Enhancement of random finite element method in reliability analysis and risk assessment of soil slopes using Subset Simulation

Random finite element method (RFEM) provides a rigorous tool to incorporate spatial variability of soil properties into reliability analysis and risk assessment of slope stability. However, it suffers from a common criticism of requiring extensive computational efforts and a lack of efficiency, particularly at small probability levels (e.g., slope failure probability P _f ?<?0.001). To address this problem, this study integrates RFEM with an advanced Monte Carlo Simulation (MCS) method called “Subset Simulation (SS)” to develop an efficient RFEM (i.e., SS-based RFEM) for reliability analysis and risk assessment of soil slopes. The proposed SS-based RFEM expresses the overall risk of slope failure as a weighed aggregation of slope failure risk at different probability levels and quantifies the relative contributions of slope failure risk at different probability levels to the overall risk of slope failure. Equations are derived for integrating SS with RFEM to evaluate the probability (P _f ) and risk (R) of slope failure. These equations are illustrated using a soil slope example. It is shown that the P _f and R are evaluated properly using the proposed approach. Compared with the original RFEM with direct MCS, the SS-based RFEM improves, significantly, the computational efficiency of evaluating P _f and R. This enhances the applications of RFEM in the reliability analysis and risk assessment of slope stability. With the aid of improved computational efficiency, a sensitivity study is also performed to explore effects of vertical spatial variability of soil properties on R. It is found that the vertical spatial variability affects the slope failure risk significantly.     

Risk Factor Based Design of Cantilever Retaining Walls

Sensitivity analysis of geotechnical random variables on potential failure modes (overturning, sliding, bearing capacity and eccentricity) of a cantilever retaining wall reveals that high sensitivity of a particular variable on a particular mode of failure does not necessarily imply a remarkable contribution to the overall failure probability. The present paper aims to combine probability of failure (P _f ) of each failure mode and sensitivity of the random variables to these failure modes and introduces a new factor, called Probabilistic Risk Factor (R _f ) for each random variable. P _f is calculated by Monte Carlo Simulation and sensitivity analysis of each random variable is calculated based on normalized F-Statistics value. R _f is a reduction factor which takes into account the variations of random variables and hence can be directly implemented in design by the designers. The random variables (friction angle and unit weight of backfill soil; and friction angle, unit weight and cohesion of foundation soil), when divided by R _f and applied in design, yield a structure which is safe against variations of the random variables. It is observed that R _f of friction angle (φ ₁ ) of backfill increases and cohesion (c ₂ ) of foundation soil decreases with an increase in variation of φ ₁ , while R _f for unit weights (γ ₁ and γ ₂ ) of both the soil and friction angle of foundation soil (φ ₂ ) remains almost constant. Finally, design guidelines for different variations of φ ₁ are provided based on the proposed methodology, which proves to be cost effective.     

Multiple response surfaces for slope reliability analysis

This paper develops a multiple response surfaces approach to approximate the limit state function for slope failure by second‐order polynomial functions, to incorporate the variation of the most probable slip surfaces, and to evaluate the slope failure probability p_f. The proposed methodology was illustrated through a cohesive soil slope example. It is shown that the p_f values estimated from multiple response surfaces agree well with those p_f values that have been obtained by searching a large number of potential slip surfaces in each Monte Carlo simulation (MCS) sample. The variation of number of the most probable slip surfaces is studied at different scale of fluctuation (λ) values. It is found that when full correlation assumed for each of random fields (i.e., spatial variability is ignored), the number of the most probable slip surfaces is equal to the number of random fields (in this study, it is 3). When the spatial variability grows significantly, the number of the most probable slip surfaces or number of multiple response surfaces firstly increases evidently to a higher value and then varies slightly. In addition, the contribution of a specific most probable slip surface varies dramatically at different spatial variability level, and therefore, the variation of the most probable slip surfaces should be accounted for in the reliability analysis. The multiple response surfaces approach developed in this paper provides a limit equilibrium method and MCS‐based means to incorporate such a variation of the most probable slip surfaces in slope reliability analysis. Copyright © 2014 John Wiley & Sons, Ltd.     

A modified normalized model for predicting effective soil thermal conductivity

Effective soil thermal conductivity (λ _eff) describes the ability of a multiphase soil to transmit heat by conduction under unit temperature gradient. It is a critical parameter for environmental science, earth and planetary science, and engineering applications. Numerous models are available in the literature, but their applicability is generally restricted to certain soil types or water contents (θ). The objective of this study was to develop a new model in the similar form of the Johansen 1975 model to simulate the λ _eff(θ) relationship of soils of various soil textures and water contents. An exponential type model with two parameters is developed and a new function for calculating dry soil thermal conductivity is presented. Performance of the new model and six other normalized models were evaluated with published datasets. The results show that the new model is able to well mimic λ _eff(θ) relationship of soils from sand to silt loam and from oven dry to full saturation. In addition, it has the best performance among the seven models under test (with root-mean-square error of 0.059 W m^?1 °C^?1, average deviations of 0.0009 W m^?1 °C^?1, and Nash–Sutcliffe efficiency of 0.994). The new model has potential to improve the reliability of soil thermal conductivity estimation and be incorporated into numerical modeling for environmental, earth and engineering studies.     

Risk de-aggregation and system reliability analysis of slope stability using representative slip surfaces

This paper develops a risk de-aggregation and system reliability approach to evaluate the slope failure probability, p_f, using representative slip surfaces together with MCS. An efficient procedure is developed to strategically select the candidate representative slip surfaces, and a risk de-aggregation approach is proposed to quantify contribution of each candidate representative slip surface to the p_f, identify the representative slip surfaces, and determine how many representative slip surfaces are needed for estimating the p_f with reasonable accuracy. Risk de-aggregation is performed by collecting the failure samples generated in MCS and analyzing them statistically. The proposed methodology is illustrated through a cohesive soil slope example and validated against results from previous studies. When compared with the previous studies, the proposed approach substantially improves the computational efficiency in probabilistic slope stability analysis. The proposed approach is used to explore the effect of spatial variability on the p_f. It is found that, when spatial variability is ignored or perfect correlation assumed, the p_f of the whole slope system can be solely attributed to a single representative slip surface. In this case, it is theoretically appropriate to use only one slip surface in the reliability analysis. As the spatial variability becomes growingly significant, the number of representative slip surfaces increases, and all representative slip surfaces (i.e., failure modes) contribute more equally to the overall system risk. The variation of failure modes has substantial effect on the p_f, and all representative surfaces have to be incorporated properly in the reliability analysis. The risk de-aggregation and system reliability approach developed in this paper provides a practical and efficient means to incorporate such a variation of failure modes in probabilistic slope stability analysis.     

Probabilistic analysis of strip footings resting on a spatially random soil using subset simulation approach

The failure probability of geotechnical structures with spatially varying soil properties is generally computed using Monte Carlo simulation (MCS) methodology. This approach is well known to be very time-consuming when dealing with small failure probabilities. One alternative to MCS is the subset simulation approach. This approach was mainly used in the literature in cases where the uncertain parameters are modelled by random variables. In this article, it is employed in the case where the uncertain parameters are modelled by random fields. This is illustrated through the probabilistic analysis at the serviceability limit state (SLS) of a strip footing resting on a soil with a spatially varying Young's modulus. The probabilistic numerical results have shown that the probability of exceeding a tolerable vertical displacement (P _e) calculated by subset simulation is very close to that computed by MCS methodology but with a significant reduction in the number of realisations. A parametric study to investigate the effect of the soil variability (coefficient of variation and the horizontal and vertical autocorrelation lengths of the Young's modulus) on P _e was presented and discussed. Finally, a reliability-based design of strip footings was presented. It allows one to obtain the probabilistic footing breadth for a given soil variability.     

Integration of root systems into a GIS-based slip surface model: computational experiments in a generic hillslope environment

Root systems of trees reinforce the underlying soil in hillslope environments and therefore potentially increase slope stability. So far, the influence of root systems is disregarded in Geographic Information System (GIS) models that calculate slope stability along distinct failure plane. In this study, we analyse the impact of different root system compositions and densities on slope stability conditions computed by a GIS-based slip surface model. We apply the 2.5D slip surface model r.slope.stability to 23 root system scenarios imposed on pyramidoid-shaped elements of a generic landscape. Shallow, taproot and mixed root systems are approximated by paraboloids and different stand and patch densities are considered. The slope failure probability (P_f) is derived for each raster cell of the generic landscape, considering the reinforcement through root cohesion. Average and standard deviation of P_f are analysed for each scenario. As expected, the r.slope.stability yields the highest values of P_f for the scenario without roots. In contrast, homogeneous stands with taproot or mixed root systems yield the lowest values of P_f. P_f generally decreases with increasing stand density, whereby stand density appears to exert a more pronounced influence on P_f than patch density. For patchy stands, P_f increases with a decreasing size of the tested slip surfaces. The patterns yielded by the computational experiments are largely in line with the results of previous studies. This approach provides an innovative and simple strategy to approximate the additional cohesion supplied by root systems and thereby considers various compositions of forest stands in 2.5D slip surface models. Our findings will be useful for developing strategies towards appropriately parameterising root reinforcement in real-world slope stability modelling campaigns.     

A New Computer Program for Pile Capacity Prediction using CPT Data

An interactive computer program “GLAMCPT” is developed for application in soil profiling and prediction of pile load capacity using cone penetration test (CPT) and laboratory soil test results. GLAMCPT calculates pile capacity according to 10 selected methods from European design codes, refereed international publications and recommendations of professional institutions. To demonstrate the capabilities of the program, a database of comprehensive ground investigation and full-scale pile tests in sand, at a Belgian site, is analysed using GLAMCPT. The database comprises 11 static tests and 12 dynamic tests on piles of different construction techniques, including driven pre-cast concrete piles and screwed cast in-situ piles, installed using 5 different procedures. Prior to pile installation, CPTs were carried out at each proposed pile location. Comparison of GLAMCPT predictions with the observed pile capacities reveals that the most accurate of the existing methods yields an average, μ, of predicted to observed pile head capacity [P_uh(p)/P_uh(m)] equal to 0.94. The most consistent method produces a coeffcient of variation (COV) of [P_uh(p)/P_uh(m)] equal to 0.1 and ranking index (RI) of 0.08. Parametric studies have been carried out using GLAMCPT to formulate an improved predictive method, which yielded: μ = 0.99, COV = 0.07 and RI = 0.04.     

The stability of a remediated bed in Hamilton Harbour,Lake Ontario,Canada

ABSTRACT In situ measurements of lakebed sediment erodibility were made on three sites in Hamilton Harbour, Lake Ontario, using the benthic flume Sea Carousel. Three methods of estimating the surface erosion threshold (τ_c(0)) from a Carousel time series were evaluated: the first method fits measures of bed strength to eroded depth (the failure envelope) and evaluates threshold as the surface intercept; the second method regresses mean erosion rate (E_m) with bed shear stress and solves for the floc erosion rate (E_f) to derive the threshold for E_m = E_f = 1 × 10^?5 kg m^?2 s^?1; the third method extrapolates a regression of suspended sediment concentration (S) and fluid transmitted bed shear stress (τ₀) to ambient concentrations. The first field site was undisturbed (C) and acted as a control; the second (W) was disturbed through ploughing and water injection as part of lakebed treatment, whereas the third site (OIP) was disturbed and injected with an oxidant used for remediation of contaminated sediment. The main objectives of this study were: (1) to evaluate the three different methods of deriving erosion threshold; (2) to compare the physical behaviour of lacustrine sediments with their marine estuarine counterparts; and (3) to examine the effects of ploughing and chemical treatment of contaminated sediment on bed stability. Five deployments of Sea Carousel were carried out at the control site. Mean erosion thresholds for the three methods were: τ_c(0) = 0·5 (±0·06), 0·27 (±0·01) and 0·34 (±0·03) Pa respectively. Method 1 overpredicted bed strength as it was insensitive to effects in the surface 1–2 mm, and the fit of the failure envelope was also highly subjective. Method 2 exhibited a wide scatter in the data (low correlation coefficients), and definition of the baseline erosion rate (E_f) is largely arbitrary in the literature. Method 3 yielded stable (high correlation coefficients), reproducible and objective results and is thus recommended for evaluation of the erosion threshold. The results of this method correlated well with sediment bulk density and followed the same trend as marine counterparts from widely varying sites. Mass settling rates, expressed as a decay constant, k, of S(t), were strongly related to the maximum turbidity at the onset of settling (S_max) and were also in continuity with marine counterparts. Thus, it appears that differences in salinity had little effect on mass settling rates in the examples presented, and that biological activity dominated any effects normally attributable to changes in salinity. Bedload transport of eroded aggregates (2–4 mm in diameter) took place by rolling below a mean tangential flow velocity (U_y) of 0·32 ms^?1 and by saltation at higher velocities. Mass transport as bedload was a maximum at U_y = 0·4 ms^?1, although bedload never exceeded 1% of the suspended load. The proportion of material moving as bedload was greatest at the onset of erosion but decreased as flow competence increased. Given the low bulk density and strength of the lakebed sediment, the presence of a bedload component is notable. Bedload transport over eroding cohesive substrates should be greater in estuaries, where both sediment density and strength are usually higher. Significant differences between the ploughed and control sites were apparent in both the erosion rate and the friction coefficient (φ), and suggest that bed recovery after disruption is rapid (< 24 h). τ_c(0) increased linearly with time after ploughing and recovered to the control mean value within 3 days. The friction coefficient was reduced to zero by ploughing (diagnostic of fluidization), but increased linearly with time, regaining control values within 6 days. No long‐term reduction in bed strength due to remediation was apparent.     

Scaling the 3-D Mohr circle and quantification of paleostress during fluid pressure fluctuation – Application to understand gold mineralization in quartz veins of Gadag (southern India)

In this study, orientations of 157 quartz veins occurring in metabasalts of the Gadag region (Dharwar craton, southern India) are used to plot the 3-D Mohr stress circle, which provides information about relative stress/fluid pressure (P_f) conditions, as well as stress state during P_f fluctuation. To scale the 3-D Mohr circle, vein orientation data are integrated with (a) available estimates from fluid inclusions of highest recorded P_f (390 MPa) and lowest recorded P_f (50 MPa) and (b) intrinsic rupture criterion that empirically quantify rock properties. Based on the scaled 3-D Mohr circle, the absolute magnitudes of the three principal stresses are quantified for high and low P_f. Of 157 veins investigated here, 14 veins are identified as having favourable orientation for dilation at high as well as low P_f. These 14 veins have a mean strike of 150°, which is similar to the orientation of the gold-bearing quartz lodes reported in the region. The effective normal stress (σ′_n = σ_n−P_f) prevalent during dilation of fracture/fabric anisotropy with 150° strike is calculated to be −11.5 MPa at high P_f, and −1.0 MPa at low P_f. Thus, it is interpreted that in the Gadag region, a change in σ′_n of 10.5 MPa prevailed during P_f fluctuation and associated separation of gold from the fluid.     

Reliability-based robust design for reinforcement of jointed rock slope

Traditional reliability-based design methodologies often involve selection of design which is of lowest cost and satisfies safety requirements. But, this design is sensitive to variation in statistics of input parameters (noise parameters) and might become unsatisfactory if an underestimation of coefficient of variation of input parameters is made. A relatively new design methodology known as robust geotechnical design (RGD) is applied for the case of reinforcement of rock slope using end-anchored rock bolts. This ensures selection of a cost-effective and safe design for which probability of failure (P_f) of reinforced rock slope is least sensitive to the noise parameters. Reliability-based RGD approach involves evaluation of P_f for each design with different possible noise parameters. Finding P_f for the complex geotechnical structure is computationally expensive, and thus an augmented radial basis function-based response surface is used as a surrogate to the finite element model of rock slope. This response surface, being very efficient, also performs well for a range of values of noise parameters. Later, minimum distance algorithm is applied to obtain a cost-effective and robust design. Finally, a comparison is made in the costs between two robust designs obtained for different target probability of failure for the same rock slope.     

Bearing capacity factors of circular foundations for a general c–ϕ soil using lower bound finite elements limit analysis

By using the lower bound limit analysis in conjunction with finite elements and linear programming, the bearing capacity factors due to cohesion, surcharge and unit weight, respectively, have been computed for a circular footing with different values of ?. The recent axisymmetric formulation proposed by the authors under ?=0 condition, which is based on the concept that the magnitude of the hoop stress (σ_θ) remains closer to the least compressive normal stress (σ₃), is extended for a general c–? soil. The computational results are found to compare quite well with the available numerical results from literature. It is expected that the study will be useful for solving various axisymmetric geotechnical stability problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Bearing capacity of strip foundations reinforced with geogrid sheets by using upper bound finite‐element limit analysis

By using the upper bound finite‐elements limit analysis, with an inclusion of single and two horizontal layers of reinforcements, the ultimate bearing capacity has been computed for a rigid strip footing placed over (i) fully granular, (ii) cohesive‐frictional, and (iii) fully cohesive soils. It is assumed that (i) the reinforcements are structurally strong so that no axial tension failure can occur, (ii) the reinforcement sheets have negligible resistance to bending, and (iii) the shear failure can take place between the reinforcement and soil mass. It is expected that the different approximations on which the analysis has been based would generally remain applicable for reinforcements in the form of geogrid sheets. A method has been proposed to incorporate the effect of the reinforcement in the analysis. The efficiency factors, η_c and η_γ, to be multiplied with N_c and N_γ , for finding the bearing capacity of reinforced foundations, have been established. The results have been obtained (i) for different values of ? in case of fully granular and cohesive‐frictional soils, and (ii) for different rates at which the cohesion increases with depth for a fully cohesive soil. The optimum positions of the reinforcements' layers have also been determined. The effect of the reinforcements' length on the results has also been analyzed. As compared to cohesive soils, the granular soils, especially with higher values of ?, cause a much greater increase in the bearing capacity. The results compare reasonably well with the available theoretical and experimental data from literature. Copyright © 2013 John Wiley & Sons, Ltd.     

Coalescence of Fractures Under Uni-axial Compression and Fatigue Loading

In this paper, white Portland cement was used as an experimental material. Prismatic specimens with pre-existing flaws at different angles of inclination (α) varying through 0°, 30°, 45°, 60°, 75° to 90° and cylindrical specimens with different numbers of pre-existing flaws (n) varying through 0, 1, 2 to 3 were tested under uni-axial compression tests. Crack initiation, propagation, coalescence, and failure were observed. The corresponding analytical expression for the stress intensity factor under uni-axial compression was derived, the coefficient of friction and the stress intensity factor of the specimens on the surfaces of the crack were analysed, and the corrective coefficient for the stress intensity factor was introduced. Fatigue tests with a loading frequency of f = 100 Hz were carried out on cylindrical specimens with constant amplitude of the cyclic load which is a proportion of the compressive load at failure (F _f) obtained from the uni-axial compression tests. The fatigue property of the specimens was analysed and the relationship (S _max − lg N _f) between the maximum stress and the number of loading cycles at failure for specimens with pre-existing flaws was proposed. The effect of pre-existing flaws on the fatigue life (N _f) and dynamic load (S _D) which can be applied was investigated.     

Seismic Bearing Capacity of Shallow Strip Footing with Coulomb Failure Mechanism using Limit Equilibrium Method

In this paper, an effort is made to evaluate the seismic bearing capacity of shallow strip footing resting on c–ф soil. The formulation is developed to get a single coefficient of bearing capacity for simultaneous resistance of weight, surcharge and cohesion. Limit equilibrium method in Pseudo-static approach with Coulomb mechanism is applied here to evaluate the seismic bearing capacity. The seismic bearing capacity of footing (q_uE) is expressed in terms of single coefficient N_γE. The effect of various parameters viz. angle of internal friction of soil (ф), angle of wall friction (δ), cohesion (c), ratio of depth to width of footing (d_f/B₀), seismic acceleration (k_h, k_v) are studied on the variation of seismic bearing capacity co-efficients.     

Evolution of fluid pressure and fracture propagation during contact metamorphism

Abstract Rock fracture enhances permeability and provides pathways through which fluids migrate. During contact metamorphism, fluids contained in isolated pores and fractures expand in response to temperature increases caused by the dissipation of heat from magmas. Heat transport calculations and thermomechanical properties of water-rich fluids demonstrate (1) that thermal energy is a viable mechanism to produce and maintain pore fluid pressure (P_f) in a contact metamorphic aureole; (2) that the magnitude of P_f generated is sufficient to propagate fractures during the prograde thermal history (cause hydrofracture) and enhance permeability; and (3) that P_f-driven fracture propagation is episodic with time-scales ranging from years to thousands of years. Because P_f dissipation is orders of magnitude faster than P, _f buildup, P_f oscillations and cyclical behaviour are generated as thermal heating continues. The P_f cycle amplitude depends on the initial fracture length, geometry and the rock's resistance to failure whereas the frequency of fracture depends on the rate of heating. Consequently, oscillation frequency also varies spatially with distance from the heat source. Time series of fluid pressures caused by this process suggest that cyclical fracture events are restricted to an early time period of the prograde thermal event near the intrusive contact. In the far field, however, individual fracture events have a lower frequency but continue to occur over a longer time interval. Numerous fracture cycles are possible within a single thermal event. This provides a provisional explanation for multiple generations of veins observed in outcrop. P _f cycling and oscillations may explain several petrological features. If pore fluids are trapped at various positions along a pressure cycle, the large amplitude of P_f variations for small fractures may account for different pressures recorded by fluid inclusions analysed from a single sample. P_f oscillations, during a single thermal episode, also drive chemical reactions which can produce complex mineral textures and assemblages for discontinuous reactions and/or zoning patterns for continuous reactions. These can mimic polymetamorphic or disequilibrium features. Temporal aspects of fracture propagation and permeability enhancement also constrain the likely timing of fluid flow and fluid-mineral interactions. These data suggest that fluid flow and fluid-mineral reactions are likely to be restricted to an early period in the prograde thermal history, characterized by high P_f coincident with relatively high temperatures, fracture propagation and consequent increases in permeability. This early prograde hydration event is followed by diffusional peak metamorphic reactions. This relationship is evident in the complex mineralogical textures common in some metamorphosed rocks.     

Parametric Monte Carlo studies of rock slopes based on the Hoek–Brown failure criterion

Probabilistic evaluation of slope failures is increasingly seen as the most appropriate framework for accounting for uncertainties in design. This paper performs reliability assessments for rock slopes based on the latest version of the Hoek–Brown failure criterion. The purpose of this study is to demonstrate the use of a new form of stability number for rock slope designs that has been recently developed from finite element upper and lower bound limit analysis methods, and to provide guidance for its use in probabilistic assessments. The analyses show that by using this newly proposed stability number, the probability of failure (P_f) obtained from case studies agrees well with the true state of the slope. In addition, this paper details a procedure to determine the magnitude of safety factor required for rock slope design.     

The Effect of Heavy Tamping on Structured Residual Clay Site

This paper examines the effect of heavy tamping (dynamic compaction) on highly porous structured residual clayey soil. The aim of this study is to analyse the feasibility of this technique when applied on lightly bonded residual soil sites, which are commonly found in tropical and subtropical regions. This soil has some interesting characteristics, such as high fine grain soil percentages (56% clay and 22% silt), a plastic index of 11%, high porosity (initial void ratio of 1.21), high hydraulic conductivity (about 10^?5 m/s) and a high stiffness at small strains (E?=?49.2-MPa). The research involves field [Cone Penetration Test (CPT) and the dynamic compaction] and laboratory (triaxial tests, characterization and hydraulic conductivity) investigation. According to laboratory tests, the void ratio decreased to 0.96, hydraulic conductivity decreased to 2.8?×?10^?7 m/s, the effective peak friction angle (?′) increased from 30.5° (in natural conditions) to about 35.5°, and the triaxial stiffness at small strains decreased to E?=?20-MPa due to dynamic compaction. CPT results have shown an improved depth in which CPT tip strength (q_t) increased from nearly 650-kPa to an average of 1700-kPa and CPT sleeve friction (f_s) increased from approximately 50-kPa to about 130-kPa. Horizontal displacements were observed up to about 4.0 m of depth (approximately the same depth at which CPT results showed soil improvement). It was concluded that heavy tamping reduces soil voids and substantially increases soil strength, but also breaks soil structure and decreases soil stiffness. It is thus not a suitable ground improvement solution for highly porous structured residual clayey soil.

     

The strain history of a dalradian slide: Using pebbles with low fluctuations in axis orientation

The strain associated with the Horn Head Slide, a major tectonic break in the Dalradian rocks of NW Ireland, is recorded by pebbles in an adjacent quartzite horizon. Mean X/Y ratios of the deformed pebble shapes in excess of 8.0 are seen closest to the slide and the field of three-dimensional shapes lies along the K = 1 line. The usual methods of separating initial shape ratio (R_i) and tectonic strain ratio (R_s) from the deformed shape ratio (R_f) of ellipsoidal markers are based on measurements of variation in fluctuation (e.g., the (R_f/φ technique). However, due to the high X/Y strains in this situation and since the pebbles initially lay parallel to bedding and to a principal plane of the finite strain ellipsoid, fluctuation is generally very low. Thus, except for the least deformed X/Y data, the R_f/φ technique is inapplicable and other methods are used. For X/Y data with mean (R_f > 4.0: R_s is calculated as the harmonic mean of R_f; maximum R_i values only are obtained from the range of R_f data. For all Y/Z and X/Z data: R_i is calculated from the logarithmic range (ω_log) of R_f; R_s is simply obtained from the geometric mean of R_f modified by R_i. It is concluded from this that a varying prolate tectonic strain (K - 1.5) reaching X/Y values in excess of 8.00 was coaxially superimposed on an initial oblate shape fabric to produce the present field of deformed pebbles in the quartzite near the slide.     

The critical scale of fluctuation for active lateral forces in spatially variable undrained clays

This study addresses the phenomenon of the critical scale of fluctuation (SOF) for active lateral force (P_a) in undrained clay when there is a spatial variability in the clay. The phenomenon is significant under shear strength (τ_f) random fields but is insignificant under unit weight (γ) random fields. It is found that the phenomenon of the critical SOF is connected to the nature of the spatial averaging, which is “line averaging” under τ_f random fields and is “area averaging” under γ random fields. The former averaging effect (line) is significantly weaker than the latter (area), so the tendency for the critical slip plane to seek for a favorable location is stronger for the τ_f random field than for the γ random field. Hence, the phenomenon of the critical SOF is more pronounced under τ_f random fields than under γ random fields. The underlying mechanisms for the phenomenon of the critical SOF will be explored in this paper.