Extrapolation of O-cell drilled shaft tests for load and resistance factor design (LRFD) calibration

Load tests of drilled shafts are often performed using Osterberg cell (O-cell) testing, a popular load test method for drilled shafts, which measures both side and tip resistance. However, it is common that only one of the resistance components can be fully mobilized. Therefore, extrapolation of the partially mobilized resistance is often required to determine the total resistance or the equivalent top-down curve. The extrapolation tends to introduce errors to the constructed total resistance values, which subsequently affect the calibrated resistance factors required for the LRFD design of drilled shafts. In this study, eight O-cell tests of drilled shafts with total measured resistances close to the failure criteria defined by FHWA, 5% of the shaft diameter (B), were collected among 64 drilled shaft load tests from Louisiana and Mississippi. For each of the eight cases, extrapolation was performed on both tip and side movement curves for the construction of the equivalent top-down load-settlement (ELT) curves. Data points from the measured side or tip movement curve were removed systematically to create a total of 80 cases with partially mobilized movement curves, and extrapolation exercises were performed on each fabricated case to obtain its equivalent top-down curve. The error of bias for each fabricated case was determined for statistical analyses. Multiple linear regression analysis was performed on the bias errors to model the bias errors caused by extrapolation. Calibrated resistance factors were determined and compared between the original database and fabricated database needing extrapolation. A correction method is proposed, based on a linear regression relationship, to estimate and minimize the extrapolation error of bias for less mobilized databases.

     

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.     

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.     

基于Copula函数的基桩荷载-位移双曲线概率分析

提出了基于Copula函数的基桩荷载-位移双曲线概率分析方法。首先将基桩标准化荷载-位移双曲线模型不确定性转化为双曲线参数不确定性,然后在Copula理论框架下建立了双曲线参数的联合分布函数。最后以钻孔现浇灌注桩试验数据为例证明了所提方法的有效性,并进行了基桩正常使用极限状态可靠度分析。结果表明：Copula函数是构造基桩标准化荷载-位移双曲线参数联合分布函数一种有效的方法,它能够更加准确地实现基桩荷载-位移双曲线的随机模拟,从而得到更为合理的可靠度结果。钻孔现浇灌注桩双曲线模型中两个参数间具有较强的负相关关系,忽略了这种负相关性将会高估基桩的失效概率。此外,常用的Gaussian Copula函数并不是拟合双曲线模型中两个参数间相关结构最优的Copula函数,采用Gaussian Copula函数将会明显低估基桩的失效概率。     

Shear Load Transfer Characteristics of Drilled Shafts Socketed in Rocks

This paper presents a shear load transfer function and an analytical method for estimating the load transfer characteristics of rock-socketed drilled shafts subjected to axial loads. A shear load transfer (f–w) function of rock-socketed drilled shafts is proposed based on the constant normal stiffness (CNS) direct shear tests. It is presented in terms of the borehole roughness and the geological strength index (GSI) so that the structural discontinuities and the surface conditions of the rock mass can be considered. An analytical method that takes into account the coupled soil resistance effects is proposed using a modified Mindlin’s point load solution. Through comparisons with load test results, the proposed methodology is in good agreement with the general trend observed in in situ measurements and represents an improvement in the prediction of the shear behavior of rock-socketed drilled shafts.     

Incorporating set-up into LRFD method for drilled shafts

The increase of pile resistance with time is referred to as ‘set-up’. This behaviour of driven piles has been widely discussed in many studies by researchers. Meanwhile, there has been little, if any, information regarding this aspect for drilled shafts. Performing a bearing capacity test for a shaft over time, however, requires higher costs and more complicated rigs compared to a driven pile. A database including results from five Osterberg cell-tested drilled shafts conducted at two different stages is considered, from which the set-up effect is statistically analysed. The reliability-based analysis technique using Monte Carlo simulation (MCS) is used to develop separate resistance factors to account for different degrees of uncertainties associated with the predicted reference resistance and the predicted set-up resistance in the framework of the load and resistance factor design (LRFD) method. By incorporating set-up into design, shaft length or number of shafts can be reduced and economical design of drilled shafts can be achieved.     

Limit equilibrium based design approach for slope stabilization using multiple rows of drilled shafts

In this paper, a limiting equilibrium based methodology, incorporating the method of slices and arching effects of the drilled shafts, is developed for optimizing the use of multiple rows of drilled shafts. This proposed method is focusing on the number of rows, the location of each row, the dimension and spacing of the drilled shafts. Three design criteria are used for optimization: target global factor of safety, the constructability and service limit. A PC-based program called M-UASLOPE has been coded to allow for handling of complex slope geometry, soil profile, and ground water conditions. A design example is presented to illustrate the application of the M-UASLOPE program in the optimized design of multiple rows of drilled shafts for stabilizing the example slope.     

基于Copula函数的CFG桩复合地基载荷-变形曲线的概率分析

CFG桩(cement-fly ash-gravel pile)复合地基是一种重要的地基处理形式,在日益增加的大面积住宅和商业开发中作用越来越突出,然而该种桩型的加卸荷-沉降变形特性仍然需深入研究,尤其在概率评估方面。根据北京星光影视股份有限公司生产科研基地项目工地中的21根CFG桩单桩静载试验和32个复合地基静载试验的原位加卸载测试成果,采用两参数的双曲线或幂曲线回归拟合了每一条加荷-变形曲线。由于土体的内在各向异性和其强度的变异性,评估整个场地的加荷-变形曲线时,其回归参数表现出了较大的离散性。将一个场地的多组回归参数组成一个随机向量,其加载-位移曲线的不确定性可由简单的两变量随机向量体现,引入双变量联结函数（Copula）描述随机回归参数间的相依性。最后,考虑正常使用极限状态,采用基于Copula函数的模拟模型计算了CFG桩复合地基的可靠度指标。研究结果有助于改进CFG桩复合地基的概率设计与评估。     

Load transfer analysis of rock-socketed drilled shafts by coupled soil resistance

The load distribution and deformation of rock-socketed drilled shafts subjected to axial loads are evaluated by a load transfer method. The emphasis is on quantifying the effect of coupled soil resistance in rock-socketed drilled shafts using 2D elasto-plastic finite element analysis. Slippage and shear-load transfer behavior at the pile–soil interface are investigated by using a user-subroutine interface model (FRIC). It is shown that the coupled soil resistance acts as pile-toe settlement as the shaft resistance is increased to its ultimate limit state. Based on the results obtained, the coupling effect is closely related to the ratio of the pile diameter to soil modulus (D/E_s) and the ratio of total shaft resistance against total applied load (R_s/Q). Through comparison with field case studies, the 2D numerical analysis reasonably estimated load transfer of pile and coupling effect, and thus represents a significant improvement in the prediction of load deflections of drilled shafts.     

Load-displacement uncertainty of vertically loaded shallow footings on sands and effects on probabilistic settlement estimation

This article focuses on the statistical characterisation and stochastic modelling of the load-displacement behaviour of shallow footings on cohesionless soils and on the probabilistic estimation of settlement for serviceability limit state design (LSD). The study relies on a field database of 30 full-scale footings subjected to vertical loading with cone penetration testing data available for each site. The performance of three load-displacement models in replicating field data is assessed comparatively through statistical analysis. Load-displacement uncertainty is subsequently modelled probabilistically to perform Monte Carlo Simulation (MCS)-based estimation of footing settlement using the best-performing power law model. The dependence among load-displacement model parameters is investigated and replicated using copula theory. Samples are generated to account for parametric uncertainties in model inputs. The simulation output samples of settlement are examined statistically in order to assess the relevance of parametric and load-displacement uncertainties in settlement estimation, as well as the importance of accounting for correlation between power law model parameters. A simple analytical model for the estimation of settlement at any target reliability level is obtained on the basis of the outputs of MCS. The model can be practically implemented in geotechnical LSD at serviceability limit states.     

特定场地下土工构筑物的几何可靠性分析

针对特定场地下土工构筑物的正常使用极限状态,采用近年发展的几何可靠性方法计算了多种构筑物的可靠度指标。考虑同一场地下的钻孔灌注桩、抗浮锚杆和CFG桩单桩加载变形测试曲线的离散性,各曲线回归参数呈现差异并可作为随机变量,进而探讨了各曲线回归参数间的相关性及联合分布特性。基于这些回归参数的联合发散概率密度等值线,即随机变量刚好达到极限承载能力状态,该几何可靠性算法可在随机变量的原始空间求得土工构筑物的可靠度指标。通过比对该几何可靠度指标与常规的一次可靠性算法成果,验证了该几何可靠性计算技术的可行性。计算表明,几何可靠性评价模型实施简便,易于被工程技术人员接受。     

Side resistance of drilled shafts in granular soils investigated by DEM

The paper presents a numerical study on the side resistance of a drilled shaft in granular materials. The numerical result is used to develop new design equations for the side resistance of drilled shafts in granular soils. The Discrete Element Method (DEM) is used to model a drilled shaft in granular material. The granular material is represented as assemblies of ellipsoidal particles. Nominal side resistance is represented as the product of a parameter (β) and vertical stress. The numerical result shows that the relationship between β and void ratio can be described by a hyperbolic function for a given vertical stress. DEM result is also compared with three design equations. Although these design equations capture the decrease of β with depth, deviation is observed between the DEM results and the design equations. Finally, new design equations based on state parameter are proposed.     

考虑应力扩散时桩端土性对单桩沉降影响分析方法

基于虚土桩模型,分析了层状地基中桩端土性对单桩沉降特性的影响。首先,以虚土桩扩散角反映桩端土层应力扩散效应,将桩端一定锥角范围内由桩端至基岩面的土体视为虚土桩,并根据其变截面特性,将虚土桩沿纵向划分为有限个微元段。然后,对桩及虚土桩桩侧土体采用理想弹塑性荷载传递模型,利用荷载传递法,推导了层状地基中以桩侧土塑性发展深度为变量的单桩荷载-沉降递推计算方法,并进一步得到了桩身轴力及桩侧摩阻力递推计算式。在此基础上,给出了荷载传递模型参数选取方法,并分析了虚土桩临界深度的影响因素及由实测荷载-沉降曲线反演虚土桩扩散角的可行性。最后,利用该方法分析了桩端沉渣和软弱下卧层对荷载-沉降曲线的影响。结果表明,考虑桩端土层应力扩散效应时,通过计算得到的桩顶及桩端荷载-沉降曲线与实测曲线吻合较好;当桩端存在沉渣或软弱下卧层时,采用虚土桩模型的单桩沉降计算方法可以在一定程度上反映沉渣特性及软弱下卧层埋深等因素对桩顶荷载-沉降曲线的影响。     

End Bearing Capacity of Drilled Shafts in Sand: A Numerical Approach

In this paper, a modeling procedure is carried out to numerically analyze the end bearing capacity of drilled shafts in sand. The Mohr–Coulomb elastic plastic constitutive law with stress dependent elastic parameters is used for all numerical analyses performed in this study. The numerical results are compared with the available experimental equations. It is seen that numerical results are in good agreement with experimental equations. The variation of the end bearing capacity of drilled shafts versus embedment depth is also studied. Numerical results show that with increase in pile embedment depth, the end bearing capacity increases. However, the rate of increase becomes smaller as the pile embedment depth increases. Also, numerical analyses show that, for equal settlement, the end bearing decreases with increase in the pile diameter. Finally, a sensitivity analysis is performed to obtain the separate effect of each sand parameter on the end bearing capacity of drilled shafts, and the parameters that are most influential are identified.     

Serviceability limit state design for uplift of helical anchors in clay

Presently, no displacement-based design methodology exists for helical anchors subjected to tensile or uplift loading. This study investigates the statistical and probabilistic aspects of the load-displacement uncertainty associated with a database of thirty-seven uplift loading tests of helical anchors founded within cohesive soils. Initially, an ultimate resistance model is identified, and the semi-empirical uplift breakout factor statistically characterized. A relationship between ultimate resistance and slope tangent capacity is established, and used to form the basis for normalizing the load-displacement response. Hyperbolic and power law models are statistically evaluated for use in serving as a reference load-displacement model; the hyperbolic curve was selected based on goodness-of-fit statistics. Monte Carlo reliability simulations are used to establish an equivalent-deterministic load factor that associates the selected load factor with a probability of exceeding a pre-determined allowable uplift displacement, given uncertainty in the undrained shear strength, ultimate resistance model, transformation uncertainty, uncertainty in the allowable displacement, and variability in uplift loading. A practical example is provided to show the intended use of this probabilistic helical anchor displacement model.     

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.     

戈壁地基扩底掏挖基础抗拔试验及其位移计算

在甘肃和新疆7个戈壁碎石土场地完成了46个扩底掏挖基础抗拔现场试验,分析了基础抗拔荷载-位移曲线的阶段特征,应用L1-L2方法确定了所有基础抗拔承载力和位移。采用归一化荷载-位移双曲线模型对实测荷载-位移曲线进行拟合,得到了各试验基础归一化荷载-位移双曲线模型拟合参数取值及其统计规律,给出了同时考虑抗拔基础极限承载力计算理论模型误差和归一化荷载-位移双曲线模型不确定性时基础荷载和位移的计算方法。结果表明：戈壁碎石土扩底掏挖基础抗拔荷载-位移曲线呈初始弹性直线段、弹塑性曲线过渡段和直线破坏的3阶段变化规律,归一化荷载-位移双曲线模型可较好拟合基础实测荷载-位移曲线,归一化荷载-位移双曲线模型不确定性可转化为该双曲线模型拟合参数的不确定性,基于强度和变形统一的工程设计更有利于工程安全。     

Three‐dimensional finite element study of arching behavior in slope/drilled shafts system

Two‐dimensional slope stability analysis for a slope with a row of drilled shafts needs a mechanism to take into account the three‐dimensional effect of the soil arching due to the spaced drilled shafts on slope. To gain a better understanding of the arching mechanisms in a slope with evenly spaced drilled shafts socketed into a stable stratum (or a rock layer), the three‐dimensional finite element modelling technique was used for a comprehensive parametric study, where the nonlinear and plastic nature of the soil and the elastic behavior of the drilled shafts as well as the interface frictions were modelled. Various factors were varied in the parametric study to include (1) the rigidity of the drilled shafts as influenced by its diameter, modulus of elasticity, and total length; (2) shafts spacing and location on the slope; (3) the material properties of rock and the socket length of shaft; and (4) the soil movement and strength parameters. Evidences of soil arching and reduction in the stresses and displacements through the load transfer mechanisms due to the presence of the drilled shafts were elucidated through the finite element method (FEM) simulation results. Design charts based on regression analysis of FEM simulation results were created to obtain a numerical value of the load transfer factor for the arching mechanism provided by the drilled shafts on the slope. Observations of the arching behavior learned from the FEM simulations provide an insight into the behavior of drilled shafts stabilized slope. Copyright © 2009 John Wiley & Sons, Ltd.