Equivalent Linear Approach Research Articles

Nonlinear lateral stiffness and bearing capacity of suction caissons for offshore wind-turbines

The paper investigates the linear-elastic and nonlinear stiffnesses of a suction caisson used as monopod foundation for an Offshore Wind-Turbine (OWT). Starting from caissons at low working stresses, in which case the linear elastic theory provides an adequate engineering model for soil, analytical expressions for the elastic stiffness matrix of a flexible skirted foundation are proposed and validated. To account for the nonlinear foundation response, the paper proposes a simplified equivalent linear iterative approach where the effective foundation stiffness is expressed in terms of deformation amplitude. To this end, utilizing results from a 3D finite element parametric study, non-dimensional charts have been produced for caissons ranging from perfectly rigid to flexible with variable embedment ratios. To deal with uncertainty on the conditions at the soil-skirt interface, three idealized interface scenarios – “fully-bonded”, “tensionless”, “frictionless” – are implemented. Reduced values of foundation stiffness are computed for a frictionless contact. On the contrary, the impact of a “tensionless” interface, whilst trivial in elastic problems, is intensified with progressing soil inelasticity resulting in severely reduced stiffnesses and capacities. Moreover, with increasing relative skirt flexibility, the elastic stiffnesses of deep suction caissons tend to recede substantially, but the rate of stiffness degradation is fairly attenuated.

Establishing Outcrop Time Series at Various Depths in a Nonlinear Soil Column: An Iterative Approach

Seismic soil-structure interaction (SSI) analysis of nuclear facilities is an important consideration during design and retrofit. SSI tools used in the nuclear industry are currently based on an equivalent linear (EL) approach. Procedures for developing input ground motion for EL approaches are well established. However, the procedures for establishing input ground motion for nonlinear soil-structure interaction (NLSSI) analysis of nuclear facilities are not well established. A collaborative research group at Idaho National Laboratory has recently developed analytical methods and numerical tools for using NLSSI analysis for nuclear facility seismic calculations. NLSSI analysis for a nuclear facility allows for calculation of seismic wave motion through a near-field soil domain using either (a) vertically propagating shear and compressive waves, which is the current industry practice, or (b) a three-dimensional nonvertical wave field. This technical note presents an iterative procedure for establishing outcrop motion at a depth in the soil column for NLSSI analysis that uses vertically propagating shear waves. The approach presented in this technical note starts with a known ground motion at the surface that is deconvolved to a depth, and then the obtained motion is convolved up to a different desired location of input for the NLSSI model. To demonstrate the validity of the approach, a finite element soil column that is representative of a nuclear facility site in the United States is used to produce compatible outcrop seismic time series for reduced nonlinear soil mesh depths. The developed approach for reducing the nonlinear soil column model depth is a two-step iterative method. The first step is establishing an outcrop time series at the lowest depth considered that produces the top-of-soil response spectrum of an actual recorded ground motion. The second step is providing compatible outcrop time series at a shallower depth based on the information from the first step. A comparison of the 5% damped response spectrum from the resulting acceleration time series based on the iterated outcrop motions and the original acceleration time series is conducted. The study shows that the proposed iterative approach produced comparable results within 1% range of the original recorded time series results when sufficient iterations were performed.

Investigation of Global Equivalent Damping and Statistical Relationship of Displacement between Nonlinear Static and Dynamic Analysis of Reinforced Concrete Frame Structures

The assessment of the seismic performance of reinforced concrete (RC) frame structures using the equivalent linearization approach requires comprehensive insight into the nonlinear response of the system, and most previous researches focused on the analysis of a single-degree-of-freedom (SDOF) system. To describe the hysteretic behavior of a multi-degree-of-freedom (MDOF) system accurately, monotonic and cyclic pushover analyses for 88 RC frames structures with various configurations and design parameters are carried out and a unified hysteresis loop expression modeling the cyclic pushover results of RC frame system is developed. Then, a global equivalent damping based on Jacobsen's approach is derived, and comparisons between the displacements obtained by nonlinear static analysis (NSA) utilizing the derived global equivalent damping and those obtained by nonlinear time history analysis (NTHA) are made. Finally, a modified global equivalent damping is presented by calibrating the derived Jacobsen's equivalent damping through NTHA results. Based on the modified equivalent damping, the statistical analysis of the ratios of the results obtained by NTHA to those obtained by NSA is implemented to predicate the probabilistic seismic displacement demands of RC frame structures.

Analysis of seismic ground response caused during strong earthquake in Northern Thailand

This paper presents a site-specific analysis of ground response during the Tarlay Earthquake on March 24, 2011 in Northern Thailand. In this study, the NGA (Next Generation Attenuation) models were selected to predict ground motions due to the earthquake event. The equivalent linear and non-linear approaches were employed in the one-dimensional ground response analysis. Furthermore, the spectral responses produced by the equivalent linear and non-linear approaches were compared with the seismic design code of Thailand. The results showed that the ground motion from the NGA models agree with the strong motion parameters of Tarlay Earthquake. Peak ground acceleration (PGA) at ground surface obtained from both equivalent linear and non-linear approaches certainly results in the high amplification factor. In general, the study results could bring an attention to the local engineer to consider the seismic design value for Northern Thailand, particularly if the stronger earthquake happens in the future.

Stochastic analysis of seismic ground response for site classification methods verification

A database of numerical ground response analyses has been generated to assess the effectiveness of site classification schemes adopted in seismic building codes. 1-D Ground models have been generated with a stochastic approach based on a Monte Carlo procedure and the corresponding seismic response has been evaluated by the equivalent linear elastic approach for a set of natural strong motion records. The study is intended to contribute to the discussion of optimal subsoil classification schemes in terms of inter-class dispersion and reliability of the predicted amplification factors. Moreover some specific analyses have been performed to show the relevance of stiff layer inclusions and of nonlinear soil behavior on the performances of classification schemes. The variability observed for each class in the different systems indicates that simple classification schemes should be preferred according to Occam's razor statement.

Site response analysis of the Kashmir valley during the 8 October 2005 Kashmir earthquake (Mw 7.6) using a geotechnical dataset

The site response analysis of Kashmir valley during the 8 October 2005 Kashmir earthquake (Mw 7.6) using the standard penetration test ‘N value’ (SPT N) geotechnical dataset is presented. Due to non-availability of the strong motion data records, synthetic ground motions at the bedrock level were generated at each borehole location by the stochastic finite fault method. The soil type-specific relationships between SPT N value and shear wave velocity were used to generate shear wave velocity profiles at each borehole site, which were later interpolated to map the shear wave velocity pattern in the Kashmir valley. The analyses revealed that site classes C and D of the NEHRP classification are dominant in the valley. The site response analysis conducted using the equivalent linear approach with DEEPSOIL showed that the local site conditions play an important role in the transmission of ground motion from the bedrock to the surface in the Kashmir valley, suggesting that it is imperative to consider the site effect in the seismic hazard assessment of the Kashmir valley. A detailed analysis of four important localities of Kashmir valley, namely, Anantnag, Baramulla, Kupwara and Srinagar, was also performed.

Site response analysis for estimating seismic site amplification in the case of Banda Aceh - Indonesia

The city of Banda Aceh is potentially exposed to a significant seismic hazard of seismic site amplification. Estimation of seismic site amplification of the city is urgently required for any mitigation efforts as the city is founded on a thick, soft layer. This study aims to estimate seismic site amplification of Banda Aceh's soil. Analytical models have demonstrated that they can simulate reasonably well the seismic ground motions amplification. The most widely used model is the equivalent linear approach. The approach computes the ground response of horizontally layered soil deposits subjected to transient and vertically propagating shear waves through a one-dimensional soil column. As aforementioned, this study focuses on Banda Aceh-Indonesia which is founded on thick alluvium. Three actual historical time histories and three developed sub-surface models were used to estimate the seismic site amplification of Banda Aceh's soft soil. The used time histories are of 2012 M8.1 Simeulue earthquake, 2013 M6.0 Mane-Geumpang earthquake and 2013 M6.2 Bener Meriah earthquake. Three sub-surface models of three separate sites across the city of Banda Aceh were developed. The site response analysis results reveal the ground motions amplification of Banda Aceh's soils of up to 4.3. Thus, applying the seismic site amplification for structural design at Banda Aceh can be further works.

Earthquake Response Analysis of Sites in State of Haryana using DEEPSOIL Software

Earthquake response analysis has been carried out for various locations in the State of Haryana (India) adopting the equivalent linear approach. For this purpose, geotechnical data have been collected from different government and private organizations. Based on the provisions of National Earthquake Hazards Reduction Program (NEHRP), all the sites have been classified using average N-value (N30) for the soil profile. The shear modulus (Gmax) values of the layers in a soil profile have been estimated using standard correlations. Cyclic Response has been accounted for using the standard shear modulus degradation and damping curves. Time histories from Himalayan Thrust System have been used as an input and soil amplification has been estimated at the surface using DEEPSOIL software. The results of the study have been formulated in terms of soil amplification map, response spectra, PGA along the depth, surface time histories and strain along the depth. These results can be used for dynamic analyses and design of structures, planning of advanced dynamic lab test etc. Moreover, it has been observed that sites in Haryana can significantly amplify ground motions and hence a site-specific design approach must be adopted for important structures.

A new fractional equivalent linearization method for nonlinear stochastic dynamic analysis

In this paper, a novel fractional equivalent linearization (NFEL) approach is developed by introducing fractional derivative term into the conventional equivalent linear equation. By appropriately formulating the fractional linearization coefficients with aid of the frequency response function of fractional equivalent linear system, an efficient iteration scheme is developed for determining these coefficients. Thus, the solution of fractional equivalent linear equation can be used to approximate the response of randomly excited nonlinear system. The contribution of the new added fractional derivative term to the classical damping force and the elastic restoring force is then quantitatively illustrated by virtue of the frequency response function of fractional system. The effectiveness of the NFEL is finally demonstrated by a nonlinear and a Duffing oscillator that is subjected to Gaussian white noise. The results are compared with the conventional equivalent linearization and exact solution.

A numerical comparison of random vibration theory and time histories based methods for equivalent-linear site response analyses

Dynamic site response is usually implemented using an equivalent-linear approach through analyses based on time-histories or random vibration theory (RVT). In the RVT approach the input motion is characterized in the frequency domain by means of Fourier Amplitude Spectra (FAS) or power spectral densities so that the need for selecting/developing multiple suitable time-histories is avoided. Nevertheless, past studies have found that RVT may produce results that differ significantly from empirically determined site amplification functions and from the time history approach. This work is aimed to further understand the potential differences in the results from RVT and time-histories based approaches by performing a comprehensive numerical evaluation that takes into account the effect of the input intensity level, input spectral shape, site conditions, and the methodology used to produce the input FAS. The results obtained corroborate that RVT over-predictions occur mainly at the site fundamental frequencies and are larger for relatively soft soil deposits with significant impedance contrast at the soil/rock interface. However, in the soft site evaluated the magnitude of the over-prediction was rather insensitive to the increase in the inelastic demand, conversely, the over-prediction in the stiffer site increased as the site softened due to the rising inelastic demand.

Effect of structural nonlinearity on probabilistic risk assessment of offshore wind turbine including inelastic soil medium

ABSTRACTThis research intends to evaluate the influence of structural nonlinearity on the seismic risk of an offshore wind turbine with respect to linear analysis. The structural nonlinearity is presented into the structure, as materially nonlinear, by calibrating plastic hinge at the end of the elastic beam-element of the structure. To guarantee ideal circumstance for seismic analysis, this study includes inelastic soil stratum by using an equivalent linear approach. Monte Carlo simulation is performed by means of seismic vulnerability of the structural system. The research presents a study of a large amount of simulation over the nonlinear and linear structures considering the random character of basic variables of soil under selective earthquakes. The earthquakes having different source-to-site (STS) distances ranging from 7 to 145 km have consistency with the soil parameters. This rigorous implementation is done to accomplish site-specific dynamic analysis. Illustrative results obtained from nonlinear and linear dynamic analysis are compared. The overall finding shows that the nonlinear structure produces highest estimated uncertainty compared to the linear structure. Another termination can be addressed that the earthquake with less STS distance (7.29 km) causes the highest level of destruction to the structure.

Seismic Microzonation of Islamabad–Rawalpindi Metropolitan Area, Pakistan

Microzonation deals with classifying seismic hazards in terms of ground motions resulting from amplification of seismic waves by nature of soil profiles underlying a site, town or city. This paper presents the results of microzonation study for Islamabad metropolitan, the capital of Pakistan. Cumulative SPT-N values from geophysical borehole and microtremor (Tromino Engy Plus) data were used to classify the soils into classes C (very dense soil profile and soft rock) and D (stiff soil profile) as devised by the National Earthquake Hazard Reduction Program (NEHRP). Soil response analyses were carried out based on scaled time histories of Kashmir earthquake (2005, 0.02 g), Mangla earthquake (2006, 0.031 g) and Haripur earthquake (2010, 0.13 g) corresponding to return periods of 150, 475, 975 and 2475 years. Spectral accelerations on the ground surface are calculated by two different approaches (1) soil response analysis performed using one dimensional shear wave propagation method (equivalent linear approach); and (2) NEHRP and Borcherdt amplification factors. Microzonation maps are produced with respect to ground shaking intensity for the return periods of 150, 475, 975 and 2475 years taking into account the variation of the spectral accelerations calculated based on these two procedures. The results show that the accelerations at the ground surface in the Islamabad–Rawalpindi metropolitan are in the range of 0.40–0.48 g (for 150 years), 0.59–0.65 g (for 475 years), 0.71–0.77 g (for 975 years), and 0.92–0.94 g (for 2475 years). The amplification factors for these four hazard levels range from 0.96 to 1.38 (150 years), 0.90–1.14 (475 years), 0.85–1.04 (975 years) and 0.84–1.00 (2475 years).

Site Specific Ground Response Analysis for Quantifying Site Amplification at A Regolith Site

DOI: 10.17014/ijog.4.3.159-167A numerical model has demonstrated that it can simulate reasonably well earthquake motions at the ground level during a seismic event. The most widely used model is an equivalent linear approach. The equivalent linear model was used to compute the free-field response of Adelaide regolith during the 1997 Burra earthquake. The aim of this study is to quantify the amplification at the investigated site. The model computed the ground response of horizontally layered soil deposits subjected to transient and vertically propagating shear waves through a one-dimensional-soil column. Each soil layer was assumed to be homogeneous, visco-elastic, and infinite in the horizontal extent. The results of this study were compared to other studies and forward computation of the geotechnical dynamic parameters of the investigated site. The amplification triggered by the 1997 Burra seismic event was deduced. This study reveals the amplification factor up to 3.6 at the studied site.

Site-specific probabilistic seismic hazard analysis for northern part of the Qeshm Island, Iran

This paper presents results of a site-specific probabilistic seismic hazard analysis for northern part of the Qeshm Island, one the most seismic prone areas of Iran. Seismotectonic and seismicity properties of seismic sources in the study area were characterized and used for evaluation of various strong ground motion parameters implementing the classical Cornell’s PSHA approach. The results show that peak rock accelerations for 475-year return period are 0.4 and 0.27 g, respectively, for 84th and 50th percentiles while being about 0.37 and 0.61 g for 2475-year return period. These values are slightly smaller than those read from national seismic zonation maps which can be attributed to the considered conservatism for development of such design maps. In order to incorporate local site conditions, a series of dynamic site response analyses based on the equivalent linear approach were also employed. The results indicate that the presence of soft subsurface deposits at the site significantly alters the fundamental characteristics of the response spectra. The obtained median (50th percentile) peak ground accelerations for 975-year return period range between 0.49 and 0.54 g at different locations in the study site showing minor amplifications relative to their corresponding bedrock acceleration of 0.48 g. Finally, the obtained site-specific spectrum was compared with the standard spectrum mandated by the design codes. In this regard, the agreement was found to be reasonable at period ranges shorter than about 0.5 s, while the differences were more obvious at longer periods. This reveals the need for implementation of site-specific design spectrum to avoid underestimation or overestimation of seismic forces for designing critically important structures especially when softer deposits are encountered.

Nonlinear system stochastic response determination via fractional equivalent linearization and Karhunen–Loève expansion

In this paper, a novel fractional equivalent linearization (EL) approach is developed by incorporating a fractional derivative term into the classical linearization equation. Due to the introduction of the fractional derivative term, the accuracy of the new linearization is improved, illustrated by a Duffing oscillator that is subjected to a harmonic excitation. Furthermore, a new method for solving stochastic response of nonlinear SDOF system is developed by combining Karhunen–Loève (K-L) expansion and fractional EL. The method firstly decomposes the stochastic excitation in terms of a set of random variables and deterministic sub-excitations using K-L expansion, and then construct sub-fractional equivalent linear system according to each sub-excitation by fractional EL, the response of the original nonlinear system is finally approximated as the weighed summation of the deterministic response of each sub-system multiplied by the corresponding random variable. The random nature of the final response comes from the set of random variables that is obtained in K-L expansion. In this way, the stochastic response computation is converted to a set of deterministic response analysis problems. The effectiveness of the developed method is demonstrated by a Duffing oscillator that is subjected to stochastic excitation modeled by Winner process. The results are compared with the numerical method and Monte Carlo simulation (MCS).

One-Dimensional Effective Stress Non-Masing Nonlinear Ground Response Analysis of IIT Guwahati

Ground response analysis (GRA) helps to assess the influence of the soil medium on the propagating shear waves and indicates about the characteristics of the waves reaching the ground surface from the bedrock level. Such a study becomes imperative for the urbanized alluvial banks of North-Eastern region of India, which is located in the highest seismic zone of the country. Conventionally, GRA is carried out based equivalent linear approach, which being a simplistic approach is unable to capture the nonlinear characteristics of saturated silty sands subjected to seismic shaking. This article presents the outcome of seismic one-dimensional nonlinear GRA of IIT Guwahati (located on a varying geology in the saturated alluvial banks of River Brahmaputra) considering pore-water pressure dissipation characteristics and non-Masing unload-reload criteria. Various ground response parameters obtained from the study helps in the accurate identification of the earthquake intensity based site amplification of the region expressed through a 2-D mapping.

Implementation of a Simplified Method in Design of Hysteretic Dampers for Isolated Highway Bridges

AbstractUsing seismic isolation systems for highway bridges modifies the structure’s principal vibration modes and effectively reduces the seismic base shear conveyed from the superstructure to the substructure. However, for some low-damping rubber isolation bearings, large displacements can be a problem. Supplemental hysteretic dampers can be introduced into the base-isolated bridge, which might nevertheless increase the structure base shear, and the merit of adding dampers has to be evaluated properly. In this paper, a simplified method was implemented for the design of a low-cost hysteretic damper, and the resulting isolator-damper system was tested experimentally. The design method used is based on an equivalent linearization approach. A full-scale elastomeric isolation bearing was characterized and used in the design of a hysteretic damper. Both the isolator and the damper went through cyclic testing and real-time dynamic substructuring (RTDS) methods to verify the capacity of the method to design ba...

Evidence of complex site effects and soil non-linearity numerically estimated by 2D vs 1D seismic response analyses in the city of Xanthi

This study discusses the effects of local site conditions on earthquake ground motion for city of Xanthi, North-Eastern Greece, focusing on the influence of complex site effects and soil non-linearity. Although city of Xanthi is characterized as a low seismicity area, documented strong earthquake events of past centuries indicate the necessity of an appropriate estimation of ground surface motion of the region. Therefore, a typical 2D cross-section along the city following an E-W direction is constructed, based on the synthesis of available geological, geotechnical and geophysical data of the broader area. 1D and 2D numerical models are generated utilizing the Finite Difference Method and site response is investigated for different seismic scenarios. Wave amplification is captured as a result of 1D resonance phenomena integrated with potential 2D complex wave effects, due to lateral propagation of the locally generated at the discontinuities surface waves. The additional, relatively to 1D, 2D amplification or de-amplification of ground shaking, in terms of its amplitude and its spectral content is quantified. The impact of soil deposits non-linearity on amplification level, as well as, on generated 2D wave fields is considered by implementing both equivalent-linear and non-linear approaches. The results reveal that, across the city, spatial variation of ground surface motion is obvious, attributed primarily to the variation of subsurface soil profiles. 2D wave effects have been generated signifying that, at particular site locations, ground motion is additionally amplified or de-amplified with respect to 1D response. However, 2D versus 1D differentiation has been calculated of ±20% in terms of PGA and spectral accelerations up to 1.0s, in the majority of the examined sites, implying that site response even though, influenced by 2D resonance phenomena, could be sufficiently captured by an 1D approach. Moreover, it is shown that if soil non-linearity is properly accounted for, it may play an important role on the estimated ground response, since non-linear approach seems to produce lower amplification levels with regard to 1D and 2D equivalent-linear approaches. As a general trend across the investigated site, for the entity of the various scenarios, the numerically calculated response spectra exceed EC8 seismic code provisions.

Earthquake Response of Base-Isolated Liquid Storage Tanks for Different Isolator Models

The effect of different isolator parameters on earthquake response of base-isolated liquid storage tanks is investigated herein. Mechanical analog, with three lumped masses, is used to model ground supported base-isolated liquid storage tank, and analyzed for recorded earthquake ground accelerations. The nonlinear force–deformation behavior of the isolator is mathematically modeled in two different ways, represented by (a) equivalent linear elastic-viscous and (b) bi-linear hysteretic behaviors. The equations of motion for the base-isolated tank are derived and solved in the incremental form using Newmark's step-by-step method of integration. Two different configurations of liquid storage tank (i.e. broad and slender) are considered to show the effect of the equivalent linear and bi-linear modeling of the isolator on the important earthquake response quantities. Effect of nonlinear hysteretic modeling of the isolator on peak response of the base-isolated liquid storage tanks is also investigated. The effect on earthquake response of the base-isolated liquid storage tank is studied for different parameters of the isolator for a range of slenderness ratio of the tank. The parameters considered include the characteristic strength of the isolator, isolation time period, isolator yield displacement etc. Significant difference is observed in the earthquake response of the base-isolated liquid storage tanks owing to the equivalent linear and bi-linear modeling approaches of the isolator. However, for bi-linear and nonlinear hysteretic modeling of the isolator, difference between the peak earthquake response of base-isolated liquid storage tanks are insignificant. The earthquake response of base-isolated liquid storage tanks is significantly influenced by the variation in the isolator parameters and slenderness ratio of the tank.

Nonlinear Response of Carbon-Carbon Laminated Panels to Random Acoustic Excitation under a Gradient Temperature Field

Future missions for aircraft will expose structures to severe thermal and acoustic loads. Efficient analysis methods for predicting nonlinear random response and fatigue life are urgently important. This paper presents a finite element model for analyzing nonlinear random dynamic behaviors of Carbon-Carbon composite panels under temperature gradients and Guassian excitations. The temperature distribution over the plate follows double sine curve. A finite element formulation combined with the equivalent linearization approach and normal mode method is established. The global system of equations is reduced to a set of nonlinear, coupled modal equations. Examples are given for an orthotropic C/C composite laminated panel at various combinations of temperatures gradients and sound pressure levels. Numerical results include RMS values of maximum deflection, time histories of deflection response and stress response, power spectrum densities, probability distribution functions and higher statistical moments. Numerical results verified all three types of panel motions for a simply supported orthotropic laminated plate: small-deflection random vibration about the initial equilibrium positions, snap-through motions between the two buckled positions, and nonlinear random response about new equilibrium positions after post-buckling. Numerical results will provide the important reference basis to aero engine structural integrity design and improving the structure dynamic strength and working life.