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.

GENERAL SOLUTION FOR DYNAMICAL PROBLEM OF INFINITE BEAM ON AN ELASTIC FOUNDATION

Based on the theory of Eider-Bernoulli beam and Winkler assumption for elastic foundation, a mathematical model is presented. By using Fourier transformation for space variable, Laplace transformation for time variable and convolution theorem for their inverse transformations, a general solution for dynamical problem of infinite beam on an elastic foundation is obtained. Finally, the cases of free vibration,impulsive response and moving load are also discussed.

A Unified Dimensionless Parameter for Finite Element Mesh for Beams Resting on Elastic Foundation

Discretising a structure into elements is a key step in finite element(FE)analysis.The discretised geometry used to formulate an FE model can greatly affect accuracy and validity.This paper presents a unified dimensionless parameter to generate a mesh of cubic FEs for the analysis of very long beams resting on an elastic foundation.A uniform beam resting on elastic foundation with various values of flexural stiffness and elastic supporting coefficients subject to static load and moving load is used to illustrate the application of the proposed parameter.The numerical results show that(a)Even if the values of the flexural stiffness of the beam and elastic supporting coefficient of the elastic foundation are different,the same proposed parameter“s”can ensure the same accuracy of the FE solution,but the accuracy may differ for use of the same element length;(b)The proposed dimensionless parameter“s”can indeed be used as a unified index to generate the mesh for a beam resting on elastic foundation,whereas the use of the same element length as a criterion may be misleading;(c)The errors between the FE and analytical solutions for the maximum vertical displacement,shear force and bending moment of the beam increase with the dimensionless parameter“s”;and(d)For the given allowable errors for the vertical displacement,shear force and bending moment of the beam under static load and moving load,the corresponding values of the proposed parameter are provided to guide the mesh generation.

Free vibration and buckling analysis of polymeric composite beams reinforced by functionally graded bamboo fbers

Natural fibers have been extensively researched as reinforcement materials in polymers on account of their environmental and economic advantages in comparison with synthetic fibers in the recent years.Bamboo fibers are renowned for their good mechanical properties,abundance,and short cycle growth.As beams are one of the fundamental structural components and are susceptible to mechanical loads in engineering applications,this paper performs a study on the free vibration and buckling responses of bamboo fiber reinforced composite(BFRC)beams on the elastic foundation.Three different functionally graded(FG)layouts and a uniform one are the considered distributions for unidirectional long bamboo fibers across the thickness.The elastic properties of the composite are determined with the law of mixture.Employing Hamilton’s principle,the governing equations of motion are obtained.The generalized differential quadrature method(GDQM)is then applied to the equations to obtain the results.The achieved outcomes exhibit that the natural frequency and buckling load values vary as the fiber volume fractions and distributions,elastic foundation stiffness values,and boundary conditions(BCs)and slenderness ratio of the beam change.Furthermore,a comparative study is conducted between the derived analysis outcomes for BFRC and homogenous polymer beams to examine the effectiveness of bamboo fibers as reinforcement materials,demonstrating the significant enhancements in both vibration and buckling responses,with the exception of natural frequencies for cantilever beams on the Pasternak foundation with the FG-◇fiber distribution.Eventually,the obtained analysis results of BFRC beams are also compared with those for carbon nanotube reinforced composite(CNTRC)beams found in the literature,indicating that the buckling loads and natural frequencies of BFRC beams are lower than those of CNTRC beams.

A complex solution on the dynamic response of sandwich graphene-reinforced aluminum-based composite beams with copper face sheets under two moving constant loads on an elastic foundation

An analytical solution was used to investigate the elastic response of a sandwich beam with a graphene-reinforced aluminum-based composite(GRAC)on an elastic foundation using copper as the face layer of the functionally graded composite beam and a simply supported boundary condition.Mantari's higher-order shear deformation theory was utilized to derive the equations,which were solved in Laplace space and then converted into space–time using Laplace inversion.The exact response of the GRAC sandwich beam was obtained by considering the displacement at the mid-span of the sandwich beam.Two moving loads with different speed ratios were applied at a single point,and the effect of various parameters,including the spring constant,the speed ratio,the percentage of graphene,the moving load speed,and the distribution pattern,was investigated.This study aimed to eliminate any overlap and improve the accuracy of the results.The exact solving method presented has not been reported in other articles so far.Additionally,due to the difficulty of solving mathematical equations,this method is only applicable to simple boundary conditions.

A method to Estimate Dynamic Responses of VLFS Based on Multi-Floating-Module Model Connected by Elastic Hinges

Based on the elastic foundation beam theory and the multi-floating-module hydrodynamic theory,a novel method is proposed to estimate the dynamic responses of VLFS(Very Large Floating Structure).In still water,a VLFS can be simplified as an elastic foundation beam model or a multi-floating-module model connected by elastic hinges.According to equivalent displacement of the two models in static analysis,the problem of rotation stiffness of elastic hinges can be solved.Then,based on the potential flow theory,the dynamic responding analysis of multi-floatingmodule model under wave loads can be computed in ANSYS-AQWA software.By assembling the time domain analysis results of each module,the dynamic responses of the VLFS can be obtained.Validation of the method is conducted through a series of comparison calculations,which mainly includes a continuous structure and a three-part structure connected by hinges in regular waves.The results of this paper method show a satisfactory agreement with the experiment and calculation data given in relative references.

Elastic Modulus Limit of Support Layer in Twin-Block Ballastless Track Based on Train Load Effect

The support layer is an important component of twin-block ballastless track. The modulus of the support layer is an important design parameter and must be carefully solved. We studied the bending stress and deformation of track slab and support layer due to train load using the beam-plate finite element model on elastic foundation. The results show that support layer type has great impact on both support layer deformation and the stress on subgrade, but has little impact on the bending stress of either track slab or support layer. The continuous support layer type, and articulated support layer type with shear transfer device at their ends, are recommended. In order to keep the stress in the support layer less than that in track slab, the modulus of the continuous, unit, and articulated types of support layer （ in unit twin-block ballastless track）, and the support layer in continuous twin-block ballastless track, should not be larger than 15, 22, 20.5 and 5 GPa, respectively. In addition, the modulus of the unit-type support layer should not be more than 20 GPa, to ensure the step in support layer remains less than 1 mm.

基于非极限土压力的基坑开挖引起隧道变形的计算方法

由于基坑开挖引起邻近隧道变形,继而影响地铁在隧道中的安全运行,因此如何准确预测邻近隧道变形是个值得研究的问题。根据非极限土压力计算方法,计算基坑开挖卸荷坑壁上的土压力,利用Mindlin求解公式,获得基坑开挖阶段旁侧隧道上的附加荷载,将隧道结构看作Winkler地基梁上的Euler梁,建立弹性地基梁微分方程,给出隧道水平位移变形的解析方法。基于该方法分析了基坑开挖引起邻近隧道的水平位移。研究结果表明:随着基坑开挖深度的增加,隧道水平位移相应增加;考虑非极限土压力计算得到的隧道水平位移结果与实测值较为接近,误差总体保持在20%以内;相较于传统隧道水平位移求解方法,笔者方法更加符合实际。

盾构注浆荷载作用下带榫管片水平轴线偏移量计算方法研究

为探明隧道盾构注浆荷载作用引起带榫管片水平轴线偏移规律,以小曲率半径盾构隧道为例,分析壁后注浆引起管片上浮的诱因,考虑管片环“椭圆化”变形、纵向连接螺栓预应力、环缝影响范围和横向刚度等综合因素的影响,建立榫杆连接条件下带榫环缝的管片单元纵向等效抗弯刚度有效率计算式,并通过计算盾构注浆荷载作用下带榫管片的静态和动态上浮力,构建盾构注浆荷载作用下带榫管片上浮力计算表达式,由此分析小曲率半径隧道盾构注浆荷载作用下带榫管片受力情况及其在文克尔弹性地基梁假定条件下受上浮力作用的挠曲变形规律,建立盾构注浆荷载作用下带榫管片水平轴线偏移量计算方法,并依托实际工程加以验证。结果表明:该文理论值与模拟值和监测值均拟合良好,该文计算方法具有良好的工程适用性,可为盾构隧道工程及其类似工程中注浆荷载作用下管片偏移及错台问题提供有效的理论指导。

A Numerical Method for Cavity Identification in Beams on an Elastic Foundation

An analytical solution for the natural frequencies of a beam containing a cavity on an elastic foundation is presented. Based on the analytical solution, a numerical method for identifying cavities in the foundation is developed. The position and size of the cavities are identified by minimizing an objective function, which is formulated according to the difference between the computed and measured natural frequencies of the system. The conjugate gradient algorithm is adopted for minimizing the objective function. Some numerical examples are presented to demonstrate the applicability of the presented cavity determination method. The results show that the presented method can be used to identify the cavity position and size conveniently and efficiently.

弹性环梁支撑下圆形深基坑支护结构变形解析解

将环梁和支护结构分别视为弹性圆环和弹性圆柱薄壳,基于环梁和圆柱壳的变形协调,得到了具有任意数目弹性环梁支撑圆形深基坑支护结构变形的解析解.在验证解析解合理性和可靠性的基础上,针对某一实际圆形基坑工程,比较了在刚性和弹性环梁支撑下支护结构变形和内力的差异,并分析了支护结构底部边界条件、环梁弹性模量、尺寸和数量以及位置等参数对支护结构变形和内力分布的影响.研究结果表明:相对于刚性环梁支撑,弹性环梁支撑处内力变化较小,环梁弹性模量和尺寸的改变只对基坑挖掘面以上支护结构变形和内力影响显著,而对挖掘面以下的支护结构变形和内力几乎没有影响.该研究成果为圆形基坑支护结构设计提供了理论依据和指导.

桩撑式支护基坑通用弹性地基梁分析法

弹性地基梁法常用于模拟分析桩撑(锚)式支护基坑在开挖、设撑等施工工况中变形和受力的变化。目前利用该方法对某工况基坑分析时,首先要将整个支护桩在支撑处划分为若干个桩段,然后利用相邻桩段交界处的位移连续条件和受力平衡条件,推导出相应的待定参数方程,之后再利用求出的待定参数分析支护体系(基坑)的变形和受力。由于不同施工工况对应的支撑数和桩段数不同,因此该方法必须根据不同工况重新推导各自的参数方程,从而导致计算过程繁杂,难以形成通用计算公式和方法,也不易编程等问题。通过将支撑与桩段归纳为3种基本连接形式(即支撑分别相连于桩顶、相邻两桩段之间和坑底),本文建立了相邻两个桩段间待定参数的递推公式,进而推导出了适用于内支撑和锚索支撑形式、任意支撑道数、开挖或设撑工况的通用待定参数方程。该方程仅含4个参数,远少于现有方法且易于求解。文中算例结果验证了本文方法的合理性和可行性。

BUCKLING LOADS AND EIGENFREQUENCIES OF A BRACED BEAM RESTING ON AN ELASTIC FOUNDATION

The eigenvalue problems of the buckling loads and natural frequencies of a braced beam on an elastic foundation are investigated. sented. The eigenvalues vary with the different The exact solutions for the eigenvalues are preparameters and are especially sensitive to the brace location. As the beam of a continuous system has infinite eigenvalues and these eigenvalues are influenced differently by a brace, the eigenvalues show rich variation patterns. Because these eigenvalues physically correspond to the structure buckling loads and natural frequencies, the study on the eigenvalues variation patterns can offer a design guidance of using a lateral brace of translation spring to strengthen the structure.

Semi-analytical approach for free vibration analysis of cracked beams resting on two-parameter elastic foundation with elastically restrained ends

In present study, free vibration of cracked beams resting on two-parameter elastic foundation with elastically restrained ends is considered. Euler-Bemoulli beam hypothesis has been applied and translational and rotational elastic springs in each end considered as support. The crack is modeled as a mass-less rotational spring which divides beam into two segments. After governing the equations of motion, the differential transform method （DTM） has been served to determine dimensionless frequencies and normalized mode shapes. DTM is a semi-analytical approach based on Taylor expansion series that converts differential equations to recursive algebraic equations. The DTM results for the natural frequencies in special cases are in very good agreement with results reported by well-known references. Also, the DTM procedure yields rapid convergence beside high accuracy without any frequency missing. Comprehensive studies to analyze the effects of crack location, crack severity, parameters of elastic foundation and boundary conditions on dimensionless frequencies as well as effects of elastic boundary conditions on cracked beams mode shapes are carried out and some problems handled for first time in this paper. Since this paper deals with general problem, the derived formulation has capability for analyzing free vibration of cracked beam with every boundary condition.

基于能量法的隧道穿越引起既有管线变形研究

针对隧道穿越引起既有管线变形的问题,本文出发点并非沿用力学平衡,而是从系统能量入手。将既有管线视为Euler-Bernoulli梁,并搁置于可考虑地基连续变形的Pasternak地基上。参考Rayleigh-Ritz法给出位移函数,写出地基梁系统各部分的功能表达式,建立关于管线变形的总势能方程。依据最小势能原理对其进行变分运算并求解,推导得到计算管线变形的理论解析。分别利用有限元软件的模拟结果和已发表文献中实际工程的监测数据验证所提方法的合理性。进一步对既有管线抗弯刚度、新旧结构穿越角度以及地层损失率展开参数分析。研究结果表明,随着既有管线抗弯刚度和新旧结构穿越角度的增大,既有管线的变形减小;而随着地层损失率的增大,既有管线的变形也随之增大。所提方法可快速评估隧道开挖对既有管线的影响。

基于横向变形约束弹性地基梁法的既有锚索桩极限承载力研究

以阿坝州岷江电化公司西侧滑坡前缘既有锚索桩为研究对象,既有锚索桩的极限承载力是滑坡综合治理中需要研究的关键。针对该问题,在桩身检测结果和锚索现场抗拉拔试验结果基础上,根据变形协调原理,对横向变形约束弹性地基梁法在桩锚固段地基水平抗力系数按梯形分布下的计算方法进行推导,并采用这种方法进行1-1'剖面既有预应力锚索桩的内力计算。把锚索极限承载力、既有桩身结构承载力和嵌固段地基强度条件共同作为限制条件,通过反分析方式不断逼近限制条件所对应的滑坡推力界限值,发现该工程中嵌固段地基强度对既有锚索桩的极限承载力起控制作用,并最终确定其极限承载力。该文所采用数据及计算方法可为同类工程研究分析提供参考。

T形地连墙码头结构计算

针对T形地连墙结构受力影响因素复杂的问题,结合工程实例,利用STAAD软件采用竖向弹性地基梁法对码头结构进行分析。研究不同模型单元划分精度、地连墙入土深度以及回填料m值选取对地连墙结构内力的影响,结果表明:随模型单元划分精度的提高,地连墙结构海侧弯矩增大,陆侧弯矩减小,锚碇墙结构弯矩变化规律与之相反,最终两者内力趋于稳定;在踢脚稳定的前提下,随地连墙底端入土深度的增加,墙体结构海侧弯矩减小、陆侧弯矩增大、位移减小;随m取值变大,锚碇墙海侧弯矩增大、陆侧弯矩减小,且敏感度较高。研究结论可为T形地连墙码头结构的设计提供参考。

非对称荷载下基坑支护结构变形计算方法

文章针对非对称受荷基坑的位移特点,基于弹性地基梁模型,以支撑轴力联系两侧墙体,提出非对称受荷基坑的整体计算模型。通过有限宽度土体的土压力求解公式计算,引入墙土摩擦角和土体内摩擦角随墙体位移的发挥公式,考虑墙体变形与坑外主动土压力的耦合作用。根据非对称受荷计算模型,得到了非对称受荷下支护结构变形控制方程,最后结合实际工程验证了算法。