Yield acceleration and permanent displacement of a slope reinforced with a row of drilled shafts
Introduction
Traditional seismic slope stability analysis is based on the use of pseudo-static approaches that apply the seismic coefficients to a conventional limit-equilibrium analysis to calculate factor of safety. This method, however, has been criticized due to lack of information of displacement of a slope [1], [2], [3]. To estimate slope displacement due to earthquake, the most often used method is the Newmark's sliding block method [4]. The Newmark method calculates the permanent displacement of a slope under earthquake excitation by integration of acceleration above yield acceleration. Calculation for yield acceleration for a slope is carried out by the limiting equilibrium method of slices. In recent years, several modifications of the Newmark's method have been made to account for the influences of site soil conditions as well as dynamic response characteristics of a slope [5], [2], [6]. Bray and Rathje [6] presented a nonlinear decoupled one-dimensional dynamic analysis method to estimate slope displacement based on dynamic response of the site, dynamic resistance, and ground motion parameters.
Stabilization of an unstable slope either in static or seismic conditions has been an important issue in geotechnical engineering. Among a variety of slope stabilization methods, a concept of using a row of drilled shafts has been well accepted and used successfully over the years. Some of the advantages offered by drilled shafts include: (a) drilled shafts are permanent structures that can provide significant resistance to earth thrust from a moving slope, (b) drilled shafts can be constructed in almost all soil and rock conditions, (c) drilled shafts can be constructed in difficult site conditions without the need for additional right-of-way in most highway applications, and (d) drilled shafts can be combined with other types of slope stabilization methods, such as tiebacks, drainages, and grade changes, etc., and finally (e) drilled shafts can be load tested to verify the resistance capacities. Design methods for using drilled shafts to stabilize an unstable slope have been developed by numerous researchers. Among them is the method based on the soil arching concept proposed by Liang and his associates [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. However, there has been no previous study related to calculation of permanent displacement of a slope reinforced by a row of equally spaced drilled shafts under an earthquake excitation. The concept of soil arching proposed by Liang could lend itself for the determination of yield acceleration of a drilled shafts reinforced slope. Once such yield acceleration is determined, then permanent displacement of a drilled shaft reinforced slope system under an earthquake could be estimated using the well-known Newmark's method.
Presented in this paper is a method for determining yield acceleration of a slope reinforced with a row of equally spaced drilled shafts under an earthquake excitation. The concept of soil arching in a drilled shafts/slope system is described. The limiting equilibrium method of slices for slope stability calculation considering soil arching effects due to drilled shafts is formulated through the load transfer factor. Parametric study results of 3-D finite element simulations using strength reduction techniques are used to obtain a semi-empirical equation for predicting load transfer factor. An example is given to show the validity of the proposed method for calculating yield acceleration. In addition, a modified Newmark method is introduced for calculating permanent displacement for a slope reinforced with a single row of equally spaced drilled shafts using the calculated yield acceleration and charts developed by Bray and Rathje [6]. A total of seven cases are performed to show good comparisons between the proposed simplified method and 3-D finite element simulation results.
Section snippets
Yield acceleration for a slope reinforced with drilled shafts
The arching mechanism occurs when the soils on a slope move through the opening between the drilled shafts, particularly when the opening is small and the drilled shafts are fixed deep enough into a stable stratum. As a result of arching, the earth pressures would transfer to the drilled shafts, which are resisted by the portion of the drilled shafts in the unyielding stratum (i.e., rock). Eventually, soil movements would slow down between the drilled shafts when equilibrium was reached. Thus,
3-D FEM simulation for load transfer factor
The determination of load transfer factor is based on the ratio of the inter-slice force on the down-slope side of the drilled shaft to the inter-slice force on the up-slope side of the drilled shaft, as depicted in Fig. 3. To quantify the load transfer factor, Liang and Zeng [7] used a 2-dimensional finite element approach to obtain an empirical equation. They found that soil arching in a drilled shaft/slope system is highly dependent upon the soil movement relative to the drilled shafts, soil
Comparison with other solutions
Li et al. [20] presented a method to compute seismic stability of a slope reinforced with a row of spaced drilled shafts. Their work was an extension of the kinematic theorem of limit equilibrium analysis by Chen [21]. However, their method can only apply to a homogeneous and isotropic soil slope. In addition, the soil is assumed to experience plastic deformation according to the normality rule associated with the Mohr–Coulomb yield condition. A log–spiral slip surface was implicitly assumed in
Permanent displacement calculated by using the improved Newmark method
Bray and Rathje [6] introduced a procedure based on the Newmark's original method to determine the permanent deviatoric displacement of a slope under a seismic excitation. It represents a major improvement over the original Newmark analysis method [4] and other similar work [22], [23], [24] due to its consideration of dynamic response of a slope, similar to the approach taken by [5]. The Bray and Rathje [6] method requires input of three sets of parameters as follows:
- 1.
Ground motion parameters,
Numerical case studies of permanent displacement of a slope reinforced with a row of drilled shafts
The inclusion of a row of equally spaced drilled shafts in a slope, as discussed previously, can increase the dynamic resistance (or yield acceleration) of the shafts/slop system. The dynamic resistance of a drilled shaft/slope system can be calculated by using Eqs. (5), (6), (7). As it can be seen, the load transfer factor, η, plays an important role in controlling the calculated yield acceleration. From the governing factors of the load transfer factor, one could conclude that the dynamic
Summary and conclusions
In this paper, a method for determining yield acceleration of a slope reinforced with a row of equally spaced drilled shafts under an earthquake excitation and a simplified method for calculating permanent displacement of a drilled shaft/slope system was presented. The limiting equilibrium method of slices for slope stability calculation, considering soil arching effects due to drilled shafts, was formulated through the utilization of the load transfer factor. A semi-empirical equation based on
Acknowledgment
The support of the 111 Project (No. B13024) is acknowledged.
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