Equivalent Linear Approach Research Articles

Development of a 3D fluid-saturated element for dynamic analysis of two-phase media in ABAQUS based on u-U formed equations

The commonly used finite element software does not provide a three-dimensional (3D) fluid-saturated element available for dynamic analysis of two-phase media, which extremely limits their advantages in solving complicated engineering problems. This paper develops a user element for 3D dynamic analysis of fluid-saturated porous media in ABAQUS based on the u-U formed equations. The dynamic governing formulations are firstly revisited, and an implicit time integration algorithm is adopted to solve the dynamic equations. Then, the major development steps, the structure of the subroutine file, and the keywords directly related to the user element in the input file are carefully presented for the application of the user element. Moreover, the equivalent linear approach is introduced with a detailed realization procedure to consider the soil nonlinearity. Finally, the effectiveness and accuracy of the developed user element are validated by comparing the results with those from four different analytical and numerical solutions. The proposed numerical method with user element has a great potential in modeling practical engineering problems due to the prominent advantages of ABAQUS in simulating complicated materials and irregular geometrics.

Simplified Criteria to Select Ground Response Analysis Methods for Seismic Building Design: Equivalent Linear versus Nonlinear Approaches

ABSTRACT The possible amplification of seismic waves in soil deposits is crucial for the seismic design of buildings and geotechnical systems. The most common approaches for the numerical simulation of seismic site response are the equivalent linear (EQL) and the nonlinear (NL). Even though their advantages and limitations have been investigated in several studies, the relative field of applicability is still under debate. This study tested both methods over a wide population of soil models, which were subjected to a set of acceleration time histories recorded from strong earthquakes. A thorough comparison of the results of the EQL and the NL approaches was carried out, to identify the conditions in which the relative differences are significant. This assessment allowed for the definition of simplified criteria to predict when the two schemes are or are not compatible for large expected shaking levels. The proposed criteria are based on simple and intuitive parameters describing the soil deposit and the ground-motion parameters, which can be predicted straightforwardly. Therefore, this study provides a scheme for the choice between the EQL and the NL approaches that can be used even at the preliminary design stages. It appears that the EQL approach provides reliable amplification estimates in soil deposits with thickness up to 30 m, except for very deformable soils, but this depth range may be extended at long vibration periods. This result reveals a good level of reliability of the EQL approach for various soil conditions encountered in common applications, even for high-intensity shaking.

Ground-borne noise and vibrations in buildings induced by pile driving: An integrated approach

The use of pile driving techniques in urban environments demands accurate methods to predict and mitigate the potential discomfort of residents of neighboring buildings. In this context, a numerical model to predict ground-borne noise and vibrations in buildings induced by pile driving is proposed in the present paper. The model is based on a sub-structuring approach, in which the entire propagation medium is considered, from the vibration source to the receiver.The pile-ground domain is simulated by an equivalent linear FEM-PML approach formulated under axisymmetric conditions taking into account the pile-hammer interaction. The building simulation is performed using a 3D FEM approach, adequately adapted to include the soil-structure interaction, while the acoustic field induced by the structural vibration is analysed using a 3D MFS model. The levels of ground-borne noise and vibration inside the building resulting from pile driving are assessed through an application example, clearly showing the potential of the proposed methodology.

A simplified procedure for predicting the soil amplification factor based on the strain compatible period

This paper proposed a simplified procedure for predicting the amplification factor based on a new empirical model for estimating the site period considering nonlinear strain range of soil. The analysis results suggested the depth to bedrock of the soil profile affects the site period at certain shaking intensity levels. Considering this finding, we first proposed a new empirical model to estimate the site period by propagating 78 input motions through 119 soil profiles. A statistical method confirmed the accuracy of the proposed empirical method, which represents the effect of both dynamic properties of soil and input motion features on site response. We then presented a simplified procedure for predicting the amplification factor by taking to account the site period shift and soil nonlinear behavior due to the interaction of damping increase and the resonance controlling effect. Finally, the accuracy of the simplified procedure results in predicting the amplification factor was evaluated by considering the equivalent linear and nonlinear approaches. The results revealed that the simplified model captures the trends of equivalent linear site response analyze in predicting the amplification factor with acceptable accuracy.

Micro-seismic hazard assessment of Ahmedabad city, Gujarat (Western India) through near-surface characterization/soil modeling

The micro-seismic hazard estimation including quantification of the ground motion amplification has been conducted at Ahmedabad city based on near-surface characterization/soil modeling. The city has experienced substantial damage in the course of the 2001 Bhuj earthquake. A total of 20 boreholes were drilled in the city up to the depths of 40–80 m. A five-fold methodology is adopted: (1) Assessment of the seismic perspective of the area under study, (2) demarcation of the engineering bed layer (EBL) through geophysical (seismic) surveys and the soil properties, (3) soil modeling using geotechnical and the geophysical parameters, (4) assessment of the strong ground motion at EBL through simulation considering far-field earthquake scenarios and near-field earthquakes scenario and (5) surface strong-motion estimation by ground response analysis based on equivalent-linear approach. The near-surface soil models were prepared from the borehole logs, shear-wave velocity estimated from the seismic survey and the soil properties like soil classification and density. The strong motion at EBL is computed by simulating seismotectonically justified scenario earthquakes through the stochastic finite-fault source modeling technique using the region-specific input parameters. The surface-strong motion is estimated by performing ground response analysis (with SHAKE) at every borehole using EBL-strong motion and prepared soil models. The EBL was found varying from 28 to 54 m in depth in Ahmedabad city. The effect of far-field and near-field earthquake sources was considered for assessing the hazard. To compensate for the uncertainty, a total of 108 and 81 input parametric combinations for near-field earthquake scenarios, and far-field earthquake scenarios, respectively have been considered for estimating the strong motion at EBL. The peak ground acceleration (PGA) of 52–111 cm/s2 and 108 cm/s2 are estimated at EBL due to near-field earthquake scenarios and far-field earthquake scenarios, respectively. The PGA through ground response analysis at surface level is found to be varying from 101 to 279 cm/s2 for near-field earthquake scenarios and, 118–161 cm/s2 for the far-field earthquake scenarios. The spectral acceleration (SA) (at surface level) has also been calculated for damping of 5%. The average SA distribution maps for 0.2 s (1–2 story), 0.55 s (4–5 story), 1 s (high rise) and 1.25 s period (large structures) have been prepared for both types of scenario earthquakes. The strong motion amplification is computed to be in the range of 1.6–3.3 for near-field earthquake scenarios and 2.2–3.0 for near-field earthquake scenarios.

An iterative equivalent linearization approach for stochastic sensitivity analysis of hysteretic systems under seismic excitations based on explicit time-domain method

The sensitivity analysis of hysteretic systems under nonstationary random excitations is of great concern to the stochastic optimal design and control of structures. The equivalent linear equation of motion is first constructed for the hysteretic system by the equivalent linearization method (ELM), and the sensitivity equation of the equivalent linear system is then derived by the direct differentiation technique. These two equations are further combined into an overall equivalent linear equation, which depends on the statistical moments and sensitivities of responses of the system and should be solved on an iterative basis. The analysis problem is thus transformed to a series of nonstationary random vibration analyses of the iterative linearized systems, which can be solved with the explicit time-domain method (ETDM) recently proposed for linear random vibration problems. The ETDM has high computational efficiency owing to the explicit formulations of statistical moments of responses. Therefore, a numerical algorithm is developed by combining ELM and ETDM for efficient stochastic sensitivity analysis of hysteretic systems. A 100-degree-of-freedom hysteretic system is investigated to illustrate the accuracy and efficiency of the proposed method, and a 10-degree-of-freedom base-isolated system is further analyzed to show the feasibility of the proposed method for stochastic structural optimization.

Design Response Spectra and Site Coefficients for Various Seismic Site Classes of Guwahati, India, Based on Extensive Ground Response Analyses

Seismicity of northeast India and its surroundings is governed by complex tectonic setting. Guwahati, the largest city of northeast India, had suffered minor to major damages during past earthquakes (EQs). Known seismic source information also suggests that Guwahati is susceptible for future EQs both from nearby as well as distance source. Taking into account the subsoil information of Guwahati, present work attempts to understand the role of Guwahati soil during probable seismic excitation. In the absence of regional ground motion records, covering a possible variation in ground motion characteristics, which can occur during potential future EQ, 30 globally recorded ground motions are used as input motions in this work. Each input motion is applied to each of the 225 boreholes resulting in 225 × 30 = 6750 ground response analyses using equivalent linear approach. Based on the response of subsoil to each input motion, site coefficients as well as average response spectra for various seismic site classes are proposed. It has to be mentioned here that in comparison to previous works which attempted to understand the subsoil of Guwahati based on specific location-based subsoil information and specific scenario EQ, present work provides a broader picture of soil response which can be applied to any level of ground shaking. Further, based on predominant period, suggestion of maximum building storeys are suggested for Guwahati. In addition, based on the work, characteristics of EQ scenario, which can trigger maximum response from local soil of Guwahati, are assessed. Based on the present analyses, design response spectra of Guwahati are also proposed. Keeping in mind Government of India’s policy towards the development of northeast India and further to develop Guwahati as “Smart city”, the present findings will be very useful in city planning, estimation of induced effects as well as for seismic design of probable infrastructure in Guwahati.

A Site-Specific Response Analysis: A Case Study in Hanoi, Vietnam

A series of one-dimensional (1-D) site response analyses were performed using the nonlinear (NL) and equivalent linear (EQL) approaches to assess the applicability of the Vietnamese earthquake-resistance design code TCVN 9386: 2012. Six soil profiles were selected from three districts in Hanoi (Vietnam). A number of ground motions compatible with the rock design spectrum were used as input for carrying out analyses. The results highlight that the calculated response is higher than the design spectrum for site class C and lower for site class D. The normalized response spectra of the EQL approach results are higher than those of the NL approach. Moreover, the peak ground accelerations at the surface from EQL analyses are greater than those of the NL method because the latter generates a higher amount of nonlinearity. The results from the NL approach also illustrate that the deamplification phenomenon occurs in the soft soils of the Hanoi region (e.g., soil profile P3 and P5 of site class D). Additionally, the shear strains calculated from the NL method are closely matched with those from the EQL method, the difference between them increasing with a decrease in soil stiffness.

Analysis of helical soil-nailed walls under static and seismic conditions

The present study investigates the behaviour of helical soil-nailed wall in a dry cohesionless medium under static and seismic conditions. Initially, results from laboratory pullout tests are used to develop a pullout capacity equation, which is subsequently used for stability analysis of helical soil-nailed wall. A detailed parametric study is conducted to evaluate the effect of angle of internal friction of soil, nail inclination, vertical spacing of nails, number of nails, helix size, number of helices, and the face angle on the stability of the soil-nailed wall. Results from the present method are compared and validated with similar existing methods available in the literature. The results suggest that for the given input parameters, the factor of safety (FoS) values from the present method are lower than the pseudo-static and pseudo-dynamic values. Further, the study clearly highlights the significance of input excitation frequency on the FoS of helical soil-nailed walls. In addition, the effects of strain-dependent dynamic properties (shear modulus and damping ratio) on the stability of helical soil-nailed walls are studied using the newly proposed equivalent linear analysis approach. The results computed from the proposed linear and equivalent linear analysis are also compared and discussed in detail.

3D seismic responses of a long lined tunnel in layered poro-viscoelastic half-space by a hybrid FE-BE method

This paper aims to study the seismic responses of a long lined tunnel embedded in a water-saturated poro-viscoelastic half-space, with focus on wave passage effects. The poro-viscoelastic half-space is modeled by the Biot's theory and the soil nonlinearity is considered via an equivalent linear approach. The seismic responses of the tunnel structure are simulated by a 2.5D hybrid FE-BE method. A total of 72 cases of different incident angles for plane SV- and P1-waves are simulated, in order to conduct a systematic study on the tunnel seismic responses, in terms of longitudinal internal forces and strains, as well as seismic pore pressure and soil stresses. It is shown that the spatial variation of earthquake ground motion due to wave passage effects results in considerable longitudinal internal forces of the tunnel lining, which cannot be overlooked for seismic design. The responses of the tunnel depend on both the horizontal and vertical incident angles of the seismic waves. It is also shown that the 2D scenario with perpendicularly incident SV-waves, which is commonly considered in engineering practice, may lead to nonconservative seismic design, especially for tunnels in saturated soil.

Site Effects of Onna during the 2009 L’Aquila (Central Italy) Seismic Sequence: Constraints on Bedrock Depth and 1D Local Velocity Structure from Aftershock Seismograms

ABSTRACTAfter the 2009 L’Aquila Mw 6.1 earthquake, particular attention was paid to the large difference of Mercalli–Cancani–Sieberg (MCS) macroseismic intensity between the nearby villages of Onna (9.5 MCS) and Monticchio (6 MCS). Several authors estimated that in Onna, settled in the Aterno river valley, ground motions were amplified at 2–3 Hz by up to a factor of 5 with respect to Monticchio, settled on more competent rocks. Although there was a general agreement that the spectral peak was caused by the resonance of the uppermost 40 m layer, a satisfactory fit of the amplitudes was not provided. Here, we apply spectral ratio techniques to 1437 aftershock seismograms (magnitude between 1.8 and 3.9) to compare ground motions within Onna and between Onna and Monticchio. Spectral amplitudes at stations located outside and inside the “red zone” of Onna show that the seismic response was uniform, confirming that vulnerability was crucial for the heavier damage of the ancient part of the village. We have also estimated the empirical transfer function of Onna through the spectral ratios between Onna and Monticchio. Although in a 1D simplification, a model with a further velocity contrast of ∼2 at 200 m of depth produces a more accurate fit of observations. Using the new velocity profile, we modeled the mainshock ground motion at Onna in an equivalent-linear approach. Accelerations are amplified by a factor of 2 and spectral ordinates increase from 0.7g at 0.2 s to 1g at 0.5 s, a shaking level that can be destructive for nonductile ancient buildings. This study shows that accurate estimates of empirical transfer functions, even in a simplified 1D approach, provide useful constraints to the deeper velocity structure where measurements are shallow or lacking.

Site-specific ground response analysis at a site in the affected area of the 2016 Pidie Jaya earthquake

Analytical models have demonstrated that they are able to simulate reasonably well the seismic motions at the ground level. The most widely used model is the equivalent linear approach. This equivalent linear model was used to compute the free-field response of Meureudu-Pidie Jaya, Aceh Indonesia’s soft soils during the 2016 Pidie Jaya earthquake. The model computes the ground response of horizontally layered soil deposits subjected to transient and vertically propagating shear waves through the one-dimensional soil column. Each soil layer is assumed to be homogeneous, visco-elastic and infinite in the horizontal extent. The equivalent linear estimation of soil properties is taken to express the nonlinearity of the soil’s shear modulus and damping values. These values are assumed to be a function of shear strain amplitude and determined by an iterative process that must be consistent with the level of the effective strain induced in each sub-layer. Starting with the highest shear modulus and a low damping value, the shear modulus and the damping ratio of each sub-layer are modified. The modification is based on the applicable relationship between both properties and the shear strain. The calculation is repeated until strain-compatible modulus and damping values converge within a tolerance of 1%. This research reveals the ground motions of Pidie Jaya’s soils. The results of the analysis are presented.

Comparison of 1-D seismic site response analysis tools for layered liquefiable deposits at an offshore windfarm site

Various offshore wind farms have been proposed in the Taiwan strait with a long-term target of installing 4.2 gigawatts by 2030. The proposed projects will be in areas with various known faults and areal seismic sources which should be accounted for in design. A reliable prediction of site response for soil deposits is crucial for seismic loading evaluation of existing energy structures and for design of new structures including offshore wind farms in seismically active regions. This paper presents generic results of 1-D site response analyses based on work performed for an offshore wind power plant development site in the Taiwan strait. The deposits in the project area generally consist of layered deposits with liquefiable layers. The site response analyses were initially performed using two different open-source tools, DEEPSOIL and Cyclic 1D. Both equivalent linear and non-linear approaches were adopted for the analyses and additional evaluations were subsequently performed using PLAXIS 2D for comparison with the results from the open source tools. The results from the different tools were systematically compared and provided useful insight on peak ground (seabed) acceleration, acceleration time histories and shear strains at specific depths and design response spectra. The paper includes a discussion of the sensitivity of the outputs to various input parameters for each of the tools utilized in the analyses and the suitability and limitations of each approach for assessing liquefaction potential are also discussed.

Study on the stiffness degradation and damping of pile foundations under dynamic loadings

This paper presents a comprehensive study on the different behaviour of batter and vertical pile foundations in terms of stiffness degradation and damping properties under dynamic loadings. Firstly, dynamic centrifuge tests were carried out and the variations of translational/rotational stiffness and the associated damping properties were identified from a series of hysteretic loops. Results show that for the rocking behaviour, the presence of batter piles causes a small decrease of the rotational stiffness and a great increase of the rotational damping; for the horizontal translation behaviour, batter pile foundation has much higher horizontal stiffness and translational damping compared to the vertical foundation. Then, a set of stiffness degradation and damping curves for the translational and rotational behaviour of batter and vertical pile foundations are proposed. Finally, the proposed stiffness degradation and damping curves were validated numerically by using equivalent linear approach. The numerical validation shows a good agreement with the experimental results.

Steady-state dynamic response of various hysteretic systems endowed with fractional derivative elements

In this paper, the steady-state dynamic response of hysteretic oscillators comprising fractional derivative elements and subjected to harmonic excitation is examined. Notably, this problem may arise in several circumstances, as for instance, when structures which inherently exhibit hysteretic behavior are supplemented with dampers or isolators often modeled by employing fractional terms. The amplitude of the steady-state response is determined analytically by using an equivalent linearization approach. The procedure yields an equivalent linear system with stiffness and damping coefficients which are related to the amplitude of the response, but also, to the order of the fractional derivative. Various models of hysteresis, well established in the literature, are considered. Specifically, applications to oscillators with bilinear, Bouc–Wen, and Preisach hysteretic models are reported, and related parameter studies are presented. The derived results are juxtaposed with pertinent numerical data obtained by integrating the original nonlinear fractional order equation of motion. The analytical and the numerical results are found in good agreement, establishing, thus, the accuracy and efficiency of the considered approach.

Linear Data-Model Based Adaptive ILC for Freeway Ramp Metering without Identical Conditions on Initial States and Reference Trajectory

Although freeway traffic system is conducted with a repeatable pattern day-to-day, the initial volume/or speed and the desired density of the traffic flow may vary with days due to the external disturbances. In this paper, a new linear data-model based adaptive ILC (LDM-AILC) is proposed to address ramp metering in a macroscopic level freeway environment. A linear data-model is developed for the nonlinear macroscopic traffic flow model by introducing an equivalent dynamical linearization approach in the time domain. Then the LDM-AILC is designed with a feedback control law and a parameter updating law. The proposed scheme is data-driven intrinsically, where only the I/O data is required for the controller design and analysis. The convergence is shown by rigorous analysis without any identical conditions exposed on both the initial state and the reference trajectory. Extensive simulation results are provided to verify the effectiveness of the proposed LDM-AILC.

2D nonlinear inversion of bedrock motion from the surface motion of a layered half-space

This paper proposes a method for two-dimensional (2D) nonlinear inversion of bedrock earthquake motion from the surface motion of a layered half-space. The exact dynamic-stiffness matrices of soil layer and half-space are used for the inversion of bedrock motion and incident angle from both the surface horizontal motion and vertical motion, and the equivalent linearization approach is used to consider the nonlinearity of the soil layers. The method of inversion is verified by an example of obliquely incident El Centro Earthquake record at bedrock. It is shown that, the bedrock motion and incident angle can be accurately inversed from both the horizontal motion and vertical motion. The one-dimensional (1D) traditional inversion of bedrock horizontal motion from only surface horizontal motion in literatures may cause large error due to ignoring wave mode conversion, because there is vertical motion accompanying two surface horizontal motions in real earthquake records anyway. The error in 1D inversion depends on the ratio of surface vertical motion to horizontal motion, and the larger the surface vertical motion, the larger the error.

Seismic isolation of surface foundations exploiting the properties of natural liquefiable soil

A novel design method has recently been proposed for the seismic protection of structures on liquefied ground using shallow (instead of deep) foundations. Contrary to conventional structural isolation approaches, which employ special mechanical devices, the proposed geotechnical means exploits the presence of natural liquefiable soil, after partial remediation of the surface ground, as a natural base isolation system which de-amplifies the seismic ground motion and, hence, reduces the seismic demand on the superstructure. This paper focuses on the comparative evaluation of the relevant Soil-Foundation-Structure Interaction (SFSI) effects. Using an equivalent-linear approach based on appropriate values for the material properties of liquefied soil, the dynamic stiffness and damping of rigid square footings on three-layer liquefiable soil under external harmonic oscillations is first numerically investigated. Results demonstrate that for common soil, foundation and seismic excitation conditions, liquefaction leads to (a) significant reduction in dynamic stiffness and (b) increase in damping of the footing over pre-liquefied conditions. Based on these results, regression formulae for estimating static stiffness of surface footings on liquefied soil were developed. In the second part of the paper, parametric numerical analyses are presented for the typical case of bridge piers on liquefiable soil, with surface foundation and a remediated surface crust. Results from both harmonic steady-state and transient analyses indicate that the effect of soil liquefaction on the vibrational characteristics of the pier-foundation system decreases drastically with increasing soil crust thickness and, consequently, the intended natural base isolation of the structural system is mainly achieved by the reduction in free-field seismic ground response.

A 2.5D equivalent linear model for longitudinal seismic analysis of tunnels in water-saturated poroelastic half-space

This paper provides an efficient and rigorous model for 3D seismic analysis of long lined tunnels considering wave passage effects. By taking advantages of the 2.5D model, computation effort is considerably reduced. The water-saturated poroelastic half-space is modelled as two-phase media and soil nonlinearity is considered via the equivalent linear approach. This improved model is realized by a finite element-indirect boundary element (FE-IBE) coupling method. The study reveals that the wave passage effect causes significant longitudinal deformation of the tunnel, and differences in the tunnel seismic responses between the two-phase half-space case and the single-phase half-space case are not negligible.

SEISMIC RESPONSE ANALYSIS ON EARTHQUAKE MOTION AT ZAFARANA WIND TURBINE FARM, EGYPT

Zafarana wind turbine towers farm, which is located 120 km south of Suez on the Red Sea, lies in not only the highest wind speed zones but also the most seismically active areas. Al-Aqaba 1995 earthquake acceleration time history, which recorded at the Eilat station, used at the bedrock level of this research. The peak ground acceleration (PGA) of the horizontal component was almost about 0.10g at the ground surface. The effect of local geologic and soil conditions on the intensity of the ground shaking discussed in this paper. Severe damage occurred due to the amplification of the earthquake motion because the site response analysis considered in the seismic response analysis. Subsurface geotechnical and geophysical (down-hole) data in eight different sites in Zafarana wind turbine farm that carried out used to assess the site response analysis on earthquake ground motion in the studied area. According to the shear wave velocities obtained from the field tests, the classification of soil at Zafarana wind turbines farm was Soil Class B and Soil Class C according to the International Building Code (IBC). Thus, the ground response analyses conducted considering the nonlinear behavior of the soil deposits using both equivalents linear and nonlinear methods. The 1-D nonlinear approach, included in DEEPSOIL v7.0, used to determine the nonlinear soil properties on seismic wave propagation through the soil column and compared to those from the equivalent linear approach, which analyzed using SHAKE 2000. The results comparison showed the same shape of spectral acceleration versus period curves for both equivalent and nonlinear analyses, and the peaks of spectral accelerations in the period range of (0.1_0.50) s. The results showed that the nonlinear amplification factors were more than the equivalent method especially in zones that classified as Soil Class C; however, the amplification of the ground motion was approximately the same for both approaches in zones that classified as Soil Class B. Finally, this paper concludes that the nonlinear analysis should be considered as the soil is softer and the equivalent linear analysis can be used at the zones that classified as Soil Class B or stiffer.