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.     

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.     

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.     

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.     

A method for designing the longitudinal spacing of slope-stabilising shafts

The work at hand deals with the design of the longitudinal spacing among rows of closely spaced large-diameter shafts used to stabilise a precarious slope. The problem under consideration is idealised through a conceptual framework where an unstable mass of an infinitely long slope pushes a stable portion of soil adjacent to shafts, leading to failure along a slip surface passing through the upper end of the reinforcement elements. By exploiting the upper bound theorem of plastic collapse, a closed-form solution is derived for the load required for the failure of the stable mass as a function of geometrical and mechanical parameters of the slope and the soil. Results are validated through physical model tests by means of geotechnical centrifuge. Given the satisfactory agreement between analytical and experimental results, the model is extended to evaluate the safety conditions of the reinforced slope.     

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 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.     

Analysis of failure of a bridge foundation under rock impact

A bridge pier supported on two drilled shafts collapsed due to the impact by a 130-ton rock in a landslide event. A series of static and dynamic numerical simulations is conducted using a nonlinear finite element analysis program to investigate the bearing behavior and responses of the bridge foundation under rock impact. The rock impact load is evaluated according to the site conditions. The deflection histories at the striking point and the internal forces in the drilled shafts during rock impacts in different directions are analyzed. The bridge pier exhibits significant system effects: the failure of the bridge pier is initiated by the failure of one pier column or one drilled shaft first, followed by the failure of the entire pier. The effects of impact loading direction, striking location, and characteristics of impact load on the behavior of the bridge pier are examined through a parametric study. The capacities of the pier along different loading directions are different due to differences in the group effects of the drilled shafts. The bridge pier is strongest when the impact load is along the 45° direction with respect to the shaft row, and weakest when the impact load is perpendicular to the shaft row.     

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.     

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.     

An Approximate Method to Determine Lateral Force on Piles or Piers Installed to Support a Structure through Sliding Soil Mass

Piles and drilled shafts are commonly used to stabilize potentially unstable natural and engineered slopes. Estimation of lateral pressures on piles or shafts used to stabilize potentially unstable sites continues to be a fundamental problem in geotechnical engineering. Several methods are available to estimate the lateral force on piles or drilled shafts used to stabilize a site. An approximate procedure to estimate the lateral force on drilled shafts supporting structures, i.e., installed at relatively large spacings on a potentially unstable site with a fixed slip surface is proposed, which is based on a widely referred theory of plastic deformation. Lateral pressures estimated using the method proposed show good agreement with a more rigorous method, which in turn has been reported to provide results close to field measurements. An approximate procedure to estimate a limiting (maximum) value of lateral pressure on shafts or piles when they are installed at close spacings is also proposed. Although, the limiting pressures estimated using the approximate procedure are in agreement with the pressures calculated using more rigorous method, verification of both methods with field measurements is recommended.     

Experimental study,numerical modeling of and axial prediction approach to base grouted drilled shafts in cohesionless soils

The pressure grouting of drilled shaft tips has become popular worldwide due to its effectiveness in mobilizing a larger portion of the available tip resistance under service displacements. This paper presents experimental and numerical studies on the load transfer mechanism and factors controlling the axial response of base grouted drilled shafts in cohesionless soils. The study found that the increased axial capacity of grout-tipped drilled shafts under service loads and displacements depended mainly on preloading effects and the increased tip area provided by the grouting process. A simple prediction approach for estimating the tip capacity of grouted shafts utilizing cone penetration resistance was suggested based on the results of the study. The validity of the proposed approach was verified by the analysis of full-scale case studies of grouted shafts reported in the literature.     

Evaluating accuracy for two empirical methods in predicting settlement of drilled shafts

Several theoretical, empirical and semi-empirical methods are available in the literature to predict settlement of drilled shafts in sandy soils. In the Arabian Gulf countries, specifically in the United Arab Emirates, equations and procedure from the rest of the world are being used in analysis and design of drilled shafts without proper validation. It is the aim of this study to assess the applicability and evaluate the accuracy of two well known, and commonly used methods for pile prediction in the United Arab Emirates (UAE), namely Vesic (1977) and Poulos (1979), via comparison with data from field pile load tests conducted on shafts drilled in the region. Some of these tests were conducted for the purpose of this study, while others were made available through the courtesy of International Piling Contractors who are active in the region (e.g. Bauer International and Swiss Borings). Pile load test data were analyzed to back-calculate the model parameters related to settlement under different loading stages. Geological data and soil properties were obtained from studies conducted at the relevant sites. An effort is made to correlate soil properties with the prediction models. Statistical analysis is conducted to assess the accuracy of the results obtained from the two methods at different stages of loading via those obtained from pile load tests. Moreover, a detailed parametric study is conducted to assess the effect of the related parameters on the predicted pile settlement and the estimated settlement at different stages of loading. The study concluded with a recommendation of the most appropriate models and procedures to be followed for predicting the settlement of drilled shafts in the UAE, together with useful charts and correlation relations. Results showed that settlement values predicted by Vesic (1977) and Poulos (1979) overestimates the true values. This revised version was published online in July 2006 with corrections to the Cover Date.     

Beckenried viaduct: Foundation problems in a creeping slope

Summary The 3150 m long Beckenried viaduct traverses an unstable creeping, slope, the surface of the sound rock lying at depths from 10 to 60 m. Altogether 44 of the total 58 piers of the bridge founded on the sound rock had to be protected by shafts against the creeping soil and loosened rock layers. Based on geological explorations and previous monitoring of the slope displacements the clearance between the rectangular concrete piers and the elliptical shafts was specified to be 1.5 m in the dip direction of the slope and up to ±1.0 m in lateral direction. In order to avoid failure of the shafts due to the unusually large slope movements likely to occur during the service life of the viaduct, an entirely new design concept was applied. It involves flexible shafts consisting of four main parts, namely a rigid shaft collar, articulated ring elements, a rigid trapezoidal cylinder and basal displacement rings resting on the pier footing. Fundamentally, this design permits both: sliding of the shaft as a whole along particular slip surfaces in the loosened rock at depth but also differential displacements due to the creep of the soil strata above it. The sliding at the base contributes to 50% and more to the total surface movements. The shafts also serve as a permanent drainage system for the highly water sensitive slope material. The paper describes the criteria upon which the novel design of the individual shafts were made and also particular constructional details. After a construction period of only four years the bridge opened to traffic in December 1980. Observations made on the completed structure indicate a behaviour of the slope and the shafts which is well in accordance with the predictions.     

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.     

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.     

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.     

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.     

Load–settlement behavior of rock-socketed drilled shafts using Osterberg-Cell tests

Osterberg-Cell (O-Cell) tests are widely used to predict the load–settlement behavior of large-diameter drilled shafts socketed in rock. The loading direction of O-Cell tests for shaft resistance is opposite to that of conventional downward load tests, meaning that the equivalent top load–settlement curve determined by the summation of the mobilized shaft resistance and end bearing at the same deflection neglects the pile-toe settlement caused by the load transmitted along the pile shaft. The emphasis is on quantifying the effect of coupled shaft resistance, which is closely related to the ratios of pile diameter to soil modulus (D/E_s) and total shaft resistance to total applied load (R_s/Q) in rock-socketed drilled shafts, using the coupled load-transfer method. The proposed analytical method, which takes into account the effect of coupled shaft resistance, was developed using a modified Mindlin’s point load solution. Through comparisons with field case studies, it was found that the proposed method reasonably estimated the load-transfer behavior of piles and coupling effects due to the transfer of shaft shear loading. These results represent a significant improvement in the prediction of load–settlement behaviors of drilled shafts subjected to bi-directional loading from the O-Cell test.