Basin boundary seismic effects in Mexico City southern region

Abstract

Rapid changes in geotechnical and geological ground conditions lead to significant ground motion variability. This condition mainly occurs at the so-called basin edges, where there is an abrupt transition between soft highly compressible soils and stiffer materials. This problem becomes more relevant in areas where ground subsidence drastically changes the dynamic response of high plasticity clay deposits, such as those found in Mexico City, due to fundamental site period evolution with time. This paper presents site response analyses at an abrupt transition area in the southeast Mexico City region, along the edges of the Xochimilco-Chalco lakes. Considerable damage associated with three-dimensional wave propagation effects was observed in this zone during the September 2017 Puebla-Mexico earthquake. A series of three-dimensional finite difference numerical models of the basin edge were developed to evaluate ground motion variability, considering topographic effects and soil non-linearities. Good agreement between the computed response and the observed damage during the 2017 Puebla-Mexico earthquake reconnaissance was found. In addition, several normal and subduction events with a return period of 250 years were considered to evaluate the effect that frequency content, and strong ground motion duration have on the soil response variability. From the results gathered here, it was established the relevance of accounting for three-dimensional wave propagation fields to assess site effects at basin-edge zones properly and to be able to implement proper risk mitigation measurements at these zones.

1 Introduction

Ground motion variability in basin-like regions, such as Mexico City, has been historically correlated to site effects, in which soil layering and geological features play a crucial role in the wave propagation patterns. Often, these motions are further modified by topographical effects on the surrounding stiffer soils (e.g., Mayoral et al. 2019a, Asimaki et al. 2018a). This variability is more significant at abrupt basin boundaries, in which drastic changes from soft to stiff soils, or rocks, associated with specific geological formations lead to three-dimensional modifications of the incoming wave field. In particular, these effects can significantly affect the amplitude, frequency content, and duration of strong ground motions associated with the large-amplitude surface waves generated through seismic wave diffraction and energy focusing (Asimaki and Gazetas 2004a). Coupled basin and site effects due to alluvial soils and sediments on the seismic motions have been studied by several authors (Assimaki et al. 2005a, Hasal and Iyisan 2014a, Garini et al. 2020; Mayoral et al. 2019b). They found that the generated local surface waves and their interaction with the body waves in the soft soil layers lead to an amplification increment with respect to the classical one-dimensional analysis often adopted in practical engineering applications. Nevertheless, the impact of topographic and three-dimensional geological features at basin edges on the seismic performance of a site located at an abrupt transition has not been fully addressed. Those effects can drastically modify the surface response.

This paper presents a numerical study of the impact on the seismic response on an abrupt basin edge studied after the Sep 2017 Puebla-Mexico earthquake, and the expected scenario for normal and subduction events with a return period of 250 years. Analyses were carried out through three-dimensional finite-difference numerical models developed with the program FLAC3D (Itasca 2009). From the results gathered herein, it can be concluded that three-dimensional effects generated in the abrupt transition zone strongly affect the expected amplification of the ground motion with respect to those obtained from one-dimensional analyses.

2 Geotechnical settings in Mexico City

A typical stratigraphic section of the Mexico City basin is presented in Fig. 1. Mexico City and its surroundings are located within an old basin that comprises the former Texcoco Lake and the Xochimilco-Chalco Lake. These lakes have disappeared mainly due to underground water extraction and land reclamation for urban development (Fig. 2). Thus, while the peripheral part of the city is underlain by rock and hard soil deposits (layer of fractured lava overlying soft rock with a shear wave velocity of 450–600 m/s), the central part is located on soft lacustrine clay deposits with variable thickness (Seed et al. 1988). The former Texcoco Lake is located towards the north of the city and is separated by a ridge of hills across the northern edge of the Xochimilco-Chalco Lake. Both lake beds are now essentially filled with clay deposits, but the clays have different characteristics. The Xochimilco-Chalco Lake clays are stiffer than the Texcoco Lake clays (Mayoral et al. 2008).

Fig. 1

A typical section of the stratigraphic sequence of the Mexico City Valley (modified from Santoyo et al. 2005)

Fig. 2

Main geotechnical zones in Mexico’s valley

Thus, from the geological and geotechnical standpoint, three main different geotechnical zones are distinguished, as depicted in Fig. 2. The lake zone consists of a 20 to more than 60 m deposit of highly compressible and high-water content clay underlayed by the so-called deep deposits, comprised of very stiff layers of cemented silty sands. The hill zone is comprised of volcanic flows and pyroclastic deposits, and the third region is called the transition zone composed of erratic stratifications of alluvial sandy and silty layers interbedded with clay.

As presented in Fig. 3, Mexico City has been divided into three main zones for geo-seismic zonation purposes according to the local Building Code (RCDF 2004): Zone I (Hills), Zone II (Transition), and Zone III (Lake). In addition, zone III has been further subdivided into zone IIIa, IIIb, IIIc, and IIId to account for the increasing depth of the clay deposits when moving from the hill zones to the center of the old lakes (Mayoral et al. 2019c).

Fig. 3

Mexico City main geotechnical zones, soil resonance periods [in seconds], and ground subsidence-induced cracking (modified from Mayoral et al. 2019b)

The variation in subsoil conditions leads to the elastic predominant soil period, T_soil, distribution presented in Fig. 3. These periods constantly evolve due to regional ground subsidence (Auvinet et al. 2017; Mayoral et al. 2017; Ovando et al. 2007), as can be seen when comparing the distribution determined by Arroyo et al. (2013), with that included in the former Mexico City building code (RCDF 2004). In addition, this ground subsidence leads to cracks along the basin edges in zones of an abrupt transition, as depicted in Fig. 3.

3 Case study

The studied site is located in the southern abrupt basin border of the Mexico City valley, close to the Chichinautzin range (Fig. 3). This area exhibits unique geotechnical and geological conditions associated with the drastic change in the stratigraphic conditions, passing from very soft high plasticity clay to very stiff cemented silty sands, sandy silts, and heavily fractured rocks. The studied site includes an irregular semicircular hill in zone I and part of the former Xochimilco-Chalco Lake (Fig. 4) in the denominated zone III. Likewise, this zone presented extensive damage during the September 19, 2017 earthquake, including ground cracking as depicted in Fig. 5, vertical and horizontal ground offset, cracking on buildings with one to three stories, and opening of preexistent transition discontinuities.

Fig. 4

Aerial view of the site studied

Fig. 5

Location of cracking generated during the September 19, 2017, earthquake, which had 20 cm of vertical offset and 4 cm of lateral displacement

3.1 Subsoil conditions

For the subsoil conditions characterization, 25 exploration borings were conducted (Fig. 6). Series of cone penetration test, CPT, standard penetration test, SPT, selective sampling recovery of undisturbed samples, core barrel, CB, and SP suspension logging test, along with a laboratory investigation, were conducted to obtain the static and dynamic properties of the materials found at the site for the strains level of interest. The depths of the exploration borings ranged from 15 to 60 m. At those depths where the ground's hardness exceeded the CPT technique's applicability, SPT and CB, were used instead. The Shelby sampler was used to obtain undisturbed soil samples at the studied site. Laboratory testing was conducted to soil samples retrieved at several depths, including water content ${w}_{L}$, Atterberg limits, plasticity index (PI), grain size distribution [gravel(G), sand (S), and fine particles (F)], and also triaxial and unconfined compression tests were carried out to undisturbed soil and rock samples to characterize deformability and strength parameters. The suspension SP logging technique was used in exploration boring S-06 and S-19 to determine in situ shear wave velocity values. Figures 7 and 8 present the two cross-sections indicated in blue and green in Fig. 6, respectively.

Fig. 6

Plan view of the exploration boring’s location

Fig. 7

Stratigraphy along the hillside

Fig. 8

Stratigraphy along the transition between zone I and zone III

The primary geotechnical units are identified as follows: U-I) filling formed by gravel and sand in a matrix of brown clay and silt, U-II) foothills comprised of gravel and rock fragments of angular shape, with clayey sand, U-III) basaltic rock with varying fracture degrees, interbedded with lenses of gravel and sand, U-IV) very dense to dense sand with some gravels, and U-V) very soft high plasticity clay with interbedded sand, volcanic ash, and silt lenses. The U-III formation was reached in all boreholes at the end of drilling, and in each case at least 4 m of rock cores were collected to ensure the material was basalt rock.

The soil strength parameters, c and ϕ, and elasticity modulus at 50% strain, E₅₀, for the geotechnical unit U-V, corresponding with the soft clay, were estimated based on triaxial tests conducted in undisturbed samples retrieved at several depths. On the other hand, the properties for granular materials were calculated based on SPT-blow counts empirical correlations for the remaining units due to the difficulty of retrieving undisturbed samples. The expressions (1) proposed by Brown and Hettiarachchi (2008), and (2) suggested by Wolff (1989a) were used to obtain the cohesive strength component, c, and the friction angle, ϕ, respectively. The elasticity modulus was estimated using expression (3), which was determined by a multiple linear regression between the mechanical strength parameters c and ϕ, and the deformability parameter, E₅₀, developed in previous research (Vital and Mayoral 2014) for the stiff soils and fractured rocks found at the northwest Mexico City area. The parameters of U-III were defined according to the Rock Quality Designation, RQD, and the unconfined compression strength, UCS, gathered from laboratory testing, based on the Hoek and Brown model, for which the Mohr–Coulomb envelope is fitted according to the confinement stress of the rock. Table 1 summarizes the material properties used in the numerical model.

$$c=\frac{1}{\alpha }\left(\frac{{p}_{a}}{8.5}{N}_{60}\right)$$

(1)

$$\phi =27.1+0.3{N}_{60}-0.00054{N}_{60}^{2}$$

(2)

$${E}_{50}=277.3 c+579.25 \phi +3902.8$$

(3)

where c is the cohesion of the soil in kN/m²; α is a constant equal to 2.82; ${p}_{a}$ is the atmospheric pressure in kN/m²; N₆₀ is the number of SPT blows, corrected by energy at 60%, and $\phi$ is the friction angle, in degrees; E₅₀ is the elasticity modulus at 50% strain, kN/m².

Table 1 Properties of main geotechnical units

3.2 Normalized modulus degradation and damping curves

Gonzalez and Romo (2011) proposed a simplified model to predict the normalized modulus degradation and damping curves for clays found in Mexico City. This was proposed from the results of several resonant column and triaxial tests. The model is able to match the shear modulus degradation and damping curves separately. This model is defined by the following expressions:

$$G=\left({G}_{min}-{G}_{max}\right){H}_{G}\left(\gamma \right)+{G}_{max}$$

(4)

$$\lambda =\left({\lambda }_{max}-{\lambda }_{min}\right){H}_{\lambda }\left(\gamma \right)+{\lambda }_{min}$$

(5)

$${H}_{G}=\frac{{\left(\gamma /{\gamma }_{rG}\right)}^{2{B}_{G}}}{1+{\left(\gamma /{\gamma }_{rG}\right)}^{2{B}_{G}}}$$

(6)

$${H}_{\lambda }=\frac{{\left(\gamma /{\gamma }_{r\lambda }\right)}^{2{B}_{\lambda }}}{1+{\left(\gamma /{\gamma }_{r\lambda }\right)}^{2{B}_{\lambda }}}$$

(7)

$${\gamma }_{rG}=2*{10}^{-5} {PI}^{1.9272}$$

(8)

$${\gamma }_{r\lambda }=0.0044 PI+0.0377$$

(9)

$${B}_{G}=-2*{10}^{-6}{IP}^{2}+0.0014 PI +0.2846$$

(10)

$${B}_{\lambda }=-7*{10}^{-6}{IP}^{2}+0.0038 PI +0.3282$$

(11)

where G_max and λ_min are the values at small strain shear modulus and damping respectively, these values are unique for each soil type; G_min and λ_max are the values that reach G and λ before dynamic failure; γ_rG and γ_rλ are the strains references corresponding to 50% of degradation of the shear modulus, and 50% of increase in damping ratio respectively, these deformations depend only on the plasticity index, PI, which is a parameter obtained from laboratory testing; B_G and B_λ are values that define the geometry of the curves G-γ and λ-γ respectively, which are a function of the plasticity index.

Due to the practical difficulty in sampling the granular material layers, the upper and lower bounds proposed by Seed and Idriss (1970) for normalized modulus degradation and damping curves, respectively, were deemed appropriate, considering that they had been used in 1-D wave propagation analysis (Romo et al. 1988; Seed et al. 1988), which predictions were in good agreement with the measured response during the 1985 Michoacan earthquake, in Mexico City. Figure 9 presents the normalized modulus degradation and damping curves used in the dynamic analyses.

Fig. 9

Normalized modulus degradation and damping curves

Nonlinear soil behavior was accounted for approximately with the Sig3 model available in FLAC3D, which is characterized by normalized stiffness degradation and damping curves. As can be seen, there are some differences among damping curves predicted by FLAC3D and those computed with the theoretical model from soil index properties as previously explained. These discrepancies are due to the fact that the Sig3 model generates more damping at high strains than experimentally derived model curves because of the well-known limitation of hysteretic-type models, which are unable to fully capture both shear stiffness degradation and damping curves developed under steady-state conditions simultaneously. However, in a nonlinear analysis, the transient ground response in each loading cycle is attempted to be characterized as a function of the evolution of shear strains during random ground shaking rather than the steady state response established in the resonant column and cyclic triaxial test, from which modulus degradation and damping curves, such as the Seed and Idriss model (1988), were developed. Therefore, it is more reliable and realistic to capture the stiffness degradation through the fitting of the G/Gmax curves. However, for the strain levels reached in the analysis for all the Vs profiles (Fig. 21), the differences in damping values among theoretical and Sig3 curves were not significant.

3.3 Shear wave velocity profiles

Shear wave velocities of the U-V were estimated based on expression (12) proposed by Ovando and Romo (1991) in terms of the penetration resistance, qc, measured with Cone Penetration Test, CPT, considering the Xochimilco-Chalco Lake parameters compiled in Table 2.

$${V}_{s}=\eta \sqrt{\frac{{q}_{c}}{{N}_{kh} {\gamma }_{s}}}$$

(12)

where V_s is the shear wave velocity, in m/s; q_c is the tip cone penetration resistance in kN/m²; γ_s is the unit weight of the soil in kN/m³; N_kh and η are parameters that depend on the soil type, which were determined for the particular conditions of the site. Typical values of these parameters are presented in Table 2.

Table 2 Typical range of values for parameters N_kh and η (Ovando and Romo 1991)

For verification purposes, in-situ shear wave velocity distributions were measured at SM-06 and SM-19 sites in the clay and abrupt transition zones, respectively, using suspension login, SP, test. Figure 10 presents a comparison of measured and estimated Vs profiles. As can be seen, estimations agree with measured Vs values. For establishing Vs distributions with depth on granular materials, the relation proposed by Seed et al. (1986) was used as follows:

$${G}_{max}(psf)=1000{K}_{2}\sqrt{{\sigma^{\prime}}_{m}}$$

(13)

where K₂ is a shear modulus coefficient, which ranges about 55–80 for sands and 80–180 for gravels, according to the grain size distribution and compactness; and σʹm is the effective mean principal stress, in psf (pounds per square foot). Due to the low RQD values of U-III, it was deemed appropriate to use the expression (13) to define its corresponding Vs distribution, thus Vs depends on the confinement stress. Although the boreholes of Fig. 10 were very close to each other, the depth of the rock basement in boring SM-06 was 15 m deeper than that of SM-19. This is due to the well documented abrupt transition zones of Mexico City between the soft clays and hard soil, associated to the old lakes that existed in the area (Mayoral et. al. 2019b).

Fig. 10

Soil profile of the boring a SM-06 and b SM-19

3.4 Ambient vibration tests

The dominant period (T_s) distribution was established through ambient vibration tests (AVT) performed at the zone. The methodology proposed by Nakamura (1989) was implemented for these tests, which uses the spectral ratio between the horizontal and vertical motion components obtained in the ambient vibration records. High-resolution broadband seismographs performed the AVT with a sampling interval of 0.01 s in the three motion components (EW, NS, and vertical). A total of 50 ambient vibration measurements were conducted to estimate the dominant period.

To calibrate the soil response analyses, with the measured shear wave velocity, Vs distribution of the soil strata, and the dominant period, the AVTs were determined in the location of every exploration boring, and the other points were used as complements to improve the precision of the estimation. The location of measuring points and the map of the dominant period obtained in the researched area is shown in Fig. 11. It can be noticed the rapid changes in the dominant period values. In the most critical case, it increases from 0.6 to 2.4 s at a distance shorter than 25 m.

Fig. 11

Map of Ts values at studied area [in seconds] from AVTs (Ambient Vibration Tests)

4 Numerical model

A three-dimensional finite-difference numerical model of the studied site was developed with the program FLAC3D (Itasca 2009) to obtain the ground motion spatial variation due to site response, topographic effects, and basin effects (Fig. 12). Figure 13 presents the sections of analysis and control points in the domain of the finite difference mesh, also shown in Fig. 11, where it can observe the corresponding dominant period.

Fig. 12

a Complete tridimensional finite difference model and b surface of the unit III

Fig. 13

Sections of analysis and control points in the tridimensional finite difference model

Following the criterion proposed by Kuhlemeyer and Lysmer (1973), to accurately represent wave transmission through the numerical model, the spatial element size of the mesh, Δl, was kept smaller than one-fifth of the shortest wavelength, λ, defined as λ = Vs/fmax, where fmax is the highest frequency component of the input wave that contains appreciable energy. For the current input waves of the synthetic acceleration time stories considered (Fig. 27) the energy is concentrated around 0.1–8 Hz, and the smallest average shear wave velocity Vs of the clay was about 80 m/s, so a maximum Δl of 2.0 m was deemed appropriate.

Although several constitutive models have been developed to account for nonlinearities in low plasticity clays and sands (Gajan et al. 2010; Carlton and Tokimatsu 2016), and medium to stiff clays (Borja and Amies 1994), there is a lack of enough experimental data to develop and calibrate a reliable constitutive model for high plasticity clays. Therefore, the stress–strain relationship of the soil was assumed to be elastoplastic following Mohr–Coulomb failure. However, due to the shaking's level, some degree of soil non-linearities associated with ground deformations are expected to occur. Thus, a fully non-linear site response analyses were also carried out to study these effects. To take into account this nonlinearity, the hysteretic model in FLAC3D, denominated as Sig3, was used to deal with modulus stiffness degradation and damping variation during the seismic event. Although hysteretic damping formulation is not a complete constitutive model, it can be used as a supplement to provide damping for those plastic models lacking intrinsic damping when not yielding, as the Mohr Coulomb model. Essentially, the formulation of hysteretic models in FLAC3D, modifies the strain-rate calculation so that the mean strain-rate tensor is calculated before any calls are made to constitutive model functions. At this stage, the hysteretic logic is called, returning a modulus multiplier, which is passed to the current constitutive model. Then, the model uses such a multiplier to adjust the apparent value of the tangent shear modulus of the full zone being processed, and regarding reversal paths, the two Masing (1926) rules are used to specify the behavior ensuring repeatable closed loops (Itasca 2009). In summary, the hysteretic models consider an ideal soil, where the stress depends only on the deformation and not the number of cycles. With these assumptions, an incremental constitutive relationship of the degradation curve can be described by τ_n/ γ = G/G_max, where τ_n is the normalized shear stress, γ is the shear strain, and G/G_max is the normalized secant modulus. The sig3 model is defined according to Eq. (14):

$$\frac{G}{{G}_{max}}= \frac{a}{1+exp\left(-\frac{L-{x}_{0}}{b}\right)}$$

(14)

where L is the logarithmic strain defined as L = log10(γ), and the parameters a, b, and × 0, used by the sig3 model, were obtained by an iterative approach in which the modulus degradation curves were fitted with the model equations. The corresponding damping is given directly by the hysteresis loop during cyclic loading. For the cases studied herein, the parameters “a,” “b,” and x_o vary from 1 to 1.014, − 0.46 to − 0.55, and 0.2 to − 1.5, respectively. Non-linear soil behavior is a function of the shaking level, which, if high, leads to shear stiffness degradation and damping increase. To avoid wave reflections during the dynamic simulation and the allowance of radiation damping, quiet boundary conditions were defined at the model's base using Lysmer & Kuhlemeyer (1969) formulation. In the case of the lateral faces of the model, Free field boundaries were considered using the formulation available in FLAC3D.

Finally, the depth of the model base was extended until reproduce the geology and shear wave velocity conditions of half-space, which was considered as the soft-rock site reference conditions observed at the National Autonomous University of Mexico with similar basaltic materials, where the CU01 seismic station, used as input motion, is located.

4.1 Seismic environment

The strong ground motion recorded during the September 19, 2017, Puebla-Mexico earthquake at station CU01 (located on basalt rock), was used as input time history in the dynamic analyses. Figure 14 shows the measured accelerations for both orthogonal components, and their corresponding response spectra. Both components of motion were considered in the seismic response analyses. Due to the long distance between the seismogenic zones and Mexico City, (i.e. around 300 km for subduction earthquakes and around 100 km for normal earthquakes), the ground motion vertical component is usually not significant, and commonly neglected in site response analyses for Mexico City. Figure 15a shows the records of the station MI15 located in zone III near the studied site, during the M7.3 September 14, 1995 Guerrero earthquake, which was a subduction event, and Fig. 15b the records of the same station during the M7.1 September 19, 2017, Puebla-Mexico earthquake, which was a normal event. As can be see the vertical component is less than 15% of the horizontal components. Nevertheless, further investigation is currently carried out to assess the effects of the vertical component on the soil response at abrupt transition zones for light one and two-story houses.

Fig. 14

a Acceleration time histories recorded during the September 19, 2017, earthquake at CU01 station and b response spectra of each recorded component

Fig. 15

a Acceleration time histories recorded during M7.3 September 14, 1995 Guerrero earthquake at MI15 station and b during the M7.1 September 19, 2017, Puebla-Mexico earthquake at same station

4.2 Free field response

Initially, for completeness, the computer code SHAKE (Schnabel et al. 1972) was used to conduct one-dimensional equivalent linear site response analyses at several points within the studied area. A total of 11 one-dimensional soil profiles, five located in firm soil (i.e., zone I) and six in the soft clay (i.e., zone III), were used to compare the results obtained from a three-dimensional non-linear finite difference numerical model. One-dimensional finite differences models were also developed with FLAC3D. These have varying depths ranging from 90 to 95 m and 79 to 81 m for zones I and III, respectively. The corresponding shear wave velocity and soil profiles at these sites are shown in Figs. 16 and 17. As can be seen, all the soil columns share the same “half-space” depth. Therefore the model’s bottom is located at the same depth in all cases, and the differences in columns high are associated with ground surface elevation variations due to surface topography.

Fig. 16

Shear wave velocity profile in points located in zone I

Fig. 17

Shear wave velocity profile in points located in zone III

Prior to the soil response analyses, the elastic site period, Ts, from AVT at each point in zone III was compared against the Ts computed based on the expression 4H/Vs, where H is the clay layer thickness and Vs the average shear wave velocity of the clay layer. Such comparison is shown in Fig. 18. As can be observed, there is a good agreement between both periods in most of the cases, so the proposed Vs profiles were considered appropriate to model the site response at zone III. Nevertheless there are some differences at control points 7 and 11, in where Ts measured from AVT results are larger than those expected based of 4H/Vs, this can be associated with the influence of the adjacent zones as depicted in Fig. 12b (i.e. 3D basin effects) where the thickness of the clay changes rapidly over distance, and the basement topography effect is expected to be large. Thus, in these cases, the commonly used ratio of 4H/Vs, based on one dimensional wave propagation is not accurate, considering that the main hypothesis of horizontally infinite layered soils is not satisfied.

Fig. 18

Comparison of elastic site periods, Ts, of points in zone II computed with 4H/Vs and those measured with AVT readings

Results gathered from frequency domain and time domain analyses conducted with SHAKE and FLAC3D, respectively, assuming one-dimensional SH waves propagating vertically, are shown in Figs. 19 and 20. Good agreement can be shown in the computed response obtained in each analyzed case. The differences observed are due to the equivalent linear method that uses the program SHAKE to compute the stiffness reduction and damping for a given shear strain, which cannot accurately represent the behavior of the soil column at high frequencies, and at high levels of shear strain, as shown in Fig. 21, during the entire duration of a seismic event (Bolisetti et al. 2014; Hashash et al. 2010). Regarding to the non-linearity, it can be observed in the G/Gmax and damping curves that for the level of maximum shear strain along the soil profiles, approximately 20% of stiffness degradation occurs, which means that significant non-linearity effects are generated. Also, the dominant periods measured in each control point are in good agreement, with the response spectra. Nevertheless, for the control points in the clay zone, the spectral accelerations obtained with SHAKE for large periods (T ≥ 1 s) don’t match correctly with the nonlinear analyses due to the facts described above.

Fig. 19

Response spectra at surface for each soil profile in zone I, free field, considering the 2017 Puebla-Mexico normal earthquake

Fig. 20

Response spectra at surface for each soil profile in zone III, free field, considering the 2017 Puebla-Mexico normal earthquake

Fig. 21

Maximum shear strain profiles reached during the soil response analysis in the FLAC3D columns of the zone III

Regarding the ground motion expected at the base of the three-dimensional finite-difference model, it was established that at this depth there is no significant ground motion variability, due to soil layering and impedance ratio, for the earthquake considered (Fig. 22). As can be seen, the time histories and response spectra of all soil columns present only minor differences between them, so the average time history was used as input movement at the base of the three-dimensional models for the two orthogonal components of recorded motion.

Fig. 22

a Input acceleration time histories, and b Response spectra at the base of the soil columns for 2017 Puebla-Mexico normal earthquake

4.3 Three-dimensional analyses

The simulation of the strong ground motion caused by the September 19, 2017, Puebla-Mexico earthquake, in the damaged area, was performed from the three-dimensional model (Fig. 12). Figure 23a shows the cracking patterns mapped on the damaged area after the September 19, 2017 earthquake, and Fig. 23b shows the contour of the maximum shear strain obtained from the numerical analysis at the end of the seismic loading. It can be seen that the zones with large values of shear strain in the model agree with the observed cracking after the earthquake. These results can be associated with the effect that both orthogonal components of motion had on the surface response, causing the maximum acceleration to occur in the direction between them, inducing longitudinal cracking on the road, as shown in Figs. 5 and 23a.

Fig. 23

Observed damage patterns a in the studied area after the 2017 Puebla-Mexico earthquake, and b in the numerical model at the end of the simulation

Figure 24 presents the displacement histories, which correspond to the perpendicular direction to the road, obtained from the numerical model at control points shown in Fig. 23b. These are located in the clay zone (CP-1), in the stiff soils zone (CP-2), and just in the abrupt transition zone (CP-3), where the most considerable shear strain was observed. The relative displacement between the lake zone and stiff soil zones induces tensional stresses during the ground shaking in the areas where the transition is more abrupt (see Fig. 11) and generates surface cracking due to the granular soils of the shallow strata (U-I) having very low tensile strength. In addition, it can be observed the instant when the failures occur in the displacement history of the CP-3 by exceeding the linear elastic range and never recovering its original position as the points CP-1 and CP-3. Likewise, Fig. 25 shows the plot contours of stress level, SL, from the numerical model in Sect. 2 (Fig. 23b) at the instant of the breaking point mentioned above. SL is defined by the ratio between the deviatoric stress developed during the simulation and the deviatoric stress at the failure, which can be expressed as:

Fig. 24

Displacement histories of the perpendicular direction to the road, from control point 1, 2 and 3 of the numerical models, obtained during the simulation

Fig. 25

Plot contours of stress level from numerical model in cross section 2, at the instant of breaking point (t = 36.5 s) indicated in Fig. 24

$$SL=\frac{\left({\sigma }_{1}-{\sigma }_{3}\right)}{{\left({\sigma }_{1}-{\sigma }_{3}\right)}_{f}}$$

(15)

$${\left({\sigma }_{1}-{\sigma }_{3}\right)}_{f}=\frac{2c\mathrm{cos}\varphi +2{\sigma }_{3}\mathrm{sin}\varphi }{1-\mathrm{sin}\varphi }$$

(16)

where c and φ are the cohesion and friction angle values from the Mohr–Coulomb failure criteria; ${\sigma }_{1}$ and ${\sigma }_{3}$ are the major and minor principal stresses, respectively, and the subindex $f$ denotes failure. Values of SL > 0.95 represent that the soil reached the failure, and the developed deviatoric stresses can induce cracking. Figure 25 shows the zones where the cracking propagation started when these reached the maximum deviatoric stresses during the simulation.

5 Parametric analyses

Once the three-dimensional numerical model was calibrated by reproducing the observed damage during the September 19, 2017, Puebla-Mexico City earthquake, it was studied the effect of the input motion's frequency content, intensity, and duration on the ground response variability on the studied area. Table 3 present the seed ground motions earthquakes considered in the analyses. The seismic environment was established based on uniform hazard spectra, UHS, determined for a return period of 250 years. Those input motions were adjusted to the target UHS by spectral matching, where each seed ground motion was modified using the method proposed by Lilhanand and Tseng (1988) and modified by Abrahamson (2000). This approach is based on a modification of the acceleration time history to make it compatible with a desired UHS, in which the long period of non-stationary phasing of the original time history is preserved. The 5% damped response spectra calculated for the modified time histories are compared with the target UHS in Fig. 26. It can be seen that the response spectrum calculated from the modified time histories reasonably matches the target spectrum. The synthetic ground motions are shown in Fig. 27.

Table 3 Original seismic parameters of seed ground motions earthquakes considered in the analyses

Fig. 26

Recommended uniform hazard spectra from Mexico City building code, and synthetic ground motion response spectra a subduction and b normal earthquakes (modified from Mayoral and Mosqueda 2021)

Fig. 27

Synthetic acceleration time histories for a subduction and b normal earthquakes (modified from Mayoral and Mosqueda 2021)

5.1 Analyses results

Site response analyses were performed considering six different synthetic acceleration time histories. Figures 28 and 29 show the spectral accelerations obtained for normal events at the control points for both parallel and transverse directions to the edge. As expected, the greater ground motion variability is presented at the control points in the lake zone (zone III, Control points P6 to P11). Their dominant period is concentrated in a range between 0.5 to 1.67 s (i.e. 0.6 to 2 Hz) for the parallel direction to the edge. For the transverse direction, the energy is concentrated around 0.7 s. On the other hand, Figs. 30 and 31 present the ground motion variability for the subduction earthquakes, obtaining similar results for the parallel direction to the edge, with a dominant period of around 0.8 to 2 s for the control points located at the lake zone. No ground motion variability was observed for the control points located in zone I.

Fig. 28

Response spectra associated to normal earthquakes in control points at zone I a parallel direction, X, and b transverse direction, Y, to the edge

Fig. 29

Response spectra associated to normal earthquake in control points at zone III a parallel direction, X, and b transverse direction, Y, to the edge

Fig. 30

Response spectra associated to subduction earthquake in control points at zone I a parallel direction, X, and b transverse direction, Y, to the edge

Fig. 31

Response spectra associated to subduction earthquake in control points at zone III a parallel direction, X, and b transverse direction, Y, to the edge

Figures 32 and 33 show the amplification ratio of the spectral accelerations (ratio of spectral acceleration at surface, Sa_surf, and spectral acceleration at base of the model, Sa_base) for control points P3 (zone I) and P8 (zone III) for both normal and subduction events. Firstly, for the control point P3, located at zone I, the amplification ratio obtained from the three-dimensional analysis in both directions (i.e., parallel and transverse directions) is very similar to the obtained from the one-dimensional analysis. According to this comparison, it can be proved that the three-dimensional seismic effects are not presented in zone I, composed principally of rigid soils and rock. However, this type of soil could show topographic effects (Mayoral et al. 2019a). On the other hand, for the control point P8, located in the Lake zone, the contrast between the seismic response of the two directions is very noticeable. The motion’s amplification corresponds to the parallel direction to the edge which is greater than the one of the perpendicular directions with similar amplification to the one obtained from the 1D analysis. Another important fact to highlight is the difference between the dominant period and the period range, where more considerable amplification is presented in the spectral accelerations corresponding to the parallel direction to the edge.

Fig. 32

Higher magnitude amplification ratios of the spectral accelerations for the 1D and 3D analyses at zone I a Normal seismic events b Subduction seismic events

Fig. 33

Higher magnitude amplification ratios of the spectral accelerations for the 1D and 3D analyses at zone III a Normal seismic events b Subduction seismic events

This effect was observed in several records from different subduction earthquakes in the seismic station CDAO (operated by RAII-UNAM network, Institute of Engineering at UNAM) (II-UNAM 2022), which is located near a zone of abrupt transition with a similar geotechnical condition to the studied site. Figure 34 shows the corresponding spectral ratios, taken as the ratio between the spectral accelerations recorded at the CDAO station (clay near abrupt transition) and the corresponding to the CU01 (basalt rock) station as the input base motion (see Fig. 3). This effect can be associated with the fact that the rigid border of the basement does not restrict motions in the parallel direction to the border line.

Fig. 34

Amplification ratios between the spectral accelerations obtained with the records of the CDAO station and the records of the CU01 station (basaltic rock) for several earthquakes (II-UNAM 2022)

Figures 35, 36 and 37 show the relation between index values (i.e., PGA, maximum spectral accelerations), and the distance between zone III (Lake zone) and zone I for normal and subduction earthquakes in both directions. Several interesting facts can be found in those relations. Initially, for the first two indices, it is observed that the values are higher in the areas close to the edge. The values decrease as they move away from it (Figs. 35 and 36). This is because as the point moves away, the interference is lower, so the frequency of the waves decreases proportionally. Instead, for this same fact, the spectral accelerations in long periods increase when the point moves away from the edge since the clay thickness gradually increases (Fig. 37), as well as the dominant period (long periods), which begins to coincide more and more with the frequency of the waves. Another fact that needs to be highlighted is that the values obtained from the normal earthquakes are more significant for the PGA and spectral accelerations at short periods. On the other hand, the spectral accelerations for long periods are more critical for subduction earthquakes, which strongly correlate to the input motions' frequency content.

Fig. 35

Relation between peak ground acceleration, PGA, at zone III and the distance to the zone I, associated to normal and subduction seismic events for a parallel and b perpendicular direction to the edge

Fig. 36

Relation between maximum spectral acceleration at short periods (i.e., 0.3 to 0.7 s) at zone III and the distance to zone I, associated to normal and subduction seismic events for a parallel and b perpendicular direction to the edge

Fig. 37

Relation between maximum spectral acceleration at long periods (i.e., 1.5 to 2.1 s) at zone III and the distance to zone I, associated to normal and subduction seismic events for a parallel and b perpendicular direction to the edge

6 Conclusions

In abrupt transition zones such as those found at basin edges of valleys, the seismic response is strongly affected by tridimensional effects associated with geologic features at the clay-stiff soil/rock interface. This paper revisits the impact of these effects on the observed damage pattern during the September 2017 Mexico-Puebla earthquake.

From the geotechnical standpoint, this area was characterized by information gathered from 25 exploratory borings, selective sampling recovery, and in-situ testing, including two suspended logging tests and 50 ambient vibration tests. Three-dimensional effects were assessed by comparing one-dimensional and three-dimensional seismic responses from numerical modeling. Strong ground motion variability was computed in the surrounding basin of stiff materials and the high plasticity clays found at the former Xochimilco-Chalco Lake.

A series of three-dimensional finite-difference numerical models were developed to simulate the southern basin edge of the Mexico’s valley. From the numerical results, it was observed that the identified surface cracking after the September 2017 Mexico-Puebla earthquake, in the studied area, was induced by the tensile stresses associated with relative displacements between the clay and the stiff soils zones developed during the ground shaking at the abrupt transition. Regarding acceleration response, three-dimensional effects were minor in sites located in the stiffer soils, having up to 5 to 10% increase due to the three-dimensional impacts in the maximum spectral acceleration for normal and subduction events, respectively, and up to 2 to 5% in the PGA. However, for the soft clay, tridimensional effects lead to an increase of up to 300% of the expected maximum spectral accelerations from the one-dimensional model, mainly for the direction parallel to the edge, which also was observed in the records from the CDAO station located in clay zone near to abrupt transition.

Therefore, there is a complex interaction between incoming wave patterns coming from the bottom of the lake and those coming from the basin lateral boundary. This fact leads to changes in the spectral shape of the computed ground motions at the clay zone, being strongly affected by both the predominant site period, which is, in turn, associated with clay thickness, and the distance from the lateral basin border. Moreover, the ground motion variability computed at the clay zone close to the edge, presents an important amplification of the spectral accelerations at short periods for normal events, induced mainly by the interference of the incoming waves, as mentioned before. This agrees with the observed damage during the Puebla-Mexico City September 2017 earthquake, mainly for structures with a dominant period close to this range, such as earth structures, road infrastructure, and buildings with 2 to 3 stories.