nonlinear behavior of their structural elements, which can give place to lengthening of the structural 25 fundamental vibration period T1. The effect of such lengthening can be beneficial for structures whose fundamental period is in the descendant branch of the acceleration response spectrum and their higher vibration modes have little influence on the structural response. On the contrary, the effect can be detrimental, for structures whose vibration period is in the ascendant branch of the response spectrum. In the latter case, the effect of "structural softening" can have severe consequences because the structure undergoes to seismic loading greater than that assumed for its design (Akkar et al., 2004;Chenouda and Ashraf, 30 2008; Chopra and Chintanapakdee, 2004;Terán-Gilmore and Espinosa Johnson, 2008). For example, during the Guerrero-Michoacán September 19, 1985 Mexican earthquake, many mid-rise buildings (5-to 10-story buildings) with T1=0.7s-1.2s approximately, located in soft soil of Mexico City, which has a vibration period around 2s, suffered severe structural damage (including collapse) because of the degrading structural effect (Montiel and Ruiz, 2007).
Seismic design guidelines for building structures recommend modifying the response-spectra ordinates by a series of factors 35 in order to include relevant structural behavior that affects the structural response. Those factors are related, for example, to seismic behavior, structural over-strength, structural irregularity, degrading behavior, etc. A common practice to derive those modification factors is by means of the ratio between specific response spectra of single-degree-of-freedom (SDOF) systems.
Indeed, most current seismic codes provisions implement simplified analyses based on these ratios. For example, the Federal Emergency Management Agency (FEMA) introduced the so-called Coefficient Method (FEMA-273, 1997;FEMA-356, 40 2000), which consists of multiplying the elastic design spectrum by several coefficients. One of them takes into account the hysteretic structural degrading behavior. More recently, FEMA-440 (2005) presented some improvements to current nonlinear analysis procedures. Accordingly, the Coefficient Method suffered slight adjustments, where the coefficient that incorporates the effect of degrading structural behavior was updated. At present, the simplified nonlinear approach is available in FEMA P-58-1 (2012) methodology. Another example is the Manual for Civil Structures Design (MCSD, 2008(MCSD, , 2015, developed by 45 the Federal Commission of Electricity of Mexico, which specifies a degrading factor that increases or decreases the design spectral ordinates, due to structural deterioration. The hysteretic degrading behavior is particularly severe for structures located in soft soil, like that in the lake bed zone of Mexico City, where there is a high-density population, and the site effects make it susceptible to severe earthquake damage (Singh et al., , 2018. In spite of that, the current Mexico City Building Code (MCBC, 2017), does not specify any 50 structural degrading factor. This study is aiming to propose a methodology for obtaining a mathematical expression corresponding to a structural degrading factor for seismic design of structures that exhibit period lengthening. The expression is a function of both the structural period and the dominant period of the soil. The methodology can be applied to any high seismic hazard region of the world. Finally, notice that the variation of the vibration periods of a structure from the undamaged to the damaged state strongly depends of 55 several parameters, and this is crucial to consider different design limit states. Although the procedure is not affected by these parameters, the variation of the structural period could be taking into account considering different values of TN (see definition of Np below); however, the assessment of this value accounting for the design limit state, structural type, interaction of the structural elements with the nonstructural ones require the study of specific structural systems such as: reinforced concrete, moment resisting steel frames, masonry, structures with eccentrically, buckling restrained braces, posttensioned, based 60 isolators among others, which is out of the scope of the present study. By the other hand, soil structure interaction (SSI) was not taken into account to compute the structural degrading modification factors for seismic design spectra; nevertheless, notice that the effect of SSI is more important for stiff structures located on soft soil, in such a way that for this type of structures, the ordinates of the response spectra tend to increase while the opposite occurs for flexible structures (Avilés and Pérez-Rocha, Building Code (MCBC, 2017) which provide recommendation about this issue.

Methodology proposed
In first place, it is necessary to perform PSHAs corresponding to a firm ground site, and then, soft soil sites located in the seismic area of interest. PSHAs are associated with Sa(T1) and alternatively to INp intensity measures; where Sa(T1) represents the spectral acceleration at the fundamental period of a structure, and INp is an intensity measure that accounts for the period 70 lengthening due to structural degrading behavior (INp is defined below). Although Sa(T1) is the most used ground-motion intensity measure (IM) around the world for PSHAs, it has some limitations. For example, it does not consider the effect of period lengthening of the structure due to its nonlinear behavior and mechanical properties degradation (Baker and Cornell, 2005;Bojorquez et al., 2008;Bojórquez et al., 2017a;Bojórquez and Iervolino, 2011;Kostinakis et al., 2018;P. P. Cordova, Gregory G. Deierlein, Sameh S.F. Mehanny, 2001;Shome et al., 1998;Tothong and Luco, 2007). 75 In second place, uniform hazard spectra (UHS) of INp and Sa(T1), which represent the response of degrading and non-degrading systems, respectively, are obtained. The UHSs are computed for several seismic recording stations located in different soil conditions. Subsequently, the effect of the structural degradation on the response of SDOF systems is characterized by the ratio between the uniform hazard spectra: INp/Sa(T1).
Finally, a mathematical expression is adjusted to the spectral ratios. In order to verify that the mathematical expression leads 80 to reasonable results it is convenient to compare these with those obtained with other expressions found in the literature.
In what follows, a description of the methodology is presented (see Fig. 1): •In first place, PSHAs are carried out for the firm ground site of interest, corresponding to Sa(T1) and, alternatively, to INp.
With the purpose of performing the analyses, the seismic tectonic zones that contribute to the seismic hazard of the site, are identified. 85 •Then, the probability distribution for earthquake magnitude and source-to-site distance are assumed. Additionally, it is necessary to define adequate ground motion prediction equations (GMPEs).
•With the total probability theorem and the information previously defined, the mean annual rates of exceedance (seismic hazard curves) corresponding to the site located in firm ground, are obtained.
•Once the hazard curves for firm ground are available, the mean annual rates of exceedance of seismic recording stations 90 located in different soil types of the seismic area of interest, are estimated (using a technique described in the following sections). The stations are grouped in different zones, which depend on the dominant period of the soil, Ts.
•For each recording station site, UHS associated with a given return period, are computed for Sa(T1), and alternatively, for INp.
•Next, the spectral ratios INp/Sa(T1) are estimated for each site. INp/Sa(T1) represents the ratio of strength demands between systems with degrading and systems with non-degrading structural behavior. 95 •Finally, a simplified mathematical expression is adjusted to the spectral ratios INp/Sa(T1). The expression contains parameters that depend on the zone of interest.
•The results of the expression proposed are compared with those obtained from other expressions found in the literature, which were obtained from time history analyses.
For illustrative purpose, in the following sections, the methodology proposed above is applied in order to find mathematical 100 expressions of structural degrading factors of the design spectra specified in MCBC however, the approach can be applied to any seismic region in the world.

Earthquake sources
The evaluation of a probabilistic seismic hazard analysis for a particular site requires identifying all possible earthquake sources 105 capable of producing a significant seismic event. For this purpose, Zúñiga et al., (2017) proposed a seismic regionalization for Mexico, which is used in the present study. Fig. 2a shows the shallow-depth seismic zones where interplate earthquakes occur due to the subduction of the Rivera and Cocos plates (SUB1, SUB2, SUB3 and SUB4). Fig. 2b illustrates the intermediatedepth seismic zones. This region corresponds to intraslab events that take place inside the subducted Rivera and Cocos plates below south-central Mexico (IN1 to IN3). Additionally, Fig. 2c displays the seismic zones for characteristic seismic events 110 (C1 to C14) proposed by Ordaz and Reyes (1999). Seismic zones in Fig. 2c are also included in the present study to compute PSHA.

Magnitude probability distribution
Earthquake sources are capable of producing different earthquake sizes. Therefore, it is crucial to define the probability distribution of the earthquake magnitudes and corresponding rates of occurrence for each source. In this sense, the distribution 115 of earthquake sizes is commonly described by the bounded Gutenberg-Richter recurrence law (Eq. 1).
where λm is the mean annual rate of exceedance for earthquakes between a minimum magnitude Mmin and a maximum magnitude Mmax, ν=exp(α-βMmin) is the mean annual number of earthquakes of magnitude Mw≥Mmin, where α=2.303p and β=2.303q. The values of p and q are indicated in Figs. 2a and 2b, according to Zúñiga et al. (2017). 120 For the seismic sources related to characteristic earthquakes ( Fig. 2c), the bounded Gutenberg-Richter recurrence law does not accurately describe the magnitude exceedance rates. Accordingly, for Mw>7, we employ a Gaussian probability distribution function (pdf) of magnitudes to account for the characteristic events in the Mexican subduction zones (see Eq. 2) (Ordaz and Reyes, 1999).
where ν7 is the mean annual number of earthquakes of magnitude Mw>7; EMw and σMw are the mean and standard deviation of the magnitude, respectively, and Φ(.) is the normal distribution function. The corresponding parameters to evaluate the distribution are shown in Fig. 2c.

Source-to-distance distribution
Once the earthquake magnitudes distribution is established, the pdf of distances from the earthquake location to the site of interest must be characterized. A uniform pdf is generally assigned to any point in the seismic zone (McGuire, 1995;Steven L. Kramer, 1996). Since the area sources, on which earthquakes can occur, are well-delimited (Figs. 2a, 2b, and 2c), it is 135 straightforward to determine the source-to-distance distribution.

Ground motion prediction equations
Attenuation relationships are fundamental for PSHA. They are commonly developed to predict the peak ground acceleration, PGA, or the spectral acceleration, Sa(T1). Unfortunately, attenuation models have not yet devised to provide INp as a function of the vibration period (as it is done with existing GMPEs); however, with GMPEs for Sa(T1) currently available, it is possible 140 to perform PSHA using INp. Here we employ the GMPEs proposed by Reyes et al., (2002) and Jaimes et al., (2015) for interplate and intraslab events, respectively. They were developed using accelerometric data recorded in Ciudad Universitaria station (CU), which is located at the hill zone (firm ground) of Mexico City, basically conformed by a surface layer of lava flows and volcanic tuffs with a shear wave velocity in the upper 30 m of 750 m/s (Ordaz and Singh, 1992;Singh et al., 2018).

Seismic hazard curves 145
The final product of a PSHA can be expressed in different forms. Seismic hazard curves are used frequently to represent the seismic hazard. They indicate the annual rate of exceeding a variety of intensity levels of a ground motion parameter at a site of interest. The procedure to compute a ground-motion hazard curve is based on the total probability theorem (Baker, 2008;Cornell, 1968;Esteva, 1968;McGuire, 1995;Steven L. Kramer, 1996).

INp Intensity measure 150
In order to overcome the limitations of traditional IMs (e.g., PGA, Sa(T1)), advanced seismic IMs have been proposed. Some researchers suggest using vector-valued ground motion IMs. By including two or more representative parameters of the ground motion, accurate evaluations of seismic performance can be achieved (Baker and Cornell, 2005;Bojorquez et al., 2008;Bojórquez et al., 2017a;Bojórquez and Iervolino, 2011;Kostinakis et al., 2018;P. P. Cordova, Gregory G. Deierlein, Sameh S.F. Mehanny, 2001;Tothong and Luco, 2007). Accordingly, Bojórquez et al., (2008) developed the vector-valued intensity 155 measure <Sa(T1), Np>, where Np is a parameter proxy for the spectral shape, this IM is an advancement in predicting the seismic response in comparison with other IMs. However, the evaluation of PSHA using vector-valued IMs is a complicated and impractical task; therefore, Bojórquez and Iervolino (2011), introduced a scalar IM based on Sa(T1) and Np, called INp, both Np scalar and vector-valued intensity measures have been effectively used (Bojórquez et al., 2012(Bojórquez et al., , 2017b. Accordingly, Buratti (2012), made an exhaustive comparison of the most influential scalar IMs available in the literature 160 respect to efficiency and sufficiency. The study concluded that the most effective intensity measure was INp. Additionally, De Biasio et al., (2014), based on a comparative study of structures with nonlinear behavior, showed the good performance of INp to predict maximum interstory drift and maximum ductility demands. Moreover, Modica and Stafford (2014) using <Sa(T1), Np>, estimated the fragility and performance of buildings with higher efficiency respect to different IMs. In this context, Minas and Galassos (2019) showed the advantages of INp comparing Sa(T1) fragility curves, for different damage states. Additionally, 165 Yakhchalian et al., (Yakhchalian et al., 2015) demonstrated the efficiency of the parameter Np. They showed that the parameter Np works appropriately, particularly in performance levels related to moderate levels of nonlinearity. Similarly, Kostinakis et al., (2016), proved the adequate efficiency of INp to reduce the uncertainty in the prediction of the response of reinforced concrete buildings. In addition, Jamshidiha et al., (2018), examined the ability of different IMs for predicting the seismic collapse capacity of steel resisting moment frames with fluid viscous dampers. They concluded that the scalar IM that resulted 170 from the combination with the parameter Np was most efficient.
Based on the literature mentioned above, the authors of the present study concluded that INp is a promising tool to perform PSHA.

Methodology to perform a PSHA using INp
In this section a methodology to perform PSHA using INp is proposed. In first place, INp is defined as follows (Bojórquez and 175 Iervolino, 2011): where INp is the scalar intensity measure, α is a parameter that should be calibrated according to the structure and the earthquake demand parameter selected (in this study α=0.5 is adopted, as recommended in Bojórquez and Iervolino, 2011); Saavg(T1…TN) 180 is the geometric mean of the spectral acceleration at N numbers of structural vibration periods considered. Saavg(T1… TN) takes into account the vibration period lengthening due to structural damage, and is expressed as: Substituting Eq. (4) and Eq. (5) into Eq. (3), applying the natural logarithm, it results: Then, the expected value and the variance of ln(INp) can be expressed as in Eq. (7) and Eq. (8), respectively.
The values of ln[Sa(Ti)] are obtained from existing attenuation models (e.g., the GMPEs described in section 3.4). On the other 190 hand, ln[Sa(Ti)] terms are commonly assumed to have joint Gaussian pdf; consequently, the summation has also Gaussian distribution. Therefore, the variance Var{ln[Saavg(T1… TN)]} and the correlation coefficient ρln[Saavg(T1… TN) ln[Sa(Ti)] can be obtained by Equations (9) and (10), respectively: where the term ρln[Sa(Ti)], ln[Sa(Tj)] represents the correlation between spectral acceleration values at periods Ti and Tj. The correlation coefficients have been obtained by the authors of the present study (Rodríguez-Castellanos et al., 2019, 2020.

Values of TN
Among the parameters that define the intensity measure INp, the geometric mean, Saavg(T1...TN), has a crucial role when computing the uniform hazard spectra (UHS). The TN value (N-th structural vibration period) takes into account the level of 200 nonlinearity developed by the structure. Bojórquez et al., (2008Bojórquez et al., ( , 2011 recommend using TN=2.0T1. Nevertheless, we consider that there is no optimal period range for Saavg(T1...TN) that meets the entire range of structural vibration periods; therefore, here we propose that TN should depend on the structural vibration period, which is in agreement with Tsantaki et al., (2012. It has been pointed out that the stiffer the structure, the larger the period lengthening. Accordingly, for structures with short vibration periods, we adopt TN=2.0T1, which agrees with recommendations made by Bianchini et al., (2009), Katsanos and 205 Sextos (2015), and , for relatively stiff structures, and assuming a ductility demand between 2 and 3.
At short-to-moderate vibration periods, the structural period lengthening diminishes somewhat linearly until it reaches a semiconstant behavior (which is independent of the level of nonlinearity developed by the structure) [44]. In this sense, Di Sarno and Amiri (2019) quantified the fundamental period lengthening of structures by the ratio of response spectra corresponding to the lengthened and the elastic structural vibration period (Tin/Tel). They suggested dividing the response spectra into two 210 main regions: the first associated with short-to-moderate period structures, whose period shift ratio Tin/Tel decreases with increasing the elastic period; and the second region related to long-period structures, where the ratio period Tin/Tel behaves practically constant. Consequently, there must be a certain bound where the period shift ratio switches to remain constant; therefore, we propose TN = Ts as that bound from which the lengthening of the structural vibration period remains almost constant. In this context, Ruiz-Garcia (2002, 2003), and independently, Terán-Gilmore and Espinoza (2008),  215 found that strength demands between degrading and non-degrading systems are similar when the structural period and dominant soil period are comparable, which means that the mean ratio value should be approximate to one when Tn ≈ Ts.
For vibration periods longer than the soil dominant period, it is assumed TN=1.25T1, which is, on average, the period shift ratiovalue for structures with a short-to-moderate nonlinearity level, that is, with ductility ratio around 2 to 3 (Katsanos and Sextos, 2015;Di Sarno and Amiri, 2019). 220 Summarizing, we used in this study: TN =2.0T1, for structural systems with short fundamental period; TN=Ts for those with intermediate period; and TN=1.25T1 for systems with long fundamental period. It is possible to get a better approximation of TN bounds, by means of a parametric study of the ratios of the equivalent period of SDOF degraded systems and that of the elastic systems (Tin/Tel), as a function of Tel, for a given ductility; such study can consider both ground motion characteristics and structural properties (such as degrading stiffness ratio, pinching factor, accumulated damage factor, etc.), as it was done 225 by Di Sarno and Amiri (2019). They proposed a mathematical expression for estimating the lengthening of the fundamental period as a function of the structural elastic period and the significant structural parameters, which is applicable to systems in sites classes D and C according to ASCE/SEI 7-10 (2010), with shear wave velocities 182.88<Vs30<365.76 m/s and 365.76<VS30<762m/s, respectively. However, the TN bounds used here lead to reasonable results, as it is verified below.

Uniform hazard spectra corresponding to firm ground
The uniform hazard spectra are computed, in first place, for the CU site, which is in firm ground. Fig. 3a shows the UHSs if only interplate, or alternatively, intraslab earthquakes occur. It also displays when both types of events are considered simultaneously (Total). Fig. 3b shows the total UHS of Sa(T1) and INp, both associated with a 250-year return period. It can be seen that the spectra are quite similar; practically, they reach the same acceleration levels, and slight differences occur at long 235 periods.

Uniform hazard spectra corresponding to soft soil sites
Estimating the seismic hazard at firm ground allows proceeding with a technique to assess the seismic hazard at soft soil sites.
In this regard, Esteva (1970) presented a formulation in which through a known hazard curve at a reference site, it is feasible to estimate a hazard curve at a recipient site. In this study, we used CU station as the reference site because, since 1964, it has 240 recorded all the significant ground motions that have struck Mexico City. In addition, different studies have taken CU as a reference site Reinoso and Ordaz, 1999;Singh et al., 1988).Therefore, it is viable to perform a hazard analysis for CU station and then to compute the annual rate of exceedance at other sites located in soft or medium soils, as follows: where: νY(y) is the mean annual rate of exceedance of a seismic IM, for the recipient site.
νx(y/τ) is the mean annual rate of exceedance of a seismic IM for the reference site, divided by the variable τ.
τ are the response spectral ratios between the response spectra corresponding to the recipient site and the reference site (Y/X). fτ(τ) is the pdf of τ. 250 Therefore, to evaluate the previous function, firstly, the spectral ratios are estimated, and then are coupled with the seismic hazard curves via Eq. (11). In this respect, Figures 4a to 4f show the mean response of the spectral ratios for Sa(T1) (solid line) and INp (dashed line) for one representative station located in each of the zones listed in Table 1. In this sense, the spectral ratios roughly represent the spectral amplification of soft soil with respect to firm ground. It is observed how the peak values shift towards increasingly longer periods, which, approximately, match with the dominant soil period (see Table 1). For this 255 analysis, more than 1100 ground-motion records corresponding to the different recording stations were used. The stations are grouped depending on the soil dominant period where these are located, as follows: Zone A: Ts<0.5s; Zone B: 0.5s<Ts<1.0s; Zone C: 1.0s<Ts<1.5s; Zone D: 1.5s<Ts<2.0s; Zone E: 2.0s< Ts<2.5s; and Zone F: 2.5s<Ts<3.0s. Additionally, Fig. 5 shows the location of the recording stations in Mexico City, which are represented with circles of different colours associated with each of the proposed zones (see Table 1). 260 Next, in order to compute the mean annual rate of exceedance of Sa(T1) and INp, the seismic hazard curves corresponding to CU station are coupled with the response spectral ratios, using Eq. (11). Figures 6a to 6f show the hazard curves (λ) of Sa(T1) and INp, associated with different vibration periods, corresponding to CU and the same recording stations of Figures 4a to 4f.
In the first place, as expected, the rates of exceedance for all the recording stations analyzed are higher than the corresponding ones of CU (up and down, respectively). Additionally, concerning the CU site, the hazard curves for both intensity measures 265 INp and Sa(T1) are very similar, and differences are barely visible at long return periods. Now, for the rest of the recording stations, Figs. 4c and 4d show noticeable variations between exceedance rates of Sa(T1) and INp; nevertheless, Figs. 4e and 4f display almost no contrast between the rates of exceedance of the two intensity measures. The previous is relative, because to fully characterize the variations between exceedance rates of Sa(T1) and INp, a wide range of periods needs to be covered; for this reason, we estimate the UHS in the following. 270 Then, having the mean rates of exceedance for each recording station site (see Table 1), the UHS are estimated for a given return interval. Figures 7a to 7f show the UHS of Sa(T1) and INp for the same stations of figures 6a to 6f, for a 250-year return period. It is observed that, at vibration periods shorter than the dominant soil period, the spectral ordinates corresponding to firm ground (Zones A, B, and C) are comparable for both IMs. However, at soft soil (Zones D, E, and F), the spectral ordinates of INp are notably higher than those of Sa(T1) (up to 30%). In contrast, at vibration periods longer than Ts, they are smaller than 275 those corresponding to Sa(T1) (5% to 20%, depending on the soil type). The same can be appreciated for different sites of the city in the maps shown in Figs. 8a to 8d, which correspond to Sa(T1) (left side) and INp (right side), for T1=0.5s (up side) and T1=1.0s (down side), for a return interval of Tr=250 years.

Degrading structural behavior effect
Once the uniform hazard spectra of Sa(T1) and INp were estimated, the degrading structural behavior effect is evaluated by 280 means of the ratio INp/Sa(T1). It represents the ratio of strength demands between a system with degrading, and the same system with non-degrading structural behavior. The ratios are obtained for each station of the zones listed in Table 1. Figures 9a to 9f show the INp/Sa(T1) ratios (thin gray lines) as a function of the normalized periods Tn/Ts, for zones A to F, respectively.
Based on these ratios, it was proposed the following spectral modification function (SMF), which is a variation of that specified by MCSD (2008MCSD ( , 2015: 285 where the values of a, b, c, d and e are shown in Table 2. It is noticed that the values of the parameters depend on the type of soil where the structure is located; on the contrary, those in MCSD function are constant values; in addition, such function is restricted only to soft soils. (approximately Ts/2), which indicates that the lateral strength demand for degrading systems is higher than the strength demand 295 for non-degrading systems. b) When the vibration period of the system is close to the soil dominant period (Tn/Ts≈1), the strength demands for degrading and non-degrading systems, are similar. c) When Tn/Ts>1, the demands of the degrading systems decrease with respect to those of the non-degrading systems. It means that for structural vibration periods longer than Ts, the degrading behavior provides a beneficial effect. 300 d) It is noticed that for zone D (Fig. 9d), the MCSD function predicts spectral modification values which are similar to the function proposed in the present study (Eq.12). It happens because MCSD function was calibrated using ground motion data recorded a station located in that zone (SCT station in zone D); however, it does not happen the same for other soil conditions, especially for Tn/Ts>1. e) Equation (12) predicts values closer to unity at sites in zones A, B, and C (firm ground and transition soil), than at zones D, 305 E and F, which means that the structural softening is not as significant as it is for zones D, E, and F. In this respect, several studies have observed that the degradation of the stiffness has little effect on the strength demands for structures located on firm sites (Akkar et al., 2004;Chenouda and Ashraf, 2008;Chopra and Chintanapakdee, 2004). Moreover, it is noticed that at very short vibration period systems (Tn/Ts<0.1), the SMF proposed here predicts amplification values very close to unity, which is consistent for extremely stiff structures. 310 f) Finally, the reduction of strength demand according to Eq. (12) fits better the observed data (thin grey lines) for each type of soil (zones A to F) than that recommended by MCSD guidelines.
With the aim of verifying the validity of the proposed expression, Figures 10a and 10b compare the results of Eq. (12) with those obtained from time-history analysis of SDOF systems. The figures show the mean ratio of strength demands of degrading and of non-degrading systems (elasto-plastic behavior) corresponding to a ductility value, μu (thin gray lines), using firm 315 ground and soft soil records, respectively (Miranda and Ruiz-Garcia, 2002;Terán-Gilmore and Espinosa Johnson, 2008).The ground motions at firm ground ( Fig.10a) correspond to synthetic accelerograms (Ts=1.0s) (Terán-Gilmore and Espinosa Johnson, 2008) and ground motions recorded in San Francisco bay area during 1989 the Loma Prieta earthquake (Ts≈1.1s) (Miranda and Ruiz-Garcia, 2002). In contrast, the ground motions at soft soil were recorded in the Lake Bed zone of Mexico City (Ts≈2.0s) (Fig.10b). 320

Conclusions
A methodology based on probabilistic seismic hazard analysis is proposed to evaluate the effect of degrading behavior on the strength demands of SDOF systems. For this aim, there are obtained uniform hazard spectra for two alternative intensity measures: INp and Sa(T1), which represent the response of degrading and non-degrading systems, respectively; so, the ratio of the hazard spectra INp /Sa(T1) characterizes the strength demands of systems with degrading behavior to those of systems with 330 non-degrading behavior. Based on the INp /Sa(T1) ratios, which correspond to systems located at different sites, grouped in different seismic zones (depending on the type of soil where the structures are located), a mathematical expression is proposed.
The methodology is applied here to structural systems located in Mexico City, but it can be applied to any seismic region of the world.
From the study the following is concluded; 335 1. For structures with vibration periods shorter than the dominant soil period (Tn/Ts<1), degrading systems exhibit strength demands up to 30% higher, than systems with non-degrading behavior.
2. For structures with vibration periods close to the soil dominant period (Tn/Ts≈1) the strength demands for degrading and non-degrading systems, are similar.
3. For systems with vibration periods longer than the soil dominant period (Tn/Ts>1), the strength demands for structures with 340 degrading behavior are lower, approximately 5% to 20%, than structures with non-degrading behavior. That reduction highly depends on the soil dominant period at the site, and it is larger for systems with longer soil dominant periods. For these cases, the structural degrading behavior produces a beneficial effect, reducing the lateral strength requirement of the structures. 4. A strength modification factor was proposed (Eq. 12). The expression was fitted according to the spectral ratios INp/Sa(T1) corresponding to different soil conditions. The value of the parameters included in the equation depends on the type of soil 345 where the structure is located.
5. The expression proposed (Eq. 12) is a useful tool for simplified nonlinear modal analyses, to incorporate explicitly the effect of degrading behavior according to the type of soil where the structure is located. It was verified that the mathematical expression proposed leads to results that are comparable to those obtained from time history analyses of SDOF systems located in soft soil. 350 6. In addition, the study presents a methodology to elaborate seismic hazard maps in terms of the intensity measure INp. Based on that methodology, it is presented the first seismic hazard map of Mexico City, in terms of INp.

Competing interests
We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.
We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who 360 satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us. We confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In so doing we confirm that we have followed the regulations of our institution concerning intellectual property. 365 We understand that the first and the second Corresponding Author are the contacts for the Editorial process (including Editorial Manager and direct communications with the office). They are responsible for communicating with the other authors about progress, submissions of revisions and final approval of proofs. We confirm that we have provided current, correct email addresses which are accessible by the Corresponding Authors and which have been configured to accept email from SruizG@iingen.unam.mx and eden@uas.edu.mx. 370    Table 1). 655 660 665 Figure 7: Uniform hazard spectra of Sa(T1) and INp, corresponding to one representative station of each zone listed in Table 1, considering 250 year-return interval.