The dynamics of a rigid foundation on the surface of an elastic half-space

The response of a rigid rectangular foundation block resting on an elastic half-space has been determined by considering first the displacement functions for any position on the surface of an unloaded half-space due to a harmonic point force. The influence of the foundation has been taken into account by assuming a relaxed condition at the interface, i.e. a uniform displacement under the foundation and that the sum of the point forces must equal the total applied force. The three motions of vertical, horizontal and rocking have been considered and numerical values for the in-phase and the quadrature components of the displacement functions are presented for a Poisson's ratio of 0.25. The effect of the mass and inertia of the foundation can be allowed for by an impedance matching technique. Response curves and non-dimensional resonant frequency curves are given for a square and a rectangular foundation for different mass and inertia ratios and for several values of Poisson's ratio. These curves are for design purposes and are an addition to similar published curves for circular and infinitely long rectangular foundations. Some of the calculated results have been verified by a laboratory experiment.     

3D seismic response of a limited valley via BEM using 2.5D analytical Green's functions for an infinite free-rigid layer

This paper presents analytical solutions for computing the 3D displacements in a flat solid elastic stratum bounded by a rigid base, when it is subjected to spatially sinusoidal harmonic line loads. These functions are also used as Greens functions in a boundary element method code that simulates the seismic wave propagation in a confined or semi-confined 2D valley, avoiding the discretization of the free and rigid horizontal boundaries.The models developed are then used to simulate wave propagation within a rigid stratum and valleys with different dimensions and geometries, when struck by a spatially sinusoidal harmonic vertical line load. Simulations are performed in the frequency domain, for varying spatial wave numbers in the axial direction of the valley. Time results are obtained by means of inverse Fourier transforms, to help understand how the geometry of the valley may affect the variation of the displacement field.     

On the performance of SCF in seismic isolation of the interior equipment of buildings

The dynamic response of equipment mounted on an isolated raised floor inside a building while the primary fixed base structure is subjected to harmonic and earthquake ground motions is numerically investigated. Sliding concave foundation (SCF) system is utilized for isolating the raised floor. The equations of motion for a MDOF shear building containing a SCF isolated raised floor with a mounted equipment are developed and the rigid link method is utilized to handle the non‐linearity of the system. The equipment, which can be modelled as a SDOF or MDOF system, may represent a critical computer unit or telecommunication processing system. SCF can be used easily to achieve the desired long period, necessary for protecting sensitive equipment. In this investigation, the ability of SCF in reducing the acceleration level experienced by the equipment inside a building is demonstrated while the lateral displacement is still within an acceptable range. The analysis considered the case of equipment housed in the upper floors of a building where the acceleration is amplified and the motion contains strong components at long periods. For this purpose, different excitations including both harmonic and real earthquake ground motions are employed and the performance of the system is evaluated. Copyright © 2007 John Wiley & Sons, Ltd.     

Estimation of rocking and torsion associated with surface waves extracted from recorded motions

By exploiting the capability of identifying and extracting surface waves existing in a seismic signal, we can proceed to estimate the angular displacement (rotation about the horizontal axis normal to the direction of propagation of the wave; rocking) associated with Rayleigh waves as well as the angular displacement (rotation about the vertical axis; torsion) associated with Love waves.For a harmonic Rayleigh (Love) wave, rocking (torsion) would be proportional to the harmonic vertical (transverse horizontal) velocity component and inversely proportional to the phase velocity corresponding to the particular frequency of the harmonic wave (a fact that was originally exploited by Newmark (1969) [15] to estimate torsional excitation). Evidently, a reliable estimate of the phase velocity (as a function of frequency) is necessary. As pointed out by Stockwell (2007) [17], because of its absolutely referenced phase information, the S-Transform can be employed in a cross-spectrum analysis in a local manner. Following this suggestion a very reliable estimate of the phase velocity may be obtained from the recordings at two nearby stations, after the dispersed waves have been identified and extracted. Synthesis of the abovementioned harmonic components can provide a reliable estimate of the rocking (torsional) motion induced by an (extracted) Rayleigh (Love) wave.We apply the proposed angular displacement estimation procedure for two well recorded data sets: (1) the strong motion data generated by an aftershock of the 1999 Chi-Chi, Taiwan earthquake and recorded over the Western Coastal Plain (WCP) of Taiwan, and (2) the strong motion data generated by the 2010 Darfield, New Zealand earthquake and recorded over the Canterbury basin. The former data set is dominated by basin-induced Rayleigh waves while the latter contains primarily Love waves.     

Reduction of steady-state forced vibrations of structures with dynamic absorbers

In the paper the problem of reduction of the steady-state response of a lightly damped structure to periodic excitation is considered. The primary structure, which has arbitrary distributions of mass, stiffness and damping, is subjected to periodic load with the fundamental frequency varying in a wide range that may include several resonant peaks of the amplitude-frequency response. A number of passive absorbers are applied simultaneously to the main structure. A general formulation of the dynamic absorber design methodology is presented, based on independent design of conventional absorbers, taking into account the selected modal systems of the original structure and the selected harmonic components of the excitation. In order to cover the losses in the vibration reduction, resulting from the couplings in the primary structure–the set of absorbers system and from the remaining harmonic components of the excitation, the mass of modal dynamic absorbers is increased properly. The methodology developed was verified on several numerical examples; one of them is presented in the study.     

A novel predictor–corrector algorithm for sub‐structure pseudo‐dynamic testing

A new predictor–corrector (P–C) method for multi‐site sub‐structure pseudo‐dynamic (PSD) test is proposed. This method is a mixed time integration method in which computational components separable from experimental components are solved by implicit time integration method (Newmark β method). The experiments are performed quasi‐statically based on explicit prediction of displacement. The proposed P–C method has an important advantage as it does not require the determination of the initial stiffness values of experimental components and is thus suitable for representing elastic and inelastic systems. A parameter relating to quality of displacement prediction at boundaries nodes is introduced. This parameter is determined such that P–C method can be applicable to many practical problems. Error‐propagation characteristics of P–C method are also presented. A series of examples including linear and non‐linear soil–foundation–structure interaction problem demonstrate the performance of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.     

Mixed boundary value problem in an elastic quarter-space

Summary The stress and displacement throughout a homogeneous, isotropic, elastic quarterspace have been evaluated when the stress is prescribed on one of the plane boundaries and the displacement on the other. The variations of the displacement and stress on the boundaries with time and distance from the origin have been shown graphically.     

Multivariate statistical analysis of deformation tensors: independent vs. correlated tensor observations

In the presence of errors in measuring a random displacement field (under the normal distribution assumption of displacement field), stochastic behaviors of principal components of deformation tensors (strain tensor and tensor of change of curvature (TCC)), based on the intrinsic assumption of geometrical modeling of surface deformation analysis, are discussed. We divided the contents into two parts: In the first, we considered independent random vectors of repeated tensor measurements. In the second step, we considered correlations among repeated measurements. Then, covariance components between tensor elements by Helmert estimator, based on prior information of variance components, are estimated. As a case study, both assumptions are applied to the estimation of principal components of deformation rate tensor observations in Zagros region (Western Iran). The results of numerical analysis showed that greatest shortening is accommodated in oblique orientation (NS) with respect to the Main Recent Fault (MRF), northwest part of North Zagros, Central Iran block and MRF, respectively. Most of the extensions occurred in the east part of the belt. The pattern of eigenspace components of TCC shows highest positive values across the NW region, nearly in orthogonal direction to the MRF and Main Zagros Fault (MZF). The pattern has insignificant values in the Central Zagros. It takes the significant negative values across the SW part, especially along the SPF and Persian Gulf shore. The effect of non-independent observations on the estimation of eigenspace components of deformation tensors (strain tensor and TCC) shows that the estimation of covariance components has influence on the confidence intervals of eigenspace components, especially in seismically active regions of the belt (along the Persian Gulf shore, NW of the belt and region between the Central Iran block and MRF). The results demonstrate the importance of considering the correlation structure among the observations on statistical behavior of principal components of deformation tensors in seismically active regions.     

Simulation of tunnel response under spatially varying ground motion

Various components including wave scattering, wave passage, and site amplification effects cause the ground motion to vary spatially. The spatially varying ground motion can significantly influence the dynamic response of longitudinal structures such as bridges and tunnels. While its effect on bridges has been extensively studied, there is a lack of study on its effect on underground tunnels. This paper develops a new procedure for simulating the tunnel response under spatially varying ground motion. The procedure utilizes the longitudinal displacement profile, which is developed from spatially variable ground motion time histories. The longitudinal displacement profile is used to perform a series of pseudo-static three-dimensional finite-element analyses. Results of the analyses show that the spatially variable ground motion causes longitudinal bending of the tunnel and can induce substantial axial stress on the tunnel lining. The effect can be significant at boundaries at which the properties of the ground change in the longitudinal direction.     

Theory and experimental study for sliding isolators with variable curvature

Because a conventional isolation system with constant isolation frequency is usually a long‐period dynamic system, its seismic response is likely to be amplified in earthquakes with strong long‐period wave components, such as near‐fault ground motions. Seismic isolators with variable mechanical properties may provide a promising solution to alleviate this problem. To this end, in this work sliding isolators with variable curvature (SIVC) were studied experimentally. An SIVC isolator is similar to a friction pendulum system (FPS) isolator, except that its sliding surface has variable curvature rather being spherical. As a result, the SIVC's isolation stiffness that is proportional to the curvature becomes a function of the isolator displacement. By appropriately designing the geometry of the sliding surface, the SIVC is able to possess favorable hysteretic behavior. In order to prove the applicability of the SIVC concept, several prototype SIVC isolators, whose sliding surfaces are defined by a sixth‐order polynomial function, were fabricated and tested in this study. A cyclic element test on the prototype SIVC isolators and a shaking table test on an SIVC isolated steel frame were all conducted. The results of both tests have verified that the prototype SIVC isolators do indeed have the hysteretic property of variable stiffness as prescribed by the derived formulas in this study. Moreover, it is also demonstrated that the proposed SIVC is able to effectively reduce the isolator drift in a near‐fault earthquake with strong long‐period components, as compared with that of an FPS system with the same friction coefficient. Copyright © 2011 John Wiley & Sons, Ltd.     

Axially equilibrated displacement‐based beam element for simulating the cyclic inelastic behaviour of RC members

Distributed plasticity beam elements are commonly used to evaluate limit state demands for performance‐based analysis of reinforced concrete (RC) structures. Strain limits are often preferred to drift limits because they directly relate to damage and are therefore less dependent on member geometry and boundary conditions. However, predicting accurately strain demands still represents a major simulation challenge. Tension shift effects, which induce a linear curvature profile in the plastic hinge region of RC columns and walls, are one of the main causes for the mismatch between experimental and numerical estimates of local level quantities obtained through force‐based formulations. Classical displacement‐based approaches are instead suitable to simulate such linear curvature profile. Unfortunately, they verify equilibrium only on an average sense due to the wrong assumption on the axial displacement field, leading to poor deformation and force predictions. This paper presents a displacement‐based element in which axial equilibrium is strictly verified along the element length. The assumed transversal displacement field ensures a linear curvature profile, connecting accurately global displacement and local strain demands. The proposed finite element is validated against two sets of quasi‐static cyclic tests on RC bridge piers and walls. The results show that curvature and strain profiles for increasing ductility demands are significantly improved when axially equilibrated rather than classical displacement‐based or force‐based elements are used to model the structural members. Copyright © 2017 John Wiley & Sons, Ltd.     

Transient signals from a buried horizontal magnetic dipole

Summary The impulse response for a horizontal magnetic dipole buried in a conducting half-space is obtained in closed form by an inverse Fourier transform of the known time harmonic solution. Since displacement currents are neglected, the response is not valid for very short times. The waveforms of the magnetic field components at the surface have geometrical dependences which may have application to source location.     

SH waves from torsional sources in semi-infinite heterogeneous media

Summary The theory of Hankel transforms is used to obtain the displacement field forSH waves generated by time harmonic, buried, torsional sources in semi-infinite heterogenous media in which material properties are functions of the depth coordinate. An application to a heterogeneous medium with exponentially varying properties has been discussed. Surface displacements from a surface source have been evaluated at large distances from the source. The results have been compared with those for the homogeneous medium.     

桥梁高墩位移延性能力的探讨

本文运用增量动力分析方法来计算高墩的屈服位移、极限位移和位移延性，给出了应用IDA方法计算屈服位移、极限位移的主要过程，探讨了高阶振型对高墩位移延性能力的影响。通过算例分析表明：现行计算桥墩位移延性系数的方法，如直接应用到高墩，将会导致较大误差；高阶振型对高墩的屈服位移、极限位移和位移延性系数都有较大的影响。     

Plastic hinge analysis for lightly reinforced and unconfined concrete structural walls

Poor performance of lightly reinforced and unconfined concrete structural walls have been observed in recent earthquake events. This research investigates the displacement capacity of such walls by comparing the results of a series of state-of-the-art finite element analyses for a range of different structural walls to that estimated using plastic hinge analyses. The common expressions used in estimating the yield curvature, yield displacement and plastic displacement are scrutinised for these types of walls. Some recommendations are given to improve the prediction of the displacement capacity of lightly reinforced and unconfined rectangular and C-shaped walls for flexural actions using a plastic hinge analysis. Importantly, a parameter has been recommended to be used in a “modified” approach for estimating the nominal yield displacement of lightly reinforced concrete walls. Different expressions are also recommended depending on the amount of longitudinal reinforcement used in the wall in comparison to that required to initiate secondary cracking. This is important for providing better estimations of the displacement capacity of RC structural wall buildings in low-to-moderate seismic regions such that vulnerability studies can be conducted.     

Seismic isolation of buildings with sliding concave foundation (SCF)

In this paper, a new base isolation system, namely the sliding concave foundation (SCF), is introduced and the behaviour of the buildings using such a system is theoretically investigated. A building supported on the new system behaves like a compound pendulum during seismic excitation. The pendulum behaviour accompanied by the large radius of foundation curvature shifts the fundamental period of the system to a high value (e.g. more than 8sec), in a frequency range where none of the previously recorded earthquakes had considerable energy. This results in a large decrease in the structural responses. Since small friction forces are essential on the contact surfaces, PTFE sheets can be used as sliding surfaces. Although the pure frictional sliding systems have the same efficiency as the SCF, in reducing the responses of the superstructure, the main advantage of the new system is a significant decrease in sliding displacement. The performance of the SCF subjected to a number of harmonic and non‐harmonic base excitations is studied and its ability to reduce the structural responses is examined. Some numerical examples are solved for a single‐degree‐of‐freedom (SDOF) structure and the responses are compared with the responses of the same SDOF structure on a fixed base or a pure frictional sliding support system. The comparisons confirm the effectiveness of the new system. Copyright © 2002 John Wiley & Sons, Ltd.     

Estimating the seismic displacement of friction pendulum isolators based on non‐linear response history analysis

A procedure for developing equations that estimate the isolator displacement due to strong ground motion is applied to buildings isolated with the friction pendulum system. The resulting design equations, based on rigorous non‐linear analysis, offer an alternative to the iterative equivalent‐linear methods used by current U.S. building codes. The governing equations of the system are reduced to a form such that the median normalized displacement of the system due to an ensemble of ground motions is found to depend on only the isolation period—a function of the curvature of the isolator—and the friction force at incipient slip normalized by peak ground velocity. The normalization is effective in minimizing the dispersion of the normalized displacement for an ensemble of ground motions, implying that the median normalized displacement is a reliable estimate of response. The design equations reflect the significant (20 to 38%) increase in displacement when the excitation includes two lateral components of ground motion instead of just one component. Equivalent‐linear methods are shown to underestimate by up to 30% the exact median displacement determined by non‐linear response history analysis for one component of ground motion, and building codes include at most a 4.4% increase for a second component. Copyright © 2003 John Wiley & Sons, Ltd.     

Experimental and theoretical investigations of drainage in horizontal rough-walled fractures with different correlation structures

Experiments on the immiscible displacement of water by air were performed in two horizontal artificial open rough-walled fractures with different correlation structures. The aperture field of the first fracture has a Gaussian variogram with a finite correlation length, while that of the second one has a power-law variogram, for which no finite correlation length can be assigned. We simulated the displacement process using an invasion percolation model, which takes the in-plane curvature of the interface between water and air into account. The correlation structure of the aperture field has no effect on the irregularity of the shape of the air clusters. However, the correlation structure in combination with the in-plane curvature influences the density of the air clusters and thus has an impact on characteristic properties of the flow. We analyze the model parameters and properties with respect to the fracture properties. The stochastic average of the air clusters for models with different curvature numbers and fields with different correlation lengths is then calculated in order to analyze their influence on continuum approach models.     

Transients in cylindrical shells

The propagation of non-axially symmetric transients in linear, elastic, isotropic and homogeneous cylindrical shells subjected to impulsive boundary loads is treated in this paper. Expressing the displacement components in the form of a Fourier series in the circumferential co-ordinate, the displacement equations of motion are written for each harmonic. The analysis is based on the concept of a wave as a carrier of discontinuities in the field variable and its derivatives. These discontinuities are determined from a set of recurrence relations which are in turn generated by the use of time-harmonic asymptotic series solutions to the equations of motion. The method is illustrated by presenting detailed numerical results pertaining to the influence of earthquake or blast-induced ground excitation on cylindrical tanks.     

Terracing potential field data

Terracing is an operator that is applied to potential field data to produce regions of constant field amplitude that are separated by sharp boundaries, as an aid to interpretation. When applied to map data, the boundaries are defined by the zero contour of the 2D Laplacian derivative operator. An improved method is described here that defines the boundaries by the zero contour of the profile curvature. This approach gives superior results because the 2D Laplacian operator is composed only of derivatives in the EW and NS directions, while the profile curvature uses the curvature in the 'uphill' direction at each point, whatever that direction may be. The method is demonstrated on gravity data from South Africa. Source code in Matlab format is available from the authors on request.