Linear Elastic Soil Research Articles

Nonlinear consolidation finite element analysis of a layered soft soil foundation under multistage loading based on the continuous drainage boundary

To comprehensively consider the impacts of stratification, residual pore water pressure, soil nonlinearity, and boundary permeability on consolidation settlement of soft soil foundations for accurate prediction, a continuous drainage boundary condition is proposed in this study that reflects the residual pore pressure under multistage loading, and a nonlinear elastic constitutive model based on the double logarithmic model is adopted to account for the nonlinear consolidation behaviour of soils. A UMAT subroutine is developed based on the proposed boundary condition and nonlinear elastic constitutive model. Subsequently, the developed subroutine is compared with the built-in linear elastic soil constitutive model in ABAQUS and engineering examples. The application of continuous drainage boundaries in stratified foundations is analysed, as well as the influence of factors such as the loading rate and soil nonlinearity on consolidation settlement. The results indicate that, compared to the built-in model, the subroutine developed in this study can be employed to more accurately calculate the nonlinear consolidation of multilayered foundations under multistage loading. By adjusting the loading rate parameter αk, consolidation under different loading conditions can be predicted. Additionally, the proposed boundary condition simplifies the calculations for soft soil foundations with sand layers, providing a novel computational approach for the design of construction loading schemes and long-term settlement predictions in soft soil foundations.

Seismic Structure‐Soil‐Structure Interaction between inelastic structures

AbstractFrequently, buildings in urban areas are designed by considering their stand‐alone response, that is, as single structures with no neighboring buildings. Nevertheless, the existence of a high density of buildings in large metropolitan areas inevitably results in the likelihood of an important seismic interaction between adjacent buildings through the underlying soil. This paper explores the effects of Structure‐Soil‐Structure Interaction (SSSI) on the seismic response of two yielding structures embedded in a linear elastic soil. A simple two‐dimensional nonlinear reduced‐order parametric model is proposed, where different building parameters are considered. A nonlinear phenomenological Bouc–Wen model is assumed for the buildings. A database of 15 strong ground motion records and an additional spectrally matched seismic ground motion are considered. An extensive parametric study comprising over two million nonlinear cases is conducted. The results show important differences between nonlinear SSSI and nonlinear SSI for particular parameter configurations. Nevertheless, due to energy dissipation and increases in damping in the nonlinear case, the effects of SSSI are less relevant compared with the linear case.

AN ALTERNATIVE PROCESS-BASED APPROACH TO PREDICTING THE RESPONSE OF WATER-SATURATED POROUS MEDIA TO HARMONIC HYDRODYNAMIC LOADS

Methods have been developed to predict how hydrodynamic loads acting on nearly saturated porous media are transmitted to the subsoil. In line with the effective stress principle of Terzaghi, these methods apply the boundary conditions that the effective stresses at the surface of a porous medium are zero, and that the pore water pressures carry the full load. Here, a new approach is presented which is based on defining a stress and a stress gradient as boundary conditions. The stress gradient follows from the momentum balance equation, thereby assuring that the solution abides by d'Alembert's principle of minimization of virtual work. The corresponding solution is in full accordance with the volume and momentum balance equations of the linear elastic soil matrix and the volume and momentum balance equations of the pore water across the computational domain. The new method is thereby able to correctly reproduce measurements of pore pressure changes due to hydrodynamic loads under the assumption of a porous medium consisting of incompressible particles and pore water which could either be compressible or incompressible. The advantage of the proposed method is that it requires one less boundary condition at the surface of the porous medium. The method is therefore able to predict the magnitude of the effective stresses on a soil surface. Due to the ability to retain the assumption of incompressible water, the method has also become independent on a calibration parameter. The results of the method induce questions with respect to the validity of Terzaghi's principle of effective stress at the boundary when porous media are subjected to hydrodynamic loads.

The mechanics of the Gentle Driving of Piles

Gentle Driving of Piles (GDP) is a new vibratory installation technology for tubular (mono)piles. It is characterized by the simultaneous application of low-frequency axial and high-frequency torsional vibrations, envisaged to achieve both high installation performance and reduced underwater noise emissions. The concept of GDP has been demonstrated experimentally in a medium-scale onshore field campaign, showcasing the potential of the method in terms of installation and post-installation performances. To further comprehend the mechanics of the GDP method, the driving process is studied by means of a novel pile–soil model; this framework has been recently developed and successfully applied to the problem of axial vibratory driving. In particular, the pile is treated as a thin cylindrical shell via a Semi-analytical Finite Element (SAFE) approach and a linear elastic layered soil half-space is considered via the Thin-Layer Method (TLM) coupled with Perfectly Matched Layers (PMLs). The pile–soil coupling is realized through a hereditary frictional interface and an elasto-plastic tip formulation, both characterized by standard geotechnical in-situ measurements. The comparison of numerical results with field data is favourable for drivability purposes, showcasing the potential of the numerical framework for the analysis of GDP. Conclusively, the mechanics of the installation process are deciphered and the redirection of the friction force vector – induced by high-frequency torsion – is identified as the main driving mechanism of GDP.

Study of Rheological-Mechanical Properties and Vibration Mechanics Bandgap of Row Pile Foundation

To analyze the rheological and mechanical properties as well as the vibration-mechanical forbidden zone effect of row pile foundations, this paper employs time-dependent modulus to examine the rheological mechanics of soil. Drawing from viscoelastic theory, we derive the expression of deformation modulus in the frequency domain to analyze the frequency dependence of the shear modulus of rheological soils. We construct a continuous medium dynamics model of the pile-soil periodic structure, taking into account soil rheology, and derive the dispersion equation of shear waves in the periodic structure using the multiple scattering method. The band gap characteristics and parameters that influence the law of shear waves in rheological soil-row pile foundations are studied through the analysis of arithmetic cases. The results show that under the loading condition, the zero-frequency shear modulus of soil is larger than the initial modulus value, and the real part of the shear modulus decreases monotonically with the increase of frequency and finally converges to the initial modulus value; under the unloading condition, the zero-frequency shear modulus of soil is smaller than the initial modulus value, and the real part of the shear modulus increases monotonically with the increase of frequency and finally converges to the initial modulus value; the larger the relaxation time of soil, the faster the convergence rate; the imaginary part of the shear modulus of soil The imaginary part of the soil shear modulus is positive under loading condition and negative under unloading condition, the value of the imaginary part increases and then decreases with increasing frequency and finally converges to 0. The imaginary part reaches the peak at the critical frequency, the larger the relaxation time the smaller the critical frequency, and the peak of the imaginary part is independent of the relaxation time. This study analyzed the dispersion curve of shear waves in a pile-soil periodic structure and found that increasing low-frequency shear wave velocity in rheological soil pile foundation shifts the band gap position to a higher frequency band, resulting in a smaller band gap width than in linear elastic soil. The relaxation time of soil affects the frequency position and width of the band gap, with larger relaxation times resulting in higher frequency positions and smaller widths. Additionally, soil rheology widens the forbidden vibration band gap of the pile periodic structure when the filling rate of the pile foundation is larger.

Rigid circular footing on the surface of a transversely isotropic linear elastic half-space

It is sometimes convenient to treat a relatively rigid footing as a single element subjected to up to six force resultants: vertical load, horizontal load in two directions, overturning moment about two axes and torsion. Applications include calculations of initial, elastic settlement; foundation responses under seismic loading or for machine vibrations; and fixity of an offshore jack-up foundation under cyclic wave loading. Present codes of practice provide formulae that assume fully isotropic and linear elastic soil and in some case assume no friction between the footing and soil. This paper uses recent theoretical advances to develop solutions for stiffnesses of rigid circular footings on transversely isotropic linear elastic soil, including with interface friction. Closed-form solutions are developed for vertical load and overturning moment on a frictionless interface and for vertical load and torsion on a frictional interface. A boundary-element analysis is presented for the cases of lateral load and overturning moment on a frictional interface. By fitting expected forms to the results, new formulae are proposed for stiffnesses for these cases. The effects of interface friction on limiting loads are calculated. Practical applications are outlined, and some limitations are briefly discussed.

Influence of Vertical Load on Lateral-Loaded Monopiles by Numerical Simulation

Monopiles are the most common foundation form of offshore wind turbines, which bear the vertical load, lateral load and bending moment. It remains uncertain whether the applied vertical load increases the lateral deflection of the pile. This paper investigated the influence of vertical load on the behaviour of monopiles installed in the sand under combined load using three-dimensional numerical methods. The commercial software PLAXIS was used for simulations in this paper. Monopiles were modelled as a structure incorporating linear elastic material behaviour and soil was modelled using the Hardening-Soil (HS) constitutive model. The monopiles under vertical load, lateral load and combined vertical and lateral loads were respectively studied taking into account the sequence of load application and pile slenderness ratio (L/D; L and D are the length and diameter of the pile). Results suggest that the sequence of load application plays a major role in how vertical load affects the deflection behaviour of the pile. Specifically, when L/D ratios obtained by lengthening the pile while keeping its diameter constant are 3, 5 and 8, the relationships between lateral load and the deflection behaviour of the pile under the effect of vertical load demonstrate a similar trend. Furthermore, the cause of increased lateral capacity of the pile under the action of applied vertical load in the common practical application case and in the VPL case was analyzed by studying the variation law of soil stress along the pile embedment. Results confirm that the confining effect of vertical load increases means effective stress of the soil around the pile, thus increasing soil stiffness and pile capacity.

Energy-Based Approach: Analysis of a Laterally Loaded Pile in Multi-Layered Non-Linear Elastic Soil Strata

Several studies have been reported in published literature on analytical solutions for a laterally loaded pile installed in a homogeneous single soil layer. However, piles are rarely installed in an ideal homogeneous single soil layer. The present study describes a new continuum-based analysis or energy-based approach for predicting the pile displacement responses subjected to static lateral loads and moments considering the soil non-linearity. This analytical analysis treats the pile as an elastic Euler–Bernoulli beam and the soil as a three-dimensional (3D) continuum in which the non-linear elastic properties are described by a modulus degradation relationship. The principle of virtual work was applied to the energy equation of a pile–soil system in order to obtain the governing differential equation for the pile and soil displacements. An iterative procedure was adopted to solve the equations numerically using a finite difference method (FDM). The pile displacement response was obtained using the software MATLAB R2021a, and the results from the energy-based method were compared with those obtained from the field test data as well as the finite element analysis (FEA) based on the software ANSYS Workbench 2021R1. The present study investigated the effect of explicit incorporation of soil properties and layering through a parametric study in order to understand the importance of predicting appropriate pile displacement responses in a linear elastic soil system. The responses indicated that the effect of soil layers and their thicknesses, pile properties and the variation in soil moduli have a direct impact on the displacements of piles subjected to lateral loading. Hence, a proper emphasis has to be given to account for the soil non-linearity. Considering the effect of soil non-linearity, it is observed that the results obtained from the energy-based method agreed well with the field measured values and those obtained from the FEA. The results indicated a difference of approximately less than 7% between the proposed method and the FEA. The approach presented in this study can be further extended to piles embedded in multi-layered soil strata subjected to the combined action of axial loads, lateral loads and moments. Furthermore, the same approach can be extended to study the response of the soil to group piles.

Energy-Based Approach: Analysis of a Vertically Loaded Pile in Multi-Layered Non-Linear Soil Strata

Numerous studies have been reported in the published literature on analytical solutions for a vertically loaded pile installed in a homogeneous single soil layer. However, piles are rarely installed in an ideal homogeneous single soil layer. This study presents an analytical model based on the energy-based approach to obtain displacements in an axially loaded pile embedded in multi-layered soil considering soil non-linearity. The developed analytical model incorporating Euler-Bernoulli beam theory proved to be an effective way in estimating the load-displacement responses of piles embedded in multi-layered non-linear elastic soil strata. The differential equations are solved analytically and numerically using the variational principle of mechanics. A parametric study investigated the effect of explicit incorporation of soil properties and layering in order to understand the importance of predicting appropriate pile displacement responses in linear elastic soil system. It is clear from the results that the analyses which consider the soil as a single homogeneous layer will not be able to produce an accurate estimation of the pile stiffnesses. Therefore, it is highly important to account for the effect of soil layering and the non-linear response. The pile displacement response is obtained using the software MATLAB R2019a and the results from the energy-based method are compared with those obtained from the field test data as well as the Finite Element Analysis (FEA) based on the software ANSYS 2019R3. The non-linear elastic constitutive relationship which described the variation of secant shear modulus with strain through a power law has shown reasonably accurate predictions when compared to the published field test data and the FEA. The developed mathematical framework is also more computationally efficient than the three-dimensional (3D) FEA.

The modified Vlasov model on a nonhomogeneous and nonlinear soil layer

This study presents a novel approach to account for the soil nonlinearity of nonhomogeneous soil deposits in foundation deflection analyses in the context of a modified Vlasov foundation model. We present an extension of the previously proposed formulation by developing a new formulation employing an improved algorithm that takes the modulus degradation curves at varying strain levels into account in an iterative manner. This new model, which takes the nonlinear soil behavior into account, was first verified against a linear elastic soil model given in the literature to ensure that the new model algorithm can capture the original solution when the soil behavior is assumed to be linear elastic. Later, the experimental data reported in the literature for a specific type of dense and loose sands were used in the example analyses. Example problems were considered for different cases, which presented (i) how the model captures nonlinear behavior and (ii) the significant effect of the nonlinear soil behavior. The result of the new model was also compared with the finite element model results, assuming elastoplastic soil. The results obtained from both models match well, especially for the maximum deflection value, provided that laterally constrained sections underneath the foundation are used.

Transverse seismic response of end‐bearing pipe piles to S‐waves

AbstractThis paper presents a method for the seismic analysis of open‐ended pipe piles subjected to vertically propagating S‐waves, that considers kinematic interaction between the pipe pile and its external and internal soil. Following the presentation of the elastodynamic continuum model, which is based on the assumptions of linear elastic soil response and uniform soil conditions, we employ the derived solution to investigate the sensitivity of the seismic response of pipe piles to certain key problem parameters, such as pile slenderness or the relative stiffness of the pipe pile compared to its surrounding soil. We demonstrate that the bending stiffness of pipe piles is governing their seismic response, and that thin‐walled pipe piles offer the best material usage versus seismic performance ratio. The presented solution offers a low‐cost alternative to complex numerical simulations for preliminary seismic design purposes, such as the selection of optimal pipe pile section geometry.

The Strength Behavior and Desiccation Crack Development of Silty Clay Subjected to Wetting–Drying Cycles

It is commonly accepted that wetting–drying cycles have an effect on the soil strength behavior. Crack development in soil is observed by many engineers during wetting–drying cycles, which may give a good explanation for the change in soil strength. A series of laboratory tests were conducted in this study to investigate the desiccation crack development and the strength change law for silty clay subjected to different numbers of wetting–drying cycles. The results show that the desiccation cracks at the end of drying process developed in two stages: the stage of rapid growth and the stage of steady state. The change law of soil strength is similar to the cracking that decreases quickly in the former stage and slowly in the latter stage, which indicates that the cracking in the soil is the main reason for strength reduction. Based on the assumption of an isotropic and linear elastic soil mass at rest earth pressure conditions, an equation for the depth of desiccation cracking after different numbers of wetting–drying cycles was obtained with soil mechanics for unsaturated soils. Finally, the applicability of the equation was verified compared with the experiment results.

Probabilistic Structural System Response to Differential Settlement Resulting from Spatially Variable Soil

The role of spatially variable soil stiffness and strength on structural performance has been increasingly recognized with the rapid development of geotechnical reliability-based design tools. Performance criteria linked to the angular distortion between foundations and the probability of exceeding certain limit states as a function of linear-elastic soil property autocorrelation have shed light on the link between foundation soils and potential structural damage. However, the role of soil-structure interaction (SSI), especially considering nonlinearity in soil and structural elements, to redistribute shear and flexure within structures that result from differential foundation movements, has been largely neglected. This study presents a critical examination of the effect of nonlinear SSI to redistribute flexural demands within a steel structure founded in soils exhibiting spatial variability and the corresponding differential settlements through a comparison of uncoupled and coupled Monte Carlo simulations (MCS). This study shows that the differential settlement, Δs, computed using coupled (i.e., SSI) analyses are significantly smaller than those derived from the uncoupled analyses for the random field models (RFMs) used to simulate soil spatial variability. The effect of SSI is to substantially reduce the probability of exceeding selected limit state criteria due to differential movements that vary in magnitude with the level of stringency of the limit state criteria; the most significant reduction corresponds to the structural ultimate limit state (ULS). Furthermore, the effect of SSI on structural performance and the critical scale of fluctuation, δcrit, becomes more apparent and beneficial as the severity of differential movement increases. Smaller and/or “allowable” angular distortions are governed by the local footing-to-footing distance, whereas the probability of exceeding the ULS (i.e., yielding of beams) is controlled by the entire structure working to redistribute the flexural demands. Limit state severity appears to control the critical soil autocorrelation length for angular distortion, indicating that the role of SSI in the performance of structures in spatially varying soils cannot be ignored.

A high-accuracy substructure method in the time domain for simulating pile-soil dynamic interaction in horizontal vibration

An accurate and efficient substructure method in the time domain is developed to simulate pile-soil dynamic interaction during horizontal vibration. The present method considers the equilibrium of a linear elastic soil in both the radial and circumferential directions to account for three-dimensional wave scattering, and considers the end bearing pile as a one-dimensional rod. First, an analytical expression for the soil resistance to a horizontal vibration pile in the frequency-domain is obtained. From this, an exact dynamic stiffness relationship between the soil resistance and the pile displacement is constructed at the pile-soil interface. Further, the dynamic stiffness relationship in the frequency domain is transformed into a high-order approximation in the time domain. Then, this high-order approximation is equivalent to a mechanical model system consisting of mass, dashpot and spring elements. Finally, the effectiveness of the present method is verified and the effects of the dynamic stiffness of the soil on the dynamic responses of the pile are investigated.

Analytical Study on the Transverse Internal Forces of Shield Tunnel Segments due to Adjacent Excavations in Soft Clays

Deep excavations unavoidably cause the changes in stress and displacement of surrounding soils, which further generate additional internal forces on the adjacent existing tunnel segments, and compromise the performance and stability of the tunnels. In this study, two-stage analysis method is used to theoretically calculate the additional internal forces in the tunnel segments due to laterally adjacent excavation. Based on an empirical deformation curve of diaphragm walls, the excavation-induced stress in a linear elastic soil is formulated using source-sink imaging method. A distribution model of additional external load acting on the tunnel segments due to adjacent excavation is further established, and the additional internal forces in the tunnel segments under the corresponding additional loads are estimated using elastic equation method. Parametric studies are then conducted to investigate the effects of excavation procedure, buried depth of shield tunnel, and excavation-tunnel horizontal distance on the additional internal forces in the tunnel segments. The results show that the studied parameters have significant effect on the additional internal forces in the tunnel segments due to laterally adjacent excavation. When the excavation-tunnel horizontal distance is less than the excavation depth, the accompanied effect on the additional internal forces in the tunnel segments is highly sensitive. This study can provide theoretical insights into the estimation of tunnel segment responses due to laterally adjacent excavation.

Effect of the Employed Material Model on the Predicted Behaviour of Corrugated Metal Pipes

In recent years, two-dimensional (2D) finite element modeling (FEM) has been used to explore how the structural integrity of a corrugated metal pipe (CMP) culvert is affected by simulation of its constituent components (i.e., the pipe metal, surrounding soil, or both). However, under various conditions, the nature of the problem is three-dimensional (3D) and 2D simulations may fall short of producing reasonable predictions. In this paper, the effect of various loading types on the structural response of round CMP culverts is investigated using 3D finite element (FE) models. The validity of the model was assessed by comparing the model output with previously published results from full-scale laboratory testing. The mechanical behaviour of real soil involves a significant degree of non-linearity. In soil-structure interaction problems such as the one considered here, it seems reasonable to expect that the non-linear behaviour of the soil, together with that of the soil-structure interface, would have a significant effect on the response of the structure. Furthermore, 2D models are limited in their ability to account for certain aspects of the actual, 3D problem. Such aspects include surface live load spreading through the cover soil and the structural behaviour of a corrugated pipe. This paper details the development of a comprehensive, 3D, non-linear FE model capable of simulating the behaviour of intact CMP culverts under backfilling and live loading. The analysis showed that the hyperbolic strain hardening model produced predicted response of the culvert that is very close to the obtained experimental data. However, the numerical results obtained using the Mohr-Coulomb or the linear elastic soil models were of less accuracy.

Basin effects and limitations of 1D site response analysis from 2D numerical models of the Thorndon basin

Plane strain (2D) finite element models are used to examine factors contributing to basin effects observed for multiple seismic events at sites in the Thorndon basin of Wellington, New Zealand. The models consider linear elastic soil and rock response when subjected to vertically-propagating shear waves. Depth-dependent shear wave velocities are considered in the soil layers, and the effects of random variations of soil velocity within layers are modelled. Various rock shear wave velocity configurations are considered to evaluate their effect on the modelled surficial response. It is shown that these simple 2D models are able to capture basin reverberations and compare more favourably to observations from strong motion recordings than conventional 1D site response models. It is also shown that consideration of a horizontal impedance contrast across the Wellington Fault affects spectral response and amplification at longer periods, suggesting the importance of this feature in future ground motion modelling studies in the Wellington region.

A Winkler model for suction caisson foundations in homogeneous and non-homogeneous linear elastic soil

Suction caisson foundations provide options for new foundation systems for offshore structures, particularly for wind turbine applications. During the foundation design process, it is necessary to make reliable predictions of the stiffness of the foundation, since this has an important influence on the dynamic performance of the overall support structure. The dynamic characteristics of the structure, in turn, influence its fatigue life. This paper describes a thermodynamically consistent Winkler model, called OxCaisson, that delivers computationally efficient estimates of foundation stiffness for caissons installed in homogeneous and non-homogeneous linear elastic soil, for general six degrees-of-freedom loading. OxCaisson is capable of delivering stiffness predictions that are comparable to those computed with three-dimensional finite-element analysis, but at a much lower computational cost. Therefore, the proposed model is suited to design applications where both speed and accuracy are essential, such as large-scale fatigue assessments of offshore wind farm structures. The paper demonstrates that the OxCaisson model can also be applied to short, rigid monopile foundations.

Modeling of pile-soil-pile interaction in laterally loaded pile groups embedded in linear elastic soil layers

This study proposes a new method to obtain the lateral response of pile groups by incorporating the pile group effect in layered soils. When a pile is loaded laterally, it creates a zone of influence in the direction of loading. In a pile group, each pile placed in the influence zone of prior piles is exposed to extra loads due to the load transfers from other piles. This mechanism results in a group effect which causes each pile in the group to have a different deflection curve compared to that of an identical isolated single pile under the same load. This study starts with a mathematical approach to model the interaction of two piles and then extends it to pile groups. The governing differential equation of a pile deflection problem is modified to take the pile-soil-pile interaction into account and solved analytically for each pile while the soil parameters and displacement fields around each pile are obtained numerically using the finite difference method written in Fortran language. The model captures the additional pile deflections induced by the group effects in pile groups and the results match well with the results of the existing methods, especially the finite element method.

Prediction of time of liquefaction using kinetic and strain energy

The time of liquefaction triggering during a strong ground motion can have a large influence on the expected level of foundation and superstructure damage. To enable simple, yet accurate estimates of the triggering time, the build-up of pore pressure needs to be understood in terms of cumulative measures of ground motion intensity. This paper develops a theoretical framework and simple procedure to predict the build-up of excess pore pressure based on the principles of conservation of energy. The liquefaction resistance is first quantified in terms of cumulative absolute change in strain energy, which is shown through the evaluation of experimental cyclic simple shear tests to be insensitive to loading amplitude. A ground motion intensity measure is presented that uniquely calculates the cumulative absolute change in kinetic energy. This intensity measure is then used to provide an exact analytical solution for the cumulative absolute change in strain energy at any depth in a homogenous linear elastic soil deposit using the novel, nodal surface energy spectrum (NSES). A simple reduction to the NSES is proposed for viscous and nonlinear soil deposits, as well as a correction for changes in stiffness between layers of soil. The estimation of strain energy and build-up of pore pressure using the simple NSES method was applied to 500 randomly generated soil deposits using a range of different ground motions and validated against nonlinear total stress and nonlinear effective stress time-history analyses, with the NSES method providing a high level of accuracy. The proposed spectrum based solution provides an efficient and physically consistent procedure for the prediction of excess pore pressure build-up.