Showing papers in "Journal of Engineering Mechanics-asce in 2006"

Generalized Bouc-Wen model for highly asymmetric hysteresis

Abstract: Bouc-Wen class models have been widely used to efficiently describe smooth hysteretic behavior in time history and random vibration analyses. This paper proposes a generalized Bouc-Wen model with sufficient flexibility in shape control to describe highly asymmetric hysteresis loops. Also introduced is a mathematical relation between the shape-control parameters and the slopes of the hysteresis loops, so that the model parameters can be identified systematically in conjunction with available parameter identification methods. For use in nonlinear random vibration analysis by the equivalent linearization method, closed-form expressions are derived for the coefficients of the equivalent linear system in terms of the second moments of the response quantities. As an example application, the proposed model is successfully fitted to the highly asymmetric hysteresis loops obtained in laboratory experiments for flexible connectors used in electrical substations. The model is then employed to investigate the effect of dynamic interaction between interconnected electrical substation equipment by nonlinear time-history and random vibration analyses.

Simulation of Crack Propagation in Asphalt Concrete Using an Intrinsic Cohesive Zone Model

Abstract: This is a practical paper which consists of investigating fracture behavior in asphalt concrete using an intrinsic cohesive zone model (CZM). The separation and traction response along the cohesive zone ahead of a crack tip is governed by an exponential cohesive law specifically tailored to describe cracking in asphalt pavement materials by means of softening associated with the cohesive law. Finite-element implementation of the CZM is accomplished by means of a user subroutine using the user element capability of the ABAQUS software, which is verified by simulation of the double cantilever beam test and by comparison to closed-form solutions. The cohesive parameters of finite material strength and cohesive fracture energy are calibrated in conjunction with the single-edge notched beam [SE(B)] test. The CZM is then extended to simulate mixed-mode crack propagation in the SE(B) test. Cohesive elements are inserted over an area to allow cracks to propagate in any direction. It is shown that the simulated crack trajectory compares favorably with that of experimental results.

Viscoelastic Modeling and Field Validation of Flexible Pavements

Abstract: The objective of this study was to characterize hot-mix asphalt (HMA) viscoelastic properties at intermediate and high temperatures and to incorporate laboratory-determined parameters into a three-dimensional finite element (FE) model to accurately simulate pavement responses to vehicular loading at different temperatures and speeds. Results of the developed FE model were compared against field-measured pavement responses from the Virginia Smart Road. Results of this analysis indicated that the elastic theory grossly underpredicts pavement responses to vehicular loading at intermediate and high temperatures. In addition, the elastic FE model could not simulate permanent deformation or delayed recovery, a known characteristic of HMA materials. In contrast, results of the FE viscoelastic model were in better agreement with field measurements. In this case, the average error in the prediction was less than 15%. The FE model successfully simulated retardation of the response in the transverse direction and rapid relaxation of HMA in the longitudinal direction. Moreover, the developed model allowed predicting primary rutting damage at the surface and its partial recovery after load application.

Experimental Investigation and Fracture Analysis of Debonding between Concrete and FRP Sheets

Abstract: The last few years have witnessed a wide use of externally bonded fiber reinforced polymer (FRP) sheets for strengthening existing reinforced and prestressed concrete structures. The success of this strengthening method relies on the effectiveness of the load-transfer between the concrete and the FRP. Understanding the stress transfer and the failure of the concrete–FRP interface is essential for assessing the structural performance of strengthened beams and for evaluating the strength gain. This paper describes an experimental investigation of the interfacial bond behavior between concrete and FRP. The strain distributions in concrete and FRP are determined using an optical technique known as digital image correlation. The results confirm that the debonding process can be described in terms of crack propagation through the interface between concrete and FRP. The data obtained from the analysis of digital images was used to determine the interfacial material behavior for the concrete–FRP interface (stress...

Unified Sand Model Based on the Critical State and Generalized Plasticity

Abstract: Based on the critical state concept and with the use of a state parameter, a unified generalized plasticity model is proposed for sand. The model uses a nonlinear critical state line. The plastic modulus, loading vectors and plastic flow direction vectors of a generalized plasticity model were modified so that they depend on the state parameter. With a single set of parameters, the model simulates the stress-deformation behavior of sand of different densities and pressure levels, under both drained and undrained conditions. A total of 12 parameters are required for monotonic loading and additional five parameters are included to consider cycling loading. The model is calibrated using the results of a minimum of two triaxial compression tests conducted on specimens of different densities and confining pressures. The model has been validated against the monotonic and cyclic test results of Toyoura sand, Nevada sand, and Fuji River sand. The comparison between simulations and test results showed that the model is capable of simulating sophisticated sand behavior. Its limitation in simulating monotonic loading following series of cyclic loadings of dense sand is discussed.

Input Parameters of Discrete Element Methods

Abstract: This paper presents a sensitivity analysis of the input parameters of a program based on the discrete element method (DEM). Triaxial compression simulations were conducted on an assembly of ellipsoids with two particle shapes. We examine four input parameters including shear modulus of particles, density of particles, time step, and damping. Generally, these parameters are chosen by calibrating the result with certain known behavior of granular materials. In dynamic simulations, these input parameters are bounded by their physical attributions that should not be altered. However, in static simulations, they do not have the same physical implication. Validity of results may be questionable when input parameters are used without justification. A sensitivity analysis of the input parameters should shed light on this issue. In this paper, we will study the effect of the input values within the range of (1∕10) –10 times the benchmark value. The benchmark values are commonly used by the writer. The results are ...

Innovative Bridge Condition Assessment from Dynamic Response of a Passing Vehicle

Abstract: An innovative approach for damage assessment of a bridge deck is proposed with the measured dynamic response of a vehicle moving on top of a structure. The simply supported bridge deck is modeled as a Euler–Bernoulli beam. The moving vehicle serves as a smart sensor and force transducer in the structural system. The damage is defined as the flexural stiffness reduction in the beam finite element. The identification algorithm is based on dynamic response sensitivity analysis, and it is realized with a regularization technique from the measured vehicle acceleration measurement. Measurement noise, road surface roughness, and model errors are included in the simulations, and the results indicate that the proposed algorithm is computationally stable and efficient, and the identified results are acceptable and not sensitive to the different parameters studied.

Fully Nonlinear Boussinesq-Type Equations for Waves and Currents over Porous Beds

Abstract: The paper introduces a complete set of Boussinesq-type equations suitable for water waves and wave-induced nearshore circulation over an inhomogeneous, permeable bottom. The derivation starts with the conventional expansion of the fluid particle velocity as a polynomial of the vertical coordinate z followed by the depth integration of the vertical components of the Euler equations for the fluid layer and the volume-averaged equations for the porous layer to obtain the pressure field. Inserting the kinematics and pressure field into the Euler and volume-averaged equations on the horizontal plane results in a set of Boussinesq-type momentum equations with vertical vorticity and z -dependent terms. A new approach to eliminating the z dependency in the Boussinesq-type equations is introduced. It allows for the existence and advection of the vertical vorticity in the flow field with the accuracy consistent with the level of approximation in the Boussinesq-type equations for the pure wave motion. Examination of...

Behavior of Corrugated Web I-Girders under In-Plane Loads

Abstract: A theoretical formulation of the linear elastic in-plane and torsional behavior of corrugated web I-girders under in-plane loads is presented. A typical corrugated web steel I-girder consists of two steel flanges welded to a corrugated steel web. Under a set of simplifying assumptions, the equilibrium of an infinitesimal length of a corrugated web I-girder is studied, and the cross-sectional stresses and stress resultants due to primary bending moment and shear are deduced. The analysis shows that a corrugated web I-girder will twist out-of-plane simultaneously as it deflects in-plane under the action of in-plane loads. In the paper, the in-plane bending behavior is analyzed using conventional beam theory, whereas the out-of-plane torsional behavior is analyzed as a flange transverse bending problem. The results for a simply supported span subjected to a uniformly distributed load are presented. Finally, finite element analysis results are presented and compared to the theoretical results for validation.

Flexibility-Based Damage Localization from Stochastic Realization Results

Abstract: Localization of damage from changes in flexibility in structures identified from operational or ambient vibration has long been hindered by the fact that the flexibility cannot be extracted exclusively from output signals. This obstacle is substantially removed here by noting that the null space of the flexibility change contains the damage localization information and that vectors in this null space can be estimated from output signals without having explicit flexibility matrices. The central pillar of the contribution is a derivation showing that although the spatial distribution of the input that generates the observed output is unknown, the state-space triplet { Ac Bc Dc −1 Cc } connected with a collocated input distribution (for a model order up to twice the number of sensors) can be extracted from output measurements. The fact that vectors in the null space of the change in flexibility can be extracted from this triplet falls from the expression for the flexibility in terms of the state-space matric...

Evaluation of Karhunen–Loève, Spectral, and Sampling Representations for Stochastic Processes

Abstract: The Karhunen–Loeve, spectral, and sampling representations, referred to as the KL, SP, and SA representations, are defined and some features/limitations of KL-, SP-, and SA-based approximations commonly used in applications are stated. Three test applications are used to evaluate these approximate representations. The test applications include (1) models for non-Gaussian processes; (2) Monte Carlo algorithms for generating samples of Gaussian and non-Gaussian processes; and (3) approximate solutions for random vibration problems with deterministic and uncertain system parameters. Conditions are established for the convergence of the solutions of some random vibration problems corresponding to KL, SP, and SA approximate representations of the input to these problems. It is also shown that the KL and SP representations coincide for weakly stationary processes.

Modal Strain Energy Decomposition Method for Damage Localization in 3D Frame Structures

Abstract: This article presents a newly developed modal strain energy decomposition method for damage localization that is capable of identifying damage to individual members of three-dimensional (3D) frame structures. This method is based on decomposing the modal strain energy of each structural member (or element) into two parts, one associated with the element’s axial coordinates and the other with its transverse coordinates. In turn, two damage indicators are calculated for each member to perform the damage localization analysis. Implementing this method requires only a small number of mode shapes identified from both the damaged and baseline structures. Numerical studies are conducted of a 3D five-story frame structure and also a complicated offshore template platform, based on synthetic data generated from finite-element models. In addition to providing theoretical insights to illustrate the advantages of using this newly developed method, this article also demonstrates numerically that the new method is capa...

Seismic Fragility Analysis of Structural Systems

Abstract: A method is presented for the evaluation of the seismic fragility function of realistic structural systems. The method is based on a preliminary, limited, simulation involving nonlinear dynamic analyses performed to establish the probabilistic characterization of the demands on the structure, followed by the solution of a general system reliability problem with correlated demands and capacities. The results compare favorably with the fragility obtained by plain Monte Carlo simulation, while the associated computational effort is orders of magnitude lower. The method is demonstrated with two applications, a steel-concrete box girder viaduct with RC piers subjected to both uniform and nonuniform excitations, and a three-dimensional RC building structure subjected to bidirectional excitation.

Bayesian State Estimation Method for Nonlinear Systems and Its Application to Recorded Seismic Response

Abstract: The focus of this paper is to demonstrate the application of a recently developed Bayesian state estimation method to the recorded seismic response of a building and to discuss the issue of model selection. The method, known as the particle filter, is based on stochastic simulation. Unlike the well-known extended Kalman filter, it is applicable to highly nonlinear systems with non-Gaussian uncertainties. The particle filter is applied to strong motion data recorded in the 1994 Northridge earthquake in a seven-story hotel whose structural system consists of nonductile reinforced-concrete moment frames, two of which were severely damaged during the earthquake. We address the issue of model selection. Two identification models are proposed: a time-varying linear model and a simplified timevarying nonlinear degradation model. The latter is derived from a nonlinear finite-element model of the building previously developed at Caltech. For the former model, the resulting performance is poor since the parameters need to vary significantly with time in order to capture the structural degradation of the building during the earthquake. The latter model performs better because it is able to characterize this degradation to a certain extent even with its parameters fixed. For this case study, the particle filter provides consistent state and parameter estimates, in contrast to the extended Kalman filter, which provides inconsistent estimates. It is concluded that for a state estimation procedure to be successful, at least two factors are essential: an appropriate estimation algorithm and a suitable identification model.

Lagrangian Approach to Structural Collapse Simulation

Abstract: Computer analysis of structures has traditionally been carried out using the displacement method combined with an incremental iterative scheme for nonlinear problems. In this paper, a Lagrangian approach is developed, which is a mixed method, where besides displacements, the stress resultants and other variables of state are primary unknowns. The method can potentially be used for the analysis of collapse of structures subjected to severe vibrations resulting from shocks or dynamic loads. The evolution of the structural state in time is provided a weak formulation using Hamilton's principle. It is shown that a certain class of structures, known as reciprocal structures, has a mixed Lagrangian formulation in terms of displacements and internal forces. The form of the Lagrangian is invariant under finite displacements and can be used in geometric nonlinear analysis. For numerical solution, a discrete variational integrator is derived starting from the weak formulation. This integrator inherits the energy and momentum conservation characteristics for conservative systems and the contractivity of dissipative systems. The integration of each step is a constrained minimization problem and it is solved using an augmented Lagrangian algorithm. In contrast to the traditional displacement-based method, the Lagrangian method provides a generalized formulation which clearly separates the modeling of components from the numerical solution. Phenomenological models of components, essential to simulate collapse, can be incorporated without having to implement model-specific incremental state determination algorithms. The state variables are determined at the global level by the optimization method.

Efficacy of Hilbert and Wavelet Transforms for Time-Frequency Analysis

Abstract: Two independently emerging time-frequency transformations in Civil Engineering, namely, the wavelet transform and empirical mode decomposition with Hilbert transform (EMD+HT) , are discussed in this study. Their application to a variety of nonstationary and nonlinear signals has achieved mixed results, with some comparative studies casting significant doubt on the wavelet’s suitability for such analyses. Therefore, this study shall revisit a number of applications of EMD+HT in the published literature, offering a different perspective to these commentaries and highlighting situations where the two approaches perform comparably and others where one offers an advantage. As this study demonstrates, much of the differing performance previously observed is attributable to EMD+HT representing nonlinear characteristics solely through the instantaneous frequency, with the wavelet relying on both this measure and the instantaneous bandwidth. Further, the resolutions utilized by the two approaches present a seconda...

Determining the Tensile Stress-Crack Opening Curve of Concrete by Inverse Analysis

Abstract: The determination of the fundamental stress versus crack opening (σ-w) response of concrete under uniaxial tension is performed in this study through inverse analysis using data from notched beam tests. The procedure used for optimizing the parameters of the σ-w relation using the load versus crack mouth opening displacement response of the notched beam is described. Satisfactory comparisons have been obtained between the σ-w curves obtained through the inverse analysis and those directly measured in uniaxial tension tests. The use of weighting functions in the inverse analysis may be necessary when large crack widths are to be considered.

Revisiting Multimode Coupled Bridge Flutter: Some New Insights

Abstract: Better understanding of the bimodal coupled bridge flutter involving fundamental vertical bending and torsional modes offers valuable insight into multimode coupled flutter, which has primarily been the major concern in the design of long span bridges. This paper presents a new framework that provides closed-form expressions for estimating modal characteristics of bimodal coupled bridge systems and for estimating the onset of flutter. Though not intended as a replacement for complex eigenvalue analysis, it provides important physical insight into the role of self-excited forces in modifying bridge dynamics and the evolution of intermodal coupling with increasing wind velocity. The accuracy and effectiveness of this framework are demonstrated through flutter analysis of a cable-stayed bridge. Based on this analysis scheme, the role of bridge structural and aerodynamic characteristics on flutter, which helps to better tailor the structural systems and deck sections for superior flutter performance, is emphasized. Accordingly, guidance on the selection of critical structural modes and the role of different force components in multimode coupled flutter are delineated. The potential significance of the consideration of intermodal coupling in predicting torsional flutter is highlighted. Finally, clear insight concerning the role of drag force to bridge flutter is presented.

Optimal vibration absorber with nonlinear viscous power law damping and white noise excitation

Abstract: A tuned mass damper with a nonlinear power law viscous damper excited by white noise is considered. The system is analyzed by statistical linearization and stochastic simulation with the objective of minimizing the standard deviation of the response. It is shown that the optimal parameters for the tuned mass damper are unaffected by the magnitude of the structural damping in the linear case. However, in the nonlinear case the structural damping influences the equivalent parameters obtained by statistical linearization and thereby indirectly the optimal values for the damper parameters. Results from stochastic simulation show good agreement with results from statistical linearization in terms of the standard deviation of the response. It is shown that the optimal damping, which can be obtained by the passive device, is the same for the linear and nonlinear damper. However, for the nonlinear tuned mass damper the optimal parameters will depend on both structural damping and excitation intensity (or vibration amplitude). The results are presented in such a way that they can be used directly for the design of a tuned mass damper with damping governed by a nonlinear viscous power law.

Experimental Verification of Smart Cable Damping

Abstract: Stay cables, such as are used in cable-stayed bridges, are prone to vibration due to their low inherent damping characteristics. Transversely attached passive viscous dampers have been implemented in many bridges to dampen such vibration. However, only minimal damping can be added if the attachment point is close to the bridge deck. For longer bridge cables, the relative attachment point becomes increasingly smaller, and passive damping may become insufficient. A recent analytical study by the authors demonstrated that “smart” semiactive damping can provide increased supplemental damping. This paper experimentally verifies a smart damping control strategy employing H2 ∕linear quadratic Gaussian (LQG) clipped optimal control using only force and displacement measurements at the damper for an inclined flat-sag cable. A shear mode magnetorheological fluid damper is attached to a 12.65 m inclined flat-sag steel cable to reduce cable vibration. Cable response is seen to be substantially reduced by the smart da...

Optimal design with probabilistic objective and constraints

Abstract: Significant challenges are associated with solving optimal structural design problems involving the failure probability in the objective and constraint functions. In this paper, we develop gradient-based optimization algorithms for estimating the solution of three classes of such problems in the case of continuous design variables. Our approach is based on a sequence of approximating design problems, which is constructed and then solved by a semiinfinite optimization algorithm. The construction consists of two steps: First, the failure probability terms in the objective function are replaced by auxiliary variables resulting in a simplified objective function. The auxiliary variables are determined automatically by the optimization algorithm. Second, the failure probability constraints are replaced by a parametrized first-order approximation. The parameter values are determined in an adaptive manner based on separate estimations of the failure probability. Any computational reliability method, including first-order reliability and second-order reliability methods and Monte Carlo simulation, can be used for this purpose. After repeatedly solving the approximating problem, an approximate solution of the original design problem is found, which satisfies the failure probability constraints at a precision level corresponding to the selected reliability method. The approach is illustrated by a series of examples involving optimal design and maintenance planning of a reinforced concrete bridge girder.

Joint First-Passage Probability and Reliability of Systems under Stochastic Excitation

Abstract: The. first-passage probability, describing the probability that a scalar process exceeds a prescribed threshold during an interval of time, is of great engineering interest. This probability is essential for estimating the reliability of a structural component whose response is a stochastic process. When considering the reliability of an engineering system composed of several interdependent components, the probability that two or more response processes exceed their respective safe thresholds during the operation time of the system is an equally essential quantity. This paper proposes simple and accurate formulas for approximating this joint first-passage probability of a vector process. The nth order joint first-passage probability is obtained from a recursive formula involving lower order joint first-passage probabilities and the out-crossing probability of the vector process over a safe domain. Interdependence between the crossings is approximately accounted for by considering the clumping of these events. The accuracy of the proposed formulas is examined by comparing analytical estimates with those obtained from Monte Carlo simulations for stationary Gaussian processes. As an example application, the reliability of a system of interconnected equipment items subjected to a stochastic earthquake excitation is estimated by linear programming bounds employing marginal and joint component fragilities obtained by the proposed formulas.

Computation of Mixed-Mode Stress Intensity Factors for Cracks in Three-Dimensional Functionally Graded Solids

Abstract: This work applies a two-state interaction integral to obtain stress intensity factors along cracks in three-dimensional functionally graded materials. The procedures are applicable to planar cracks with curved fronts under mechanical loading, including crack-face tractions. Interaction-integral terms necessary to capture the effects of material nonhomogeneity are identical in form to terms that arise due to crack-front curvature. A discussion reviews the origin and effects of these terms, and an approximate interaction-integral expression that omits terms arising due to curvature is used in this work to compute stress intensity factors. The selection of terms is driven by requirements imposed by material nonhomogeneity in conjunction with appropriate mesh discretization along the crack front. Aspects of the numerical implementation with (isoparametric) graded finite elements are addressed, and examples demonstrate the accuracy of the proposed method.

Failure and Dilatancy Properties of Sand at Relatively Low Stresses

Abstract: Analysis of geotechnical problems concerned by low confinement such as design of shallow foundations and analysis of slope stability and soil liquefaction requires modeling of the soil behavior at low stresses. This note includes a laboratory study of the behavior of Hostun RF sand at low cell pressure (20–50 kPa) . Isotropic and triaxial compression drained tests were performed. Drained tests show that both failure and dilatancy angles at low stresses are stress dependent. The contractive/dilative phase transition is observed for loose sand, which may result from the overconsolidated nature of this sand for low values of cell pressure.

Hyperelasticity Model for Finite Element Analysis of Natural and High Damping Rubbers in Compression and Shear

Abstract: Rate-independent monotonic behavior of filled natural rubber and high damping rubber is investigated in compression and shear regimes. Monotonic responses obtained from tests conducted in both regimes demonstrate the prominent existence of the Fletcher- Gent effect, indicated by high stiffness at low strain levels. An improved hyperelasticity model for compression and shear regimes is proposed to represent the rate-independent instantaneous and equilibrium responses including the Fletcher-Gent effect. A parameter identification scheme involving simultaneous minimization of least-square residuals of uniaxial compression and simple shear data is delineated. The difficulties of identifying a unique set of hyperelasticity parameters that hold for both compression and shear deformation modes are thus overcome. The proposed hyperelasticity model has been implemented in a general purpose finite element program. Finite element simulations of experiments have shown the adequacy of the proposed hyperelasticity model, estimated parameters, and employed numerical procedures. Finally, numerical experiments were conducted to further explore the potential of the proposed model, and estimated parameters in analyzing rubber layers of a base isolation bearing subjected either to compression or to a combination of compression and shear.

Effects of Environmental Action on Thermal Stress Analysis of Karaj Concrete Arch Dam

Abstract: A three-dimensional finite element analysis is carried out to determine the thermal stresses of a concrete arch dam. Appropriate heat transfer boundary conditions in the dam body are used for air and reservoir temperature as well as solar radiation variations. A finite element model is used to determine annual variation of temperature and thermal stress in the body of Karaj arch dam in Iran as a case study. The rate of convergence of the numerical solution is examined. The temperatures predicted by the model are satisfactorily compared with the instrumentation records at Karaj Dam. Results of the finite element analysis show that probable cracks occur in a very narrow region of the downstream face. Thermal loads have the most significant effects for causing downstream cracks in comparison with self-weight and hydrostatic loads. The cracked areas of the downstream face conform to the regions that have the highest temperature in the downstream face. This can be associated with the solar radiation, which shows that two-dimensional analysis of an arch dam cannot yield accurate results and three-dimensional analysis is necessary.

Safety Analysis of Moving Road Vehicles on a Long Bridge under Crosswind

Abstract: This paper presents a safety analysis of high-sided road vehicles running on a long span cable-stayed bridge when the road vehicle enters a sharp-edged crosswind gust while the bridge is oscillating under fluctuating winds. Road vehicle accidents, including overturning, excessive sideslip, and exaggerated rotation, are defined first. The mathematical model and the equation of motion of coupled road vehicle–bridge systems under crosswind are then established, which include road surface roughness, vehicle suspension, and the sideslip of the vehicle tire relative to the bridge deck in the lateral direction. A case study using a real long cable-stayed bridge and a high-sided road vehicle is finally conducted, and an extensive computational work is performed to obtain a series of accident vehicle speed against mean crosswind speed, by which the decision on the threshold of mean wind speed above which the bridge should be closed to the road vehicle can be made. The obtained accident vehicle speeds are also compared with those for the same vehicles running on the ground. It is shown that the oscillation of the cable-stayed bridge will lower the accident vehicle speed when wind speed reaches a certain level.

Influence of Functionally Graded Material on Buckling of Skew Plates under Mechanical Loads

Abstract: In this technical note, the critical buckling of simply supported functionally graded skew plate subjected to mechanical compressive loads is evaluated using first-order shear deformation theory in conjunction with the finite element approach. The material properties are assumed to vary in the thickness direction according to the power-law distribution in terms of volume fractions of the constituents. The effective material properties are estimated from the volume fractions and the properties of the constituents using the Mori–Tanaka homogenization method. The effects of aspect ratio, material gradient index, and skew angle on the critical buckling loads of functionally graded material plates are highlighted.

Review of The Finite Element Method for Solid and Structural Mechanics, 6th Edition, by O. C. Zienkiewicz and R. L. Taylor

Abstract: The 6th edition of the classic text on The Finite Element Method by O. C. Zienkiewicz has come a long way since it was published first in 1967 by McGraw-Hill, Berkshire, England. The original 272 pages of text on The Finite Element Method in Structural and Continuum Mechanics written in collaboration with Y. K. Cheung, University of Hong Kong, has increased more than six-fold and turned into a comprehensive treatment comprised of three substantial volumes of more than 600 pages each: • The FEM: Its Basis & Fundamentals by Zienkiewicz, Taylor & Zhu • The FEM for Solid & Structural Mechanics by Zienkiewicz & Taylor • The FEM for Fluid Dynamics by Zienkiewicz, Taylor & Nithiarasu This review is limited to the first two volumes of the sixth edition with focus on the second volume because of the interest and background of the reviewer. The first volume on The FEM: Its Basis and Fundamentals of FEM sets the stage for linear applications. Although introductory in nature, this volume is not a text book in the usual sense of classroom use with homework assignments and solution manuals. It is a powerful review of current findings, with numerous historical anecdotes and references to recent and past publications at the end of each of the 19 chapters. Among the classical topics on finite-element approximations in 1D, 2D, and 3D, the first volume addresses mixed element formulations, error estimates, PUM approximations and extended finite-element methods as well as time discretization methods. The second volume on The FEM for Solid and Structural Mechanics contains 19 chapters and two appendices. Roughly speaking, Chapters 1, 2, and 3 set the stage for nonlinear analysis of solids in Chapters 4, 5, and 6 which deal with nonlinear material behavior, nonlinear geometric problems and the combination of these two sources of nonlinearity. Chapter 4 provides a concise

Normal Transmission of S-Wave Across Parallel Fractures with Coulomb Slip Behavior

Abstract: When an elastic wave propagates through a rock mass, its amplitude is attenuated and velocity is slowed due to the presence of fractures. During wave propagation, if the shear stress at a fracture interface reaches the fracture shear strength, the fracture will experience a large shear displacement. This paper presents a study of the normal transmission of S-waves across parallel fractures with Coulomb slip behavior. In our theoretical formulation, the method of characteristics combined with the Coulomb slip model is used to develop a set of recurrence equations with respect to particle velocities and shear stress. These equations are then solved numerically. In a comparison with the theoretical study, numerical modeling using the universal distinct element code (UDEC) has been conducted. A general agreement between UDEC modeling and theoretical analysis is achieved. The magnitude of the transmission coefficient is calculated as a function of shear stress ratio, nondimensional fracture spacing, normalized shear stiffness, and number of fractures. The study shows that the shear stress ratio is the most important factor influencing wave transmission, and the influence of other factors becomes more apparent when the shear stress ratio is small.