A mathematical model to study the soil arching effect in stone column-supported embankment resting on soft foundation soil

Soil arching is a common phenomena in pile or columnar (vibroconcrete columns, soil–cement columns by mixing or grouting, stone columns) supported geosynthetic-reinforced or unreinforced embankments resting on soft soil. Due to soil arching, stress acting on soft soil or geosynthetic reinforcement decreases and stress on piles or columns increases. In this paper, using mechanical elements (such as spring, dashpot), a generalized mathematical model has been developed to study the soil arching effect in stone column-supported geosynthetic-reinforced and unreinforced embankments resting on soft soil. Pasternak model concept has been used to model the embankment soil. The soft soil has been idealized by spring-dashpot system to include the time-dependent behavior. The stone columns and geosynthetic reinforcement are idealized by stiffer nonlinear springs and rough elastic membrane, respectively. The consolidation effect of soft soil due to inclusions of stone columns has also been included in the model to study its effect on soil arching. Plane strain condition has been considered in the analysis. A finite difference scheme has been used to solve the governing differential equations and results are presented in non-dimensional form. It has been observed that the height of embankment, degree of consolidation of soft soil, stiffness of the stone column material, spacing between the stone columns, use of geosynthetic reinforcement and properties of soft and embankment soils (such as ultimate bearing capacity of soft soil, shear modulus and ultimate shearing resistance of embankment soil) significantly influence the degree of soil arching.     

Energy method as solution for deformation of geosynthetic-reinforced embankment on Pasternak foundation

To more accurately analyze the settlement of geosynthetic-reinforced embankments on soft soil foundation, we simplified the ground surface structures as an Euler–Bernoulli beam, and the geosynthetic-reinforced layer as a Timoshenko beam. The granular fill and the soft soil foundation were both modelled using two-parameter Pasternak foundation models. We used energy method to establish energy balance equations for the system, and then we used the principle of resident potential energy to derive the governing differential equations of the settlement of the ground surface structures and the geosynthetic-reinforced layer. The MATLAB solver bvp4c was used to obtain numerical solutions for the settlement of the ground surface structures and the geosynthetic-reinforced layer on Pasternak foundations. By comparing to test results and existing models, the validity of the proposed solution is verified. This study analyzed the effects of parameters, such as flexural rigidity of the geosynthetic-reinforced layer, shear modulus of the granular fill, thickness of the soft soil foundation, and horizontal shear modulus of the Timoshenko beam, on the settlement of the ground surface structures and the geosynthetic-reinforced layer. The results showed that the model using a Pasternak foundation and a Timoshenko beam is more accurate at predicting settlement than that using a Winkler foundation and an Euler–Bernoulli beam.     

A nonlinear model for footings on granular bed–stone column reinforced earth beds

In the present study, an attempt has been made to model and analyze a combined footing supporting column, which is to be constructed on very soft soil. In view of small bearing capacity and very large deflections, foundations are constructed after improving the original ground. Here, the ground has been improved by providing the stone columns in the soft soil and on top of this improved ground; a granular fill layer has been placed just below the footing. The footing has been modeled as a beam having finite flexural rigidity. Granular fill layer, soft soil and stone columns have been represented by Pasternak shear layer, Kelvin–Voigt body and the Winkler springs, respectively. Nonlinear behavior of these has been considered by means of hyperbolic constitutive relationships. Governing differential equations for response of the system have been derived and presented in non-dimensional form. These equations have been solved using appropriate boundary conditions by means of an iterative Gauss Elimination technique.     

Some problems connected with 2D modeling of geosynthetic tubes

Some problems connected with 2D modeling of geosynthetic tubes on rigid foundations are studied. Basic equations are derived and analyzed. The analysis of the equations is based on the implicit function theorem. Geosynthetic tubes are made of a special fabric and then filled with water or slurry. After being filled tubes take certain shapes and tensions are induced in the geosynthetic. The modeling is based on the following hypotheses: the problem is two dimensional; the geosynthetic is flexible, inextensible and has negligible weight; the foundation is rigid; and there is no friction between the foundation and the geosynthetic.     

Closed-form solution considering the tangential effect under harmonic line load for an infinite Euler–Bernoulli beam on elastic foundation

The dynamic response of an infinite Euler–Bernoulli beam resting on an elastic foundation, which considers the tangential interaction between the beam and foundation under harmonic line loads, is developed in this study in the form of a closed-form solution. Previous studies have focused on elastic Winkler foundations, wherein the tangential interaction between the bottom of the beam and the foundation is not considered. In this study, a series of separate horizontal springs is diverted to the contact surface between the foundation and beam to simulate the horizontal tangential effect. The horizontal spring reaction is assumed proportional to the relative tangential displacement. As the geometric equation and linear-elastic constitutive equation of beam under the condition of small deformation have been presented based on the basic principle of elasticity mechanics, the analysis model is built and the governing differential equations about normal and tangential deflections of beam are deduced. Double Fourier transformation and the residue theorem are used to derive the closed-form solution to this problem. The proposed solution is then validated by comparing the degraded solution with the known results and comparing the numerical solution with the analytical solution. We also discuss the case in which the load direction is not vertical to the beam. Results can be used as a reference for engineering design.     

车路耦合非线性振动高阶Galerkin截断研究

将移动车辆模型化为运动的两自由度质量-弹簧-阻尼系统,道路模型化为立方非线性黏弹性地基上的弹性梁,并将路面不平度设定为简谐函数．通过受力分析,建立车路非线性耦合振动高阶偏微分方程．采用高阶Galerkin截断结合数值方法求解耦合系统的动态响应．首次研究不同截断阶数对车路耦合非线性振动动态响应的影响,确定Galerkin截断研究车路耦合振动的收敛性．研究结果表明,对于软土地基的沥青路面,耦合振动的动态响应,需要150阶以上的截断才能达到收敛效果．并通过高阶收敛的Galerkin截断研究了系统参数对车路耦合非线性振动动态响应的影响.     

Analysis of stone column-supported geosynthetic-reinforced embankments

In the present paper, the behavior of stone column-supported geosynthetic-reinforced embankments has been studied. The soil arching effect is incorporated in the study to determine the stresses acting on stone columns as well as soft soil. The effect of stiffness of the stone column is also incorporated in the present study. Based on the stress action in the improved ground, stress concentration ratio, axial strain of geosynthetic reinforcement, tension developed in it and settlement of the improved soft ground are determined by using the developed methodology. Present analytical method is also verified with several current design methods. It is observed from parametric studies that modular ratio or stiffness of the stone columns, spacing to diameter ratio, height of the embankment, depth of the soft soil, stiffness of the geosynthetic reinforcement significantly affect the behavior of geosynthetic-reinforced stone column-supported embankments resting on soft soil.     

The variability of dynamic responses of beams resting on elastic foundation subjected to vehicle with random system parameters

This paper investigates the variability of dynamic responses of a beam resting on an elastic foundation, which is subjected to a vehicle with uncertain parameters, such as random mass, stiffness, damping of the vehicle and random fields of mass density, and the elastic modulus of the beam and stiffness of elastic foundation. The vehicle is modeled as a two-degree-of-freedom spring-damper-mass system. The equations of motion of the beam was constructed using a finite element method. The mass and elastic properties of the beam, and the stiffness of foundation are assumed to be Gaussian random fields and were simulated by the spectral represent method. Masses, stiffness of the spring, and the damping coefficient of the vehicle are assumed as Gaussian random variables. The numerical analyses were performed using the finite element method (FEM) in conjunction with the Monte Carlo simulation (MCS). The variability of dynamic responses of the beam were investigated with various cases of random parameters. For each sample, the equations of motions were solved with the Wilson-q integral method to find dynamic responses. The influence of random system parameters and their correlation on the response variability is discussed in detail.     

Bending analysis of elastically connected Euler–Bernoulli double-beam system using the direct boundary element method

Double and multiple-Beam System (BS) models are structural models that idealize a system of beams interconnected by elastic layers, where beam theories are assumed to govern the beams and elastic foundation models are assumed to represent the elastic layers. Many engineering problems have been studied using BS models such as double and multiple pipeline systems, sandwich beams, adhesively bonded joints, continuous dynamic vibration absorbers, and floating-slab tracks. This paper presents for the first time a direct Boundary Element Method (BEM) formulation for bending of Euler–Bernoulli double-beam system connected by a Pasternak elastic layer. All of the mathematical steps required to establish the direct BEM solution are addressed. Discussions deriving explicit solutions for double-beam fundamental problem are presented. Integral and algebraic equations are derived where influence matrices and load vectors of double-beam systems are explicitly shown. Finally, numerical results are presented for differing cases involving static loads and boundary conditions.     

Nonlinear vibrations of piezoelectric microcantilevers for biologically-induced surface stress sensing

In this paper, nonlinear vibrations of a piezoelectrically-driven microcantilever beam in presence of a biological monolayer are investigated and the corresponding equations of motion are derived and simulated. A part of the microcantilever beam surface is covered by a piezoelectric layer, which acts as an actuator. Inextensibility condition and the coupling between electrical and mechanical properties in piezoelectric materials are considered in the bending vibrations of the beam. The adsorbed biological layer is considered to be a monolayer and its adsorption induced surface stress is formulated from the molecular viewpoint. The nonlinear terms in the governing equations of motion of the beam appear in the quadratic form due to the presence of the piezoelectric layer, and the cubic form due to geometry of the beam and the adsorbed biological layer. Through extensive numerical simulations, it is demonstrated that the nonlinear effect of piezoelectric layer is significant in the microcantilever resonance sensing range. It is also shown that the effect of intermolecular attraction–repulsion on the surface stress is less dominant than other sources of surface stress (e.g., the electrostatic forces). Finally, it is observed that piezoelectrically-actuated microcantilever provides the ability of indirect measurement of vibrations and frequency response characteristics, instead of using bulky laser sensor.     

横观各向同性饱和地基与中厚圆板的非轴对称动力相互作用

研究横观各向同性饱和土地基上中厚弹性圆板的非轴对称振动问题,即首先利用Fourier展开和Hankel变换技术,求解了简谐激励下横观各向同性饱和土地基的非轴对称Biot波动方程,然后按混合边值问题建立地基与弹性中厚圆板非轴对称动力相互作用的对偶积分方程,并将对偶积分方程化为易于数值计算的第二类Fredholm积分方程．文末给出了算例．数值结果表明,在一定频率范围内,地基表面的位移幅值随激振频率增加而增大,随距离的增大以振荡形式衰减变化．     

On One method for the numerical solution of a system of parabolic equations

An algorithm for the numerical solution of a system of two parabolic equations of a special form has been presented, and its stability has been proved. Estimates of the numerical solutions obtained have been made. Conditions guaranteeing the applicability of the sweep method are determined, and the time step at which the iterative method has the inner convergence has been estimated. An example of the application of the algorithm in soil science has been presented.     

上覆单相弹性层的饱和地基上刚性基础竖向振动的轴对称混合边值问题

运用作者提出的饱和土弹性波动方程,从理论上研究了上覆单相弹性土层的饱和地基上刚性基础的竖向振动轴对称问题,即采用Hankel积分变换技术并按混合边值条件建立了部分饱和地基上刚性基础竖向振动的对偶积分方程,并将其蜕化为完全饱和地基的情形;该对偶积分方程可化为易于数值计算的第二类Fredholm积分方程。文末的算例给出了地基表面动力柔度系数C_v随无量纲频率a₀的变化曲线。     

Heat and mass transfer in MHD non-Darcian flow of a micropolar fluid over a stretching sheet embedded in a porous media with non-uniform heat source and thermal radiation

A mathematical analysis has been carried out to study magnetohydrodynamic boundary layer flow, heat and mass transfer characteristic on steady two-dimensional flow of a micropolar fluid over a stretching sheet embedded in a non-Darcian porous medium with uniform magnetic field. Momentum boundary layer equation takes into account of transverse magnetic field whereas energy equation takes into account of Ohmic dissipation due to transverse magnetic field, thermal radiation and non-uniform source effects. An analysis has been performed for heating process namely the prescribed wall heat flux (PHF case). The governing system of partial differential equations is first transformed into a system of non-linear ordinary differential equations using similarity transformation. The transformed equations are non-linear coupled differential equations which are then linearized by quasi-linearization method and solved very efficiently by finite-difference method. Favorable comparisons with previously published work on various special cases of the problem are obtained. The effects of various physical parameters on velocity, temperature, concentration distributions are presented graphically and in tabular form.     

Study of a boundary value transmission problem for two-dimensional flows in a piecewise anisotropic inhomogeneous porous layer

We consider a boundary value transmission problem for two-dimensional filtration flows in an anisotropic porous layer consisting of adjacent domains in which the media have essentially different conductivities (permeability and thickness). In general, the layer conductivity is specified by a nonsymmetric second rank tensor whose components are modeled by continuously differentiable functions of coordinates. To study the problem, we use two complex planes, the physical plane and an auxiliary plane, which are related by a homeomorphic (one-to-one and continuous) transformation satisfying an equation of the Beltrami type. On the physical plane, we pose a transmission problem for a rather complicated elliptic system of equations. This problem is reduced on the auxiliary plane to canonical form, which dramatically simplifies the analysis of the problem. Then the problem is reduced to a system of boundary singular integral equations with generalized kernels of the Cauchy type, which are expressed via the fundamental solutions of the main equations. The boundary value transmission problem studied here can be used as a mathematical model of processes arising in the recovery of fluids (water and oil) from natural soil formations of complicated geological structure.     

Modified pasternak model for reinforced soil

The paper studies the influence of introducing a geosynthetic to into a geotechnical structure composed of a granular fill overlying a soft soil strata. It is shown that the geosynthetic component effectively reduces the magnitude of the settlement due to loads carried by the system through increasing the lateral confining pressure in the granular fill. The study is an extension of a previous study by the same authors, which presents a general model of the layered system and which reduces to Winkler, Filenko—Borodich and Pasternak models for particular cases.     

Modeling of AFM with a piezoelectric layer based on the modified couple stress theory with geometric discontinuities

AFM has been one of the most accurate instruments for measuring intermolecular forces and surface topography in the nano-scale. Micro-cantilever (MC) with piezoelectric layer has been used to improve the AFM performance. The Classic Continuum Mechanics (CCM) which currently used to develop the governing equation leads to noticeable errors. Hence, the accuracy of the governing equations for examining the MC vibrational behavior needs to be improved by using a modified model. In response to this need, the Modified Couple Stress theory (MCS) based on the Timoshenko beam model has been employed in this research. The governing equations have been derived using the Hamilton's principle and solved using the Differential Quadrature (DQ) method. In the modeling, the geometric discontinuities resulting from the presence of a piezoelectric layer enclosed between electrodes and MC surface area variations resulting from the connection of the probe to the MC have been considered. Moreover, the coupling effects of piezoelectric on MC stiffness have been taken into account. The results have revealed that the size parameter not only affects the frequency and amplitude but also improves the accuracy of the results when compared with the CCM theory. Moreover, the effects of geometric parameters on the piezoelectric MC frequency have been examined.     

Analytical solution considering the tangential effect for an infinite beam on a viscoelastic Pasternak foundation

In this study, the dynamic response of an infinite beam resting on a Pasternak foundation subjected to inclined travelling loads was developed in the form of the analytical solution wherein the tangential effect between the beam and foundation and the damping were taken into consideration. Three parameters were used to model the mechanical resistance of the viscoelastic Pasternak foundation, one of them accounts for the compressive stress in the soil, the other accounts for the shearing effect of soils, and the last one accounts for the damping of the foundation. By contrast, the Pasternak model is more realistic than the Winkler model that just considers the compressive resistance of soil. In the paper, the tangential effect between the beam and foundation was simulated by a series of separate horizontal springs, the damping was also considered to obtain the dynamic response under forced vibration. The theory of elasticity and Newton's laws were used to derive the governing equation. To simplify the partial-differential equation to an algebraic equation, the double Fourier transformation was used wherein the analytical solution in the frequency domain for the dynamic response of the beam is obtained. And its inversion was adopted to convert the integral representation of the solution into the time domain. The degraded solution was then utilized to verify the validity of the proposed solution. Finally, the Maple mathematical software was used for further discussion. The solution proposed in this study can be a useful tool for practitioners.     

Dynamics of a two-layer beam with layer slip

A possible type of structural damping in mechanical systems when there is a distinct dependence of the oscillation decrement on the amplitude is investigated, using the example of the solution of a model problem on the oscillations of a two-layer beam. It is assumed that layer slip only occurs along the beam axis and that the layers together in the transverse direction. The interaction between the layers is of an elastic-friction form. The equations of motion of the beam in Timoshenko's form are numerically integrated using Godunov's difference scheme.     

Stability analysis of cantilever carbon nanotubes subjected to partially distributed tangential force and viscoelastic foundation

Dynamic instability of cantilever carbon nanotubes conveying fluid embedded in viscoelastic foundation under a partially distributed tangential force is investigated based on nonlocal elasticity theory and Euler–Bernouli beam theory. The present study has incorporated the effects of nonlocal parameter, Knudsen number, surface effects and magnetic field. And two main parameters have also considered, namely partially distributed tangential force and foundation. It is assumed that viscoelastic foundation has modeled as Kelvin–Voigt, Maxwell and Standard linear solid types. The size-dependent governing equation of transverse vibration is derived using Hamilton’s variational principle and discretized by the Galerkin truncation method. A detailed parameter study is carried out, indicating the stability behavior of the nanotubes. In the light of numerical results, it is shown that variables considered in nondimensional equations have significant effects on natural frequencies and flutter velocities, especially for the foundation distribution length and model as well as the partially distributed tangential force.