Complex variable solution for boundary value problem with X-shaped cavity in plane elasticity and its application

Zhou, Hang

doi:10.1007/s10483-017-2235-8

Cited by 9 publications

(4 citation statements)

References 19 publications

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“…where R and c2n-1 are conformal mapping parameters, which can be determined through the method of least squares (Zhou, 2017b) c  is instead used, the cavity becomes asymmetric. Figure 3 plots the conformal mapping coordinate system obtained from published classical solutions for circular, elliptical and square shapes and through iterative calculation using the method of least squares (Zhou, 2017b) for the X-shape.…”

Section: Conformal Mapping Equationmentioning

confidence: 99%

“…For non-circular cavity expansion (N-CCE) in elastic media, theoretical solutions are feasible using complex variable elasticity (CVE) developed by Muskhelishvili (1954). Zhou et al (2016Zhou et al ( , 2017b explored the application of both displacement-controlled and pressurecontrolled N-CCE to elastic soil and proposed a series of closed-form solutions using CVE. However, CVE is no longer suitable if soil plasticity is allowed to develop because the biharmonic stress function is often non-existent.…”

Section: Introductionmentioning

confidence: 99%

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Noncircular Cavity Expansion in Undrained Soil: Semi-Analytical Solution

2022

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The cavity expansion approach has been a popular tool to interpret a wide range of geotechnical problems over the last several decades. Most previous research focused on the expansion of cylindrical and/or spherical cavities whereas 'non-standard' cavities have received much less attention. To address this shortcoming, this paper presents a general theoretical framework for two-dimensional (2D) displacementcontrolled undrained non-circular cavity expansion (N-CCE) in undrained soil. The new approach combines strain path method (SPM) concepts and conformal mapping to determine the soil velocity and strain rate fields analytically. The soil displacement and strain are subsequently determined by integrating the soil velocities and strain rates along the strain path using a series of transformed ordinary differential equations. In this study, the modified Cam clay (MCC) effective stress constitutive model is used to determine the soil stress-strain relationship while consolidation effects are captured using finite difference calculations. The proposed methodology is validated by comparing the reduced solution for a circular cavity with traditional circular cavity expansion theory. A parametric analysis is subsequently undertaken to explore the influence of three non-circular cavity shapes on expansion-induced soil deformation mechanisms, shear strains, effective stresses, and pore water pressure development and consolidation. The proposed solution can be implemented with any critical state-based soil model and can be applied to arbitrary non-circular cavity problems.

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Section: Conformal Mapping Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Noncircular Cavity Expansion in Undrained Soil: Semi-Analytical Solution

2022

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“…Hence, in this paper, a conformal mapping technique is used to address the boundary value problem of the noncircular boundary in the z ‐plane. Following Muskhelishvili, 39 the general mathematical formula of conformal transformation could be expressed in the form of infinite power series; if the outer region of a noncircular boundary in the z ‐plane is mapped to the outer region of a unit circle in the ζ ‐plane, then the general conformal transformation equation could be written in the following form:

z = ω (ζ) = c_{0} ζ + \sum_{n = 1}^{\infty} c_{2 n - 1} ζ^{1 - 2 n} false| ζ false| \geq 1

where

z = x + i y = r {normale}^{i θ}

denotes a complex variable in the z ‐plane;

ζ = ξ + i η = ρ {normale}^{i ϑ}

denotes a complex variable in the ζ ‐plane; x and y denote the variables in the Cartesian coordinate system of the z ‐plane; r and θ denote the variables in the polar coordinate system of the z ‐plane; ξ and η denote the variables in the Cartesian coordinate system in the ζ ‐plane; ρ and ϑ denote the variables in the polar coordinate system in the ζ ‐plane;

i = \sqrt{- 1}

; c 0 and c 2n‐1 are constant coefficients; and the sign

| ζ |

is the modulus of complex variable

ζ

40 …”

Section: Conformal Transformation Techniquementioning

confidence: 99%

Three‐dimensional analytical continuum model for axially loaded noncircular piles in multilayered elastic soil

Zhou

Liu

et al. 2021

Num Anal Meth Geomechanics

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To investigate the settlement response, load transfer behavior, and geometric effect of noncircular piles, a general three‐dimensional (3D) analytical continuum model for axially loaded noncircular pile‐multilayered homogeneous soil systems is presented. In the analysis, both the vertical and horizontal soil displacement fields are considered, and the variations of the soil displacement components are described by the three displacement transfer functions along the x, y, and z directions. The governing differential equations of noncircular piles and soil are derived by applying the principle minimum potential energy and variational method to the pile‐soil system. A complex variable function conformal mapping technique is introduced to overcome the complex boundary conditions of the transfer functions. Then, the numerical solutions of all the governing differential equations are simultaneously solved following an iterative algorithm. The reliability of this approach is validated by comparing some pile and soil responses with the results from existing analytical solutions or a rigorous finite element analysis. Differences in the settlement response and the axial force distribution between noncircular and circular piles are observed. In addition, the pile shaft resistance along the perimeter of the cross‐sections shows that a greater load is required in order to fully mobilize the ultimate resistance of the noncircular piles. We also compare the stress (σ1−σ3) contours of some typical noncircular piles with a circular pile, which provides useful insights into the geometric effect.

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“…Notably, a general analytical solution of a two-dimensional laterally loaded noncircular section pile soil system based on the complex variable elasticity theory was proposed by Zhou et al [13]. Zhou [14] also developed a simplified theoretical model that captures the non-uniform deformation effect of XCC piles. Liu et al [15] proposed that, under the same cross-sectional area, the ultimate axial resistance of a circular pile was 45% lower than that of an XCC pile.…”

Section: Introductionmentioning

confidence: 99%

Effect of Expanded Body Diameter on the Soil Surrounding a Pile Based on the Half-Face Pile Model Test of Undisturbed Soil

Niu

et al. 2023

Buildings

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The effect of expanded body diameter on the displacement field of soil surrounding a pile under different vertical loads was investigated using the half-face pile model test of undisturbed soil. Digital image correlation technology was used to record the displacement characteristics of soil around the pile in real time. The displacement and failure characteristics of the soil around the pile were analyzed. The results show that with an increased load, the soil below the expanded body is compressed, and the soil at both ends will slip, leading to the continuous development of cracks. In a horizontal direction, the soil surrounding the pile first moves close to the pile and then tends to stabilize or move away from the pile. The horizontal and vertical displacement of the soil decreases as the distance from the pile increases. The main area of influence on the soil is below the expanded body, in which the increased diameter of the expanded body results in a gradual increase in the area of influence. Furthermore, all of the load-settlement curves show a slow decline and the bearing capacity increases with the increased diameter of the expanded body. Therefore, the research in this paper can provide an experimental method for the study of soil displacement around drill-expanded concrete piles.

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Complex variable solution for boundary value problem with X-shaped cavity in plane elasticity and its application

Cited by 9 publications

References 19 publications

Noncircular Cavity Expansion in Undrained Soil: Semi-Analytical Solution

Noncircular Cavity Expansion in Undrained Soil: Semi-Analytical Solution

Three‐dimensional analytical continuum model for axially loaded noncircular piles in multilayered elastic soil

Effect of Expanded Body Diameter on the Soil Surrounding a Pile Based on the Half-Face Pile Model Test of Undisturbed Soil

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