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
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (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|MATHS 7103

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|MATHS 7103


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

根据固结历史、当前孔隙比和受力状态,黏土可分为正常固结和超固结黏土,而砂土分为松散和致密砂。大量三轴剪切试验结果如图(图 1.6)所示
(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。

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


在哪里和参考 是对应于参考平均有效应力的参考空隙率p参考 ; 和λ是临界状态线 (CSL) 的斜率和-日志p′飞机。因此,两个参数(和参考 和λ) 是定义 CSL 所必需的。这个公式的优点是它的形式简单。然而,实验结果表明,CSL 并不总是线性的和−日志⁡p′平面,在数学上,临界空隙率和C对于高压力水平可能会变成负数,这是没有意义的。请注意,岩土结构中仍然存在非常高的应力水平,例如在安装过程中的桩尖。

最近,Li 和 Wang [169] 提出了另一个公式,他们假设在和-日志p′平面,表示带有一个附加参数的线性公式的扩展X :

其中附加参数X控制临界状态线的非线性,根据实验数据给出更灵活和准确的描述,特别是对于非常低到中等的应力水平。然而,对于高应力水平,临界空隙率的正值和C不能保证,这可能会导致有限元建模中某些局部单元出现数值问题。为了克服这个困难,Gudehus [170] 提出了 CSL 的第三个公式,它也是一个非线性公式,但具有“s”通过考虑在非常高的应力水平下的最终临界空隙率来形成。这个公式可以表示如下:

在哪里和和 是临界空隙率,当p′→∞. 该表达式消除了在高应力水平下临界空隙率出现负值的可能性。然而,必须确定另外两个参数。

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

本章回顾了岩土工程中的不确定性,主要讨论了土壤性质和岩土模型估计中涉及的不确定性。总结了不确定性的影响因素及相关研究。指出在估算某一土壤性质时应考虑和分析土壤参数的不确定性。贝叶斯概率方法作为一种有用的工具从两个应用方面进行了概述,即参数识别和模型类别选择。鉴于复杂的更新 PDF,本章回顾了几种可用的数值模拟方法。

本章接着回顾了前人对岩土工程两个问题的研究,即非饱和土的土壤保水性和软土的蠕变行为。在非饱和土的土壤保水剖面中,首先阐述了土壤吸力作为非饱和土力学发展的重要因素,并通过回顾现有研究讨论了其对土壤抗剪强度、渗透性和压缩性的贡献。然后解释了土壤-水特征曲线,并总结了其影响因素。最后给出了估算 SWCC 的四种常用方法。提出了研究 SWCC 的目标,并提出了一种新模型,该模型可以考虑初始孔隙比对相同质地土样 SWCC 的影响,

在软土的蠕变行为部分,简要介绍了土体蠕变变形的机理,并回顾了现有关于描述蠕变行为的时变模型的研究。作为本研究的基础,说明了 Bjerrum [114] 提出的概念时间线模型,以及 Yin 和 Graham [122, 123, 149] 基于 Bjerrum 模型和开发的一维弹性粘塑性模型。审查了 EVP 模型。总结了几种确定EVP模型参数的方法,提出了蠕变行为研究的目标,即利用贝叶斯概率方法分析模型参数,选择合适的模型进行蠕变行为预测。的软土。

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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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