Kinematic interaction factors of deep foundations with inclined piles

The beneficial or detrimental role of battered piles on the dynamic response of piled foundations has not been yet fully elucidated. In order to shed more light on this aspect, kinematic interaction factors of deep foundations with inclined piles, are provided for single‐battered piles, as well as for 2 × 2 and 3 × 3 groups of piles subjected to vertically incident plane shear S waves. Piles are modelled as linear‐elastic Bernoulli beams, whereas soil is assumed to be a linear, isotropic, homogeneous viscoelastic half‐space. Different pile group configurations, pile‐soil stiffness ratios, and rake angles are considered. The relevance and main trends observed in the influence of the rake angle on the kinematic interaction factors of the analysed foundations are inferred from the presented results. An important dependence of the kinematic interaction factors on the rake angle is observed together with the existence of an inclination angle at which cap rotation and excitation become out of phase in the low‐to‐mid frequency range. The existence of a small batter angle that provides minimum cap rotation is also shown. Copyright © 2014 John Wiley & Sons, Ltd.     

Effects of fault movement and material properties on deformation and stress fields of Tibetan Plateau

We compare the factors which affect the movement of Tibetan Plateau by building three types of finite element models: an elastic materials (M-EC), a continuous model composed by non-linear materials (M-PC), and an elastic model with discontinuous fault movements (M-ET). Both in M-ET and M-EC, the materials in Qiangtang and Lhasa block are elastic, and in M-ET, discontinuous movement of faults is considered for evaluating the effects of strike-slip faults. In model M-PC Druker-Prager plastic materials are us...     

The Elastic Strain Energy of Damaged Solids with Applications to Non-Linear Deformation of Crystalline Rocks

Laboratory and field data indicate that rocks subjected to sufficiently high loads clearly deviate from linear behavior. Non-linear stress–strain relations can be approximated by including third and higher-order terms of the strain tensor in the elastic energy expression (e.g., the Murnaghan model). Such classical non-linear models are successful for calculating deformation of soft materials, for example graphite, but cannot explain with the same elastic moduli small and large non-linear deformation of stiff rocks, such as granite. The values of the third (higher-order) Murnaghan moduli estimated from acoustic experiments are one to two orders of magnitude above the values estimated from stress–strain relations in quasi-static rock-mechanics experiments. The Murnaghan model also fails to reproduce an abrupt change in the elastic moduli upon stress reversal from compression to tension, observed in laboratory experiments with rocks, concrete, and composite brittle material samples, and it predicts macroscopic failure at stress levels lower than observations associated with granite. An alternative energy function based on second-order dependency on the strain tensor, as in the Hookean framework, but with an additional non-analytical term, can account for the abrupt change in the effective elastic moduli upon stress reversal, and extended pre-yielding deformation regime with one set of elastic moduli. We show that the non-analytical second-order model is a generalization of other non-classical non-linear models, for example “bi-linear”, “clapping non-linearity”, and “unilateral damage” models. These models were designed to explain the abrupt changes of elastic moduli and non-linearity of stiff rocks under small strains. The present model produces dilation under shear loading and other non-linear deformation features of the stiff rocks mentioned above, and extends the results to account for gradual closure of an arbitrary distribution of initial cracks. The results provide a quantitative framework that can be used to model simultaneously, with a small number of coefficients, multiple observed aspects of non-linear deformation of stiff rocks. These include, in addition to the features mentioned above, stress-induced anisotropy and non-linear effects in resonance experiments with damaged materials.     

Dynamic response of 3-D rigid surface foundations by time domain boundary element method

The dynamic response of three-dimensional rigid surface foundations of arbitrary shape is numerically obtained. The foundations are placed on a linear elastic, isotropic and homogeneous half-space representing the soil medium and are subjected to either external dynamic forces or seismic waves of various kinds and directions, with a general transient time variation. The problem is formulated in the time domain by the boundary element method and the response is obtained by a time step-by-step integration. Two examples dealing with three-dimensional rectangular foundations are presented in detail, together with comparisons with other methods, in order to document the accuracy of the method. The main advantages of the proposed method are that, unlike frequency domain techniques, it provides directly the transient response and forms the basis for extension to the case of non-linear behaviour.     

Markov chain Monte Carlo algorithms for target-oriented and interval-oriented amplitude versus angle inversions with non-parametric priors and non-linear forward modellings

In geophysical inverse problems, the posterior model can be analytically assessed only in case of linear forward operators, Gaussian, Gaussian mixture, or generalized Gaussian prior models, continuous model properties, and Gaussian-distributed noise contaminating the observed data. For this reason, one of the major challenges of seismic inversion is to derive reliable uncertainty appraisals in cases of complex prior models, non-linear forward operators and mixed discrete-continuous model parameters. We present two amplitude versus angle inversion strategies for the joint estimation of elastic properties and litho-fluid facies from pre-stack seismic data in case of non-parametric mixture prior distributions and non-linear forward modellings. The first strategy is a two-dimensional target-oriented inversion that inverts the amplitude versus angle responses of the target reflections by adopting the single-interface full Zoeppritz equations. The second is an interval-oriented approach that inverts the pre-stack seismic responses along a given time interval using a one-dimensional convolutional forward modelling still based on the Zoeppritz equations. In both approaches, the model vector includes the facies sequence and the elastic properties of P-wave velocity, S-wave velocity and density. The distribution of the elastic properties at each common-mid-point location (for the target-oriented approach) or at each time-sample position (for the time-interval approach) is assumed to be multimodal with as many modes as the number of litho-fluid facies considered. In this context, an analytical expression of the posterior model is no more available. For this reason, we adopt a Markov chain Monte Carlo algorithm to numerically evaluate the posterior uncertainties. With the aim of speeding up the convergence of the probabilistic sampling, we adopt a specific recipe that includes multiple chains, a parallel tempering strategy, a delayed rejection updating scheme and hybridizes the standard Metropolis–Hasting algorithm with the more advanced differential evolution Markov chain method. For the lack of available field seismic data, we validate the two implemented algorithms by inverting synthetic seismic data derived on the basis of realistic subsurface models and actual well log data. The two approaches are also benchmarked against two analytical inversion approaches that assume Gaussian-mixture-distributed elastic parameters. The final predictions and the convergence analysis of the two implemented methods proved that our approaches retrieve reliable estimations and accurate uncertainties quantifications with a reasonable computational effort.     

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.     

Dynamics of elastic–plastic shear frames with secondary structures: shake table and numerical studies

Results from experimental and numerical studies of earthquake‐excited small‐scale primary–secondary structures are presented. The primary structure considered is a plane three‐storey shear frame with a fundamental frequency of 5.5 Hz. The columns of the first floor are built with soft aluminium and they are stressed beyond its linear range of behaviour. After each test the elastic–plastic columns are replaced by a new set of undeformed virgin aluminium bars. The elastic–plastic shear frame is tested with and without an attached secondary structure. The secondary structure is modelled as an elastic SDOF oscillator, and its natural frequency is tuned to the fundamental frequency of the shear frame. Alternatively, the oscillator is mounted on the horizontal beam of the second and third floor. The base excitation of the structural model is characterized by a broad band random process with constant spectral density in a frequency range between 3 and 30 Hz. In the numerical study, the digital recorded acceleration of the base excites the mechanical model of the investigated structures. Numerical outcomes assuming fictitious unlimited elastic material behaviour of the shear frame are set in contrast to results from experiments and computational simulations where the measured non‐linear force displacement relation of the elastic–plastic floor is approximated by a piecewise linear curve. The effect of elastic–plastic materials on the dynamic interaction between primary and secondary structure is shown and the difference to unlimited elastic material behaviour is worked out in detail. Copyright © 2001 John Wiley & Sons, Ltd.     

Axisymmetric transients in shells of revolution

The propagation of axisymmetric transients in shells of revolution subjected to impulsive boundary loads is treated in this paper. Consideration is restricted to linear, elastic, isotropic, homogeneous shells of revolution with straight line generators. 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 asymptotic series solutions to the equations of motion. A numerical superposition technique which enables the calculation of long time response is developed. Several numerical examples are presented in order to illustrate the method.     

Transversely isotropic media equivalent to thin isotropic layers1

Any set of isotropic layers is equivalent, in the long wavelength limit, to a unique transversely isotropic (TI) layer; to find the elastic moduli of that layer is a solved problem. The converse problem is to find a set of isotropic layers equivalent to a given TI media. Here, explicit necessary and sufficient conditions on the TI stiffness moduli for the existence of an equivalent set of isotropic layers are found by construction of a minimal decomposition consisting of either two or three isotropic constituent layers. When only two constituents are required, their elastic properties are uniquely determined. When three constituents are required, two have the same Poisson's ratio and the same thickness fraction, and even then there is a one-parameter family of satisfactory minimal decompositions. The linear slip model for fractured rock (aligned fractures in an isotropic background) yields a restricted range of transverse isotropy dependent on only four independent parameters. If the ratio of the normal to tangential fracture compliance is small enough, the medium is equivalent to thin isotropic layering and in general its minimal decomposition consists of three constituents.     

Elasto-plastic earthquake shear-response of one-dimensional earth dam models

A simplified analysis procedure for the non-linear hysteretic earthquake-response of earth dams is presented. The dam is modelled as a one-dimensional hysteretic shear-wedge subjected to base excitation. The hysteretic stress-strain behaviour of the dam materials is modelled by using elasto-plastic constitutive equations based on multi-surface kinematic plasticity theory. The method is based on a Galerkin formulation of the equations of motion in which the solution is expanded using eigenmodes of the linearized problem defined over the spatial domain occupied by the dam. The technique is applied to analyse the non-linear dynamic response of an earth dam subject to two very different input ground motions. The following investigations are presented: (i) comparison between the results obtained using two soil models depicting different nonlinear properties, (ii) comparison between the results of the one-mode and the multi-mode solution expansions, (iii) comparison with the results obtained through an elaborate finite element representation of the dam, and finally, (iv) comparison with the results obtained through the Makdisi-Seed¹¹ iterative procedure for earth dam analysis. The comparisons show that the proposed technique can be used to determine adequately the transient earthquake response of long earth dams. Furthermore, the efficiency and low computational cost make the technique very attractive; it can easily and systematically be extended to two- and three-dimensional calculations of earth dam response.     

Kinematic internal forces in deep foundations with inclined piles

This paper presents a parametric study that looks into the influence of pile rake angle on the kinematic internal forces of deep foundations with inclined piles. Envelopes of maximum kinematic bending moments, shear forces and axial loads are presented along single inclined piles and 2 × 2 symmetrical square pile groups with inclined elements subjected to an earthquake generated by vertically incident shear waves. Inclination angles from 0° to 30° are considered, and three different pile–soil stiffness ratios are studied. These results are obtained through a frequency–domain analysis using a boundary element–finite element code in which the soil is modelled by the boundary element method as a homogeneous, viscoelastic, unbounded region, and the piles are modelled by finite elements as Euler–Bernoulli beams. The rotational kinematic response of the pile foundations is shown to be a key factor on the evolution of the kinematic internal forces along the foundations. Copyright © 2015 John Wiley & Sons, Ltd.     

The impact of different clay minerals on the anisotropy of clay matrix in shale

Although clay is composed of disconnected anisotropic clay platelets, many rock physics models treat the clay platelets in shale as interconnected. However, the clay matrix in shales can be modelled as anisotropic clay platelets embedded within a soft isotropic interplatelet region, allowing the influence of disconnected clay platelets on the elastic properties of the clay matrix to be analysed. In this model, properties of the interplatelet region are governed by its effective bulk and shear moduli, whereas the effective properties of the clay platelets are governed by their volume fraction, aspect ratio and elastic stiffness tensor. Together, these parameters implicitly account for variations in clay and fluid properties, as well as fluid saturation. Elastic stiffnesses of clay platelets are obtained from the literature, including both experimental measurements and first-principles calculations of the full anisotropic (monoclinic or triclinic) elastic stiffness tensors of layered silicates. These published elastic stiffness tensors are used to compile a database of equivalent transverse isotropic elastic stiffness tensors, and other physical properties, for eight common varieties of layered silicates. Clay matrix anisotropy is then investigated by examining the influence of these different elastic stiffnesses, and of varying model parameters, upon the effective transverse isotropic elastic stiffness tensor of the clay matrix. The relationship between the different clay minerals and their associated anisotropy parameters is studied, and their impact on the resulting anisotropy of the clay matrix is analysed.     

Dynamic stiffness and kinematic response of single piles in inhomogeneous soil

The effect of soil inhomogeneity on dynamic stiffness and kinematic response of single flexural elastic piles to vertically-propagating seismic SH waves is explored. A generalized parabolic function is employed to describe the variable shear wave propagation velocity in the inhomogeneous stratum. A layered soil with piece-wise homogeneous properties is introduced to approximate the continuous inhomogeneity in the realm of a Beam-on-Dynamic-Winkler-Foundation model. The problem is treated numerically by means of a layer transfer-matrix (Haskell–Thompson) formulation, and validated using available theoretical solutions and finite-element analyses. The role of salient model parameters such as pile-head fixity conditions, pile-to-soil stiffness ratio, surface-to-base shear wave velocity ratio and rate of inhomogeneity is elucidated. A new normalization scheme for inertial and kinematic response of such systems is presented based on an average Winkler wavenumber. With reference to long piles in moderately inhomogeneous soils, results indicate that: (a) kinematic pile response is essentially governed by a single dimensionless frequency parameter accounting for pile-to-soil stiffness ratio, pile slenderness and soil inhomogeneity and (b) definition of a characteristic pile wavelength allows an approximate estimation of pile elastodynamic response for preliminary design or analysis. Issues related to domain discretization and Winkler moduli are discussed.     

An efficient wave extrapolation method for anisotropic media with tilt

Wavefield extrapolation operators for elliptically anisotropic media offer significant cost reduction compared with that for the transversely isotropic case, particularly when the axis of symmetry exhibits tilt (from the vertical). However, elliptical anisotropy does not provide accurate wavefield representation or imaging for transversely isotropic media. Therefore, we propose effective elliptically anisotropic models that correctly capture the kinematic behaviour of wavefields for transversely isotropic media. Specifically, we compute source‐dependent effective velocities for the elliptic medium using kinematic high‐frequency representation of the transversely isotropic wavefield. The effective model allows us to use cheaper elliptic wave extrapolation operators. Despite the fact that the effective models are obtained by matching kinematics using high‐frequency asymptotic, the resulting wavefield contains most of the critical wavefield components, including frequency dependency and caustics, if present, with reasonable accuracy. The methodology developed here offers a much better cost versus accuracy trade‐off for wavefield computations in transversely isotropic media, particularly for media of low to moderate complexity. In addition, the wavefield solution is free from shear‐wave artefacts as opposed to the conventional finite‐difference‐based transversely isotropic wave extrapolation scheme. We demonstrate these assertions through numerical tests on synthetic tilted transversely isotropic models.     

Probabilistic critical excitation for MDOF elastic–plastic structures on compliant ground

Earthquake ground motions and their effects on structural responses are very uncertain even with the present knowledge. It is therefore desirable to develop a robust structural design method taking into account these uncertainties. Critical excitation approaches are promising and a new random critical excitation method is proposed for MDOF elastic–plastic shear‐building structures on compliant ground. The power (area of power spectral density (PSD) function) and the intensity (magnitude of PSD function) are fixed and the critical excitation is found under these restrictions. In contrast to linear elastic structures, transfer functions and simple expressions for response evaluation cannot be defined in elastic–plastic structures and difficulties arise in describing the peak responses except by laborious elastic–plastic time‐history response analysis. Statistical equivalent linearization is used to estimate the elastic–plastic stochastic peak responses approximately. The critical excitation responses are obtained for several examples and compared with those of the corresponding recorded earthquake ground motion. Copyright © 2001 John Wiley & Sons, Ltd.     

S‐wave kinematics in acoustic transversely isotropic media with a vertical symmetry axis

Acoustic transversely isotropic models are widely used in seismic exploration for P‐wave processing and analysis. In isotropic acoustic media only P‐wave can propagate, while in an acoustic transversely isotropic medium both P and S waves propagate. In this paper, we focus on kinematic properties of S‐wave in acoustic transversely isotropic media. We define new parameters better suited for S‐wave kinematics analysis. We also establish the travel time and relative geometrical spreading equations and analyse their properties. To illustrate the behaviour of the S‐wave in multi‐layered acoustic transversely isotropic media, we define the Dix‐type equations that are different from the ones widely used for the P‐wave propagation.     

Simulation of full-waveform log in saturated cracked formations using Hudson's approach

We study the propagation of elastic waves that are generated in a fluid‐filled borehole surrounded by a cracked transversely isotropic medium. In the model studied the anisotropy and borehole axes coincide. To obtain the effective elastic moduli of a cracked medium we have applied Hudson's theory that enables the determination of the overall properties as a function of the crack orientation in relation to the symmetry axis of the anisotropic medium. This theory takes into account the hydrodynamic mechanism of the elastic‐wave attenuation caused by fluid filtration from the cracks into a porous matrix. We have simulated the full waveforms generated by an impulse source of finite length placed on the borehole axis. The kinematic and dynamic parameters of the compressional, shear and Stoneley waves as functions of the matrix permeability, crack orientation and porosity were studied. The modelling results demonstrated the influence of the crack‐system parameters (orientation and porosity) on the velocities and amplitudes of all wave types. The horizontally orientated cracks result in maximal decrease of the elastic‐wave parameters (velocities and amplitudes). Based on the fact that the shear‐ and Stoneley‐wave velocities in a transversely isotropic medium are determined by different shear moduli, we demonstrate the feasibility of the acoustic log to identify formations with close to horizontal crack orientations.     

Mechanical formulations for bilinear and trilinear hysteretic models used in base isolators

The best known model for numerically simulating the hysteretic behavior of various structural components is the bilinear hysteretic system. There are two possible mechanical formulations that correspond to the same bilinear model from a mathematical viewpoint. The first one consists of a linear elastic spring connected in series with a parallel system comprising a plastic slider and a linear elastic spring, while the second one comprises a linear elastic spring connected in parallel with an elastic-perfectly plastic system. However, the bilinear hysteretic model is unable to describe either softening or hardening effects in these components. In order to account for this, the bilinear model is extended to a trilinear one. Thus, two trilinear hysteretic models are developed and numerically tested, and the analysis shows that both exhibit three plastic phases. More specifically, the first system exhibits one elastic phase, while the second one exhibits two elastic phases according to the level of strain amplitude. Next, the change of slope between the plastic phases in unloading does not occur at the same displacement level in the two models. Furthermore, the dissipated energy per cycle in the first trilinear model, as proven mathematically and explained physically, decreases in the case of hardening and increases in the case of softening, while in the second trilinear model the dissipated energy per cycle remains unchanged, as is the case with the bilinear model. Numerical examples are presented to quantify the aforementioned observations made in reference to the mechanical behavior of the two trilinear hysteretic models. Finally, a set of cyclic shear tests over a wide range of strain amplitudes on a high damping rubber bearing is used in the parameter identification of the two different systems, namely (a) trilinear hysteretic models of the first type connected in parallel, and (b) trilinear hysteretic models of the second type also connected in parallel. The results show that the complex nonlinear shear behavior of high damping rubber bearings can be correctly simulated by a parallel system which consists of only one component, namely the trilinear hysteretic system of the first type. The second parallel system was not able to describe the enlargement of the dissipated hysteresis area for large strain amplitudes.     

Axisymmetric vibration of an elastic circular plate bonded on a transversely isotropic multilayered half-space

Based on the analytical layer-element method, an analytical solution is proposed to determine the dynamic interaction between the elastic circular plate and transversely isotropic multilayered half-space. The dynamic response of the elastic circular plate is governed by the classical thin-plate theory with the assumption that the contact surface between the plate and soil is frictionless. The total stiffness matrix of the transversely isotropic multilayered half-space is acquired by assembling the analytical layer-element of each soil layer with the aid of the continuity conditions between adjacent layers. According to the displacement condition of coordination between the plate and soil, the dynamic interaction problem is reduced to that of multilayered transversely isotropic half-space subjected to axisymmetric harmonic vertical loading. Some numerical examples are given to study the vertical vibration of the plate, and the results indicate that the dynamic response of elastic circular plate depends strongly on the material properties of the soils, the rigidity of the plate, the frequency of excitation and the external load form.     

Prevailing-frequency approximation of the coupling ray theory along the SH and SV reference rays in a heterogeneous generally anisotropic medium which is approximately uniaxial

The behaviour of the actual polarization of an electromagnetic wave or elastic S–wave is described by the coupling ray theory, which represents the generalization of both the zero–order isotropic and anisotropic ray theories and provides continuous transition between them. The coupling ray theory is usually applied to anisotropic common reference rays, but it is more accurate if it is applied to reference rays which are closer to the actual wave paths. In a generally anisotropic or bianisotropic medium, the actual wave paths may be approximated by the anisotropic–ray–theory rays if these rays behave reasonably. In an approximately uniaxial (approximately transversely isotropic) anisotropic medium, we can define and trace the SH (ordinary) and SV (extraordinary) reference rays, and use them as reference rays for the prevailing–frequency approximation of the coupling ray theory. In both cases, i.e. for the anisotropic–ray–theory rays or the SH and SV reference rays, we have two sets of reference rays. We thus obtain two arrivals along each reference ray of the first set and have to select the correct one. Analogously, we obtain two arrivals along each reference ray of the second set and have to select the correct one. In this paper, we suggest the way of selecting the correct arrivals. We then demonstrate the accuracy of the resulting prevailing–frequency approximation of the coupling ray theory using elastic S waves along the SH and SV reference rays in four different approximately uniaxial (approximately transversely isotropic) velocity models.