Vertical Scale Of Fluctuation Research Articles

Reliability Analysis of Safety Assessment of Buildings above Twin Shield Tunnels Based on Random Field Theory

Aiming at the problem of deformation of adjacent masonry structures caused by the surrounding soil disturbance in the construction of twin shield tunnels, taking the spatial variability of soil parameters as the breakthrough point and the surface deformation mode as the basis, a program is compiled to search the position of characteristic points of the settlement curve, and a safety assessment method of masonry buildings above twin tunnels based on the random field theory is obtained. According to the reliability theory, the concept of the potential damage zone is defined. The probability that the maximum tensile strain of the building exceeds the allowable ultimate tensile strain is not less than 5% in the potential damage zone. The assessment method mentioned above, together with the Monte Carlo simulation, is used to evaluate the safety of buildings and the range of potential damage zones induced by twin tunneling with different relative locations and different construction sequences, taking into account configurations and construction sequences of twin tunnels, the scales of fluctuation of elastic modulus of the soil, and the size of the building. The research results show that, in the parallel and overlapping twin tunnels, the randomness of the surface deformation mode caused by construction increases with the decrease of the transverse scales of fluctuation or the increase of the vertical scales of fluctuation. In overlapping and shoulder-mounted twin tunnels, the effect of the construction sequence on the safety of buildings is the opposite. When the overlying tunnel is excavated first in shoulder-mounted condition, the safety of buildings is more excellent. Considering the effect of the spatial variability of parameters, the probability of higher damage levels in the buildings is significantly increased. The greater the scales of fluctuation, the greater the damage probability. The potential damage zones of the buildings will be expanded, and the safety distance of the buildings will be wider than the deterministic analysis result by 1.5∼2.5 m.

Numerical investigation of pile responses induced by adjacent tunnel excavation in spatially variable clays

The pile responses induced by adjacent tunnel excavation have been a hot research topic in geotechnical engineering. Tunnel excavation may exert disturbance to the surrounding soil mass and then influence the adjacent pile foundations. In this paper, the random finite difference analysis considering the spatial variations of soil properties is conducted to explore the effect of tunnel excavation on the adjacent pile response by varying the distance of pile away from the tunnel centerline (D), the pile length (L), the pile diameter (dp), the tunnel depth (h), and the anisotropic ratio between the horizontal and the vertical scales of fluctuation (δx/δy). A set of pile response curves are developed to assess the influence of tunnel excavation on the adjacent passive pile foundations in spatially variable clays and provide guidelines for the tunnel excavation in the complex constructed environment.

Novel approach to estimate vertical scale of fluctuation based on CPT data using convolutional neural networks

The inherent spatial variability of soil properties is the main sauces of uncertainties in the site investigation, and it is commonly characterized using random field theory. In the context of random fields, the scale of fluctuation (SOF) is a significant parameter to reflect the spatial correlation between the two points of soil properties. However, it is challenging to estimate the SOF value accurately, especially when there are only limited project-specific test results, such as cone penetration test (CPT) data. This study aims to develop a convolutional neural network (CNN) approach to estimate the vertical SOF based on limited CPT data. The CNN model was constructed and trained by the simulated 15,000 CPT samples using random fields. The results show that the CNN model has excellent performance for estimating vertical SOF. The approach is validated and illustrated through newly simulated CPT data, eight real CPT data obtained from the literature, and three CPT data collected from the Shanghai site. The proposed scale factor method can solve the mismatch between the actual CPT depth and the required depth for input data of the CNN model, making the CNN model more widely applicable.

Probabilistic stability assessment of tunnel-support system considering spatial variability in weak rock mass

Analysis of tunnel-support system stability with uncertain rock mass properties is conducted in this paper. A horseshoe shaped tunnel driven in weak rock mass is analyzed through deterministic, random variable and random field approaches. Performance of the tunnel in probabilistic analysis was assessed by defining three limit states which ensure that tunnel convergence remained below a safe threshold level, the rock bolts installed are embedded sufficiently beyond the depth of yielded rock mass and load induced on liner support does not exceed its capacity. Both unsupported and supported tunnels were analyzed with deterministic, random variable and random field approaches. Fourier series method was applied to discretize the random fields, and random finite difference analysis was conducted using Monte Carlo simulations. Scale of fluctuation (SOF) for isotropic random fields and horizontal and vertical SOF ratios for anisotropic random fields were varied to study their effect on performance of the tunnel and failure mechanisms involved. It was found that SOF significantly influences the output statistics and Pf of the limit states. It was observed that, random variable approach underestimates the performance of the tunnel-support system; however, it can be adopted as conservative option in absence of data required for random field characterization.

Probabilistic characteristics analysis for the time-dependent deformation of clay soils due to spatial variability

Consolidation and creep are two types of time-dependent deformation of clay soil. However, the soil deformation presents probabilistic behaviours when the spatial variability is considered. The Monte–Carlo method is used to sample anisotropic random fields that are mapped into two-dimensional random finite element models (RFEM) to analyse the influences of spatial variability on the time-dependent deformation of clay soil. The anisotropic random fields of soil parameters are generated by a spectral representation method (SRM) to describe the spatial variability of clay soil. The consolidation and the creep problems are modelled by Biot’s consolidation theory and the creep model proposed by Yin et al. (2010) respectively. The spatial variability of Young modulus and permeability coefficient in probabilistic consolidation analysis and the secondary consolidation coefficient in probabilistic creep analysis are emphasized by the RFEM simulation. The results show that the scale of fluctuation (SOF) in vertical direction and the coefficient of variation (COV) of Young modulus and permeability have significant influences on the probabilistic characteristics of consolidation behaviour. Besides, the bunched region of low permeability soil produces a blockage effect, which seriously affects the consolidation characteristics. It is found that the uncertainty of creep deformation will accumulate over time until it remains constant. The variability of creep deformation increases with the increase in the vertical SOF and COV of the secondary consolidation coefficient. The research also provides insightful clues for the time-varying reliability design related to the time-dependent deformation of clay soil.

Probabilistic analysis of tunnel convergence on spatially variable soil: The importance of distribution type of soil properties

Accurate prediction of tunnel convergence is important to ensure the stability of tunnel structures. Reliability analysis of tunnel convergence on spatially variable soils was performed based on Monte Carlo simulations. The Young’s modulus of soils was shown to be spatially varied. The influence of the coefficient of variation, the vertical scale of fluctuation and the distribution type (i.e. beta distribution, gamma distribution and lognormal distribution) of soils on the tunnel convergence were systematically discussed. From measuring the mean and standard deviation of tunnel convergence, and the probability of exceeding some thresholds, it was found that the distribution type significantly influenced the results. The results were accompanied by a detailed analysis of the mechanisms of tunnel convergence in spatially random soils.

3D bearing capacity probabilistic analyses of footings on spatially variable c\u2013\u03c6 soil

The paper examines three-dimensional (3D) analyses of the load bearing capacity of square and strip footings on a spatially variable cohesive–frictional (c–φ) soil. Embedded, rigid and smooth footings are considered. Cohesion and friction angle are modelled using random fields. The vertical scale of fluctuation (SOF) is based on earlier studies of natural soils. For the square footing problem, the horizontal SOF is the subject of a parametric study. For the strip footing, only one value of the horizontal SOF is assumed, and the parametric study concerns the modelled length of the foundation. To generate the fields of strength parameters, the Fourier series method is employed. For both problems analyzed, the influence of the parameters on the probabilistic characteristics of the bearing capacity of the soil is investigated. For the square footing problem, both the mean value of bearing capacity and its coefficient of variation increase with an increasing horizontal fluctuation scale, but they stabilize at higher values of SOF. For the strip footing problem, it appears that 2D modelling is a very conservative approach, and for the precise assessment of the probability of foundation failure, the footing should be modelled in 3D.

Effect of spatial variability of shear strength parameters on critical slip surfaces of slopes

Critical slip surface (CSS) of slopes is useful information in system reliability analyses and risk assessments of slopes. This paper investigates the effect of spatial variability of shear strength parameters on the CSS distribution of a two-dimensional slope using a finite element stress-based slope stability method. The failure mechanics for typical CSSs in a heterogeneous slope are analyzed and distributions of CSS for various combinations of horizontal and vertical scales of fluctuation (SOFs) are evaluated. The results show that local failures may occur in statistically homogeneous slopes, which cannot be observed in deterministic slope stability analyses. The distribution range of CSS first increases and then decreases with an increasing scale of fluctuation. As the ratio of horizontal SOF to vertical SOF is large, local failures have entry points located at the top of a slope. By contrast, as the ratio of horizontal SOF to vertical SOF is relatively small, local failures have entry points located either at the top of a slope or on the slope surface. The findings facilitate a more accurate evaluation of the system reliabilities or risks of slopes.

Probabilistic assessment of tunnel convergence considering spatial variability in rock mass properties using interpolated autocorrelation and response surface method

This study aims at the probabilistic assessment of tunnel convergence considering the spatial variability in rock mass properties. The method of interpolated autocorrelation combined with finite difference analysis is adopted to model the spatial variability of rock mass properties. An iterative procedure using the first-order reliability method (FORM) and response surface method (RSM) is employed to compute the reliability index and its corresponding design point. The results indicate that the spatial variability considerably affects the computed reliability index. The probability of failure could be noticeably overestimated in the case where the spatial variability is neglected. The vertical scale of fluctuation has a much higher effect on the probabilistic result with respect to the tunnel convergence than the horizontal scale of fluctuation. And the influence of different spacing of control points on the computational accuracy is investigated.

Influence of spatial variability of soil Young's modulus on tunnel convergence in soft soils

Field data have shown the fact that soil spatial variability could aggravate the uncertainty of tunnel convergence ΔD (a key indicator for serviceability and safety of tunnels). This paper presents a detailed numerical analysis to investigate the probabilistic response of tunnel convergence in spatially varied soft soils. The soil Young's modulus Es is highlighted and modeled with isotropic and horizontally stratified anisotropic random fields, respectively. The influence of scale of fluctuation (SOF) δ of the Es on convergence ΔD is discussed in detail with respect to different directions, i.e., the vertical and horizontal directions both for δ and ΔD. It is observed that ignoring the spatial variability of Es, i.e., disregarding the possibility of unfavorable soft soil (low stiffness soil) locally around tunnel, can underestimate the mean value of ΔD. The horizontally stratified anisotropic random field is more appropriate than isotropic random field in the sense of an accurate prediction, especially when extreme tunnel convergence occurs. In horizontally stratified anisotropic random fields, the influence of horizontal and vertical SOF is different on tunnel convergence. The surrounding soils near tunnel crown and invert or across tunnel horizontal diameter are very critical to the tunnel convergence. In addition, the effect of horizontal SOF δx on failure probability of the calculated ΔD exceeding the specified allowable ΔDlim is limited when the δx is larger than 4.84 times of tunnel outer diameter.

Assessment of Sandy Soil Variability Based on CPT Data

This paper presents a statistical analysis to characterize soil variability based on the cone penetration tests (CPTs) data obtained from a site near to Jijel port located in northern of Algeria.Ten CPTs were performed to a depth between 10 m and 13 m in sandy soils within an area of 120x86 m2. Only the vertical variability of the cone tip resistance (probability density function, coefficient of variation and scale of fluctuation) is considered and the data are analysed within a homogenous layer using the random field theory.The results in terms of coefficient of variation and vertical scaleof fluctuation are compared to the corresponding values available in the literature. The comparison showed different ranges of the vertical scale of fluctuation that may be due to the variation of the cone tip resistance in cohesionless soils. This variability will be integrated in a study of the potential liquefaction of the site.

Influence of spatially random soil on lateral thrust and failure surface in earth retaining walls

ABSTRACTThe present study investigates the influence of spatial variability of soil properties on the lateral thrust and failure surface of a 6 m high frictionless rigid earth retaining wall in active condition. The drained friction angle of the soil is modelled as a log-normal anisotropic random field in two dimensions, using the Cholesky decomposition technique. The effect of horizontal and vertical scale of fluctuation is observed on the lateral thrust. The failure surface in the backfill is found to be more or less similar for different combinations of spatial variability in vertical and horizontal directions. Monte-Carlo simulation technique is used to compute the probability of failure and to obtain the worst-case spatial variability configuration.

Spatial correlation for transformation uncertainty and its applications

ABSTRACTThis paper shows that the transformation uncertainty varies in space and can be decomposed into (a) a slowly fluctuating component contributed by the systematic bias for a local site and (b) a rapidly fluctuating component partly contributed by the measurement error. Based on a piezocone database, the slowly fluctuating component is found to have a vertical scale of fluctuation (SOF) of 17–60 m, larger than the vertical SOFs for most soil properties and characteristic dimensions for many geotechnical structures. This paper provides an empirical basis to justify the assumption that the transformation uncertainty is fully correlated in space.

Undrained strength for a 3D spatially variable clay column subjected to compression or shear

The mobilized shear strength (τfm) of a soil is the most important design parameter governing ultimate limit states. The authors have proposed some models and equations to characterize several useful aspects of the mean, variance, and probability density function of τfm for a two dimensional (2D) spatially variable clay specimen. This paper demonstrates that the models and equations developed for 2D can be extended to three-dimensional (3D) cases. An important class of 3D cases are examined in detail: two horizontal scales of fluctuation (SOF) are equal, and both are larger than the vertical SOF. A key observation for 3D cases is that the averaging effect over the potential slip planes is significantly stronger than that for 2D cases. The upshot is that the phenomenon of the critical SOF (e.g., SOF producing the lowest mean value of τfm) is less clear for 3D cases.

Reliability-Based Analysis of Bearing Capacity of Strip Footings Considering Anisotropic Correlation of Spatially Varying Undrained Shear Strength

AbstractA reliability-based analysis of the bearing capacity of strip footings considering the spatial variability of undrained shear strength of clay was performed using the random FEM. The effect of anisotropic correlation of spatially varying undrained shear strength on the mean and standard deviation of bearing capacity was obtained using the random FEM. It was found that the anisotropic correlation of undrained shear strength significantly affects the bearing capacity. The mean bearing capacity was found to be minimal for a specific combination of horizontal and vertical scales of fluctuation. The coefficient of variation of the bearing capacity increased with increases in horizontal or vertical scales of fluctuation, or both. The factor of safety required to achieve a specified target failure probability for different scales of fluctuation and coefficients of variation of the undrained shear strength of clay is presented.

Effective Young's modulus for a spatially variable soil mass subjected to a simple stress state

ABSTRACTThe effect of spatial variability on the effective Young's modulus of a soil mass is investigated. The soil mass is a two-dimensional plane strain square domain subjected to a simple stress state, with Young's modulus being modelled as a stationary lognormal random field. The effective Young's modulus is simulated by random field finite-element analysis. It is found that under the condition of horizontal scale of fluctuation (SOF) = vertical SOF, the effective Young's modulus can be satisfactorily approximated as the geometric average over the square domain. However, the conclusion changes dramatically when the spatial variability is highly anisotropic, e.g. horizontal scale vertical SOF. In this case, the effective Young's modulus can be approximated as the arithmetic average or harmonic average, depending on the direction of loading. A unified spatial averaging model is further proposed in this study. It is shown that the effective Young's modulus can be satisfactorily approximated by this unified model without the need to switch among arithmetic, geometric and harmonic averages.

Simplified reliability method for spatially variable undrained engineered slopes

A random finite element analysis (RFEA) is employed to study the effect of shear strength spatial variability on the stability of undrained engineered slopes. Based on the RFEA simulation results for 34 (idealized) real engineered slope cases, it is concluded that the coefficient of variation in the spatially variable shear strength and the ratio (vertical scale of fluctuation)/(length of failure curve) have a major effect on the stability reliability. The RFEA simulation results are further used to derive simplified equations for the reliability analysis and the reliability-based design of undrained engineered slopes.

The Effect of Spatial Correlation of Cone Tip Resistance on the Bearing Capacity of Shallow Foundations

The effect of directional behaviour of correlation structure of cone tip resistance on the bearing capacity of shallow strip footing resting on cohesionless soil deposit in 2-D random field is analysed using probabilistic approach. The results obtained from the analysis show that the assumption of perfect (or infinite) correlation of cone tip resistance data leads to lower values of probability of failure. In contrast, the isotropic assumption of correlation behaviour based on vertical scale of fluctuation leads to higher values of probability of failure. The results also show that the transformation model would play a major role in the evaluation of variability of design property. In conclusion, the need for a proper evaluation methodology for calculation of correlation lengths of soil properties and their influence in foundation design is highlighted.

Influence of heterogeneity on undrained clay slope stability

This paper concentrates on the fact that natural materials are variable, and that representation of this variability appears crucial to getting a realistic understanding of certain geotechnical problems. In this case, finite elements have been used to assess the influence of heterogeneity on the stability of a clay slope, characterized by a spatially varying undrained shear strength c u . For low to intermediate values of the coefficient of variation (e.g. 0.1–0.3), as are often observed in practice, pointwize variability can be approximated by a normal distribution. Data on spatial variability are more scarce, although vertical scales of fluctuation are usually much smaller than horizontal scales of fluctuation, and will often be small relative to the height of a slope. The results show that, for a given factor of safety, reliability is greatest for c u constant with depth. For c u increasing linearly with depth, reliability decreases due to the greater range of possible rupture surfaces: hence, in conventional (deterministic) design, higher factors of safety (or lower characteristic values) would be needed. In this study, horizontal anisotropy (of the heterogeneity) causes modest changes in reliability. However, problems are identified in which anisotropy is likely to have a significant influence.