Axisymmetry Conditions Research Articles

Reliability analysis of bolt‐supported tunnels regarded as homogenized structures

AbstractThis paper presents a reliability analysis of deep tunnels reinforced by fully grouted bolts using a homogenization approach. From the constitutive viewpoint, the anisotropic elastoplastic behavior of bolted rock mass is formulated in the context of the homogenization method. Reliability design analysis is undertaken considering the configurations of both radial and face bolts used separately or in association. The combination of rock bolting associated with classical shotcrete lining is also addressed. Restricting to deep tunnels and axisymmetry conditions, a simplified analytical approach followed by finite element numerical modeling are developed in the framework of the homogenization method. Reliability analyzes are performed using the Monte Carlo simulation (MCS) and the first‐order reliability method (FORM). The response surface method (RSM) provides analytical approximations to enable reliability analyses without time‐consuming computational models. The results show excellent correlation between the obtained reliability parameters, which emphasizes the relevancy of the adopted reliability methods. Structural failure probabilities below or close to 1.0% were obtained in all cases studied for the best structural tunnel conditions.

Study of stationary rigidly rotating anisotropic cylindrical fluids with new exact interior solutions of GR. IV. Radial pressure

This article belongs to a series where the influence of anisotropic pressure on gravitational properties of rigidly rotating fluids is studied using new exact solutions of GR constructed for the purpose. For mathematical simplification, stationarity and cylindrical symmetry implying three Killing vectors are considered. Moreover, two pressure components are set to vanish in turn. In Papers I and II, the pressure is axially directed, while it is azimuthal in Paper III. In present paper (Paper IV), a radially directed pressure is considered. Since a generic differential equation, split into three parts, emerges from field equations, three different classes of solutions can be considered. Two could only be partially integrated. The other one, which is fully integrated, yields a set of solutions with a negative pressure. Physical processes where a negative pressure is encountered are depicted and give a rather solid foundation to this class of solutions. Moreover, these fully integrated solutions satisfy the axisymmetry condition, while they do not verify the so-called “regularity condition.” However, since their Kretschmann scalar does not diverge on the axis, this feature must be considered as reporting a mere coordinate singularity. Finally, the matching of these solutions to an exterior appropriate vacuum enforces other constraints on the two constant parameters defining each solution in the class. The results displayed here deserve to be interpreted in light of those depicted in the other four papers in the series.

Finite Element Investigation of Anisotropic Circular Plates

The mathematical modeling and finite element analysis of solid circular plates, analyzed using continuum finite elements under conditions of axisymmetry, is investigated in this paper. Both isotropic and anisotropic material characterizations are considered. Once a suitable “baseline” model is developed, parametric analyses are performed so as to gain insight into the behavior of anisotropic solid circular plates.

Buoyant heat transfer of nanofluids in a vertical porous annulus: a comparative study of different models

PurposeThis study aims to numerically study the buoyant convective flow of two different nanofluids in a porous annular domain. A uniformly heated inner cylinder, cooled outer cylindrical boundary and adiabatic horizontal surfaces are considered because of many industrial applications of this geometry. The analysis also addresses the comparative study of different porous media models governing fluid flow and heat transport.Design/methodology/approachThe finite difference method has been used in the current simulation work to obtain the numerical solution of coupled partial differential equations. In particular, the alternating direction implicit method is used for solving transient equations, and the successive line over relaxation iterative method is used to solve time-independent equation by choosing an optimum value for relaxation parameter. Simpson’s rule is adopted to estimate average Nusselt number involving numerical integration. Various grid sensitivity checks have been performed to assess the sufficiency of grid size to obtain accurate results. In this analysis, a general porous media model has been considered, and a comparative study between three different models has been investigated.FindingsNumerical simulations are performed for different combinations of the control parameters and interesting results are obtained. It has been found that the an increase in Darcy and Rayleigh numbers enhances the thermal transport rate and strengthens the nanofluid movement in porous annulus. Also, higher flow circulation rate and thermal transport has been detected for Darcy model as compared to non-Darcy models. Thermal mixing could be enhanced by considering a non-Darcy model.Research limitations/implicationsThe present results could be effectively used in many practical applications under the limiting conditions of two-dimensionality and axi-symmetry conditions. The only drawback of the current study is it does not include the three-dimensional effects.Practical implicationsThe results could be used as a first-hand information for the design of any thermal systems. This will help the design engineer to have fewer trial-and-run cases for the new design.Originality/valueA pioneering numerical investigation on the buoyant convective flow of two different nanofluids in an annular porous domain has been carried out by using a general Darcy–Brinkman–Forchheimer model to govern fluid flow in porous matrix. The results obtained from current investigation are novel and original, with numerous practical applications of nanofluid saturated porous annular enclosure in the modern industry.

Active Earth Pressure Acting on Circular Shafts Using Numerical Approach

Retaining walls in axi-symmetric conditions and in plane strains have been widely treated in the literature using different approaches (limit equilibrium, limit analysis, slip line, and numerical techniques by finite elements or finite differences). The finite element or finite difference method provides more accurate solutions to the problem than the limit equilibrium method. In this paper, a new model of retaining wall in the axi-symmetry conditions under outward pressure is considered, this case can be widely used in the design of grain silos, buildings and road constructions. Numerical calculations using FLAC are reported to evaluate the evolution of the earth pressure distribution on a cylindrical wall filled with granular material and subjected to radial displacement. A parametric study is carried out in order to evaluate the distribution of the active earth pressure on the wall according to the radius, the angle of friction of the granular material, and the angle of friction of the interface granular material-wall. This study shows that there is an effect of the circular shape and the inclination of the wall on the active earth pressures. Doi: 10.28991/CEJ-2022-08-04-09 Full Text: PDF

Optimal perturbations in viscous round jets subject to Kelvin–Helmholtz instability

Abstract

Three-dimensional sharp Hardy–Leray inequality for solenoidal fields

This is a complement of the former works by Costin and Maz’ya (2008) [2] and Hamamoto and Takahashi (2019) [6], on sharp Hardy–Leray inequality for solenoidal (i.e., divergence-free) fields in R3 with some axisymmetry conditions. Here we derive the same best constant, with no assumption of axial symmetry.

축대칭 조건 및 경계면 요소를 이용한 Piled Raft 기초의 유한차분 모델링 연구

본 연구에서는 수치해석을 이용한 piled raft 기초의 복합적인 거동평가를 위해 축대칭 조건 및 경계면 요소를 적용한 유한차분해석 모델링 방법이 합리적인지 검증하였다. 이를 위해 실내모형실험 결과와 수치해석 결과를 비교 분석하여 piled raft 기초의 모델링 방법의 적합성을 평가하였다. 그리고 실제 현장조건을 고려한 기초 매개변수에 대한 민감도분석을 수행하여 raft의 하중분담율을 분석하였다. 연구 결과, 실내모형실험과 수치해석 결과에서 지지력-수직변위의 상관관계가 동일한 경향을 보였으며, piled raft 기초의 극한지지력과 raft의 하중분담율이 비교적 유사하게 산정되었다. 그리고 민감도분석을 통해 raft의 하중분담율은 약 33%~52% 사이에서 산정되어, 기존 연구 결과와 유사함을 확인하였다. 따라서 piled raft 기초는 축대칭 조건 및 경계면 요소를 이용하여 효과적으로 모델링될 수 있을 것으로 판단된다. In this study, FDM modelling with axisymmetry condition and interface element was verified whether it is reasonable to estimate compositive behavior of a piled raft foundation. To this end, the modelling validity of piled raft foundations was estimated by comparing and analyzing numerical analysis results and laboratory model test results. Also, load bearing ratio of a raft is analyzed by performing sensitivity analysis of foundation parameters with the actual field conditions. As a result of this study, correlation between bearing capacity and vertical displacement of numerical results turned out to be similar with that of a laboratory model test. In addition, ultimate bearing capacity of piled rafts and load bearing ratio of the raft is calculated to be similar in both cases. The load bearing ratio of the raft was also estimated to be in the range of 33% to 52% from the sensitivity analysis. The results were confirmed to be similar to the previous studies. Therefore, it can be inferred that piled rafts can be effectively modelled applying axisymmetry condition and interface element.

Elastic and viscoelastic response of heterogeneous annular structures under arbitrary transient pressure

A novel approach is employed in the forced-vibration analysis of functionally-graded annular structures under conditions of axisymmetry. The material is heterogeneous in the sense that it is functionally-graded in the radial direction. The grading function may be an arbitrary continuous function of the radial coordinate. Forcing functions applied on the inner boundary are dynamic pressures which may be harmonic, an arbitrary continuous function of time, or impulsive. These conditions result in governing differential equations with variable coefficients. Analytical solutions of such equations cannot be obtained except for certain simple grading functions and pressures. Numerical approaches must be adopted to solve the problem on hand. The novelty of the present study lies in the fact that a combination of Laplace transform and Complementary Functions Method (CFM) is employed in the analysis. Laplace transformation gives a time-independent boundary-value problem in spatial coordinate which is then solved by CFM. Inverse transformation of the results into the time domain is performed by modified Durbin׳s method. In the Laplace domain, viscoelasticity is modeled easily by a simple change of variable. Benchmark solutions available in the literature are used to validate the results and to observe the convergence of the numerical solutions. The solution procedure is well-structured, simple and efficient and it can be readily applied to cylinders, disks and spheres. It is also well suited for problems in which the graded mechanical properties and applied pressures are supplied point-by-point in a discretized manner.

A Study on the Effect of Bottom-hole Differential Pressure on the Rock Stress Field

Bottom-hole differential pressure is one of the most important factors that affects the drilling penetration rate. The purpose of the article is to study the effect of bottom-hole differential pressure on the vertical bottom-hole stress field. On the basis of mechanical analysis of the bottom-hole rock, the fluid-solid coupling model with the bottom-hole differential pressure is established under axisymmetry condition and is solved by using a numerical method. The comparison of the distribution of borehole wall stress between the numerical and theoretical solution is performed to verify the feasibility and rationality of the model. Then analysis of the stress state of the whole bottom-hole rock to break is carried out. The results show that the numerical solution is consistent with the theoretical solution and the fluid-solid coupling model is reasonable. With the increasing of the bottom-hole differential pressure, the bottom-hole tensile stress increases and compressive stress decreases and tensile region becomes larger. The whole bottom-hole rock to break is divided into three regions: triaxial tension region, biaxial compression and unidirectional tension region, and triaxial compression region. Qualitative and quantitative analysis of the bottom-hole stress field under different differential pressure provides the theoretical basis for rock breaking mechanism and faster and more efficient drilling.

On the toroidal current density flowing across a poloidal-magnetic-field null in an axisymmetric plasma

The axisymmetry condition and two of Maxwell's equations are used to show that, in general, there are no nested magnetic surfaces around a poloidal-magnetic-field null for a sufficiently small value of the toroidal current density flowing there. Hence, the toroidal current density at the axis of a magnetic configuration with extreme shear reversal cannot continuously approach zero unless nested surfaces are first broken or particular values are assigned to boundary conditions and other plasma parameters. The threshold of the toroidal current-density at which the topology changes is shown to be set by such parameters, and some examples of the predicted topology transition are presented using analytical solutions of the Grad-Shafranov equation.

A Unified Formulation for the Analysis of Temperature Field in a Thick Hollow Cylinder Made of Functionally Graded Materials With Various Grading Patterns

In this article, a unified formulation is presented to analyze the heat conduction and temperature field in a functionally graded thick hollow cylinder. To solve the heat transfer equation in a functionally graded thick hollow cylinder, an analytical method is presented using a nonlinear volume fraction law with variable exponent for mechanical properties. The new unified formulation is obtained based on various thermal boundary conditions such as heat convection at bounding surfaces and constant temperature of bounding surfaces. The functionally graded thick hollow cylinder is considered in axisymmetry condition and infinite length. The temperature field is discussed for various boundary conditions and different grading patterns in functionally graded materials. The presented results and data are verified with published literature.

Stochastic Assessment of Thermo-Elastic Wave Propagation in Functionally Graded Materials (FGMs) with Gaussian Uncertainty in Constitutive Mechanical Properties

The main objective of this article is focused on stochastic analysis of wave propagation and effects of uncertainty in mechanical properties on transient behaviors of displacement and temperature fields in functionally graded materials under thermo-mechanical shock loading. The problem is studied in a cylindrical domain and the governing equations of a functionally graded thick hollow cylinder are solved. To assess the wave propagation, the generalized coupled thermoelasticty equations based on Green-Naghdi theory (without energy dissipation) are analyzed in a FG thick hollow cylinder. The FG cylinder is considered to have infinite length and axisymmetry conditions. The constitutive mechanical properties of FGM are assumed as random variables with Gaussian distribution and also the mechanical properties are considered to vary across thickness of FG cylinder as a nonlinear power function of radius. The FG cylinder is divided into many elements across thickness of cylinder and hybrid numerical method (Galerkin finite element and Newmark finite difference methods) along with the Monte Carlo simulation are employed to solve the statistical coupled equations. The effects of uncertainty in functionally graded materials on thermal and elastic waves, transient behaviors of radial displacement and temperature fields and variance and maximum values of displacement and temperature are discussed in details for various grading patterns in FGMs and various points on thickness at several times.

Reliability of stress field in Al–Al 2O 3 functionally graded thick hollow cylinder subjected to sudden unloading, considering uncertain mechanical properties

In this paper, the reliability analysis and safety evaluation of dynamic stresses are presented for Al–Al 2O 3 functionally graded (FG) thick hollow cylinder subjected to sudden unloading as a mechanical shock loading. The FG cylinder is considered to have infinite length and axisymmetry conditions. The constitutive mechanical properties of Al and Al 2O 3 are assumed as random variables with Gaussian distribution and also the mechanical properties are considered to vary across thickness of FG cylinder as a non-linear power function of radius. The radial and hoop stresses are obtained by solving Navier equation in displacement form and stress–displacement equations. The FG cylinder is divided to many linear functionally graded elements across thickness of cylinder and hybrid numerical method (Galerkin finite element and Newmark finite difference methods) along with the Monte Carlo simulation are employed to solve the statistical problem. The reliability of radial and hoop stresses are calculated in various points across thickness for different grading patterns in functionally graded material (FGM) and several yield stresses. The variability of the dynamic stress reliability of the FG cylinder to the values of coefficients of variation (COVs) is examined and discussed in details.

General analytical solution for elastic radial wave propagation and dynamic analysis of functionally graded thick hollow cylinders subjected to impact loading

An analytical method is proposed for the dynamic response analysis of functionally graded thick hollow cylinders under impact loading. The wave motion equation is solved using an analytical method that is based on the composition of Bessel functions. The mechanical properties are considered as power functions of the radius across the thickness of FG cylinder. The FG cylinder is excited by an impact loading at the inner surface of the cylinder, and the plane strain and axisymmetry conditions are assumed for the problem. The time histories of radial displacement and radial and hoop stresses are presented. Also the dynamic response of the FG cylinder is obtained and discussed for various kinds of power function exponents.

Analytical and Numerical Study of the Mechanics of Rockbolt Reinforcement around Tunnels in Rock Masses

This paper addresses the problem of quantifying the mechanical contribution of rockbolts installed systematically around tunnels excavated in rock masses. The mechanical contribution referred to here is that of increased stress confinement and decreased tunnel convergences as compared with corresponding stresses and displacements obtained for non-reinforced tunnels. The problem is treated analytically first by presenting a closed-form solution for stress and displacement distributions around a circular tunnel excavated in elastic material and reinforced by grouted or anchored rockbolts. The analytical solution assumes that rockbolts are regularly spaced around the tunnel and that axi-symmetry conditions of geometry and loading apply. The results obtained with the closed-form solution are shown to be equivalent to the results of the same problem solved with traditional numerical methods. Based on the analytical and numerical results and by introducing dimensionless ratios that allow to quantify the increase of radial stresses and the decrease of radial displacements in the reinforced region of the tunnel, the paper shows that reinforcement can have a significant mechanical effect (i.e., increasing the confinement and decreasing the convergences) in tunnels excavated in rock masses of poor to very poor quality. The paper analyzes then the mechanical contribution of rockbolt reinforcement when the rock mass is assumed to behave elasto-plastically. For this case, it is shown that rockbolt reinforcement can also have a critical effect in controlling the extent of the plastic failure zone and the convergences of the tunnel.

Nada: A new code for studying self-gravitating tori around black holes

We present a new two-dimensional numerical code called $\mathtt{N}\mathtt{a}\mathtt{d}\mathtt{a}$ designed to solve the full Einstein equations coupled to the general relativistic hydrodynamics equations. The code is mainly intended for studies of self-gravitating accretion disks (or tori) around black holes, although it is also suitable for regular spacetimes. Concerning technical aspects the Einstein equations are formulated and solved in the code using a formulation of the standard $3+1$ Arnowitt-Deser-Misner canonical formalism system, the so-called Baumgarte-Shapiro Shibata-Nakamura approach. A key feature of the code is that derivative terms in the spacetime evolution equations are computed using a fourth-order centered finite difference approximation in conjunction with the Cartoon method to impose the axisymmetry condition under Cartesian coordinates (the choice in $\mathtt{N}\mathtt{a}\mathtt{d}\mathtt{a}$), and the puncture/moving puncture approach to carry out black hole evolutions. Correspondingly, the general relativistic hydrodynamics equations are written in flux-conservative form and solved with high-resolution, shock-capturing schemes. We perform and discuss a number of tests to assess the accuracy and expected convergence of the code, namely, (single) black hole evolutions, shock tubes, and evolutions of both spherical and rotating relativistic stars in equilibrium, the gravitational collapse of a spherical relativistic star leading to the formation of a black hole. In addition, paving the way for specific applications of the code, we also present results from fully general relativistic numerical simulations of a system formed by a black hole surrounded by a self-gravitating torus in equilibrium.

Transient heat conduction in functionally graded thick hollow cylinders by analytical method

In this article, transient heat conduction in a cylindrical shell of functionally graded material is studied by using analytical method. The shell is assumed to be in axisymmetry conditions. The material properties are considered to be nonlinear with a power law distribution through the thickness. The temperature distribution is derived analytically by using the Bessel functions. To verify the proposed method the obtained numerical results are compared with the published results. The comparisons of temperature distribution between various time and material properties are presented.

Thermocapillary motion of deformable fluid particles in tubes

The boundary integral technique is used to study the effect of deformation on the steady, creeping, thermocapillary migration of a fluid particle under conditions of axisymmetry, negligible thermal convection and an insulated tube wall. The spherical radius of the fluid particle (i.e. the radius as if the particle were a sphere, a ′=(3 V p r /4π) 1/3 is the particle volume) and that of the tube are denoted, respectively, by a ′and b ′ . For small capillary number Ca = 0.05, only for a large fluid particle ( a ′/ b ′ = 0.8) is deformation significant. For a′/b′ = 0.8, hydrodynamic stresses squeeze the particle, reduce the interaction of the particle with the wall and thereby increase the terminal velocity. For small particles a ′/ b ′ Ca = 0.05 the fluid particles translate as spheres, due to the fact that the fluid particle is too far away from the wall to be subject to distending hydrodynamic stresses. The deformable particle moves faster than a spherical one in the thermocapillary migration. The increase in velocity with capillary number is larger for thermocapillary motion than for buoyancy.

Detection of Metal Defects on Gas Distribution Pipeline by Remote Field Eddy Current (Rfec) Using Finite-Element Analysis

It is necessary to find out whether there are metal defects on underground gas distribution pipelines without excavation in order to establish safety strategies for replacement or maintenance. The metal defects are classified into general corrosion, stress corrosion cracking, lamination, pits, and metal loss, which cause leak or partial damages to a gas pipeline. Therefore, it is required to develop an effective method in the form of an in-line inspection concept that could be implemented internally into a gas pipeline. In this study, theoretical formulations of the magnetic vector potential and magnetic flux density including axial and radial wave numbers based on Maxwell equations are presented analytically to find out the changes of amplitude and phase in the magnetic flux density and the consequent induced voltage which can be a criterion in the remote field eddy current principle for detecting the metal defects in gas pipelines. Three-dimensional finite-element analysis is also presented to analyze the physical phenomenon in metal defects according to each defect size, excitation frequency and moving velocity, which can overcome inaccuracy of the two-dimensional approach, using axisymmetry condition, and simulate local pit conditions occurred severely in real gas pipelines; otherwise only metal loss such as whole circumferential decrease in wall thickness can be modeled. Some experimental works are performed to validate the analytical and finite elemental results regarding the magnetic flux density and induced voltage in the detector.