### 数学代写|概率论代写Probability theory代考|MATHS 7103

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|概率论代写Probability theory代考|Shear Behavior of Granular Soils

The shear properties of soil refer to the contraction/dilatancy and yield strength properties or friction properties during shearing. The conventional testing method includes (1) shear tests for uneven deformation of soil samples, such as direct shear tests and simple shear tests, and (2) triaxial shear tests that lead to relatively uniform deformation of soil samples. According to different design requirements, methods such as slow shearing or fast shearing and drained or undrained test conditions are usually adopted, and different deformation and strength indexes are obtained. Triaxial shear tests are widely used due to the advantage of sample uniformity during shearing.
Based on the consolidation history, the current void ratio and the stress state, clay can be divided into normally consolidated and overconsolidated clay, while sandy soil is divided into loose and dense sand. The results of a large number of triaxial shear tests are shown in (Fig. 1.6):
(a) Normally consolidated and slightly overconsolidated clays and loose sand exhibit volumetric contraction during shearing; the void ratio decreases under drained conditions, and the mean effective stress decreases under undrained conditions.
(b) Highly overconsolidated clay and dense sand exhibit volumetric expansion during shearing, that is, dilative characteristics (the void ratio becomes larger under drained conditions, and the mean effective stress becomes larger under undrained conditions), along with the peak stress ratio above the critical stress ratio.

## 数学代写|概率论代写Probability theory代考|Critical State Line of Granular Soils

Several formulae for expressing the critical state concept have been proposed. The most typical formula is the linear formula in the $e$-log $p^{\prime}$ plane. The relationship has traditionally been written as follows:

$$e_{c}=e_{r e f}-\lambda \ln \left(\frac{p^{\prime}}{p_{r e f}}\right)$$
where $e_{\text {ref }}$ is a reference void ratio corresponding to a reference mean effective stress $p_{\text {ref }}$; and $\lambda$ is the slope of the critical state line (CSL) in the $e$-log $p^{\prime}$ plane. Therefore, two parameters ( $e_{\text {ref }}$ and $\lambda$ ) are required for the definition of the CSL. The advantage of this formula is the simplicity of its form. However, experimental results have shown that the CSL is not always linear in the $e-\log p^{\prime}$ plane, and mathematically, the critical void ratio $e_{c}$ could become negative for high stress levels, which is meaningless. Note that a very high stress level still exists in geotechnical structures, such as at the pile tip during its installation.

More recently, another formula was proposed by Li and Wang [169], who assumed a nonlinear critical state line in the $e$-log $p^{\prime}$ plane, representing an extension of the linear formula with one additional parameter $\xi$ :
$$e_{c}=e_{c 0}-\lambda\left(\frac{p^{\prime}}{p_{a t}}\right)^{\xi}$$
where the additional parameter $\xi$ controls the nonlinearity of the critical state line, giving a more flexible and accurate description according to experimental data, especially for very low to moderate stress levels. However, for high stress levels, the positiveness of the critical void ratio $e_{c}$ could not be guaranteed, which could possibly cause numerical problems in some local elements in finite element modeling. To overcome this difficulty, Gudehus [170] suggested a third formula for the CSL, which is also a nonlinear formula but with an ” $\mathrm{s}$ ” form by considering an ultimate critical void ratio at very high stress levels. This formula can be expressed as follows:
$$e_{c}=e_{c u}+\left(e_{c 0}-e_{c u}\right) \exp \left(-\left(\frac{p^{\prime}}{p_{a t} \cdot \lambda}\right)^{\xi}\right)$$
where $e_{\text {cu }}$ is the critical void ratio when $p^{\prime} \rightarrow \infty$. This expression eliminates the possibility of a negative value of the critical void ratio at high stress levels. However, two more parameters have to be determined.

## 数学代写|概率论代写Probability theory代考|Summary

This chapter reviewed the uncertainty in geotechnical engineering, and mainly discussed the uncertainties involved in the estimation of soil properties and geotechnical models. The influencing factors on the uncertainties and relevant studies were summarized. It was pointed out that the uncertainty of soil parameters should be considered and analyzed in estimating a certain soil property. Bayesian probabilistic approach as a useful tool was outlined from two application aspects, that is, parametric identification and model class selection. In view of the complex updated PDF, the chapter reviewed several available numerical simulation methods.

The chapter then reviewed previous studies on two problems of geotechnical engineering, that is, soil water retention property of unsaturated soil and creep behavior of soft soil. In the sections of soil water retention of unsaturated soil, the soil suction as an important factor for the development of the unsaturated soil mechanics was explained first, and its contribution on the soil shear strength, permeability and compressibility was discussed by reviewing the existing studies. The soil-water characteristic curve was then explained, and its influencing factors were summarized. The four commonly used methods for estimating SWCC were finally presented. The objective in the study of SWCC was mentioned, and a new model, which can consider the effect of initial void ratio on the SWCC of same textured soil sample, was required to be constructed and the relevant uncertainty analysis should also be conducted.

In the sections of creep behavior of soft soil, the mechanism of soil creep deformation was presented briefly, and the existing studies on the time-dependent models for describing the creep behavior were reviewed. As the basis of this study, the conceptual time line model proposed by Bjerrum [114] was illustrated, and the 1-D elastic viscoplastic model developed by Yin and Graham [122, 123, 149] based on the Bjerrum’s model and the development of EVP models were reviewed. Several methods for determining the parameters of EVP model were summarized, and the objectives in the study of creep behavior were proposed, that is, to analyze the model parameters by using the Bayesian probabilistic method and to select the suitable model for the predictions of creep behavior of soft soil.

## 数学代写|概率论代写Probability theory代考|Shear Behavior of Granular Soils

：排水条件下空隙率降低，不排水条件下平均有效应力降低。
(b) 高超固结黏土和致密砂在剪切过程中表现出体积膨胀，即膨胀特征（排水条件下孔隙比变大，不排水条件下平均有效应力变大），峰值应力比高于临界应力比。

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## MATLAB代写

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