International Journal of Geomechanics

Tensile cracking of soil is an important issue that affects the strength and permeability of soil in many geotechnical applications, such as in soil dams and landfill liners. When the size of the fracture process zone is not negligible compared to specimen dimensions, a nonlinear fracture mechanics approach, such as a cohesive crack model, should be used to simulate the experimental results. A critical issue in the simulation of cohesive failure mechanism is the definition of cohesive interactions along fracture surfaces. In this study, bilinear and trilinear cohesive laws are examined based on data sets available in the literature. The inverse method is used for the definition of the cohesive parameters. As a result of this study, some conclusions are established to assess the importance of using a suitable cohesive law shape for a given soil that accurately predicts the cohesive fracture parameters under different material conditions. Finally, based on a tensile strength model of granular soil, a modified model is proposed to interpret the tensile strength of compacted clayey soils.

An analytical solution is presented to predict ground movement induced by a nonuniformly deforming circular tunnel in a heavy elastic half-plane. Simple expression is proposed to describe the nonuniform soil deformation around the tunnel. The tunnel-deforming components (i.e., ground loss, ovalization, vertical translation) can be determined by the solution-inherent mechanism and by fitting from measured field data. The effects of tunnel-deforming components on the resulting displacement field are discussed. Case studies from various ground conditions were used to check the capacity and applicability of the proposed solution. Although elasticity is a rough representation of the soil behavior, the proposed analytical solution can serve as a simple tool for predicting a reasonable ground settlement profile in the preliminary design of tunnels.

The present work aims to study the propagation of Stoneley wave at the common interface of two distinct homogeneous viscoelastic semi-infinite media. The closed-form expression of the velocity equation has been obtained. The real part of the velocity equation corresponds to the phase velocity of Stoneley wave, whereas the imaginary part corresponds to the damped velocity associated with the Stoneley wave. The Lamé ratio associated with Lamé’s first constant for the upper/lower viscoelastic semi-infinite medium and the frictional parameter associated with the shear strain for the upper/lower viscoelastic semi-infinite medium have significant effects on the phase velocity and the damped velocity of Stoneley wave associated with a longitudinal/transversal wave in the upper/lower viscoelastic semi-infinite medium. Numerical computation and graphical illustration were carried out to highlight the important peculiarities of the problem. Moreover, as a special case, it was observed that the velocity equation of propagation of Stoneley wave in the present problem is in good agreement with the velocity equation obtained earlier for the Stoneley wave propagation at the common interface of two distinct homogeneous isotropic semi-infinite media.

A hydromechanical model with explicit fracture flow is presented for the fully coupled analysis of flow and deformation in fractured porous media. Extended finite-element method (XFEM) was utilized to model the fracture discontinuity in the two-dimensional plane-strain mechanical model. Two flow models, a one-dimensional laminar flow within the fracture and a two-dimensional Darcy flow through porous media, were considered. The flow domains were coupled through a mass exchange term (leak-off) accounting for discontinuous Darcy flow velocity across the fracture. Particular attention was given to the coupling of the flow domains with the mechanical model. Spatial and temporal discretization was achieved using the standard Galerkin method and the finite-difference technique, respectively. Unlike the successive coupled models in which the results of the mechanical model are used to update the fracture flow model and vice versa, the fully coupled hydromechanical formulation is solved simultaneously. The model was verified against several closed-form solutions from the literature. The impact of coupled fracture flow and formation flow on the fracture response to external mechanical loading as well as internal hydraulic loading from fluid injection was investigated, and the results are discussed.

Landfill stability and deformation analysis is one of the most important concerns in the design and operation of municipal solid waste (MSW) landfills. Estimation of landfill capacities based on the deformation characteristics of MSW assumes considerable significance in the estimation of additional landfill capacity before closure. In such analysis, it is important to consider mechanisms that occur in the MSW system, such as primary compression, time-dependent mechanical creep, and biodegradation effects. In this article, a simple case of landfill slope representing the MSW landfill of Bangalore city is considered, and the results of a stability and deformation analysis based on the constitutive modeling approach are presented. The implications for the slope stability and deformation response of MSW for a 5-m-high landfill cell are explained in terms of vertical and horizontal deformations, volume changes, and factors of safety, considering the effects of loading, mechanical degradation, and biodegradation. The effects of time on the strength and stability of the landfill cell are also observed. The study shows that analysis using the constitutive modeling approach provides useful insights for landfill operations. The model predicts that the effect of different mechanisms in a MSW landfill is to increase the overall stability of the landfill with time, and the horizontal and vertical deformations help in estimating landfill volume changes. The influence of lift thickness on the stability and performance of the landfill is also studied.

Many soil-structure interaction problems can be idealized as foundation beam problems. In the present work, the tensionless contact problem of an Euler-Bernoulli beam of finite length resting on an elastic-plastic foundation and carrying arbitrarily complex static loads was investigated. Fourth-order difference equations dealing with the vertical displacement of a beam under three different soil support conditions were presented for each beam segment. On the basis of the continuity conditions at the junctions of two adjacent segments, the response of the whole beam was expressed through the response of the initial beam segment in a matrix form. A comparison with the results from the transfer displacement function method (TDFM) showed the expected complete agreement. A simple case was used to illustrate the influence of the foundation models (elastic model, tensionless-elastic model, and tensionless and elastic-plastic model), and the tensionless and nonlinear behavior of the soil on the beam responses. The lift-off length of the beam depends on both the load $P$ and the property of the foundation model regardless of whether it is assumed to be tensionless. The maximum displacement of the beam and the maximum bending moment within the beam are influenced mainly by the acting load and the soil characteristics manifested by the coefficient of soil reaction and the yielding displacement.

The skeleton of soil, which consists of soil particles at various scales, is a complex granular material and displays multiscale and hierarchical mechanical properties. The coupling effects of deformations at different scale levels of the soil structures have great influence on the macroscale mechanical behaviors of the soil. According to the scale divisions of soil and the physical and mechanical effects generated by the interactions between soil particles at different scales, a soil cell element that can describe the internal material information and particle characteristics of soil was constructed. On the basis of this soil cell element, a soil cell element model that can characterize the multiscale mechanical properties of soil is proposed. A series of unconsolidated and undrained triaxial compression tests on saturated, remolded soil samples with a variety of particle combinations was designed to analyze the proposed soil cell element model. The results show that the macrostrength of soil increased with an increase in the density of coordinated microcracks and effective strain gradient. The relationship between the macrostrength of soil and each of these two parameters can be presented as a parabolic function, respectively. The soil cell element model, which establishes the relationship between macrostrength and the intrinsic length scale and the effective strain gradient, can reproduce and predict the multiscale mechanical properties of soil. In the soil cell element model, the intrinsic length scale is a reflection of the geometrical morphology of microcracks, and the effective strain gradient is a reflection of the shape distortion of the mesoscale soil cell element. The experimental data can be well fitted to the soil cell element model. These research results are significant for the development of a multiscale theoretical framework that links different coupling scales.

The prediction of lateral spreading is an important task because of the complexities of lateral-spreading behavior. The aim of this work is to improve an accurate liquefaction-induced lateral-spreading prediction by using multiple regression methods, such as multilinear regression (MLR), multilayer perceptrons (MLPs), and the adaptive neuro-fuzzy inference system (ANFIS). Predictions of lateral spreading from the developed MLR, MLP, and ANFIS models in tractable (susceptible) equation form are obtained and compared with the value predicted using traditional methods. Principal-component analysis is used to evaluate the effects of each input variable on the lateral spreading. On the basis of the comparisons, it is found that the MLP is better than the ANFIS, MLR, and Youd equation for estimating maximum lateral displacement of free-face conditions. For gently sloping ground conditions, however, similar results are obtained with MLP and ANFIS, which are better than the MLR and Youd equation. The MLP model was also tested with data obtained from Adapazari, Turkey, to estimate total lateral displacement.

A novel, meshless numerical method, called general particle dynamics (GPD), is proposed to simulate the initiation, propagation, and coalescence of three-dimensional (3D), pre-existing penetrating and embedded flaws as well as size effects and large deformations of rock materials. On the basis of the nonlinear unified strength criterion, an elastic–brittle–plastic damage model was developed to reflect the initiation, growth, and coalescence of the 3D flaws and the macrofailure of rocklike materials by tracing the propagation of the cracks. Then, growth paths of cracks were captured through the sequence of such damaged particles. In this paper, the GPD code is applied to simulate the macrofailure, large deformation, and size effects of the heterogeneous rocklike materials. The present numerical simulations focus on the effects of sample sizes, the nonoverlapping length and types of flaws on the failure, and the complete stress–strain curves of the rocklike materials. The initiation, propagation, and coalescence processes of the wing cracks, the antiwing cracks, the oblique secondary cracks, the out-of-plane shear cracks, and the quasi-coplanar shear crack in a rocklike sample subjected to uniaxial compression is numerically simulated using GPD3D. The numerical results indicate that the nonoverlapping lengths and types of flaws significantly influence the coalescence types. The numerical results are in good agreement with the experimental results. It is proven that the GPD3D can adequately simulate the failure processes, large deformation, and size effects of the rocklike materials.

This study examined the effects of small-scale airblast experiments on clay soils and compared experimental results with numerical solutions obtained through finite-element simulations. Thirty-three suspended explosive blasts were conducted above clay soils with explosive masses ranging from 0.9 to 100.9 g and suspended heights ranging from 2.5 to 7.6 cm. The experiments were instrumented with airblast sensors and subsurface triaxial geophones to measure vibration energy and air overpressure from the blast events. Laboratory tests were conducted on the experimental soils to obtain geotechnical and shear strength soil properties. Two-dimensional (2D), arbitrary Lagrangian Eulerian (ALE) finite-element simulations were performed using a finite-element software program and compared with the experimental results. Soils were modeled using the Federal Highway Administration (FHWA) soil material model. Air overpressure, ground vibration, and crater geometry data obtained from the experimental blasts were compared with the numerical simulation results. The first-order simulated results compared fairly well with the experimental results, with the exception of simulated crater diameters, which were 1.5 times larger than experimental results. However, stress-response instabilities were observed in the model after the initial stress pulse had propagated through the soil, and the model did not appear to capture postpeak behavior. Therefore, the soil model used in the study is recommended for use only as a first estimate for capturing the response of airblast loading of clay soils in a 2D ALE analysis. More recent models, such as a cap plasticity model or the disturbed state concept (DSC) model, are more applicable if stress path and postpeak behaviors are to be adequately captured. In addition, a three-dimensional analysis of ALE coupled with Lagrange elements should be considered for capturing a more accurate strength response.

Wave propagation in a two-dimensional unbounded domain with rigid bedrock was studied. The study was based on the scaled-boundary finite-element method (SBFEM) combing the continued fraction method. The SBFEM is a novel semianalytical technique, combining the advantages of the finite-element and the boundary-element methods. Meanwhile, SBFEM has unique properties of its own. In this paper, the modified SBFEM was introduced. The original scaling center was replaced by a scaling line, which is more suitable for analyzing the multilayered soil model. As a key point, the modified SBFEM and the dual system, which is used to solve a proposed time domain problem, were combined to derive the governing equations. The derivation process was based on the framework of the Hamiltonian system by introducing dual variables. By using the continued fraction solution and introducing auxiliary variables, the equation of motion was built. More significantly, the efficient, precise time-integration method was first used to solve the global equation of motion. This technology improved the stability of the modified SBFEM greatly. Because the integral interval was divided into very small pieces, five terms of the Taylor expansion were used for each piece. This precision can achieve computer precision. Therefore, an extremely efficient and accurate solution of the SBFEM in time domain was obtained. The substructure method was proposed for solving the multilayered inclined model in engineering practice. Numerical examples demonstrated the accuracy and high efficiency of proposed methods for the complex shape models.

This paper deals with the reflection and refraction (transmission) phenomena that result from a three-dimensional plane-wave incident at the plane interface between two distinct triclinic media under the influence of initial stresses. Analytical expressions of all the phase velocities of reflected and transmitted quasi-compressional qP and quasi-shear waves ${\text{qS}}_1$ and ${\text{qS}}_2$ (or qSV and qSH) under initial stresses were obtained. Closed-form expressions for the amplitude ratios or reflection and refraction coefficients of the three quasi-waves were derived for suitable boundary conditions. It has been shown that reflection and transmission coefficients depend on the angle of incidence, the material constants, and the initial stresses present in the media. For numerical discussion, the variations of reflection and transmission coefficients with the initial stresses and angle of incidence are presented by means of graphs using mathematical software. Some particular remarkable cases are also considered.

The thermomechanical response of geothermal energy pile groups in sand was studied in the present work by using a three-dimensional nonlinear finite-element analysis procedure. A raft was considered on the piles. The stress-strain responses of the piles and raft were considered linear elastic. The stress-strain response of sand was reproduced by using the state parameter-based constitutive clay and sand model (CASM). The CASM was implemented in finite-element software through a user-defined material subroutine. The geothermal energy pile group was analyzed for different combinations of thermal and nonthermal piles in the group in a single layer of sand. Analysis results were studied for the displacement at the base of the pile and the axial stresses in the pile. Parametric sensitivity studies were performed by varying the spacing between the piles. It can be observed from the results that the geothermal piles, when subjected to heating, exhibit higher displacement and axial stress than the piles under only mechanical loading. Differential settlement at the pile base was observed for the groups that contained both thermal and nonthermal piles.

Downward movement of sliding soil mass in earthquake-induced earth slopes may gradually improve the system's stability. Hence, significant change in critical (or yield) acceleration of the slope is anticipated. The existing simplified coupled procedures, however, assume a constant critical acceleration in stick-slip sliding analysis. Fully coupled sliding block analyses with continuously increased critical acceleration were conducted to estimate earthquake-induced permanent displacement of earth slopes. The sliding mass was assumed as a flexible chain moving along the planes with gradually gentler inclinations. The measured data of a slope failure test in a shaking table were used to validate performance of the model. The proposed approach was compared with the previous procedures, which ignored critical acceleration changes due to downward movement of the sliding mass. Accordingly, earth slopes with smaller slip length were more influenced by the downward movement modification, especially in small amounts of initial yield acceleration. A semi-empirical equation is presented based on more than 425,000 modified-coupled analyses of several hundred strong ground motion records. An equation was then developed as a function of acceleration ratio, period ratio, slip length, peak ground velocity, Arias intensity, and earthquake magnitude. Predictions of the developed equation were compared with the available equations, which ignore the role of increased critical acceleration. It is shown that a majority of the previous models predict conservative estimates of seismic permanent displacements compared with that of new equation. The proposed semi-empirical model can be used as an alternative equation of seismic displacement for deterministic and probabilistic evaluations of earthquake-induced landslide hazard.

This work investigated the behavior of embankment models resting on soft soil reinforced with ordinary and encased stone columns (ESCs). Model tests were performed with different spacing distances between stone columns and two length-to-diameter ratios $(L/d)$ of the stone columns, in addition to different embankment heights. A total of 39 model tests were performed on soil with an undrained shear strength of $\text{≈}10 \text{kPa}$. The system consisted of a stone column–supported embankment at different spacing-to-diameter ratios (s/d) of stone columns. Earth pressure cells were used to measure directly the vertical stress on the column for all models, and another cell was placed at the base of the embankment between two columns to measure directly the vertical stress in reinforced soft soil. For embankment models constructed on soft clay reinforced with ESCs, it was found that whether a column was floating or end bearing (resting on a rigid stratum), encasement of the column by a geogrid was most effective in improving the bearing ratio of reinforced soil by approximately 1.29, 1.39, and 1.63 times and 1.4, 1.57, and 1.83 times that of untreated soil, reducing the settlement by approximately 0.71, 0.67, and 0.62 times and 0.63, 0.6, and 0.45 times that of untreated soil for 200-, 250-, and 300-mm embankment heights with $L/d$ = 5 and 8, respectively, and spacing $s=2.5d$. The bearing improvement ratio (bearing capacity of treated-to-untreated soil) increased with decreasing spacing of stone columns for a given embankment. A higher improvement ratio was achieved for the models reinforced with stone columns at $s=2.5d$ at any embankment height.

This paper reports the results of a numerical parametric study focused on the prediction of vertical load distribution and vertical gap compression between precast concrete facing panel units in steel-reinforced soil walls ranging in height from 6 to 24 m. The vertical compression was accommodated by polymeric bearing pads placed at the horizontal joints between panels during construction. This paper demonstrates how gap compression and magnitude of vertical load transmitted between horizontal joints are influenced by joint location along the height of the wall, joint compressibility, and backfill and foundation soil stiffness. The summary plots in this study can be used to estimate the number and type (stiffness) of the bearing pads to ensure a target minimum gap thickness at the end of construction, to demonstrate the relative influence of wall height and different material component properties on vertical load levels and gap compression, or as a benchmark to test numerical models used for project-specific design. The paper also demonstrates that although the load factor (ratio of vertical load at a horizontal joint to weight of panels above the joint) and joint compression are relatively insensitive to foundation stiffness, the total settlement at the top of the wall facing is very sensitive to foundation stiffness. This paper examines the quantitative consequences of using a simple linear compressive stress–strain model for the bearing pads versus a multilinear model that is better able to capture the response of bearing pads taken to greater compression. The study demonstrates that qualitative trends in vertical load factor are preserved when a more advanced stress-dependent stiffness soil hardening model is used for the backfill soil as compared with the simpler linear elastic Mohr–Coulomb model. Although there were differences in vertical loads and gap compression with the use of both soil models for the backfill, the differences were small and not of practical concern.

In this paper, equations that were proposed in the literature over the past four decades to estimate or predict the variation of the resilient modulus with respect to soil suction for pavement base-course materials and subgrade soils are summarized into three groups: (1) empirical relationships, (2) constitutive models incorporating the soil suction into applied shearing or confining stresses, and (3) constitutive models extending the independent stress state variable approach. Two equations selected from each of the groups (a total of six equations) are used to predict the resilient modulus–soil suction correlations for three compacted subgrade soils. Strengths and limitations of these widely used equations are discussed based on the comparisons between the measurements and predictions. The key objective of the state-of-the-art research summarized in this paper is for assisting practicing engineers to choose suitable equations for the rational prediction of the resilient modulus taking into account the influence of the soil suction.

In this study, cyclic triaxial tests were performed on Zipingpu gravel, and they were followed by a discrete element study that was designed to investigate the stress-dilatancy relationship of gravelly soils under cyclic loading in triaxial stress states. Several conclusions emerged from the results. (1) A nearly linear relationship was found between the stress ratio $\eta =q/p$ and the dilatancy ratio $D^p=d{\epsilon }_v^p/d{\epsilon }_s^p$ under both conventional compression and extension monotonic loading. The slope parameter $\alpha$, which relates $\eta$ and $D^p$, was smaller during triaxial compression; (2) The stress-dilatancy relationship was different during the virgin and cyclic loading. The dilatancy line under cyclic loading was located inside the virgin/monotonic loading dilatancy lines, and the dilatancy relationship was related to the location of the most recent load reversal point; and (3) A nearly parallel linear relationship was found between $\eta$ and $D^p$ in the $d\eta >0$ and $d\eta <0$ paths under cyclic loading, with $\alpha$ smaller under cyclic loading than under virgin loading. Numerical simulations were performed using discrete element software, and the results of these simulations showed that the stress-dilatancy characteristics under two-way cyclic loading were similar to the stress-dilatancy characteristics under one-way cyclic loading. The study also found that a hyperbolic relationship exists between the input plastic work and the particle breakage for Zipingpu gravel, regardless of the initial void ratio or the confining pressure under both monotonic and cyclic loading.

This paper discusses the torsional surface wave in a Voigt-type layer placed over an initially stressed heterogeneous semi-infinite substratum. The heterogeneities in the semi-infinite substratum are thought to emerge as a result of exponential variation in rigidity, density, and initial stress. A closed-form dispersion relation for the torsional surface waves under the considered geometry was obtained. The effect of various technical constants, such as internal friction parameter, stress parameter, heterogeneity parameter, dissipation factor, complex wave number, and thickness of the layer, were studied numerically by plotting various graphs between the nondimensional angular frequencies against the nondimensional real wave number for the torsional surface wave propagation. The numerical solution for the limiting case of torsional surface waves is also presented. In the absence of the considered technical constants, the obtained results are in close agreement with the results obtained by other relevant researchers. The present work exhibits scientific results applicable to the field of earthquake engineering and applied informatics.

Hypothesis A in clay rheology is widely used in practical engineering. It is important to explore its theoretical basis and proper scope of application in practical engineering. To build the relation between Hypotheses A and B, a quasi-plastic viscous concept is defined on the basis of hyperviscoplasticity. An equivalent-time nonlinear rheological model is proven to be one of the elastic quasi-plastic viscous models. From the perspective of the quasi-plastic viscous concept and this model, modified Hypothesis A is proven to be approximately equivalent to Hypothesis B under constant vertical stress in one-dimensional compression when the separation condition is satisfied. The consolidation behavior of clay with large quasi-preconsolidation pressure and vertical drains is numerically analyzed. A new phenomenon is predicted qualitatively and quantitatively in this clay, indicating that the larger the secondary consolidation coefficient is, the higher the degree of consolidation is when the original effective stress, the original void ratio, the original absolute equivalent time, the swelling index, and the compressive index remain constant.

A new method for nonlinear analysis of Euler-Bernoulli beams resting on heterogeneous, multilayered soil is presented. The governing differential equations for beam and soil displacements are obtained using the principle of virtual work, and these equations are solved using one-dimensional finite-element and finite-difference methods. Using the analysis, beam displacements can be obtained as a nonlinear function of applied loads if the beam and soil geometry and properties are known. The distinct feature of the analysis is that beam responses with accuracy comparable with those obtained from equivalent two-dimensional finite-element analysis are obtained within seconds. Examples illustrate how the proposed method can be applied in practice to obtain beam displacements in soil deposits that are characterized with nonlinear stress-strain relationships, have multiple layers, and have a modulus varying spatially within each soil layer. Parametric studies are performed that highlight the effects of soil layering and nonlinearity on the beam response.

Gravity-type caisson walls are a type of popular but easily damaged waterfront construction structure, especially in seismic regions. Various forms of mitigation have been successfully applied to improve their performance when subjected to liquefaction. Establishment of an effective, reliable, and easily implemented liquefaction remedial design procedure based on commonly used ground improvement technology is important. In this study, following the philosophy of performance-based design, a novel optimization process was developed and applied to a well-known case study. The vibrocompaction method was examined within a framework of a well-calibrated case history to hypothetically improve the seismic performance of a damaged caisson quay wall to an acceptable level. This study includes three major steps: first, the constitutive model used in this study for describing liquefaction was calibrated based on the published data; second, a caisson quay wall case history damaged by liquefaction was simulated as the unimproved benchmark; finally, the vibrocompaction method was applied as the mitigation technique to provide improvement to an acceptable level, and the mitigation design was optimized. The remedial program design characteristics, such as improved soil zone lateral extent in backfill soils and improved zone depth in foundation soils, were optimized. Through a parametric study, insights and recommendations on the establishment of liquefaction remediation design and optimization process for gravity-type caisson wall are provided for further study.

The primary objective of this study was to demonstrate that a significant degree of anisotropy may occur in particulate materials that have nearly spherical aggregates (i.e., are typically considered as isotropic) provided there is a bias in the distribution of pore space as a result of the initial densification process. The study included both an experimental and a theoretical component. An experimental program was conducted on dynamically compacted samples of Ottawa standard sand (C109) and incorporated a series of tests performed at different orientations of the specimens. Following the experimental part, a plasticity formulation based on the critical-plane approach was presented. A systematic procedure for identification of material parameters was outlined, and some numerical simulations of a series of triaxial tests were conducted. The last part of this work dealt with a much weaker load-induced anisotropy. Here, the mathematical framework was enhanced by an evolution law for the material fabric. Again, some numerical simulations were conducted, and it was demonstrated that the proposed framework can, at least in a qualitative manner, account for various manifestations of induced anisotropy.

The modified friction circle method is still an important tool in slope-stability investigations. The theoretical soundness of this method was investigated, and it is shown here that there is theoretical inconsistency and that the radius of the friction circle is incorrect. Using fundamental relationships of classical mechanics, the exact solution is derived from first principles.

This study presents a mathematical model for the injection of Bingham fluid as common grout material into a horizontal slot formed between parallel plates. The proposed model consisted of a grout reservoir model, a slot model, and an intermediate model that bridged the former two models. The authors showed that it is necessary to include a grout reservoir model in the injection model to avoid singularity of velocity when the grout penetration length is zero. This approach also allowed investigation of the initial stage flow condition during the injection process, especially at the inlet of the slot. Based on the proposed model, the approximate solution for the grout penetration length at the initial stage of injection was also derived. The performance of the proposed model was compared against a well-established model under laminar flow condition.

This article explores the propagation of torsional type surface waves in an initially stressed inhomogeneous layer of finite thickness lying over an inhomogeneous half-space. Inhomogeneity in this layer is caused by hyperbolic variation in directional rigidities, density, and initial stress. Inhomogeneity in the half-space is caused by linear variation in rigidity and density. The inhomogeneity parameter and the initial stress play major roles in the propagation of torsional type surface waves. This research derives the dispersion relation of the phase velocity in a concrete form using separation of variables and numerically calculates the velocities of torsional type waves as a function of dimensionless wave number.

The soils surrounding foundation pits are commonly unsaturated, but routine anti-overturning calculations of retaining walls are conducted assuming saturated soil mechanics. The primary objective of this paper is to present a simple method for the anti-overturning design of a rigid retaining wall for foundation pits in unsaturated soils. The earth pressure of the Rankine theory for unsaturated soils is adopted to derive the resisting and driving moments with respect to the toe of the retaining wall. The closed-form equations of embedment depth are derived with two different distributions of matric suction. The obtained result is easy to use and has wide applicability and excellent extensibility by considering comprehensive effects from many factors. The validity of the proposed simple method is demonstrated by comparing it with a lower bound solution. Engineering application procedures for this simple method are detailed, and parametric studies about a new defined overturning buried coefficient are discussed. It is found in this paper that the anti-overturning design ignoring matric suction is too conservative; the overturning buried coefficient is significantly influenced by matric suction and its distribution, the suction angle, and effective shear strength.

In this paper, two very simple and accurate equations are introduced that enable geotechnical engineers to determine the undrained ultimate bearing capacity of shallow strip foundations resting on two-layered clays. Results of the analyses show that in the case of weak-over-strong clay, usually general shear failure occurs, and in the case of strong-over-weak clay, the soil will most likely experience punching failure with a large amount of plastic settlement before it reaches its ultimate bearing capacity. This study applied the finite-difference method using a computer program to evaluate the undrained bearing capacity for all cases analyzed. The Mohr-Coulomb failure criterion was used for modeling the plastic behavior of soil. Simple equations are presented for cases of both strong-over-weak clay and weak-over-strong clay representing the bearing capacity of shallow foundations resting on two-layered clays. Results of this study are verified by acceptable statistical methods, and advantages of the presented equations in this paper are discussed.