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 (PSHAs), is proposed in order to obtain strength modification factors of design spectra which consider structural degradation through the spectral-shape intensity measure

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 fundamental vibration period

Seismic design guidelines for building structures recommend modifying the response-spectra ordinates by a series of factors 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 code 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, 2000), which consists of multiplying the elastic design spectrum by several coefficients. One of them takes into account the hysteretic structure-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 (2012) methodology. Another example is the Manual for Civil Structures Design (MCSD, 2015), developed by 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., 1988, 2018). In spite of that, the current Mexico City Building Code (MCBC, 2017) does not specify any structure-degrading factor.

This study is aiming to propose a methodology for obtaining a mathematical
expression corresponding to a structure-degrading factor for seismic design
of buildings 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 in the vibration periods of a structure from the undamaged to the damaged state strongly depends of several parameters, and this is crucial to consider different design limit states. Although the procedure is not affected by these parameters, the variation in the structural period could be taken into account considering different values of

First, it is necessary to perform probabilistic seismic hazard analyses (PSHAs) corresponding to a firm-ground site and then soft-soil sites located in the seismic area of
interest. PSHAs are associated with Sa(

Second, uniform hazard spectra (UHS) of

Finally, a mathematical expression is adjusted to the spectral ratios. In order to verify that the mathematical expression leads to reasonable results, it is convenient to compare these with those obtained with other expressions found in the literature.

Block diagram of the proposed methodology.

In what follows, a description of the methodology is presented (see Fig. 1).

First, PSHAs are carried out for the firm-ground site of interest, corresponding to Sa(

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 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,

For each recording station site, UHS associated with a given return period are computed for Sa(

Next, the spectral ratio

Finally, a simplified mathematical expression is adjusted to the spectral ratio

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.

The evaluation of a probabilistic seismic hazard analysis for a particular site requires identification of all possible earthquake sources 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. Figure 2a shows the shallow-depth seismic zones where interplate earthquakes occur due to the subduction of the Rivera and Cocos plates (SUB1–SUB4). Figure 2b illustrates the intermediate-depth 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 (C1 to C14) proposed by Ordaz and Reyes (1999). Seismic zones in Fig. 2c are also included in the present study to compute PSHA.

Seismic 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 regard, the distribution of earthquake sizes is commonly described by the bounded Gutenberg–Richter recurrence law (Eq. 1).

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

The present study assumes

Once the earthquake magnitude 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; Kramer, 1996). Since the area sources, where earthquakes can occur, are well-delimited (Fig. 2a–c), it is straightforward to determine the source-to-distance distribution.

Attenuation relationships are fundamental for PSHA. They are commonly developed to predict the peak ground acceleration, PGA, or the spectral
acceleration, Sa(

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, 2013; Cornell, 1968; Esteva, 1968; McGuire, 1995; Kramer, 1996).

In order to overcome the limitations of traditional IMs (e.g., PGA, Sa(

Accordingly, Buratti (2012) made an exhaustive comparison of the most influential scalar IMs available in the literature with respect to efficiency and sufficiency. The study concluded that the most effective intensity measure was

Based on the literature mentioned above, the authors of the present study
concluded that

In this section a methodology to perform PSHA using

Among the parameters that define the intensity measure

It has been pointed out that the stiffer the structure, the larger the period lengthening. Accordingly, for structures with short vibration periods, we adopt

At short to moderate vibration periods, the structural period lengthening
diminishes somewhat linearly until it reaches a semi-constant behavior (which is independent of the level of nonlinearity developed by the structure) (Katsanos and Sextos, 2015). In this regard, 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 (

For vibration periods longer than the dominant soil period, it is assumed

Summarizing, we used

The uniform hazard spectra are computed, first, for the CU site,
which is in firm ground. Figure 3a shows the UHS if only interplate, or
alternatively intraslab, earthquakes occur. It also displays when both types
of events are considered simultaneously (Total). Figure 3b shows the total UHS of Sa(

Zones of Mexico City grouped in accordance with the dominant soil period.

Estimating the seismic hazard at firm ground allows us to proceed 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 recorded all the significant ground motions that have struck Mexico City. In addition, different studies have taken CU as a reference site (Ordaz et al., 1988; 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:

Mean response spectral ratios for Sa(

Therefore, to evaluate the previous function, firstly, the spectral ratios
are estimated, and then they are coupled with the seismic hazard curves via Eq. (11). In this respect, Fig. 4a–f show the mean response of the
spectral ratios for Sa(

Locations of seismic recording stations in Mexico City (see Table 1).

Mean annual rate of exceedance (

Next, in order to compute the mean annual rate of exceedance of Sa(

Then, having the mean rates of exceedance for each recording station site
(see Table 1), the UHS are estimated for a given return interval. Figure 7a
to f show the UHS of Sa(

Uniform hazard spectra of Sa(

Intensity maps corresponding to Sa(

Once the uniform hazard spectra of Sa(

Spectral ratios between the uniform hazard spectra of

Based on these ratios, the following spectral modification
function (SMF) was proposed, which is a variation of that specified by MCSD (2015):

Numerical coefficients for SMF expression (Eq. 12).

Figure 9a to f show the equation proposed here (Eq. 12) (thick dashed line),
as well as the MCSD (2015) function (thick solid line). In the figures, the horizontal and vertical dotted lines, aligned at

The figures show the following.

The highest

When the vibration period of the system is close to the dominant soil period (

When

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). This happens because the MCSD function was calibrated using ground motion data recorded at a station located in that zone (SCT station in zone D); however, it does not happen the same for other soil conditions, especially for

Equation (12) predicts values closer to unity at sites in zones A–C (firm ground and transition soil) than at zones D–F, which means that the structural softening is not as significant as it is for zones D–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 (

Finally, the reduction of strength demand according to Eq. (12) fits the observed data (thin gray lines) better for each type of soil (zones A to F) than that recommended by MCSD guidelines.

Mean ratios of strength demands of degrading and of non-degrading
systems corresponding to

Figure 10a and b also include the

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, uniform hazard spectra are obtained for two
alternative intensity measures:

From the study the following is concluded.

For structures with vibration periods shorter than the dominant soil period (

For structures with vibration periods close to the dominant soil period (

For systems with vibration periods longer than the dominant soil period (

A strength modification factor was proposed (Eq. 12). The expression was fitted according to the spectral ratios

The expression proposed (Eq. 12) is a useful tool for simplified nonlinear modal analyses, to explicitly incorporate 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.

In addition, the study presents a methodology to elaborate seismic hazard maps in terms of the intensity measure

The data are available for free at

ARC developed the theoretical framework, performed the computations and wrote the original manuscript. SER and EB conceived the study and contributed to the development and design of the methodology. MAO and ARS contributed to the sample preparation and review of the final manuscript.

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 paper has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the paper 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.

We understand that the first and the second corresponding authors 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.

Thanks are given to DGAPA-UNAM (project PAPIIT IN100320) and to Instituto para la Seguridad de las Construcciones de la Ciudad de Mexico, for their support. The first and fourth authors acknowledge the scholarship given by Consejo Nacional de Ciencia y Tecnología (CONACyT) during their graduate studies. The authors wish to thank CIRES for having provided the seismic records used in this study.

This research has been supported by the Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México (grant no. PAPIIT-IN100320).

This paper was edited by Filippos Vallianatos and reviewed by two anonymous referees.