With the intent of letting the variability in the basin geometry and the soil profile be the only independent variable of the problem, all the other attributes of the numerical assessments are kept constant. In simpler terms, the same set of software, constitutive model, parameter sets and element sizes along with the seismic input motions, were used for the analyses. Non-linear, finite element-based numerical assessments were performed by using Plaxis 2-D and 3-D software. The Small strain hardening model (HS-small) with Mohr Coulomb failure criterion were used as part of the constitutive model. After a brief discussion of the steps followed in the development of idealized 3-D basin model, the constitutive model will be introduced along with the set of input parameters used as part of the assessments.
2.1 Idealized basin soil profiles
To quantitatively assess the effects of 1-, 2- and 3-D site conditions on the observed amplifications and prolonging in seismic demand, 2- and 3-D basin models were developed based on available site investigation studies. Benefitting from the results of geotechnical and geophysical investigations (Akgun et al. 2011; Pamuk et al. 2017, 2018a, 2018b; Kıncal et al. 2009; Ozdag et al. 2015; Akgun 2012), soil and shear wave velocity profiles were developed for 11 east-west (E-W) and 7 north-south (N-S) cross-sections. The locations of critical sections that form the basis of 1-D, 2D and 3D analyses in the basin are shown in Fig. 4. Figure 5 presents a sample from these geophysical investigations, based on which the 1-, 2- and 3-D models were developed, as given in Fig. 6–8.
As already shown in Fig. 3, five east-west and six north-south sections were analyzed in the 1- and 2-D models. For the 3-D model, the geometry was downsized to contain the soft alluvial soil boundary (model dimensions = 4100 m x 3100 m x 1794 m), where maximum spectral accelerations are expected.
While 1-D models have 461 elements, 2- and 3-D models have 16,684 and 1,799,783 elements, respectively. A personal computer (PC) with core i7-7700HQ CPU at 2800 MHz and 16GB RAM was used for 1- and 2-D models, on which a single set of analysis lasts approximately 1 and 20 hours, respectively. 3-D model was performed with a PC core AMD Ryzen 7 3700X 8-core processor at 4050 MHz, which lasts for about 504 hours (21 days).
2.2 Constitutive model and input parameters
A hardening soil (HS) model with small-strain stiffness (HS-small) is used as available in Plaxis 2-D and 3-D software. HS is an advanced model for simulating the behavior of both soft and stiff, cohesive and cohesionless soils (Schanz et al. 1999). Some basic characteristics of the model are: i) stress-dependent stiffness according to a power law (input parameter m), ii) plastic straining due to primary deviatoric loading (input parameter E50,ref), iii) plastic straining due to primary compression (input parameter Eoed,ref), iv) elastic unloading/reloading (input parameters Eur,ref, νur), v) failure according to the Mohr-Coulomb failure criterion (parameters c, φ and ψ). HS-small in Plaxis software is based on the HS model and uses almost entirely the same parameters. Only two additional parameters are needed to describe the variation of stiffness with strain: i) the initial or very small-strain shear modulus, G0, and ii) the shear strain level (γ0.7) at which the secant shear modulus, GS, is reduced to about 70% of G0. A list of input parameters for HS-small constitutive model are presented in Table 1.
The E-W and N-S accelerograms of 2020 Samos-Aegean Sea Earthquake at SGMS #3514 were deconvolved to the ones at seismic bedrock level, which are located at approximately 1500 to 1800 m depth from the ground surface (Cetin et al. 2022a). These within bedrock accelerograms were used in 1-D, 2-D, and 3-D finite element dynamic response analyses.
As for boundary conditions, for 1-, 2- and 3-D models, “tied-degree of freedoms” and “free-field” lateral boundaries were used, respectively, while the mesh window base boundary conditions were assigned as “compliant base”. The “tied-degree of freedoms”, which is recommended for 1-D site response analysis models, is used to connect nodes on the same elevation at the left and right boundaries so that they are affected by the same horizontal and vertical displacements. The “free-field” lateral boundary elements used in the 2-D and 3-D analyses enable to extent the boundaries to infinity, simulating far-field conditions by applying equivalent normal and shear forces. In the normal and shear directions, at each node of the lateral boundary, 2 dashpots are added to adsorb the reflected secondary waves from the internal structures. The “compliant-base” boundary is a combination of line prescribed displacement and a viscous boundary, and is used at seismic bedrock interface (Plaxis 2D Reference manual 2021; Joyner and Chen 1975). More detailed discussions on the use of boundary elements are available in Plaxis 2D Reference manual (2021), Joyner and Chen (1975) and Amorosi et al. (2010), and will not be repeated herein.
The mesh and element sizes used in finite element models were determined considering the natural period range of the site (i.e.: 0.7–1.4 s). As noted earlier that all input parameters were tried to be kept constant in 1-D, 2-D and 3-D models so that any changes in results can be reasonably attributed to dimensional differences (i.e.: basin geometry).
Table 1
Input parameters for HS-small constitutive model in Plaxis Software
|
Parameter
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Definition
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c
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Cohesion
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ɸ
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Internal friction angle
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y
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Dilation angle
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\({E}_{50}^{ref}\)
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Secant stiffness in standard drained triaxial test
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\({E}_{oed}^{ref}\)
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Tangent stiffness for primary oedometer loading
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\({\text{E}}_{\text{u}\text{r}}^{\text{r}\text{e}\text{f}}\)
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Unloading/ reloading stiffness from drained triaxial test
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\({\text{G}}_{0}^{\text{r}\text{e}\text{f}}\)
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Reference shear modulus at very small strains (e < 10− 6)
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υ0.7
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Poisson's ratio for unloading-reloading
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γ0.7
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Threshold shear strain at which GS = 0.722 G0
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2.3 Validation of site response models
Developed finite element models, the adopted constitutive models and its input parameters, were validated against the results of 1-D total stress-based equivalent linear dynamic analysis performed by DEEPSOIL software. As discussed in the first and the second papers of the trilogy (Cetin et al. 2022a, b), the 1-D DEEPSOIL model was calibrated and validated by using SGMS #3513 and #3514 seismic soil and rock site responses, recorded during 26 historical events including the 2020 Samos earthquake. Moreover, as part of validation process, probability-based analytical procedures were also used. Hence, a well calibrated and validated equivalent linear 1-D model is available, and the developed 3-D model is calibrated and validated against it. For the purpose, 1-D site profile at SGMS #3513 site location is developed from the 3-D model as shown in Fig. 6. This 1-D model was used for the site response assessments performed by Plaxis software. Figure 9 shows the recorded and the resulting elastic response spectra evaluated by equivalent linear and nonlinear (Plaxis 1-D) models. Despite the widely known differences in the adopted constitutive models (equivalent linear vs. nonlinear), the match among the recorded, equivalent linear and nonlinear model responses is judged to be satisfactory and mutually supportive. Hence, the same set of models and their parameters will now be extended to 2-D and 3-D space, and the analyses results will be jointly used to assess the basin geometry effects.
2.4 Quantification of 1-, 2- and 3-D site effects on seismic demand
The estimated overall spectral amplification, \(A\left(T\right)\), defined as the ratio of 3-D (or 2-D) soil basin spectral accelerations normalized by reference rock spectral accelerations, is decomposed into the product of two amplification factors, aiming to quantify site amplifications due to soil stratigraphy and 3-D (or 2-D) basin geometry, as given in Eq. 1. The first of these amplification factors, \({A}_{1D, Rock-Soil}\left(T\right)\), represents the spectral amplification of a 1-D soil columns as compared to a reference rock column (site). Hence, it is defined consistently, as presented in Eq. 2. The latter of the two amplification factors, \({A}_{Soil,1D-3D or 2D}\left(T\right)\), simply represents the spectral soil amplifications estimated by a 3-D (or 2-D) normalized by a 1-D soil stratigraphy model (i.e.: 1-D soil columns vs. 3-D (or 2-D) soil basin). Similarly, it is given in Eq. 3.
|
\({{A\left(T\right)}_{Rock-3D\left(or 1D or 2D\right) Soil}=\frac{{S}_{A, 3D\left(or 1D or 2D\right) Soil }\left(T\right)}{{S}_{A,Rock }\left(T\right)}=A}_{Rock-1D Soil}\left(T\right)\bullet {A}_{Soil,1D-3D}\left(T\right)\) | (1) |
\({A}_{Rock-1D Soil}\left(T\right)=\frac{{S}_{A,1D Soil }\left(T\right)}{{S}_{A,Rock }\left(T\right)}\) | (2) |
\({A}_{Soil,1D-3D\left(or 2D\right)}\left(T\right)=\frac{{S}_{A, 3D\left(or 2D\right) Soil }\left(T\right)}{{S}_{A,1D Soil }\left(T\right)}\) | (3) |
Consistent with these definitions, as given in Table 2, \({A\left(T\right)}_{Rock-3D\left(or 1D or 2D\right) Soil}\) values were presented for spectral periods, \(T\), of 0.0, 0.85, 1.0 and 1.4 seconds. As stated earlier, for the normalization, reference rock site was adopted as SGMS #3514 site. Note that \({A\left(T\right)}_{Rock-3D\left(or 1D or 2D\right) Soil}\) values jointly represent the amplifications due to soil stratigraphy and multi-dimensional basin geometry.
As illustrated by the figures presented in Table 2, peak ground accelerations (PGA’s) were assessed to be amplified by a factor of 1.5-3.0, as compared to the “rock” PGA level which was recorded at SGMS #3514. PGA amplification contour maps were observed to be significantly different in 1-D, 2-D and 3-D models. PGA amplification predictions by 1-D models suggest that PGA levels were mostly amplified at the northern and southern edges of the basin, whereas 3-D model suggests maximum PGA response occurring at the deepest part of the alluvial basin, confined by the sections N1-3 and E4-6. Similar non-consistent amplification predictions by 1-D and 2-D and 3-D models were also observed at spectral period of 1.0 second, and other periods are no exception. These inconsistent amplification predictions by 1-D, 2-D and 3-D models suggest that simplifications of a 3-D basin geometry through 1-D or even 2-D idealized models may not guarantee consistent predictions with the ones by 3-D models. It should be noted once again that in all three model sets (i.e.: 1-D, 2-D and 3-D models), exactly the same set of soil profiles and parameters were used, and the differences in predictions are attributed to only basin geometry-induced effects.
The highest amplification factors reaching 5 to 6 range, were observed at the spectral periods of 0.85 to 1 second range. The highest amplified region (shaded by warmer colors) correlates well with the scatter of moderate to highly damaged buildings, shown with yellow dots. Note that as discussed elsewhere (Cetin et al. 2022a; Cetin et al. 2021), majority of these moderate to highly damaged buildings were 7–9 story residential buildings with natural periods falling into the spectral period range of 0.7–1.1 seconds. Their natural periods coincide with most-amplified spectral periods suggesting a resonating soil-structure response. Hence, it can be concluded that increased seismic demand levels due to site effects (including both soil site effects and basin geometry) could be listed as one of the major contributors to the observed overall structural damage, along with other structural design and construction factors discussed elsewhere (Yakut et al. 2021). The correlation between the amplified seismic demand and the structural damage scatter is shown to be more pronounced in the N-S direction than E-W direction. This observation is not surprising since Bayrakli basin is two-way confined by Mounts Yamanlar and Nif in the N-S direction. In the E-W direction, the basin is only confined in the East, and opens to Izmir Bay in the West (i.e.: one way confined in the E-W direction). Please revisit Fig. 2 for visual inspection of topographical confinement of the basin. Consistently, induced seismic waves were trapped more inside the basin in the N-S direction, producing higher levels of N-S amplifications.
With the intent of identifying 3-D basin geometry-induced amplifications only, \({A\left(T\right)}_{Soil,1D-3D\left(or 2D\right)}\) values were assessed after normalizing 3-D basin model results with those of 1-D model. Table 3 presents these amplification factors.
After studying Table 3, it can be concluded that 3-D basin geometry did not change PGA levels significantly; moreover, PGA levels can be shown to be slightly de-amplified in 3-D models at the edges of the basin as compared to 1-D model predictions. This observation clearly supports the reconnaissance findings documenting lack of major retaining structure failures after Samos event, as opposed to significant number of residential building damages observed in Bayrakli. Note that the seismic performance of retaining structures is governed by PGA levels as opposed to those of elastic response spectra.
3-D basin geometry induced amplification factors reach a maximum of 1.40 and 1.60 in the E-W and N-S directions, respectively. Again, geometric amplifications were more pronounced in the N-S direction for the same reasons discussed earlier. Moreover, geometric amplification factors reach their highest value at spectral period range of 0.7-1.0 seconds. Hence, both the soil stratigraphy- and basin geometry-induced amplifications reach their maximum values at the period range of mostly damaged 7–10 story residential buildings of Bayrakli. It should be noted that these basin amplification factors are event specific due to nonlinear nature of basin soil response.