Equivalent Linear Approach Research Articles

Numerical modeling of site response at large strains with simplified nonlinear models: Application to Lotung seismic array

Abstract Site response analyses at large strains are routinely carried out neglecting the shear strength of soil and the stiffness degradation due to the increase in pore pressures, leading to unrealistic predictions of the seismic response of soil deposits. The study investigates the performance of a simplified nonlinear (NL) approach, implemented in the Deepsoil code, constituted by coupling a hyperbolic model incorporating shear strength with a strain-based semi-empirical pore pressure generation model. The first part of the study, based on a large one-dimensional parametric study, shows that above a shear strain of 0.1%, it is necessary to include shear strength in the site response modelling to get more realistic results. Then, the approach has been evaluated with reference to the well-known downhole Large-Scale Seismic Test array located in Lotung (Taiwan): numerical results have been compared with recordings in terms of acceleration response spectra and pore water pressure time histories at different depths along the soil profiles. The comparison shows that the NL simplified model is characterized by an accuracy comparable with more sophisticated advanced elasto-plastic NL analyses adopting essentially the same input data of the traditional equivalent linear approaches(shear modulus and damping curves) and simple physical-mechanical properties routinely determined during geotechnical surveys (i.e., shear strength, relative density, fine content). This approach is therefore recommended for site response analyses reaching large strains (i.e., soft soil deposits and moderate-to-high input motions).

2D equivalent linear ground response analysis for randomized site profiles

AbstractLocal soil conditions play a crucial role in shaping ground surface responses and impacting the intensity of ground shaking. In this study, the influence of different site profiles on computed ground motion was investigated using a 2D equivalent linear analysis approach. Four distinct site profiles: sand, clay, sand-clay-sand-clay, and clay-sand-clay-sand profiles were considered. The results were presented using multiple metrics, including surface acceleration, displacement, modulus decreasing ratio, and coherence analysis. Notably, the clay profile significantly influenced ground motion, while the sand profile exhibited relatively lower seismic activity. This suggests that softer sites significantly influence ground motion, leading to potentially high levels of shaking.

Identification of Ground Response Parameters of Itanagar City, Arunachal Pradesh, India, Using Varying Seismic Intensities and Equivalent Linear Analysis Approach

Identification of Ground Response Parameters of Itanagar City, Arunachal Pradesh, India, Using Varying Seismic Intensities and Equivalent Linear Analysis Approach

Neural network prediction model for site response analysis based on the KiK-net database

Currently, the most widely used site response analysis methods include linear, equivalent linear and nonlinear approaches. However, the results of these analyses gradually deviate from the true values as the site softens or the earthquake intensity increases. To overcome this issue, this study employs a 1D Convolutional Neural Network model to simulate the relationships between the surface and downhole motions. To ensure comprehensive coverage of various sites and earthquake conditions, a careful selection of stations and seismic intensities resulted in 56 stations and 34,538 seismic records from KiK-net. Additionally, 30 seismic intensity measures and three site condition parameters were chosen. The convolutional operation was utilized to consider the correlations between the parameters, enabling the prediction of surface motion intensity measures. The performance of the neural network model was assessed based on residual analysis, and the model’s generalization ability was tested using samples that were not included in the training dataset. The results demonstrate that the neural network can effectively predict surface motion intensity measures. Furthermore, in comparison with the equivalent linear and nonlinear methods that do not rely on site-specific detailed soil data, the 1D Convolutional Neural Network exhibits superior performance and maintains high prediction accuracy even for soft soil and strong earthquake scenarios, confirming the enhanced predictive capability of the proposed model.

Transformation of the Equivalent Linear Method for solving soil-structure interaction problems in the time domain

Transformation of the Equivalent Linear Method for solving soil-structure interaction problems in the time domain

One Dimensional Ground Response analysis in Patan: Implications to Damage Pattern due to the 2015 Mw 7.8 Gorkha Earthquake

The 2015 Mw 7.8 Gorkha earthquake caused extensive damages in Kathmandu Valley located about 78 km SE from the epicenter. The damage patterns in the city clearly indicated the subsurface geology of the city had strongly modified the ground motion causing typical damages to tall structures. In this contribution, following one-dimensional approach, a ground response analysis is performed in Patan utilizing the deep borehole, shear wave profile and dynamic soil properties adopting both equivalent linear and non-linear approaches. The results of both equivalent linear and non-linear analyses were compared with the measured ground motions at soil site. It is found that the non-linear analysis better simulates the deamplification of the peak ground acceleration and strong shaking at longer period than the equivalent linear analysis. The obtained results confirm that the deamplification of PGA was due to the strong non-linear behavior of the fluvio-lacustrine deposits.

Futuristic study on seismic response analysis of North Indian soils– A sustainable approach

India's growing population and extensive unscientific structures put the country at risk of suffering major casualties during earthquakes. To make buildings more earthquake-resistant, and develop safe and durable structures for future, we need seismic sustainability. For this purpose, Cities should adopt ‘seismic microzonation’, which includes Site response analysis (SRA). In this paper, SRA for eight places in Varanasi (India) was performed using the equivalent linear technique subjected to four significantly different strong motions (Bhuj, Uttarkashi, Chamba, and Kobe motion). Geotechnical data from various government and private entities have been gathered for this purpose. Analysis has been conducted using equivalent linear approach in DEEPSOIL software. The results of the study have been formulated in terms of surface time histories, response spectra, Fourier spectra and PGA along the depth. These findings may be applied to dynamic analysis and structural design, as well as the preparation of advanced dynamic lab tests. Furthermore, sites in Varanasi have been found to considerably enhance ground motion, thus a site-specific design approach must be implemented for important constructions.

NC92Soil: A computer code for deterministic and stochastic 1D equivalent linear seismic site response analyses

The extensive evaluation of the impact of local seismo-stratigraphic configurations on seismic ground motion presents significant challenges due to the necessity of considering the combined effects of uncertainty and small-scale lateral variability of the relevant parameters. To effectively explore these sources of uncertainty, a new Python-based computer program is proposed for one-dimensional seismic site response simulations, adopting the equivalent linear viscoelastic approach in the frequency domain. With respect to existing software, the code introduces new pre- and post-processing features, which also meet the specific requirements of seismic microzonation studies. Within the code, the complete spectrum of uncertainties related to local seismo-stratigraphic configurations, including lithotype successions, layer thicknesses, and seismic and geotechnical properties for the considered lithotypes, is managed by considering user-defined constraints and statistical properties of the relevant parameters. Additionally, a batch approach is offered, enabling the application of the procedure to an unlimited number of different scenarios. To demonstrate the potentiality of the proposed code, a comprehensive set of 90,000 local seismic site response analyses was conducted, showing a clear correlation between the amplification factors, the mean shear wave velocity in the upper 30 m, the fundamental frequency of the deposit and the depth to the seismic bedrock.

Dynamic behavior analysis of systems with friction and hysteretic effects

This paper investigates the steady-state response of a nonlinear system that incorporates both a hysteretic element and frictional contact under harmonic excitation. While previous studies have separately explored hysteretic oscillators and frictional systems, the simultaneous consideration of both elements has received limited attention. In this study, an analytical solution for the steady-state response is proposed, employing the equivalent linearization approach. This method enables the determination of equivalent stiffness and equivalent damping coefficients for the linearized system, which are directly linked to the amplitude of the steady-state response. To validate the derived analytical solution, a numerical model is utilized, demonstrating the agreement between the analytical and numerical results. Moreover, this paper identifies noteworthy dynamic characteristics of the designed system, notably the occurrence of peak amplitude at excitation frequencies lower than the pre-yield natural frequency. Furthermore, the apex point can be adjusted by manipulating the nonlinear parameters. Building upon these observations, a new mounting system is examined, revealing distinctive behavior.

Ataköy Gözlem Kuyusunda Saha Tepki Tahminleri

Site response analyses are seen to be the reliable way of reproducing and predicting earthquake input motions. The analyses are generally performed by adopting equivalent linear or nonlinear approaches solving the problem in time or frequency domains. Instrumented geotechnical downhole arrays, in this regard, are very important as to obtaining earthquake data through the soil deposits. This data can eventually be used to verify the approaches developed for site response analyses. In this study, the input motions of the 24.05.2014 (Aegean) earthquake event recorded at relatively recently installed Atakoy geotechnical downhole array are assessed. Moreover, the recorded input motions at the bottom bedrock level of the downhole array are simulated in the East-West and North-South directions. The site response analyses are conducted based on frequency domain equivalent linear approach. The peak ground acceleration and the spectral accelerations of the predicted input motions are compared with the recorded ones at 70 m, 50 m, 25 m and at the ground surface. The results indicate that the spectral acceleration predictions can be simulated well until the depth of 50 m. At 25 m and at ground surface, the predictions are always greater than the recorded one. However, the predictions still exhibits good indication of actual values in the North-South direction. In terms of peak ground acceleration and shear strain profiles, the predictions display the soil layers featured with different soil properties. The equivalent linear approach appears to be suited reasonably well in site response analysis.

Pre and Post Earthquake Slope Stability Analysis Using Equivalent Linear Method – a Case Study of Jure Landslide

Slope stability analysis is one of the most important tools to determine the stability of both natural as well as man-made slopes. The main objective of this study is to compare safety factor of the Jure landslide before and after an earthquake occurrence using Geostudio QUAKE/W module. Three different positions of water table (top, middle and bottom of debris) were considered for the study. The safety factors of both deterministic and probabilistic case are determined. The Gorkha Earthquake's time history data were taken into consideration. Both the bedrock and debris are modeled using the equivalent linear approach for dynamic analysis. The findings indicate that the slope was on the verge of failing prior to the earthquake and is predicted to fail following the earthquake event that was taken into account.

Optimization-oriented exponential-polynomial-closure approach for analyzing nonlinear stochastic oscillators

A novel method named optimization-oriented exponential-polynomial-closure approach is proposed in this article. The main idea of this attempt is to extend the original exponential-polynomial-closure solution procedure methodologically by minimizing the resulted residual error square of the governing equation, which is achieved after an exponential polynomial is adopted as the approximate solution. The objective function for computing the parameters in the approximate solutions of nonlinear random oscillators is then formulated. The probabilistic solutions of the oscillators obtained by the presented approach are verified by the exact solutions in some special cases or by Monte Carlo simulation. Numerical examples indicate that the solutions attained by the presented approach match with the exact or Monte Carlo simulation solutions. The advantage of the presented solution procedure is that it can provide a much more accurate solution than the Equivalent Linearization approach and it is much more efficient than Monte Carlo simulation as demonstrated by the numerical examples.

Nonlinear Dynamic Soil–Foundation–Superstructure Interaction Analysis for a Reactor Building Supported on a Combined Piled–Raft System

There has been a rapid development of the nuclear industry on account of climatic change caused by global warming. The most favorable rocky sites for nuclear power plants are exhausted worldwide, and now nuclear facilities are proposed on soil sites. In a soil site, a combined piled raft foundation (CPRF) system can provide the desired level of performance for a nuclear structure under seismic loading conditions. The combined system of the nuclear structure with the CPRF involves complex dynamic interactions between the pile, soil, raft, and the superstructure. In the existing literature, the dynamic behavior of the CPRF system supporting massive and stiff structures is not well studied and investigated. Due to the highly nonlinear dynamic characteristic of the CPRF system supporting massive and stiff reactor buildings, this paper adopts interface nonlinearities (separation and sliding) through the master and slave concept, concrete and reinforcement material nonlinearities through the concrete damage plasticity model for piles and superstructures, and an equivalent linear approach considering soil degradation and damping with Mohr–Coulomb postyielding hardening behavior for the soil. After the successful validation of the modeling approach with the results of the experimental testing, the different aspects of the CPRF of reactor building are investigated, such as amplification, pile and raft behavior, pile–raft coefficient, separation and sliding, contact pressure, nonlinearity in the piles, and the response of the superstructure. This study indicates that the peripheral piles of the CPRF system could prevent the separation of the raft from the soil. The numerical result shows that damage in the CPRF system occurs near the top portion of the pile. The presented procedure to develop the combined model for the seismic analysis in the time domain could help to formulate guidelines for safety-related nuclear structures resting on the CPRF system.

Maximum aggregate size effects on the evolution of the FPZ and crack extensions in concrete – Experimental and numerical investigation

In this paper, an experimental/numerical investigation is proposed to study the effect of maximum aggregate size (Dmax) on the fracture process zone (FPZ) length variations and the crack extension. Experimentally, three-point bending tests have been realized on plain concrete with different maximum aggregate sizes under crack mouth opening displacement (CMOD) control. The crack growth is monitored using the acoustic emission (AE) technique. The numerical investigation is conducted at a mesoscale level. From the evolution of cohesive tangential stresses along the crack path, the variation of FPZ is explored. The crack extensions are investigated using the equivalent linear elastic fracture mechanics approach (LEFM) to investigate the R-curves. The results show a great influence of the maximum aggregate size both on the FPZ variation and on the global behavior.

The design of nonlinear damped building isolation systems by using mobility analysis

Additional power law nonlinear damping has shown advantages in seismic isolation of buildings. The design of nonlinear damped building isolation systems can be conducted in the frequency domain based on the output frequency response function (OFRF) of the building’s frequency output responses. But this often requires many runs of simulations of nonlinear building systems, which will spend a long time when the finite element (FE) model or differential equation model of the building system is complex. In this study, the issue will be resolved by an integrated mobility analysis and equivalent linearization approach. In this approach, the complex linear building system without additional nonlinear damping is represented by several data-driven autoregressive with exogenous input (ARX) models. Mobilities of the building system can be evaluated from these ARX models, so that a mobility-based frequency domain representation of the linear building system can be achieved. After that, an equivalent linearization approach is applied to simulate the building system with additional nonlinear damping. Finally, the OFRF-based design can be conducted based on the proposed mobility analysis and equivalent linearization approach. A 4-degree of freedom (DoF) building system is discussed to demonstrate the advantages of the proposed method. The results indicate that the new approach can significantly increase the efficiency of the design and be effectively applied to the design of complex nonlinear building isolation systems.

Amplification analysis at Nador city and surrounding area (NE of Morocco)

This study presents a ground response assessment performed at Nador city (Morocco) and surrounding areas. We carried out the one-dimensional equivalent linear approach using the known geotechnical parameters to derive amplification analysis at studied sites, using the shear wave velocity (VS(30)) derived from boreholes. Seismic hazard assessment is performed using two gridded seismicity models and four ground motions attenuation equations, in order to address the epistemic uncertainty in a logic-tree framework. Seismic hazard maps depicting the mean estimated hazard values and including the uncertainty are presented in terms of spectral acceleration at 0.0, 0.1 and 0.2 s, with 10% and 5% probabilities of exceedance in 50 years, and for B, B/C and C soil conditions, as defined by the National Earthquake Hazards Reduction Program classification and for local site conditions as well. Typical values of Peak Ground Acceleration (PGA) equal to 0.18g±0.03 and 0.26g±0.05 are obtained at the Nador city for 475 and 975 return period respectively and for B/C soil conditions. The deaggregation analysis for both returns periods (475 and 975 years) in terms of magnitude-distance (named 2D deaggregation) and in azimuth (longitude and latitude named 3D deaggregation) allows us to derive the contribution of the different couple of magnitude-distance (M,D) and the different sources, the controlling earthquake are also determined at the five considered cities, for both return periods. From the site-specific seismic hazard analysis performed at five main cities located in the studied area, the seismic hazard curves and the uniform hazard spectra (UHS) are obtained with their uncertainties for the considered return periods and soil conditions, as well as the design spectra proposed by Malhotra (2005). Real accelerogram time-history is used to generate compatible acceleration time-history with the design spectra proposed by Malhotra (2005) and the EC-8 type 2 (Eurocode 8.2004), for both return periods and for B/C soil conditions. The lower root mean square error value (RMSE) is archived by using the Malhotra design spectra for the considered return periods. The results display a clear amplification in silty and clayey soils, greater than in silt-gravely soils, as well as no significant amplification in volcanic soils. Moreover, the resulting fundamental frequency is in the range of 1.4–2.8 Hz, and the maximum amplification factor observed in Nador is about 3.5. These values are consistent with the results derived from ambient noise, seismic records using HVSR technique and with the values given by Moroccan Seismic Code in the 2000 and 2011 regulations.

Kinematic Soil–Pile Interaction under Earthquake-Induced Nonlinear Soil and Pile Behavior: An Equivalent-Linear Approach

Kinematic Soil–Pile Interaction under Earthquake-Induced Nonlinear Soil and Pile Behavior: An Equivalent-Linear Approach

Prediction of ground-borne vibrations induced by impact pile driving: experimental validation of an equivalent linear approach

Impact pile driving is one interesting possibility to install deep foundations from the economic and technical point of view. However, it is necessary to ensure that the ambiental drawbacks, namely ground-borne vibrations, are adequately met. In that way, the authors propose an axisymmetric finite element method-perfectly matched layer approach, where the nonlinear behavior of the soil is attended by an equivalent linear methodology. Given the complexity of the problem, an experimental test site was developed and fully characterized. The experimental work comprises in situ and laboratorial soil characterization as well as the measurement of vibrations induced during pile driving. The comparison between experimental and numerical results achieved a very good agreement, making the proposed numerical approach suitable for the prediction of vibrations induced by impact pile driving.

Framework for dynamic response analysis of monopile supported offshore wind turbine excited by combined wind-wave-earthquake loading

This study develops a framework to perform integrated analysis for the seismic response of offshore wind turbines (OWTs). The ground motions are introduced in the analysis as depth-varying excitations that are obtained from free-field site response analysis involving equivalent linearization approach. The calculated accelerations at different elevations are used in a beam on nonlinear Winkler foundation model, which is then used to simulate the pile-soil interaction (PSI). These models are combined with the FAST code to develop an integrated seismic response analysis framework (SRAF) for the system consisting of OWT, along with one-dimensional site response analysis and PSI elements. Considering the NREL 5 MW OWT as an example, the reliability of SRAF is validated by comparisons with simulation results obtained from GL-Bladed and ADINA software. Subsequently, the validated model was used to investigate the influence of three earthquake excitation models on the seismic response of OWTs. We found that the uniform earthquake excitation method significantly underestimates the seismic response of OWT. In addition, the validated model was used to evaluate the seismic responses of OWTs under different operational conditions. The results indicate that the seismic responses of OWTs differ significantly under running, parked and emergency shutdown states.

A computationally efficient algorithm for computing convex hull prices

Electricity markets worldwide allow participants to bid non-convex production offers. While non-convex offers can more accurately reflect a resource’s capabilities, they create challenges for market clearing processes. For example, system operators may be required to execute side payments to participants whose costs are not covered through energy sales as determined via traditional locational marginal pricing schemes. Convex hull pricing minimizes this and other types of side payments while providing uniform (i.e., locationally and temporally consistent) prices. Computing convex hull prices involves solving either a large-scale linear program or the Lagrangian dual of the corresponding non-convex scheduling problem. Further, the former approach requires explicit descriptions of market participants’ convex hulls.While linear programs for computing convex hull prices are large, their structure is naturally decomposable by generators. Here, we propose and empirically analyze a Benders decomposition approach to computing convex hull prices that leverages recent advances in convex hull formulations for thermal generating units. We demonstrate across a large set of test instances that our decomposition approach only requires modest computational effort, obtaining solutions at least an order of magnitude faster than the equivalent large-scale linear programming approach. Overall, we provide a computationally feasible method for computing convex hull prices for industrial scale market clearing problems, enabling the possibility of practical adoption of this advanced pricing mechanism.