Reliability analysis of piles in spatially varying soils considering multiple failure modes

In most limit state design codes, the serviceability limit checks for drilled shafts still use deterministic approaches. Moreover, different limit states are usually considered separately. This paper develops a probabilistic framework to assess the serviceability performance with the consideration of soil spatial variability in reliability analysis. Specifically, the performance of a drilled shaft is defined in terms of the vertical settlement, lateral deflection, and angular distortion at the top of the shaft, corresponding to three limit states in the reliability analysis. Failure is defined as the event that the displacements exceed the corresponding tolerable displacements. The spatial variability of soil properties is considered using random field modeling. To illustrate the proposed framework, this study assesses the reliability of each limit state and the system reliability of a numerical example of a drilled shaft. The results show the system reliability should be considered for the serviceability performance. The importance measures of the random variables indicate that the external loads, the performance criteria, the model errors of load transfer curves and soil strength parameter are the most important factors in reliability analysis. Moreover, it is shown that the correlation length and coefficient of variation of soil strength can exert significant impacts on the calculated failure probability.     

Importance sampling based algorithm for efficient reliability analysis of axially loaded piles

In reliability analysis, the crude Monte Carlo method is known to be computationally demanding. To improve computational efficiency, this paper presents an importance sampling based algorithm that can be applied to conduct efficient reliability evaluation for axially loaded piles. The spatial variability of soil properties along the pile length is considered by random field modeling, in which a mean, a variance, and a correlation length are used to statistically characterize a random field. The local averaging subdivision technique is employed to generate random fields. In each realization, the random fields are used as inputs to the well-established load transfer method to evaluate the load–displacement behavior of an axially loaded pile. Failure is defined as the event where the vertical movement at the pile top exceeds the allowable displacement. By sampling more heavily from the region of interest and then scaling the indicator function back by a ratio of probability densities, a faster rate of convergence can be achieved in the proposed importance sampling algorithm while maintaining the same accuracy as in the crude Monte Carlo method. Two examples are given to demonstrate the accuracy and the efficiency of the proposed method. It is shown that the estimate based on the proposed importance sampling method is unbiased. Furthermore, the size of samples can be greatly reduced in the developed method.     

Analysis of compressible axially loaded piles in rock

Analytical solutions are developed for the prediction of the load–settlement response of a compressible axially loaded cast in situ pile in rock. The principal input parameters are derived from the τ–z curve which applies to the contact zone between the pile shaft and the embedment material. For larger diameter piles in rock, these τ–z curves can differ markedly from curves that apply to piles in clay and may include a significant strain hardening region prior to reaching peak strength. In addition to the complete analyses which would normally require the use of a computer, simplified solutions suitable for hand calculations have been derived for the peak load and the pile head displacement to peak. An example of the application of these simplified design calculations is presented.     

Variational elastic solution for axially loaded piles in multilayered soil

Most analytical or semi‐analytical solutions of the problem of load‐settlement response of axially loaded piles are based on the assumption of zero radial displacement. These solutions also are only applicable to piles embedded in either a homogeneous or a Gibson soil deposit. In reality, soil deposits consist of multiple soil layers with different properties, and displacements in the radial direction within the soil deposit are not zero when the pile is loaded axially. In this paper, we present a load‐settlement analysis applicable to a pile with circular cross section installed in multilayered elastic soil that accounts for both vertical and radial soil displacements. The analysis follows from the solution of the differential equations governing the displacements of the pile–soil system obtained using variational principles. The input parameters needed for the analysis are the pile geometry and the elastic constants of the soil and pile. We compare the results from the present analysis with those of an analytical solution that considers only vertical soil displacements. The analysis presented in this paper also provides useful insights into the displacement and strain fields around axially loaded piles. Copyright © 2011 John Wiley & Sons, Ltd.     

Interpreting the behavior of laterally loaded drilled shaft from measured deflection data

The drilled shafts have been widely used to support lateral loads (active load case) or as a means to stabilize an unstable slope (passive load case) due to their large lateral load resistance and structural capacity for shear and bending moments. However, there is a need to develop an analytical procedure that can use the actual measured deflection data of a drilled shaft subject to either active or passive load case to interpret the soil‐drilled shaft interaction behavior. The mathematical formulation and the accompanied numerical procedure based on the principle of superposition were developed in this paper to allow for deducing the relevant soil‐drilled shaft interaction behavior under the applied lateral load (i.e. net soil reaction force on the drilled shaft, the shear and bending moment in the shaft) from the measured deflection data. Both compatibility and force equilibrium conditions were utilized in formulating the mathematical equations for common single drilled shaft boundary conditions (free head and fixed bottom). The current application is limited to small deformation to meet the requirement that the drilled shaft responds in a linear elastic range. A total of three theoretical cases, along with two actual field cases, were used to demonstrate the validity of the proposed method and its engineering applications. Copyright © 2009 John Wiley & Sons, Ltd.     

Reliability analysis of drilled shaft behavior using finite difference method and <Emphasis Type="Italic">Monte Carlo</Emphasis> simulation

Load displacement analysis of drilled shafts can be accomplished by utilizing the “t-z” method, which models soil resistance along the length and tip of the drilled shaft as a series of springs. For non-linear soil springs, the governing differential equation that describes the soil-structure interaction may be discretized into a set of algebraic equations based upon finite difference methods. This system of algebraic equations may be solved to determine the load–displacement behavior of the drilled shaft when subjected to compression or pullout. By combining the finite difference method with Monte Carlo simulation techniques, a probabilistic load–displacement analysis can be conducted. The probabilistic analysis is advantageous compared to standard factor of safety design because uncertainties with the shaft–soil interface and tip properties can be independently quantified. This paper presents a reliability analysis of drilled shaft behavior by combining the finite difference technique for analyzing non-linear load–displacement behavior with Monte Carlo simulation method. As a result we develop probabilistic relationships for drilled shaft design for both total stress (undrained) and effective stress (drained) parameters. The results are presented in the form of factor of safety or resistance factors suitable for serviceability design of drilled shafts.     

Mobilized side and tip resistances of piles in clay

This paper presents a procedure to assess the mobilized pile side and tip resistance versus pile head and tip settlement under axial load in clay soil. The load transfer (t–z) curve is evaluated at any point on the loaded pile based on the combined pile tip/side resistance–displacement mechanisms along the length of the pile. Unlike current methods that assume the pile settlements as a percentage of the pile/shaft diameter, the presented technique determines the side and tip resistance of the pile and the associated pile settlement under existing load based on the current stress/strain level in the surrounding soil up to failure (excessive settlement). The technique employs the concepts of the elastic theory and Ramberg–Osgood characterization of the stress–strain behavior of the clay soil. Case studies are also presented to exhibit the capabilities of the proposed procedure. The good agreement between measured and calculated load transfer curves along the pile and pile tip resistance versus pile head, side, and tip displacements shows the consistency of the proposed procedure. A computer code is developed to employ the presented technique.     

Axial service limit state analysis of drilled shafts using probabilistic approach

Drilled shafts are, typically, designed by considering the axial ultimate limit state. In this design methodology, the axial displacement requirements are verified once the design is completed. As an alternative, drilled shafts may be designed by considering the axial service limit state. Service limit state foundation design is more efficient when done using the load and resistance factor design (LRFD) approach. Furthermore, reliability may be rationally incorporated into the design process that utilizes the LRFD method. In this paper, we develop probabilistic approaches for axial service limit state analysis of drilled shafts. The variability of shaft-soil interface properties is modeled by lognormal probability distribution functions. The probability distributions are combined with a closed-form analytical relationship of axial load-displacement curves for drilled shafts. The closed-form analytical relationship is derived based upon the “t–z” approach. This analytical relationship is used with the Monte Carlo simulation method to obtain probabilistic load-displacement curves, which are analyzed to develop methods for determining the probability of drilled shaft failure at the service limit state. The developed method may be utilized to obtain resistance factors that can be applied to LRFD based service limit state design.     

A NEW MODEL FOR THE ANALYSIS OF SETTLEMENT OF DRILLED PIERS

A variational model for the analysis of axially loaded piers is presented. A closed-form solution technique employing an iterative procedure, is developed to obtain the displacement and forces in the pier along its axial direction. The method is suitable for similar analyses of pile foundations. It is shown that displacements and the load distribution along the axis of the pier compare well with a more sophisticated finite element solution. Furthermore, the new model complements the well-known Reese model employing t–z curves for the analysis of settlement of axially loaded piers. This new formulation using continuum mechanics principles, distributes the work done by the applied load as compressive strain energy in the pier, and as shear strain energy in the soil, as well as, the compressive strain energy in the soil surrounding the pier and at the bottom of the pier.     

Limiting equilibrium method for slope/drilled shaft system

The use of drilled shafts to stabilize an unstable slope has been a widely accepted practice. There are two basic design and analysis issues involved: one is to determine the global factor of safety of the drilled shafts stabilized slope and the other one is to determine the design earth thrust on the drilled shafts for structural design of the shafts. In this paper, a limiting equilibrium method of slices based solution for calculating global factor of safety (FS) of a slope with the presence of a row of drilled shafts is developed. The arching mechanisms due to the presence of the drilled shafts on slope were taken into account by a load transfer factor. The method for calculating the net force applied to the drilled shaft from the soil mass was also developed. The interrelationships among the drilled shaft location on the slope, the load transfer factor, and the global FS of the slope/shaft system were derived utilizing the developed numerical closed‐form solution. An illustrative example is presented to elucidate the use of the solution in optimizing the location of the drilled shafts on slope to achieve the desired global factor of safety of the slope/shaft system. Copyright © 2009 John Wiley & Sons, Ltd.     

A numerical approach to estimate shaft friction of bored piles in sands

A new approach to estimate shaft capacity of bored piles in sandy soils, based on numerical analysis, is presented. The topic is relevant as current design methods often largely underestimate the shaft capacity of piles in sands, thus resulting in an over-conservative design. The proposed approach is based on explicitly modelling the thin cylinder of soil surrounding the pile, where strain localization concentrates (shear band), and on the fundamental mechanic behaviour of sandy soils (e.g. dilatancy, softening). This approach is both simple and easy to apply. Results of a broad parametric study involving axially loaded single piles embedded in different sandy soils are presented, highlighting that relative density and grain size distribution mainly affect the shaft capacity. The capability of the procedure to predict shaft friction is checked against data from a well-documented full-scale axial load test on instrumented pile. Some suggestions for calibration and application of the method are also reported.     

Prediction of the shaft resistance of nondisplacement piles in sand

Using pile segment analysis, the mobilized shaft resistance of axially loaded nondisplacement piles in sand is investigated here. It is accepted that the shaft capacity of piles constructed in granular soils is highly influenced by the mechanical behavior of soil–structure interfaces forming adjacent the piles skin. Adopting the thin interface layer as a load transfer mechanism, a simple but accurate critical state compatible interface constitutive model is introduced. After evaluation, the interface model in conjunction with the pile segment analysis is applied for the prediction of the shaft resistance mobilized in nondisplacement piles. The proposed approach takes into account the influences of pile diameter and surface roughness together with the effects of the surrounding soil density and stiffness on the mobilized shaft resistance. The performance of the proposed method is verified by comparing its predictions with the experimental data of various model piles covering wide ranges of length, diameter, roughness, and surrounding soil properties. Copyright © 2012 John Wiley & Sons, Ltd.     

纵横向受荷基桩变形内力的矩阵传递解

针对地基的土体屈服性状,将桩侧土体分为弹性变形区域与塑性变形区域两种情况。假定地基反力系数为3参数的一般形式,同时考虑桩身 效应并计入桩身自重、桩侧摩阻力的影响,根据地基反力法分别建立桩身弹性段和塑性段挠曲线微分方程。在解微分方程的过程中,采用矩阵传递法结合Laplace正逆变换的方法解得桩顶作用轴向力、水平力、弯矩时桩身内力和变形的矩阵传递解,并用Fortran语言编制了相应计算程序。最后将试验数据对上述方法进行验证,结果表明计算值与模型试验的实测值吻合很好,采用文中3参数地基反力法反算所得的地基参数离散性很小,研究结果具有较高的应用价值。     

Bivariate simulation using copula and its application to probabilistic pile settlement analysis

This paper aims to propose a procedure for modeling the joint probability distribution of bivariate uncertain data with a nonlinear dependence structure. First, the concept of dependence measures is briefly introduced. Then, both the Akaike Information Criterion and the Bayesian Information Criterion are adopted for identifying the best‐fit copula. Thereafter, simulation of copulas and bivariate distributions based on Monte Carlo simulation are presented. Practical application for serviceability limit state reliability analysis of piles is conducted. Finally, four load–test datasets of load–displacement curves of piles are used to illustrate the proposed procedure. The results indicate that the proposed copula‐based procedure can model and simulate the bivariate probability distribution of two curve‐fitting parameters underlying the load–displacement models of piles in a more general way. The simulated load–displacement curves using the proposed procedure are found to be in good agreement with the measured results. In most cases, the Gaussian copula, often adopted out of expedience without proper validation, is not the best‐fit copula for modeling the dependence structure underlying two curve‐fitting parameters. The conditional probability density functions obtained from the Gaussian copula differ considerably from those obtained from the best‐fit copula. The probabilities of failure associated with the Gaussian copula are significantly smaller than the reference solutions, which are very unconservative for pile safety assessment. If the strong negative correlation between the two curve‐fitting parameters is ignored, the scatter in the measured load–displacement curves cannot be simulated properly, and the probabilities of failure will be highly overestimated. Copyright © 2011 John Wiley & Sons, Ltd.     

A simplified analysis method for piled raft foundations in non‐homogeneous soils

A simplified method of numerical analysis based on elasticity theory has been developed for the analysis of axially and laterally loaded piled raft foundations embedded in non‐homogeneous soils and incorporated into a computer program “PRAB”. In this method, a hybrid model is employed in which the flexible raft is modelled as thin plates and the piles as elastic beams and the soil is treated as springs. The interactions between structural members, pile–soil–pile, pile–soil–raft and raft–soil–raft interactions, are approximated based on Mindlin's solutions for both vertical and lateral forces with consideration of non‐homogeneous soils. The validity of the proposed method is verified through comparisons with some published solutions for single piles, pile groups and capped pile groups in non‐homogeneous soils. Thereafter, the solutions from this approach for the analysis of axially and laterally loaded 4‐pile pile groups and 4‐pile piled rafts embedded in finite homogeneous and non‐homogeneous soil layers are compared with those from three‐dimensional finite element analysis. Good agreement between the present approach and the more rigorous finite element approach is demonstrated. Copyright © 2002 John Wiley & Sons, Ltd.     

The modelling of anchors using the material point method

The ultimate capacity of anchors is determined using the material point method (MPM). MPM is a so‐called meshless method capable of modelling large displacements, deformations and contact between different bodies. A short introduction to MPM is given and the derivation of the discrete governing equations. The analysis of a vertically loaded anchor and one loaded at 45° is presented. The load–displacement curves are compared to that obtained from experiments and the effect of soil stiffness and anchor roughness is investigated. The results of the vertically loaded anchor are also compared to an analytical solution. The displacement of the soil surface above the anchor was measured and compared to the numerical predictions. Convergence with mesh refinement is demonstrated and the effect of mesh size and dilatancy angle on the shear band width and orientation is indicated. The results show that MPM can model anchor pull out successfully. No special interface elements are needed to model the anchor–soil interface and the predicted ultimate capacities were within 10% of the measured values. Copyright © 2005 John Wiley & Sons, Ltd.     

Evaluation of Lateral Interpretation Criteria for Drilled Shaft Capacity in Gravels

Representative interpretation criteria are examined in detail to evaluate the lateral response of drilled shaft foundations in gravels. A database of field lateral load tests is used for the analysis. The representative criteria are subdivided into absolute displacement, displacement as a function of shaft diameter, rotation limit, mathematical model, and graphical construction method. Comparison of these criteria shows that the initial elastic region ends at approximately 3–5 mm, 0.50 %B, and 0.08° rotation, which represent the serviceability limit state. The final region begins at about 25 mm, 2.5 %B, and 0.33° rotation, which represent the ultimate limit state. The graphical construction methods, Q_L and Q_S&W, are located within the transition region of the curve. Among these criteria, the displacement limits, Q_L, and Q_S&W methods demonstrate more reliable results. The typical load–displacement curves for lateral capacity in gravels are developed for engineering practical use. Further, recommendations for the use of these interpretation criteria are also given.     

Numerical integration method for computing reliability index of geotechnical system

To account for the uncertainties in the design of a geotechnical system, reliability-based design approach is often adopted, in which the main task is to evaluate reliability index of the system based on a performance function (or limit state function). In this paper, we propose a new method for computing the reliability index, based upon the numerical integration of the cumulative distribution function (CDF) of the performance function. This numerical integration method requires only a deterministic evaluation of the system performance and the joint probability of the uncertain input parameters. The effectiveness and the efficiency of the proposed method, measured in terms of the accuracy and the computational effort, respectively, are demonstrated with two geotechnical problems: a drilled shaft in sand and a semi-gravity retaining wall. The new method is found valid regardless of the type of distribution of uncertain input parameters, whether the correlations exist among these input parameters, whether the system involves single or multiple failure modes, and how the performance function is formulated.