Hermite Finite Element Method for Vibration Problem of Euler-Bernoulli Beam on Viscoelastic Pasternak Foundation

Viscoelastic foundation plays a very important role in civil engineering. It can effectively disperse the structural load into the foundation soil and avoid the damage caused by the concentrated load. The model of Euler-Bernoulli beam on viscoelastic Pasternak foundation can be used to analyze the deformation and response of buildings under complex geological conditions. In this paper, we use Hermite finite element method to get the numerical approximation scheme for the vibration equation of viscoelastic Pasternak foundation beam. Convergence and error estimation are rigourously established. We prove that the fully discrete scheme has convergence order O(τ2+h4), where τis time step size and his space step size. Finally, we give four numerical examples to verify the validity of theoretical analysis.

Unified two-phase nonlocal formulation for vibration of functionally graded beams resting on nonlocal viscoelastic Winkler-Pasternak foundation

A nonlocal study of the vibration responses of functionally graded(FG)beams supported by a viscoelastic Winkler-Pasternak foundation is presented.The damping responses of both the Winkler and Pasternak layers of the foundation are considered in the formulation,which were not considered in most literature on this subject,and the bending deformation of the beams and the elastic and damping responses of the foundation as nonlocal by uniting the equivalently differential formulation of well-posed strain-driven(ε-D)and stress-driven(σ-D)two-phase local/nonlocal integral models with constitutive constraints are comprehensively considered,which can address both the stiffness softening and toughing effects due to scale reduction.The generalized differential quadrature method(GDQM)is used to solve the complex eigenvalue problem.After verifying the solution procedure,a series of benchmark results for the vibration frequency of different bounded FG beams supported by the foundation are obtained.Subsequently,the effects of the nonlocality of the foundation on the undamped/damping vibration frequency of the beams are examined.

Vibration of axially moving beam supported by viscoelastic foundation

In this paper, transverse vibration of an axially moving beam supported by a viscoelastic foundation is analyzed by a complex modal analysis method. The equation of motion is developed based on the generalized Hamilton＇s principle. Eigenvalues and eigenfunctions are semi-analytically obtained. The governing equation is represented in a canonical state space form, which is defined by two matrix differential operators. The orthogonality of the eigenfunctions and the adjoint eigenfunctions is used to decouple the system in the state space. The responses of the system to arbitrary external excitation and initial conditions are expressed in the modal expansion. Numerical examples are presented to illustrate the proposed approach. The effects of the foundation parameters on free and forced vibration are examined.

Theoretical investigation of interaction between a rectangular plate and fractional viscoelastic foundation

The interaction between plates and foundations is a typical problem encountered in geotechnical engineering. The long-term plate performance is highly dependent on the theological characteristics of ground soil. Compared with conventional linear theology, the fractional calculus-based theory is a more powerful mathematical tool that can address this issue. This paper proposes a fractional Merchant model （FMM） to investigate the time-dependent behavior of a simply supported rectangular plate on viscoelastic foundation. The correspondence principle involving Laplace transforms was employed to derive the closed-form solutions of plate response under uniformly distributed load. The plate deflection, bending moment, and foundation reaction calculated using the FMM were compared with the results obtained from the analogous elastic model （EM） and the standard Merchant model （SMM）. It is shown that the upper and lower bound solutions of the FMM can be determined using the EM. In addition, a parametric study was performed to examine the influences of the model parameters on the time- dependent behavior of the plate-foundation interaction problem. The results indicate that a small fractional differential order corresponds to a plate resting on a sandy soil foundation, while the fractional differential order value should be increased for a clayey soil foundation. The long-term performance of a foundation plate can be accurately simulated by varying the values of the fractional differential order and the viscosity coefficient. The observations from this study reveal that the proposed fractional model has the capability to capture the variation of plate deflection over many decades of time.

Solving Nonlinear Differential Equation Governing on the Rigid Beams on Viscoelastic Foundation by AGM

In the present paper a vibrational differential equation governing on a rigid beam on viscoelastic foundation has been investigated. The nonlinear differential equation governing on this vibrating system is solved by a simple and innovative approach, which has been called Akbari-Ganji＇s method （AGM）. AGM is a very suitable computational process and is usable for solving various nonlinear differential equations. Moreover, using AGM which solving a set of algebraic equations, complicated nonlinear equations can easily be solved without any mathematical operations. Also, the damping ratio and energy lost per cycle for three cycles have been investigated. Furthermore, comparisons have been made between the obtained results by numerical method （Runk45） and AGM. Results showed the high accuracy of AGM. The results also showed that by increasing the amount of initial amplitude of vibration （A）, the value of damping ratio will be increased, and the energy lost per cycle decreases by increasing the number of cycle. It is concluded that AGM is a reliable and precise approach for solving differential equations. On the other hand, it is better to say that AGM is able to solve linear and nonlinear differential equations directly in most of the situations. This means that the final solution can be obtained without any dimensionless procedure Therefore, AGM can be considered as a significant progress in nonlinear sciences.

A closed-form solution for a double infinite Euler-Bernoulli beam on a viscoelastic foundation subjected to harmonic line load

The dynamic response of a double infinite beam system connected by a viscoelastic foundation under the harmonic line load is studied. The double infinite beam system consists of two identical and parallel beams, and the two beams are infinite elastic homogeneous and isotropic. A viscoelastic layer connects the two beams continuously. To decouple the two coupled equations governing the response of the double infinite beam system, a variable substitution method is introduced. The frequency domain solutions of the decoupled equations are obtained by using Fourier transforms as well as Laplace transforms successively. The time domain solution in the generalized integral form are then obtained by employing the corresponding inverse transforms, i.e. Fourier transform and inverse Laplace transform. The solution is verified by numerical examples, and the effects of parameters on the response are also investigated.

Dynamic Analysis of Submarine Pipelines on an Elastic / Viscoelastic Foundation

This paper presents a new finite element method for solving static and dynamic problems in laying operation of pipelines. The effect of the viscoelastic soil behavior is considered by using the Pasternak foundation model. Some examples are also presented.

An Improved Model for Bending of Thin Viscoelastic Plate on Elastic Foundation

An improved model for bending of thin viscoe-lastic plate resting on Winkler foundation is presented. The thin plate is linear viscoelastic and subjected to normal distributed loading, the effect of normal stress along the plate thickness on the deflection and internal forces is taken into account. The basic equations for internal forces and stress distribution are derived based on the general viscoelastic theory under small deformation condition. The reduced equations for elastic case are given as well. It is shown that the proposed model reveals a larger flex-ural rigidity compared to that in classic models, in which the normal stress along the plate thickness is neglected.

Wave propagation analysis of porous functionally graded piezoelectric nanoplates with a visco-Pasternak foundation

This study investigates the size-dependent wave propagation behaviors under the thermoelectric loads of porous functionally graded piezoelectric(FGP) nanoplates deposited in a viscoelastic foundation.It is assumed that(i) the material parameters of the nanoplates obey a power-law variation in thickness and(ii) the uniform porosity exists in the nanoplates.The combined effects of viscoelasticity and shear deformation are considered by using the Kelvin-Voigt viscoelastic model and the refined higher-order shear deformation theory.The scale effects of the nanoplates are captured by employing nonlocal strain gradient theory(NSGT).The motion equations are calculated in accordance with Hamilton’s principle.Finally,the dispersion characteristics of the nanoplates are numerically determined by using a harmonic solution.The results indicate that the nonlocal parameters(NLPs) and length scale parameters(LSPs) have exactly the opposite effects on the wave frequency.In addition,it is found that the effect of porosity volume fractions(PVFs) on the wave frequency depends on the gradient indices and damping coefficients.When these two values are small,the wave frequency increases with the volume fraction.By contrast,at larger gradient index and damping coefficient values,the wave frequency decreases as the volume fraction increases.

On wave dispersion of rotating viscoelastic nanobeam based on general nonlocal elasticity in thermal environment

The present research focuses on the analysis of wave propagation on a rotating viscoelastic nanobeam supported on the viscoelastic foundation which is subject to thermal gradient effects.A comprehensive and accurate model of a viscoelastic nanobeam is constructed by using a novel nonclassical mechanical model.Based on the general nonlocal theory(GNT),Kelvin-Voigt model,and Timoshenko beam theory,the motion equations for the nanobeam are obtained.Through the GNT,material hardening and softening behaviors are simultaneously taken into account during wave propagation.An analytical solution is utilized to generate the results for torsional(TO),longitudinal(LA),and transverse(TA)types of wave dispersion.Moreover,the effects of nonlocal parameters,Kelvin-Voigt damping,foundation damping,Winkler-Pasternak coefficients,rotating speed,and thermal gradient are illustrated and discussed in detail.

Deformation response of roof in solid backfilling coal mining based on viscoelastic properties of waste gangue

Solid backfill mining(SBM)is a form of green mining,the core of which is to control and minimize the deformation and movement of strata above longwall coal mines.Establishing a mechanical model that can reliably describe roof deformation by considering the viscoelastic properties of waste gangue is important as it assists in improving mine designs and reducing the environmental impact on the surface.In this paper,the time-dependent deformation characteristics of gangue under different stress levels were obtained by using lateral confinement compression,that reliably represents the compaction of goaf.The viscoelastic foundation model for gangue mechanical response is different from the traditionally used elastic foundation model,as it considers the time factor and viscoelasticity.A mechanical model using a thin plate on a fractional viscoelastic foundation was established,and the roof deflection,bending moment,time-dependent,viscous and other characteristics of SBM were included and analyzed.Compared with the existing elastic foundation model,the proposed fractional order viscoelastic foundation model has higher accuracy with laboratory data.The plate deflection increases by 50.9%and the bending moment increases by 37.9%after 100 days,which the elastic model would not have been able to predict.

STABILITY ANALYSIS OF VISCOELASTIC CURVED PIPES CONVEYING FLUID

Based on the Hamilton' s principle for elastic systems of changing mass, a differential equation of motion for viscoelastic curved pipes conveying fluid was derived using variational method, and the complex characteristic equation for the viscoelastic circular pipe conveying fluid was obtained by normalized power series method. The effects of dimensionless delay time on the variation relationship between dimensionless complex frequency of the clamped-clamped viscoelastic circular pipe conveying fluid with the Kelvin-Voigt model and dimensionless flow velocity were analyzed. For greater dimensionless delay time, the behavior of the viscoelastic pipe is that the first, second and third mode does not couple, while the pipe behaves divergent instability in the first and second order mode, then single-mode flutter takes place in the first order mode.

Influence of Soil-Foundation Interaction Properties on Oscillations of the System “Building-Building” and “Building-Stack-Like Structure”

Seismic oscillations of the “building-building” system which is interconnected buildings built close to each other, and “building-stack-like structure” system which is adjacent and connected in different ways to existing building are considered in the paper. Different types of connections, such as dampers, including the ones suggested by the authors, are studied. Seismic impact is given as a harmonic function and various existing accelerograms, including synthesized ones. Distinctive feature of this paper from previously published ones [1] [2] is the fact that the emphasis falls on the influence of soil-foundation interaction properties, which are described using various models of load-displacement connections. Calculation results are compared in the case of representation of the building as concentrated masses and spatial systems. Ways to reduce seismic response of buildings during the earthquakes are pointed out. Results of experimental studies are given in the paper and are compared with calculations.

考虑蠕变影响的堆载作用下被动排桩受力变形分析

为分析堆载作用下被动桩受力变形的时间效应,首先引入分数阶Merchant模型描述土体的蠕变特性,然后根据对应性原理和Laplace变换推导Boussinesq黏弹性解,计算堆载引发水平附加应力,并通过Terzaghi土拱模型将附加应力传递到桩上,得到堆载作用在桩上的被动荷载.其次,将桩简化为黏弹性Pasternak地基上的欧拉梁,建立桩身挠曲微分方程,并采用有限差分法及Laplace逆变换求解,随后通过与已有试验结果的对比分析,验证了本文方法的可行性.最后,对分数阶Merchant模型参数(虎克体的弹性模量Eh、Kelvin体的弹性模量Ek及黏滞系数η、分数阶阶次α)、堆载-桩身水平距离以及堆载荷载对桩身水平位移的影响进行了分析,分析结果表明:Eh越大,桩身初始水平位移越小;Ek越大,桩身水平位移随时间的增量越小;η越大,桩身到达最终变形量的时间越长;α越大,Kelvin体的黏滞性越大;堆载-桩身水平距离越小、堆载荷载越大,桩身水平位移越大;且随时间的增加,桩身水平位移对Eh的敏感度降低,对堆载-桩身水平距离和堆载荷载的敏感度会增大.

分数阶粘弹性地基上裂纹梁的自由振动

本文研究了分数阶粘弹性Pasternak地基上含多裂纹Euler梁的自由振动问题.首先,引入分数阶导数概念,建立粘弹性Pasternak地基模型,推导出地基应力-应变本构的复模量以及地基反力.其次,将裂纹等效成无质量扭转弹簧,借助于裂纹能量释放率和应力强度因子的关系,推导出梁的局部柔度.然后,基于传递矩阵方法,建立含多裂纹梁的分段传递矩阵进而得到总体传递矩阵.最后,基于梁的边界条件,建立求解裂纹梁的复固有频率及振型的线性代数方程组,数值求解得到含多裂纹梁的固有频率及振型.以含双裂纹的两端简支Euler梁为例,数值计算了复固有频率和振型.并基于曲率模态分析裂纹位置对复固有频率和振型的影响.数值结果揭示了粘弹性地基的分数阶系数、粘性系数以及裂纹位置和裂纹深度对复固频率和振型的影响规律.

基于广义Voigt模型的半渗透边界隧道周围饱和软土固结分析

引入衬砌半渗透边界条件,使用广义Voigt黏弹性模型描述隧道周围饱和软黏土的流变特性,基于Terzaghi-Rendulic理论建立以超孔隙水压力为变量的软土二维流变固结控制方程。采用保角变换和分离变量方法,得到超孔隙水压力关于时间和空间变量的两个独立方程,依次使用Laplace变换法与部分分式展开法得到时域内超孔隙水压力消散解。将广义Voigt模型退化为已有四元件Burgers模型和五元件模型并与其对比,验证解答的可靠性。基于建立的解答,分析衬砌透水程度、Kelvin体数量、独立Newton黏壶黏滞系数、隧道埋深等因素对超孔隙水压力消散与分布的影响规律。结果表明:衬砌与土体的相对渗透系数比越大,超孔隙水压力的消散起始时间越早,消散速度越快。Kelvin体数量越多,超孔隙水压力消散越慢;独立黏壶黏滞系数越大,其产生的黏滞变形越小,超孔隙水压力消散速率越快;隧道埋深越大,土体渗流路径越长,超孔隙水压力消散越慢;当相对渗透系数比为0.01时,超孔隙水压力随着到隧道外壁距离的增大而逐渐减小,当相对渗透系数比为1.00时,超孔隙水压力随着距离的增大而出现先增大后减小的现象;随着相对渗透系数比的增加,Kelvin体数量对超孔压消散的影响越小,不同独立黏滞系数带来的超孔压消散差异逐渐减小,隧道衬砌作为土层内部新透水边界的作用越突出。

软土地区隧道-土体-地表结构筏板基础地震响应分析

以上海地区典型的下穿隧道-土体-地表结构筏板基础系统为研究对象,建立了相应的非线性动力有限元模型。考虑隧道埋深、土体本构模型以及地表结构3个因素,采用ABAQUS软件,系统研究了在地震荷载作用下隧道对地表结构筏板基础的影响,同时也揭示了地表结构对隧道横截面变形及衬砌动态内力的影响特征。结果表明,地震荷载作用下筏板基础会产生显著的沉降及转角,基础两端的土体产生显著的塑性应变区,且浅埋隧道所产生的影响大于深埋隧道;考虑筏板基础会引起隧道结构变形和衬砌动态内力的显著增加,筏板基础的存在对浅埋隧道的地震响应影响要大于深埋隧道,而选用黏弹性土体本构模型进行分析的隧道地震响应显著小于黏弹塑性模型的分析结果,说明相互作用的复杂体系,其地震响应分析中考虑土体塑性的重要性。

黏弹性Pasternak地基上修正Timoshenko梁横向自振特性分析

基于修正Timoshenko梁理论,建立黏弹性Pasternak地基上修正Timoshenko梁的横向振动控制方程,运用回传射线矩阵法推导出黏弹性Pasternak地基上两端简支修正Timoshenko梁自振频率和衰减系数的解析解,结合二分法和黄金分割法计算了经典边界条件下黏弹性Pasternak地基上修正Timoshenko梁的自振特性,对比分析了考虑剪切变形引起的转动惯量、梁长及不同的边界条件对结构自振特性的影响。研究表明:黏弹性Pasternak地基上修正Timoshenko梁的各阶自振频率和衰减系数小于经典Timoshenko梁的各阶自振频率和衰减系数;梁越短,剪切变形引起的转动惯量对结构自振频率和衰减系数的影响越大,且对高阶的影响明显大于低阶;边界约束条件越强,振动能量衰减越明显。

层状异性流变地层桩基长时位移的理论预测

针对层状异性流变地层中桩基长时沉降位移预测计算难题,基于荷载传递模式、Poyting-Thomson模型和土体一维虚拟结构等效模型,建立了该类地层中桩基承载模式判别以及桩基服役全过程长时沉降位移的理论预测方法。研究了稳定蠕变地层桩基承载模式演变及其沉降位移时效特性。结果表明:摩擦承载状态下桩基沉降呈现瞬态弹性;土体的层状异性特型导致桩体压缩变形具有明显的台阶突变;摩擦+桩端共同承载时,桩基总体位移等于桩体形变位移和桩端沉降位移的叠加,具有显著时效特性并随时间逐步趋于稳定。

Vibration of an Axially Moving String Supported by a Viscoelastic Foundation

The transverse vibration of an axially moving string supported by a viscoelastic foundation is analysed using the complex modal method. The equation of motion is developed using the generalized Hamilton principle. The exact closed-form solution of eigenvalues and eigen- functions are obtained. The governing equation is represented in a canonical state space form defined by two matrix differential operators, and the eigenfunctions and adjoint eigenfunctions are proved to be orthogonal with respect to each operator. This orthogonality is applied so that the response to arbitrary external excitations and initial conditions can be expressed in modal expansion. Numerical examples are presented to validate the proposed approach.