Use of Exact Solutions of Wave Propagation Problems to Guide Implementation of Nonlinear Seismic Ground Response Analysis Procedures

One-dimensional nonlinear ground response analyses provide a more accurate characterization of the true nonlinear soil behavior than equivalent-linear procedures, but the application of nonlinear codes in practice has been limited, which results in part from poorly documented and unclear parameter selection and code usage protocols. In this article, exact (linear frequency-domain) solutions for body wave propagation through an elastic medium are used to establish guidelines for two issues that have long been a source of confusion for users of nonlinear codes. The first issue concerns the specification of input motion as “outcropping” (i.e., equivalent free-surface motions) versus “within” (i.e., motions occurring at depth within a site profile). When the input motion is recorded at the ground surface (e.g., at a rock site), the full outcropping (rock) motion should be used along with an elastic base having a stiffness appropriate for the underlying rock. The second issue concerns the specification of viscous damping (used in most nonlinear codes) or small-strain hysteretic damping (used by one code considered herein), either of which is needed for a stable solution at small strains. For a viscous damping formulation, critical issues include the target value of the viscous damping ratio and the frequencies for which the viscous damping produced by the model matches the target. For codes that allow the use of “full” Rayleigh damping (which has two target frequencies), the target damping ratio should be the small-strain material damping, and the target frequencies should be established through a process by which linear time domain and frequency domain solutions are matched. As a first approximation, the first-mode site frequency and five times that frequency can be used. For codes with different damping models, alternative recommendations are developed.     

On-Site Nonlinear Hysteresis Curves and Dynamic Soil Properties

Strong motion records at five vertical array sites in Japan are used to examine soil shear modulus and material damping as a function of shear strain during large earthquakes. Acceleration data from the sites are processed directly for evaluation of site shear stress-strain hysteresis curves for different time windows of the record. Results of the analysis demonstrate a significant nonlinear ground response at the sites with surface peak ground accelerations exceeding 90 gal. The results of shear stress-strain hysteresis curves are also used to estimate variation of soil shear modulus and material damping characteristics with shear strain amplitude at each site. The identified shear modulus-shear strain and damping ratio-shear strain relationships are in general agreement with published laboratory results. These response interpretations are also compared with the results of a frequency-domain analysis by using the spectral ratio (uphole∕downhole) technique. There is general agreement between the time- and frequency-domain results. The results illustrate the significance of the site nonlinearity during strong ground motions as well as the accuracy of the dynamic soil properties obtained from laboratory tests.     

Optimal Control of Earthquake Response Using Semiactive Isolation

The responses of two, low-rise, 2-degree-of-freedom base isolated structures with different isolation periods to a set of near-field earthquake ground motions are investigated under passive linear and nonlinear viscous damping, two pseudoskyhook semiactive control methods, and optimal semiactive control. The structures are isolated with a low damping elastic isolation system in parallel with a controllable damper. The optimal semiactive control strategy minimizes an integral norm of superstructure absolute accelerations subject to the constraint that the nonlinear equations of motion are satisfied and is determined through a numerical solution to the Euler–Lagrange equations. The optimal closed-loop performance is evaluated for a controllable damper and is compared to passive viscous damping and causal pseudoskyhook control rules. Results obtained from eight different earthquake records illustrate the type of ground motions and structures for which semiactive damping is most promising.     

Damping in Shear Beam Structures and Estimation of Drift Response

Under pulse-type ground motions modal analysis is not quite efficient for estimating the elastic response of multi-degree-of-freedom systems, in particular when the effects of higher modes are significant. This paper first shows that the assumption of nondispersive damped waves for shear beams leads to inconsistent response estimation. Subsequently, a closed form time domain dispersive damped wave solution to the partial differential equation of motion is presented and it is verified with frequency domain solutions. Finally, using the solutions to the differential equation of motion, the response of frame structures with energy dissipating devices is studied.     

Influence of Input Motion and Site Property Variabilities on Seismic Site Response Analysis

Seismic site response analysis evaluates the influence of local soil conditions on earthquake ground shaking. There are multiple sources of potential uncertainty in this analysis; the most significant pertaining to the specification of the input motions and to the characterization of the soil properties. The influence of the selection of input ground motions on equivalent-linear site response analysis is evaluated through analyses performed with multiple suites of input motions selected to fit the same target acceleration response spectrum. The results indicate that a stable median surface response spectrum (i.e., within ±20% of any other suite) can be obtained with as few as five motions, if the motions fit the input target spectrum well. The stability of the median is improved to ±5 to 10% when 10 or 20 input motions are used. If the standard deviation of the surface response spectra is required, at least 10 motions (and preferably 20) are required to adequately model the standard deviation. The influence of soil characterization uncertainty is assessed through Monte Carlo simulations, where variations in the shear-wave velocity profile and nonlinear soil properties are considered. Modeling shear-wave velocity variability generally reduces the predicted median surface motions and amplification factors, most significantly at periods less than the site period. Modeling the variability in nonlinear properties has a similar, although slightly smaller, effect. Finally, including the variability in soil properties significantly increases the standard deviation of the amplification factors but has a lesser effect on the standard deviation of the surface motions.     

Learning of Dynamic Soil Behavior from Downhole Arrays

An increasing number of downhole arrays are deployed to measure motions at the ground surface and within the soil profile. Measurements from these arrays provide an opportunity to improve site response models and to better understand underlying dynamic soil behavior. Parametric inverse analysis approaches have been used to identify constitutive model parameters to achieve a better match with field observations. However, they are limited by the selected material model. Nonparametric inverse analysis approaches identify averaged soil behavior between measurement locations. A novel inverse analysis framework, self-learning simulations (SelfSim), is employed to reproduce the measured downhole array response while extracting the underlying soil behavior of individual soil layers unconstrained by prior assumptions of soil behavior. SelfSim is successfully applied to recordings from Lotung and La Cienega. The extracted soil behavior from few events can be used to reliably predict the measured response for other events. The field extracted soil behavior shows dependencies of shear modulus and damping on cyclic shear strain level, number of loading cycles, and strain rate that are similar qualitatively to those reported from laboratory studies but differ quantitatively.     

Hydraulic Damping of Saturated Poroelastic Soils During Steady-State Vibration

A theoretical study of the steady-state response of a saturated poroelastic soil column during compressional and rotational harmonic vibrations is presented. Hydraulic damping due to Biot flow is evaluated for top-drained and double-drained boundary conditions and for compressional and rotational motions using the theory of a damped single-degree-of-freedom system. For compressional motions, the dynamic response of gravels and sands is highly influenced by the compressibility of the pore fluid. More hydraulic damping occurs as soil hydraulic conductivity increases and as the column boundary conditions change from top drained to double drained. On the other hand, hydraulic damping for rotational motions is significantly less than that for compressional motions and is dependent on a dimensionless hydraulic conductivity parameter Ks. For Ks within the range of 10?3–100, hydraulic damping may have an important contribution to total soil damping, especially at small strain levels.     

Site-Specific Validation of Random Vibration Theory-Based Seismic Site Response Analysis

Seismic site response analysis is typically performed using a suite of rock acceleration-time histories prescribed at the base of a soil column and propagated to the ground surface. To develop statistically stable estimates of the site response, a large number of input motions are required. Alternatively, random vibration theory (RVT) can be used to predict statistically stable estimates of the surface response spectrum in one analysis without the need to prescribe the input rock motion in the time domain. Thus, the critical and time consuming activity of choosing appropriate input ground motions and fitting them to a target spectrum is avoided. This paper describes the RVT approach, its analytical background and input requirements, and provides a site-specific validation of the procedure against traditional site response predictions. The single-corner frequency Brune source spectrum is used in the RVT procedure to describe the input motion in the frequency domain. RVT site response predictions using the Brune spectrum as input are compared with those from traditional site response analyses that incorporate different suites of input rock motions. Results indicate that RVT site response analysis can provide a response spectrum that is similar to the median response spectrum from analyses performed using a suite of input rock motions. However, the favorable comparison is obtained only when the seismological parameters used to describe the RVT input motion are carefully chosen to be consistent with the suite of input rock motions.     

Effect of Nonlinear Soil Behavior on Inelastic Seismic Response of a Structure

The characteristics of the earthquake motions at the base of a structure are affected by the properties of the underlying soil through the soil amplification and soil–structure interaction phenomena. In this paper the effect of nonlinear soil behavior on the elastic and inelastic response spectra of the motions that would be recorded at the free surface of a soft soil deposit or at the base of each structure is investigated. The analyses are conducted for a soil layer by itself and for a complete soil structure system using a finite element discretization of the soil in cylindrical coordinates and an approximate linear iterative procedure to simulate nonlinear behavior. Studies are conducted for structures, with a constant base and variable height modeled as equivalent linear or nonlinear single degree of freedom systems and an input motion at the base of the soil deposit representative of rock outcrop motions. Both mat and pile foundations are considered. The results illustrate clearly the importance of the nonlinear soil behavior.     

Numerical Modeling of Seismic Response of Rigid Foundation on Soft Soil

A numerical model was developed to simulate the response of two instrumented, centrifuge model tests on soft clay and to investigate the factors that affect the seismic ground response. The centrifuge tests simulated the behavior of a rectangular building on 30?m uniform and layered soft soils. Each test model was subjected to several earthquakelike shaking events at a centrifugal acceleration level of 80g. The applied loading involved scaled versions of an artificial western Canada earthquake and the Port Island ground motion recorded during the 1995 Kobe Earthquake. The centrifuge model was simulated with the three-dimensional finite-difference-based fast Lagrangian analysis of continua program. The results predicted with the use of nonlinear elastic–plastic model for the soil are shown to be in good agreement with measured acceleration, soil response, and structural behavior. The validated model was used to study the effect of soil layering, depth, soil–structure interaction, and embedment effects on foundation motion.     

Dynamic Ratcheting in Elastoplastic Single-Degree-of-Freedom Systems

In dynamic analysis, hysteretic damping often provides a reasonable model of the inelastic behavior of a structure. Nonlinearity presented by hysteretic damping, however, introduces the possibility of developing complicated motions not expected in linear dynamics. In this study, motions of a single-degree-of-freedom system with hysteretic damping under dual-frequency sinusoidal excitations are investigated through numerical simulation. Hysteretic damping behavior is represented by three different plasticity models: the elasto-perfectly-plastic model; the linear kinematic hardening model; and the two-surface model. Under certain conditions, the resultant motions from the elasto-perfectly-plastic model and the two-surface model exhibit a continual increment of plastic deformation in successive cycles. Parametric study shows that this dynamic ratcheting develops when applied frequencies are commensurable (i.e., related to each other with integer ratio), and the product of terms comprising the ratio is an even number. In the Poincaré section, motion from commensurable frequencies shows limit cycle behavior, whereas the boundedness of motion for incommensurable frequencies is depicted by having quasi-periodicity. On the other hand, the response of the linear kinematic hardening model is qualitatively different and, in particular, dynamic ratcheting does not develop, irrespective of the frequency commensurability. These findings suggest that model selection may have unanticipated consequences for the analysis and design of structural systems subjected to severe dynamic loadings, such as major earthquakes.     

Kinematic Soil-Structure Interaction from Strong Motion Recordings

Earthquake strong motion recordings from 29 sites with instrumented structures and free-field accelerographs are used to evaluate variations between foundation-level and free-field ground motions. The focus of the paper is on buildings with surface and shallowly embedded foundations. The foundation/free-field ground motion variations are quantified in terms of frequency-dependent transmissibility function amplitude ∣H∣. Procedures are developed to fit to ∣H∣ analytical models for base slab averaging for the assumed conditions of a rigid base slab and a vertically propagating, incoherent incident wave field characterized by ground motion incoherence parameter κ. The limiting assumptions of the model are not strictly satisfied for actual structures, and the results of the identification are apparent κ values (denoted κa) that reflect not only incoherence effects, but also possible foundation flexibility and wave inclination effects. Nonetheless, a good correlation is found between κa values and soil shear wave velocity for sites with stiff foundation systems. Based on these results, recommendations are made for modifying free-field ground motions to estimate base slab motions for use in response analyses of buildings.     

Online Earthquake Response Test for Stratified Layers of Clay and Sand

Artificial islands often consist of layers of alluvial clay and reclaimed soil of varying order and thickness. Soft clay layers have nonlinear characteristics and can both amplify and attenuate earthquake ground motions. Liquefied ground impedes propagation of shear waves and thus attenuates the earthquake accelerations. Online testing is a method of feeding soil response characteristics directly from soil samples into a modeling algorithm. The effects of the layer thickness, configuration, and degree of consolidation on the earthquake response characteristics of alternating layers of clay and sand have been investigated. The degree of liquefaction and strain generated in sand adjacent to clay layers increased with the degree of consolidation. Clay layers attenuate the motions of sand layers for short period vibrations but amplify the long period motions, increasing the strain in overlying liquefied sand layers. Clay layers which were closer to the ground surface or of greater thickness tended to increase the surface accelerations. Normalized cumulative energy loss was larger in clay than in sand increasing with a decreasing degree of consolidation.     

Energy Balance Assessment of Base-Isolated Structures

This paper explores the use of energy concepts in the analysis of base-isolated structures subject to severe earthquake ground motions. We formulate the energy balance equations in moving- and fixed-base coordinate frames and provide new physical insight into the time-dependent behavior of individual terms. Conventional wisdom in earthquake engineering circles is that systems with base isolation devices should be economically competitive and designed to: (1) minimize input energy, and (2) maximize the percentage of input energy dissipated by damping and inelastic mechanisms. Through the nonlinear time-history analysis of a base-isolated mass-spring system subject to an ensemble of severe ground motion inputs, we demonstrate that improvements in objective (2) often need to be balanced against increases in input energy. Hence, by itself, objective (1) presents an overly simplified view of desirable behavior.     

Energy Dissipation in Surface Layer due to Vertically Propagating SH Wave

Vertical array data recorded during the 1995 Kobe earthquake are used to calculate the upward and downward energy flow based on one-dimensional SH-wave multireflection theory, from which the energy dissipation in a surface layer is evaluated as their residual. The dissipated energy thus evaluated in a liquefied site is found to reach about 70% of the upward input energy, which indicates that soil nonlinearity and liquefaction serve as effective energy absorbers. In contrast, more energy returns to deeper ground in sites without strong nonlinear behavior. Furthermore, the dissipated energy in the surface layer tends to increase nonlinearly in a convex shape with increasing equivalent damping ratio of the soil there. A simplified two-layer system indicates that the energy dissipation is influenced not only by the soil damping in the surface layer but also by the impedance ratio between the base and surface layers and the input frequency. The same convex relationship is also obtained in the two-layer system, indicating that the simplified system may reflect some important aspects of the energy dissipation mechanisms in the ground.     

Semianalytical Solution of Wave-Controlled Impact on Composite Laminates

Based on a structural model for wave-controlled impact, a modified Hertzian contact law was used to investigate the impact responses of composite laminates. The original nonlinear governing equation was transformed into two linear equations using asymptotic expansion. Closed-form solution can be derived for the first linear homogeneous equation, which is the equation of motion for single degree of freedom system with viscous damping. The second linear nonhomogeneous equation was solved numerically. The overall impact responses for wave-controlled impacts can be obtained semianalytically and agree well with the numerical solutions of nonlinear governing equations. The proposed methodology is useful for providing guidance to numerical simulation of impact on complex composite structures with contact laws fitting from experimental data.     

Seismic Site Response for Near-Fault Forward Directivity Ground Motions

Forward directivity effects in the near-fault region produce pulse-type motions that differ significantly from ordinary ground motions that occur at greater distances from the causative fault. Current code site factors are based on empirical observations and analyses involving less intense nonpulse ordinary ground motions. Nonlinear site response analyses with bidirectional shaking are performed using representative site profiles to quantify seismic site response effects for intense near-fault motions resulting from forward directivity. Input rock motions are represented with simplified velocity pulses that characterize the amplitude and period of forward directivity motions. Results indicate that site response affects both the amplitude and period of forward directivity pulses, and hence, local site conditions should be considered when evaluating seismic designs in the near-fault region. Stiff soil sites tend to amplify the peak ground velocity and increase the period of pulse-type motions, particularly, when the period of the rock motion coincides with the degraded period of the site. Amplification is limited at soft soil sites by the dynamic strength of the weak soil, so attenuation occurs for intense input motions. This nonlinearity is not reflected in the site factors in current building codes. Guidance is provided for estimating the amplitude and pulse period for velocity pulses at soil sites.     

Wave Passage and Ground Motion Incoherency Effects on Seismic Response of an Extended Bridge

This paper investigates the implications of ground motion spatial variability on the seismic response of an extended highway bridge. An existing 59-span, 2,164-meter bridge with several bearing types and irregularity features was selected as a reference structure. The bridge is located in the New Madrid Seismic Zone and supported on thick layers of soil deposits. Site-specific bedrock input ground motions were selected based on a refined probabilistic seismic hazard analysis of the bridge site. Wave passage and ground motion incoherency effects were accounted for after propagating the bedrock records to the ground surface. The results obtained from inelastic response-history analyses confirm the significant impact of wave passage and ground motion incoherency on the seismic behavior of the bridge. The amplification in seismic demands exceeds 150%, whereas the maximum suppression of these demands is less than 50%. The irregular and unpredictable changes in structural response owing to asynchronous earthquake records necessitate in-depth seismic assessment of major highway bridges with advanced modeling techniques to realistically capture their complex seismic response.