scholarly journals The Longitudinal Deformation Profile of a Rock Tunnel: An Elastic Analysis

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Yonghong Wang ◽  
Wen Du ◽  
Guohui Zhang ◽  
Yang Song
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Form Solution ◽  
Elastic Model ◽  
Longitudinal Deformation ◽  
Tunnel Face ◽  
Wall Displacement ◽  
In Situ Experiments ◽  
Rock Tunnel

The longitudinal deformation profile (LDP) is the profile of wall displacement versus the distance from the tunnel face. To develop LDP equations, numerical methods and in situ experiments have been used to obtain the deformation of a tunnel in three-dimensional space. However, extant approaches are inadequate in terms of explaining the mechanical relation between the wall displacement and the conditions of a tunnel (e.g., properties of rock). In this paper, an analytical approach is proposed to develop a new LDP equation. First, on the basis of the axisymmetric elastic model of a tunnel, a closed-form solution of wall displacement is derived. Then, a new LDP equation is presented according to the solution developed above; the coefficient β, defined as the ratio of the effective range of the “face effect” to the radius of the tunnel, is proposed for the first time. Finally, a case study is proposed to validate the practicability of this equation.

Induction Proof for a General Exponential Integral Formula

1995 ◽  
Vol 80 (2) ◽  
pp. 424-426
Author(s):  
Frank O'Brien ◽  
Sherry E. Hammel ◽  
Chung T. Nguyen
Keyword(s):  
Closed Form ◽  
Integral Formula ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Form Solution ◽  
Probability Method ◽  
Exponential Integral ◽  
Need To Evaluate ◽  
Three Dimensional Space

The authors' Poisson probability method for detecting stochastic randomness in three-dimensional space involved the need to evaluate an integral for which no appropriate closed-form solution could be located in standard handbooks. This resulted in a formula specifically calculated to solve this integral in closed form. In this paper the calculation is verified by the method of mathematical induction.

UWB Positioning Performance Tests Based on Chan Algorithm in IEEE 802.15.4a Channel Model

2012 ◽  
Vol 433-440 ◽  
pp. 2663-2669 ◽  
Author(s):  
Xiao Long Mu ◽  
Xue Rong Cui ◽  
Hao Zhang ◽  
T. Aaron Gulliver
Keyword(s):  
Channel Model ◽  
Dimensional Space ◽  
Moving Average ◽  
Weighted Least Squares ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Form Solution ◽  
Ultra Wideband ◽  
Time Of Arrival ◽  
High Degree

Chan algorithm is a closed form solution to the non-recursive equation set. This algorithm needs only a small amount of calculations but has a high degree of precision on positioning. It is valuable for academic reference. Firstly, it obtains the preliminary solution by using WLS (Weighted Least Squares) twice. Then, it uses the preliminary solution to linearise the nonlinear equation and finally makes the estimation of the position. The channel model can provide the model of indoor office environment ranging from 2 GHz to 10 GHz. Through the UWB (Ultra WideBand) positioning system of the channel model, the LOS(line-of-sight) environment can be simulated and TOA(Time-Of-Arrival) data measured by distance can also be obtained by sampling. However, small LOS errors included in the TOA data may lead to big ones in the positioning of 3D(three-dimensional) space and the precision of positioning may be undermined, when the data are directly applied to the Chan algorithm which is based on the TOA. In order to solve this issue, the TOA data obtained can be processed with MA(Moving Average) algorithm and the precision can be improved.

WSN Localization Using RSS in Three-Dimensional Space—A Geometric Method With Closed-Form Solution

2016 ◽  
Vol 16 (11) ◽  
pp. 4397-4404 ◽  
Author(s):  
Yue Ivan Wu ◽  
Hao Wang ◽  
Xiujuan Zheng
Keyword(s):  
Closed Form ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Form Solution ◽  
Geometric Method ◽  
Three Dimensional Space

Analytical and Numerical Studies of a Thick Anisotropic Multilayered Fiber-Reinforced Composite Pressure Vessel

2018 ◽  
Vol 141 (1) ◽  
Author(s):  
Isaiah Ramos ◽  
Young Ho Park ◽  
Jordan Ulibarri-Sanchez
Keyword(s):  
Finite Element ◽  
Pressure Vessel ◽  
Structural Damage ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Form Solution ◽  
Fiber Reinforced ◽  
Element Analysis ◽  
Analytical Results ◽  
Composite Pressure Vessel

In this paper, we developed an exact analytical 3D elasticity solution to investigate mechanical behavior of a thick multilayered anisotropic fiber-reinforced pressure vessel subjected to multiple mechanical loadings. This closed-form solution was implemented in a computer program, and analytical results were compared to finite element analysis (FEA) calculations. In order to predict through-thickness stresses accurately, three-dimensional finite element meshes were used in the FEA since shell meshes can only be used to predict in-plane strength. Three-dimensional FEA results are in excellent agreement with the analytical results. Finally, using the proposed analytical approach, we evaluated structural damage and failure conditions of the composite pressure vessel using the Tsai–Wu failure criteria and predicted a maximum burst pressure.

scholarly journals SOLUTION OF THE MULTIDIMENSIONAL DIFFUSION EQUATION USING LIE SYMMETRIES: SIMULATION OF POLLUTANT DISPERSION IN THE ATMOSPHERE

2005 ◽  
Vol 4 (2) ◽  
Author(s):  
J. R. Zabadal ◽  
C. A. Poffal
Keyword(s):  
Differential Operators ◽  
Hybrid Methods ◽  
Formal Solution ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Pollutant Dispersion ◽  
Form Solution ◽  
Lie Symmetries ◽  
Advection Equation ◽  
Mean Values

Several analytical, numerical and hybrid methods are being used to solve diffusion and diffusion advection problems. In this work, a closed form solution of the three-dimensional diffusion advection equation in a Cartesian coordinate system is obtained by applying rules, based on the Lie symmetries, to manipulate the exponential of the differential operators that appear in its formal solution. There are many advantages of applying these rules: the increase in processing velocity so that the solution may be obtained in real time, the reduction in the amount of memory required to perform the necessary tasks in order to obtain the solution, since the analytical expressions can be easily manipulated in post-processing and also the discretization of the domain may not be necessary in some cases, avoiding the use of mean values for some parameters involved. These rules yield good results when applied to obtain solutions for problems in fluid mechanics and in quantum mechanics. In order to show the performance of the method, a one-dimensional scenario of the pollutant dispersion in a stable boundary layer is simulated, considering that the horizontal component of the velocity field is dominant and constant, disregarding the other components. The results are compared with data available in the literature.

scholarly journals SOLUTION OF THE MULTIDIMENSIONAL DIFFUSION EQUATION USING LIE SYMMETRIES: SIMULATION OF POLLUTANT DISPERSION IN THE ATMOSPHERE

2005 ◽  
Vol 4 (2) ◽  
pp. 197
Author(s):  
J. R. Zabadal ◽  
C. A. Poffal
Keyword(s):  
Differential Operators ◽  
Hybrid Methods ◽  
Formal Solution ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Pollutant Dispersion ◽  
Form Solution ◽  
Lie Symmetries ◽  
Advection Equation ◽  
Mean Values

Several analytical, numerical and hybrid methods are being used to solve diffusion and diffusion advection problems. In this work, a closed form solution of the three-dimensional diffusion advection equation in a Cartesian coordinate system is obtained by applying rules, based on the Lie symmetries, to manipulate the exponential of the differential operators that appear in its formal solution. There are many advantages of applying these rules: the increase in processing velocity so that the solution may be obtained in real time, the reduction in the amount of memory required to perform the necessary tasks in order to obtain the solution, since the analytical expressions can be easily manipulated in post-processing and also the discretization of the domain may not be necessary in some cases, avoiding the use of mean values for some parameters involved. These rules yield good results when applied to obtain solutions for problems in fluid mechanics and in quantum mechanics. In order to show the performance of the method, a one-dimensional scenario of the pollutant dispersion in a stable boundary layer is simulated, considering that the horizontal component of the velocity field is dominant and constant, disregarding the other components. The results are compared with data available in the literature.

scholarly journals Reconstruction of three-dimensional maps based on closed-form solutions of the variational problem of multisensor data registration

2019 ◽  
Vol 484 (6) ◽  
pp. 672-677
Author(s):  
A. V. Vokhmintcev ◽  
A. V. Melnikov ◽  
K. V. Mironov ◽  
V. V. Burlutskiy
Keyword(s):  
Closed Form ◽  
Large Scale ◽  
Three Dimensional ◽  
Dynamic Environment ◽  
Closed Form Solution ◽  
Point Clouds ◽  
Accurate Method ◽  
Form Solution ◽  
Closed Form Solutions ◽  
Reconstruction Methods

A closed-form solution is proposed for the problem of minimizing a functional consisting of two terms measuring mean-square distances for visually associated characteristic points on an image and meansquare distances for point clouds in terms of a point-to-plane metric. An accurate method for reconstructing three-dimensional dynamic environment is presented, and the properties of closed-form solutions are described. The proposed approach improves the accuracy and convergence of reconstruction methods for complex and large-scale scenes.

On Saint-Venant's Problem for Helicoidal Beams

2015 ◽  
Vol 83 (2) ◽  
Author(s):  
Shilei Han ◽  
Olivier A. Bauchau
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Fem Analysis ◽  
Original System ◽  
Form Solution ◽  
Hamiltonian Matrix ◽  
Jordan Structure ◽  
Rigid Body Motions

This paper proposes a novel solution strategy for Saint-Venant's problem based on Hamilton's formalism. Saint-Venant's problem focuses on helicoidal beams and its solution hinges upon the determination of the subspace of the system's Hamiltonian matrix associated with its null and pure imaginary eigenvalues. A projection approach is proposed that reduces the system Hamiltonian matrix to a matrix of size 12 × 12, whose eigenvalues are identical to the null and purely imaginary eigenvalues of the original system, with the same Jordan structure. A fundamental theoretical result is established: Saint-Venant's solutions exist because rigid-body motions create no strains. Indeed, the solvability conditions for the governing equations of the problem are satisfied because a matrix identity holds, which expresses the fact that rigid-body motions create no strains. Because it avoids the identification of the Jordan structure of the original system, the implementation of the proposed projection for large, realistic problems is straightforward. Closed-form solutions of the reduced problem are found and three-dimensional stress and strain fields can be recovered from the closed-form solution. Numerical examples are presented to demonstrate the capabilities of the analysis. Predictions are compared to exact solutions of three-dimensional elasticity and three-dimensional FEM analysis.

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