Soil has spatial variability, which means that soil properties at different locations are different but correlated. To represent the spatial variability of soil surrounding a pile, the random field method (RFM) is usually adopted to discretize a random field into a number of random variables. Then, the first-order reliability analysis method (FORM) is modified and employed to perform reliability analysis, and the load-transfer method (LTM) is adopted to compute the bearing capacity of the pile. To reduce the computation cost of the reliability analysis and random field simulation, a FORM-LTM-variance reduction method (VRM) method is proposed to conduct reliability analysis for single pile in spatially variable soil, in which VRM is adopted to transfer a random field into a random variable over a characteristic length. By comparing the reliability indices using FORM-LTM-RFM and FORM-LTM-VRM, analytical formulas of the characteristic lengths under different pile lengths, coefficients of variation (COVs), and autocorrelation distances (ACDs) are computed. Benefitting from the computation accuracy and efficiency of the FORM-LTM-VRM with analytical formulas of characteristic length, resistance factors in LRFD for the reliability-based design of single pile in spatially variable soil can be easily computed for different safety levels. The accuracy and efficiency of the FORM-LTM-VRM with analytical formulas of characteristic length are demonstrated by a case study of a vertically loaded pile.

This study develops a practical copula-based FORM in an original physical space for an efficient slope reliability analysis involving correlated non-normal variables. First, the copula theory for modeling the joint probability distribution of cohesion and friction angle of soils is briefly introduced. Second, the traditional expanding dispersion ellipse perspective of the FORM is reviewed. Then, a new expanding dispersion contour perspective of the copula-based FORM is formulated in detail. Finally, two illustrative slope examples are presented to illustrate and demonstrate the developed copula-based FORM and its expanding dispersion contour perspective. The results indicate that the developed copula-based FORM has good accuracy and efficiency in deriving the reliability index and design point for a typical slope reliability problem. It operates in an original physical space and thus is suitable for a complex slope reliability problem with an implicit performance function. The copulas for characterizing various dependence structures between the cohesion and friction angle of soils have a significant impact on slope reliability index, especially under a high reliability level and a strong degree of negative correlation. The proposed expanding dispersion contour perspective facilitates the understanding of the copula-based FORM, which can readily explain the slope reliability results produced by various copulas. The derived design points are obtained by simultaneously considering the marginal distributions, correlation as well as various non-Gaussian dependence structures of soil parameters.

In the random field model's consideration of the spatial variability of soil, soil properties at different locations play different roles in the reliability analysis of the foundation. Investigating the importance distribution of the random field through reliability sensitivity analysis (RSA) is beneficial for understanding how the random field affects the reliability of the foundation. However, many existing RSA methods for the random field model are deficient in terms of efficiency, accuracy, and applicability under complex engineering conditions. Consequently, this study proposes an efficient RSA method for the random field model based on the Karhunen-Loeve (KL) expansion method and the first-order reliability method (FORM) to identify the important random field domain in foundation engineering. In the proposed method, the mean reliability sensitivity index (MRSI) is extended to a random field model of continuous form to characterize the importance distribution of the random field. The MRSI is analytically derived based on the results of the KL expansion method and the FORM without additional limit state function (LSF) calculations. Subsequently, the important random field domain, in which the variation of the mean of the soil property contributes significantly to the reliability index, is identified based on the MRSI. Last, two foundation engineering examples that consider the cross-correlated random fields of cohesion and friction angle, including strip footing on single-layer soil and pile in multiple-layer soil, were used to verify the proposed method. The results showed that an important random field domain with a small area dominates the variation of the reliability index of a foundation, and important random field domain area increases with autocorrelation distance (ACD). This innovative identification method holds great engineering significance, because it allows geotechnical practitioners to gain a comprehensive understanding of the failure modes and foundation treatment areas of foundations in spatially varying soil. In the random field model's consideration of the spatial variability of soil, soil properties at different locations play different roles in the reliability analysis of the foundation. Investigating the importance distribution of the random field through RSA is beneficial for understanding how the random field affects the reliability of the foundation. However, many existing RSA methods for the random field model are deficient in terms of efficiency, accuracy, and applicability under complex engineering conditions. Consequently, this study proposes an efficient RSA method for the random field model to identify the important random field domain, in which the variation of the mean of the soil property contributes significantly to the reliability index. Two foundation engineering examples that consider the cross-correlated random fields of cohesion and friction angle, including strip footing on single-layer soil and pile in multiple-layer soil, were used to verify the proposed method. The results showed that the innovative identification will allow geotechnical practitioners to gain a comprehensive understanding of the failure modes and foundation treatment areas of foundations in spatially varying soil.

Probabilistic analysis has been widely used to assess the inherent uncertainty of variables in laterally loaded pile systems, but the calculation is still difficult and time-consuming. The present study presents an efficient probabilistic analysis framework for a laterally loaded pile system. The performance of the system is defined as the lateral deflection at the pile head and maximum bending moment of the pile shaft, corresponding to two failure modes. Within this framework, the spatial variability of the soil and the correlation between failure modes are considered by the random field theory and the First-Order Reliability Method, respectively. Moreover, the Sequential Compounding Method is used as an efficient tool to determine the system reliability indexes. The framework is confirmed by comparing the reliability indexes of failure modes and systems with those of the Monte Carlo Simulation Method. Furthermore, a parametric analysis and system sensitivity analysis are performed. The results show that the auto-correlation distance, allowable lateral displacement at the pile head, and allowable bending moment of the pile shaft have a great influence on reliability indexes of failure modes and system, and the major parameter of soil in affecting pile is the elastic modulus compared with the undrained shear strength.

At present, the reliability analysis and design method of vertically loaded piles embedded in spatially variable soils is difficult to be applied in practical engineering due to the huge computation effort required. To improve computational efficiency, this paper proposes a new method called the FORM-KL-LTM, which integrates the advantages of the first-order reliability method (FORM), the Karhunen-Loeve (KL) expansion method, and the load transfer method (LTM). The main framework of the FORM-KL-LTM is the FORM, which is used to perform reliability analysis for the pile. The KL expansion method is adopted to carry out random discretization to generate the discrete soil parameters required by each iterative computation of the reliability index using the FORM, and the LTM is employed to evaluate the nonlinear load-settlement behavior of the pile head and to compute the values of limit state functions required by the FORM. The proposed method is computationally efficient because the number of random variables is controlled by the limit number of KL expansion terms. Based on the FORM-KL-LTM, a reliability sensitivity analysis method is proposed, which can compute the sensitivity index for measuring the relative sensitivity of the reliability index with respect to soil properties. Furthermore, a procedure for the reliability-based design (RBD) of piles embedded in spatially variable soils is established for the design of pile geometry, and a design ratio is defined to select the controlling limit state in the RBD of pile for both the ultimate limit state and the serviceable limit state. The procedure, accuracy, and efficiency of the proposed methods are demonstrated by providing an example of the reliability analysis and design of a vertically load pile in spatially variable soils.

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In the case of DFIG (double-fed induction generator) with a low sampling frequency, the current controller designed in continuous-time domain will be affected by discretization errors and digital control delays in the process of digital realization. In response to this problem, this paper first establishes a complex vector model on the rotor side of DFIG to analyze the influence of digital implementation on control effect. Then, by establishing an accurate discretization model, designing a discrete-time current controller with delay considered. Finally, the effect of designed controller is verified by simulation.

Many uncertainties exist in geotechnical engineering. Spatial variability is a type of uncertainty that has received more and more attention in recent years. Scale of fluctuation (SOF) is a parameter for measuring the spatial variability of soil properties. To investigate the proper method for computing the SOF of soil properties, two types of methods (auto-correlation function method (ACM), and variance reduction method (VRM)) are compared theoretically and numerically, and the influence factors, such as number of data points involved in curve fitting for the auto-correlation function (ACF) or variance reduction function (VRF) and sampling interval, on the SOFs are analyzed. To improve the accuracy of curve fitting for the ACF or VRF, a weighted least squares (WLS) method is proposed, in which the weight of each data point of sample function decreases with the increasing of lag distance between two points in a random field. Three types of soil parameters, including the CPT, the physical, and the hydraulic parameters, are tested by in-situ and laboratory tests, and the SOFs of these parameters are computed. Computation of the SOFs proves that the VRM combined with the suggested WLS is a good method for computing the SOFs of soil properties. Comparison of the SOFs shows that there are negligible differences among the three types of soil parameters of clay layers in Hefei, which is further proved by the Wilcoxon test from a statistical point of view. By using the VRM combine with WLS, the SOFs of clay are computed based on 509 sets of CPT data from 34 sites in Hefei. The statistics of SOFs can give a reference for the selection of SOF for reliability analysis or reliability-based design of geo-structures with spatial variability.

Geomaterial has spatial variability, which can be described by the random field theory. A powerful tool for realizing random fields is the Karhunen–Loève series expansion (K–L expansion) where the number of random variables depends on the number of K–L expansion terms rather than on the number of grids of geo-structure model. However, the K–L expansion requires the solution of an integral eigenvalue problem whose analytical form exists only in the special case. Hence, the Galerkin method is usually employed to calculate the approximate solution, which especially for multi-dimensional random field inevitably causes the huge computational cost of multi-fold integrals and the approximate error of the solution. For quickly and accurately simulating the random field with fewer random variables, a Jacobi–Lagrange–Galerkin (JLG) method is proposed where the huge amounts of multi-fold integrals are transformed into simple matrix multiplications which are implemented quickly in a MATLAB environment. Furthermore, the discretization error and its influence factors are discussed to determine the parameters of the JLG method, and the procedure of the JLG method is proposed. Finally, a two-dimensional shallow foundation and a three-dimensional slope are employed to demonstrate computational efficiency, accuracy, and applicability of the JLG method.

To analyze the influences of parameter uncertainty on system reliability, a system reliability sensitivity analysis method based on the sequential compounding method (SCMSA) is proposed. The SCMSA makes use of the principle of SCM combination element, and further calculates the equivalent correlation coefficient between the two components and other remaining components in the system on the basis of calculating the reliability and sensitivity of a simple system with two components in parallel or in series, so as to achieve the purpose of combining the two components and simplifying the complex system. The advantage of SCMSA is that it integrates the calculation of the relative sensitivity index into the system reliability analysis, so that the sensitivity analysis can be calculated together as a byproduct of the reliability analysis, and this method can be applied to the system reliability sensitivity analysis of the relative non-normal variables. Finally, a simple numerical example is used to illustrate the calculation process, calculation accuracy and calculation advantage of SCMSA, and it is applied to the sensitivity analysis of a system reliability of semi-gravity retaining wall, indicating that the SCMSA can provide a theoretical basis for the risk analysis and prevention of geotechnical engineering.

Most of the pile's vertical static load tests in construction sites are the proof load tests, which is difficult to accurately estimate the ultimate bearing capacity and analyze the reliability of piles. Therefore, a reliability analysis method based on the proof load-settlement (Q-s) data is proposed in this study. In this proposed method, a simple ultimate limit state function based on the hyperbolic model is established, where the random variables of reliability analysis include the model factor of the ultimate bearing capacity and the fitting parameters of the hyperbolic model. The model factor M = R-uR / R-uP is calculated based on the available destructive Q-s data, where the real value of the ultimate bearing capacity (R-uR) is obtained by the complete destructive Q-s data; the predicted value of the ultimate bearing capacity (R-uP) is obtained by the proof Q-s data, a part of the available destructive Q-s data, that before the predetermined load determined by the pile test report. The results demonstrate that the proposed method can easy and effectively perform the reliability analysis based on the proof Q-s data.

In this article, a mathematical analysis model of economics of prefabricated building structure based on improved neural network algorithm is proposed in order to solve the low analysis accuracy in traditional methods. Firstly, by means of analyzing the costs of materials, labor, and equipment, the economic characteristics of the cost of fabricated building structures are determined. Secondly, the single neuron is analyzed and the weight coefficient is adjusted in accordance with the multilayer neural network structure, so as to minimize the construction error of the economic analysis model of the assembled building structure. Meanwhile, the weight vector is obtained, error-weighted square sum is calculated through choosing an adaptive filter and obtained, and the weight vector is updated by the least squares algorithm. -irdly, the neural network algorithm training and learning process is designed and improved, the dependent variable is selected, the number of input points is determined, and then, the training and learning process of the improved neural network algorithm is completed. Finally, a fitness function is set to measure the authenticity of dataset, which is further defined as a combination of different weights to construct an economic mathematical analysis model.-e experimental results indicate that the analysis results of this method can reach an accuracy up to 96%, so it has a broader application prospect in low-rise buildings.

The spatial variability is an inherent uncertainty of soils. The random field theory is used to represent the spatial variability of soils, and the random field discretization is performed by the Karhunen-Lobve (KL) expansion method. Using the slope upper bound analysis based on the discrete mechanism, the discretization results of the internal friction angle random field at each point in the space are considered when generating the velocity discontinuity surface. The upper bound analysis, shear strength reduction technique, bisection searching, and sequential quadratic programming method are combined to solve the safety factor of slopes. The first-order reliability method (FORM) and subset simulation (SS) are employed for slope reliability analysis. Given the characteristics of SS and the shear strength reduction technique, an optimization algorithm coupling the two is proposed to improve computational efficiency. By calculating and analyzing an earth slope, the similarities and differences between FORM and SS based on the KL expansion method in solving the slope reliability index and failure consequence are clarified. The influence of the variation coefficient of soil strength parameters on the slope reliability index and failure consequence is investigated, providing a theoretical basis for risk analysis and prevention of slopes.