Bending of Timoshenko beam with effect of crack gap based on equivalent spring model

Considering the effect of crack gap,the bending deformation of the Timoshenko beam with switching cracks is studied.To represent a crack with gap as a nonlinear unidirectional rotational spring,the equivalent flexural rigidity of the cracked beam is derived with the generalized Dirac delta function.A closed-form general solution is obtained for bending of a Timoshenko beam with an arbitrary number of switching cracks.Three examples of bending of the Timoshenko beam are presented.The influence of the beam’s slenderness ratio,the crack’s depth,and the external load on the crack state and bending performances of the cracked beam is analyzed.It is revealed that a cusp exists on the deflection curve,and a jump on the rotation angle curve occurs at a crack location.The relation between the beam’s deflection and load is bilinear,each part corresponding to an open or closed state of crack,respectively.When the crack is open,flexibility of the cracked beam decreases with the increase of the beam’s slenderness ratio and the decrease of the crack depth.The results are useful in identifying non-destructive cracks on a beam.     

STEADY-STATE RESPONSE OF A TIMOSHENKO BEAM ON AN ELASTIC HALF-SPACE UNDER A MOVING LOAD

By introducing the equivalent stiffness of an elastic half-space interacting with a Timoshenko beam, the displacement solution of the beam resting on an elastic half-space subjected to a moving load is presented. Based on the relative relation of wave velocities of the half-space and the beam, four cases with the combination of different parameters of the half-space and the beam, the system of soft beam and hard half-space, the system of sub-soft beam and hard half-space, the system of sub-hard beam and soft half-space, and the system of hard beam and soft half-space are considered. The critical velocities of the moving load are studied using dispersion curves. It is found that critical velocities of the moving load on the Timoshenko beam depend on the relative relation of wave velocities of the half-space and the beam. The Rayleigh wave velocity in the half-space is always a critical velocity and the response of the system will be infinite when the load velocity reaches it. For the system of soft beam and hard half-space, wave velocities of the beam are also critical velocities. Besides the shear wave velocity of the beam, there is an additional minimum critical velocity for the system of sub-soft beam and hard half-space. While for systems of (sub-) hard beams and soft half-space, wave velocities of the beam are no longer critical ones. Comparison with the Euler-Bernoulli beam shows that the critical velocities and response of the two types of beams are much different for the system of (sub-) soft beam and hard half-space but are similar to each other for the system of (sub-) hard beam and soft half space. The largest displacement of the beam is almost at the location of the load and the displacement along the beam is almost symmetrical if the load velocity is smaller than the minimum critical velocity (the shear wave velocity of the beam for the system of soft beam and hard half-space). The largest displacement of the beam shifts behind the load and the asymmetry of the displacement along the beam increases with the increase of the load velocity due to the damping and wave radiation. The displacement of the beam at the front of the load is very small if the load velocity is larger than the largest wave velocity of the beam and the half space. The results of the present study provide attractive theoretical and practical references for the analysis of ground vibration induced by the high-speed train.     

POWER REFLECTION AND TRANSMISSION IN BEAM STRUCTURES CONTAINING A SEMI-INFINITE CRACK

Wave reflection and transmission in a beam containing a semi-infinite crack are studied analytically based on Timoshenko beam theory., Two kinds of crack surface conditions： non-contact （open） and fully contact （closed） cracks, are considered respectively for an isotropic beam. The analytical solution of reflection and transmission coefficients for a semi-infinite crack is obtained. The power reflection and transmission ratios depend on both the frequency and the position of the crack. Numerical results show the conservation of power transport. The transmitted energy among various wave modes is also investigated. A finite element method is used to verify the validity of the analytical results.     

Closed-form solution of beam on Pasternak foundation under inclined dynamic load

The dynamic response of an infinite Euler–Bernoulli beam resting on Pasternak foundation under inclined harmonic line loads is developed in this study in a closed-form solution.The conventional Pasternak foundation is modeled by two parameters wherein the second parameter can account for the actual shearing effect of soils in the vertical direction.Thus,it is more realistic than the Winkler model,which only represents compressive soil resistance.However,the Pasternak model does not consider the tangential interaction between the bottom of the beam and the foundation;hence,the beam under inclined loads cannot be considered in the model.In this study,a series of horizontal springs is diverted to the face between the bottom of the beam and the foundation to address the limitation of the Pasternak model,which tends to disregard the tangential interaction between the beam and the foundation.The horizontal spring reaction is assumed to be proportional to the relative tangential displacement.The governing equation can be deduced by theory of elasticity and Newton’s laws,combined with the linearly elastic constitutive relation and the geometric equation of the beam body under small deformation condition.Double Fourier transformation is used to simplify the geometric equation into an algebraic equation,thereby conveniently obtaining the analytical solution in the frequency domain for the dynamic response of the beam.Double Fourier inverse transform and residue theorem are also adopted to derive the closed-form solution.The proposed solution is verified by comparing the degraded solution with the known results and comparing the analytical results with numerical results using ANSYS.Numerical computations of distinct cases are provided to investigate the effects of the angle of incidence and shear stiffness on the dynamic response of the beam.Results are realistic and can be used as reference for future engineering designs.     

CRACK PROPAGATION IN STRUCTURES SUBJECTED TO PERIODIC EXCITATION

In the present paper,a simple mechanical model is developed to predict the dynamic response of a cracked structure subjected to periodic excitation,which has been used to identify the physical mechanisms in leading the growth or arrest of cracking.The structure under consideration consists of a beam with a crack along the axis,and thus,the crack may open in Mode I and in the axial direction propagate when the beam vibrates.In this paper,the system is modeled as a cantilever beam lying on a partial elastic foundation,where the portion of the beam on the foundation represents the intact portion of the beam.Modal analysis is employed to obtain a closed form solution for the structural response.Crack propagation is studied by allowing the elastic foundation to shorten(mimicking crack growth)if a displacement criterion,based on the material toughness,is met.As the crack propagates,the structural model is updated using the new foundation length and the response continues.From this work,two mechanisms for crack arrest are identified.It is also shown that the crack propagation is strongly influenced by the transient response of the structure.     

Force-displacement characteristics of simply supported beam laminated with shape memory alloys

As a preliminary step in the nonlinear design of shape memory alloy(SMA) composite structures,the force-displacement characteristics of the SMA layer are studied.The bilinear hysteretic model is adopted to describe the constitutive relationship of SMA material.Under the assumption that there is no point of SMA layer finishing martensitic phase transformation during the loading and unloading process,the generalized restoring force generated by SMA layer is deduced for the case that the simply supported beam vibrates in its first mode.The generalized force is expressed as piecewise-nonlinear hysteretic function of the beam transverse displacement.Furthermore the energy dissipated by SMA layer during one period is obtained by integration,then its dependencies are discussed on the vibration amplitude and the SMA’s strain(Ms-Strain) value at the beginning of martensitic phase transformation.It is shown that SMA’s energy dissipating capacity is proportional to the stiffness difference of bilinear model and nonlinearly dependent on Ms-Strain.The increasing rate of the dissipating capacity gradually reduces with the amplitude increasing.The condition corresponding to the maximum dissipating capacity is deduced for given value of the vibration amplitude.The obtained results are helpful for designing beams laminated with shape memory alloys.     

On the limitations of linear beams for the problems of moving mass-beam interaction using a meshfree method

This paper deals with the capabilities of linear and nonlinear beam theories in predicting the dynamic response of an elastically supported thin beam traversed by a moving mass. To this end, the discrete equations of motion are developed based on Lagrange’s equations via reproducing kernel particle method (RKPM). For a particular case of a simply supported beam, Galerkin method is also employed to verify the results obtained by RKPM, and a reasonably good agreement is achieved. Variations of the maximum dynamic deflection and bending moment associated with the linear and nonlinear beam theories are investigated in terms of moving mass weight and velocity for various beam boundary conditions. It is demonstrated that for majority of the moving mass velocities, the differences between the results of linear and nonlinear analyses become remarkable as the moving mass weight increases, particularly for high levels of moving mass velocity. Except for the cantilever beam, the nonlinear beam theory predicts higher possibility of moving mass separation from the base beam compared to the linear one. Furthermore, the accuracy levels of the linear beam theory are determined for thin beams under large deflections and small rotations as a function of moving mass weight and velocity in various boundary conditions.     

Method of reverberation ray matrix for static analysis of planar framed structures composed of anisotropic Timoshenko beam members

Based on the method of reverberation ray matrix(MRRM), a reverberation matrix for planar framed structures composed of anisotropic Timoshenko(T) beam members containing completely hinged joints is developed for static analysis of such structures.In the MRRM for dynamic analysis, amplitudes of arriving and departing waves for joints are chosen as unknown quantities. However, for the present case of static analysis, displacements and rotational angles at the ends of each beam member are directly considered as unknown quantities. The expressions for stiffness matrices for anisotropic beam members are developed. A corresponding reverberation matrix is derived analytically for exact and unified determination on the displacements and internal forces at both ends of each member and arbitrary cross sectional locations in the structure. Numerical examples are given and compared with the finite element method(FEM) results to validate the present model. The characteristic parameter analysis is performed to demonstrate accuracy of the present model with the T beam theory in contrast with errors in the usual model based on the Euler-Bernoulli(EB) beam theory. The resulting reverberation matrix can be used for exact calculation of anisotropic framed structures as well as for parameter analysis of geometrical and material properties of the framed structures.     

Free vibration of functionally graded beams based on both classical and first-order shear deformation beam theories

The free vibration of functionally graded material （FGM） beams is studied based on both the classical and the first-order shear deformation beam theories. The equations of motion for the FGM beams are derived by considering the shear deforma- tion and the axial, transversal, rotational, and axial-rotational coupling inertia forces on the assumption that the material properties vary arbitrarily in the thickness direction. By using the numerical shooting method to solve the eigenvalue problem of the coupled ordinary differential equations with different boundary conditions, the natural frequen- cies of the FGM Timoshenko beams are obtained numerically. In a special case of the classical beam theory, a proportional transformation between the natural frequencies of the FGM and the reference homogenous beams is obtained by using the mathematical similarity between the mathematical formulations. This formula provides a simple and useful approach to evaluate the natural frequencies of the FGM beams without dealing with the tension-bending coupling problem. Approximately, this analogous transition can also be extended to predict the frequencies of the FGM Timoshenko beams. The numerical results obtained by the shooting method and those obtained by the analogous transformation are presented to show the effects of the material gradient, the slenderness ratio, and the boundary conditions on the natural frequencies in detail.     

The method of variation on parameters for integration of a generalized Birkhoffian system

This paper focuses on studying the integration method of a generalized Birkhoffian system.The method of variation on parameters for the dynamical equations of a generalized Birkhoffian system is presented.The procedure for solving the problem can be divided into two steps:the first step,a system of auxiliary equations is constructed and its general solution is given;the second step,the parameters are varied,and the solution of the problem is obtained by using the properties of generalized canonical transformation.The method of variation on parameters for the generalized Birkhoffian system is of universal significance,and we take a nonholonomic system and a nonconservative system as examples to illustrate the application of the results of this paper.     

Analysis and optimization against buckling of beams interacting with elastic foundation

We consider an infinite continuous elastic beam that interacts with linearly elastic foundation and is under compression. The problem of the beam buckling is formulated and analyzed. Then the optimization of beam against buckling is investigated. As a design variable (control function) we take the parameters of cross-section distribution of the beam from the set of periodic functions and transform the original problem of optimization of infinite beam to the corresponding problem defined at the finite interval. All investigations are on the whole founded on the analytical variational approaches and the optimal solutions are studied as a function of problems parameters.     

Winkler弹性地基上梁的精化理论

将Cheng精化理论推广到winkler弹性地基上梁的研究当中,对winkler弹性地基上的梁进行了精确的分析,给出其精化理论。首先将板内的位移利用中面上位移及其沿梁厚方向的梯度表示出来,并获得梁内应力张量。再利用winkler弹性地基条件和Lur'e算子方法,获得弹性地基上梁的控制方程。若略去控制方程中的高阶项,与弹性地基上欧拉-伯努利梁的挠度控制方程一致。     

用无网格局部Petrov-Galerkin法分析非线性地基梁

利用无网格局部Petrov-Galerkin法求解了非线性地基梁。在Petrov-Galerkin方法中，采用移动最小二乘（MLS）近似函数作为场主量挠度的试函数并取移动最小二乘近似函数中的体验函数作为近似场函数的加权函数，采用罚因子法施加本质边界条件。文末给出了两个计算实例，算例的结果表明，Petrov-galerkin法不仅能成功地分析线性地基梁，而且也适用于解非线性地基梁，在分析非线性地基梁时具有收敛快，稳定性好的优点。     

A new semi-analytical method for the non-linear static analysis of an infinite beam on a non-linear elastic foundation: A general approach to a variable beam cross-section

We propose a new non-linear method for the static analysis of an infinite non-uniform beam resting on a non-linear elastic foundation under localized external loads. To this end, an integral operator equation is newly formulated, which is equivalent to the original differential equation of non-uniform beam. By using the integral operator equation, we propose a new functional iterative method for static beam analysis as a general approach to a variable beam cross-section. The method proposed is fairly simple as well as straightforward to apply. An illustrative example is presented to examine the validity of the proposed method. It shows that just a few iterations are required for an accurate solution.     

基于遗传算法的弹性地基加肋板肋梁无网格优化分析

基于遗传算法及一阶剪切理论, 提出一种弹性地基上加肋板肋条位置优化的无网格方法. 首先, 通过一系列点来离散平板及肋条, 并用弹簧模拟弹性地基, 从而得到加肋板的无网格模型; 其次, 基于一阶剪切理论及移动最小二乘近似原理导出位移场, 求出弹性地基加肋板总势能; 再次, 根据哈密顿原理导出结构的弯曲控制方程, 并通过完全转换法处理边界条件; 最后, 引入遗传算法和改进遗传算法, 以肋条的位置为设计变量、弹性地基板的中心点挠度最小值为目标函数, 对肋条位置进行优化达到地基板控制点挠度最小的目的. 以不同参数、载荷布置形式的弹性地基加肋板为例, 与ABAQUS有限元结果及文献解进行比较. 研究表明, 采用所提出的无网格模型可有效求解弹性地基上加肋板弯曲问题, 结果易收敛, 同时基于遗传算法与改进混合遗传算法所提出的无网格优化方法均可有效优化弹性地基加肋板肋条位置, 后者计算效率相对较高, 只进行了三次迭代便可获得稳定的最优解, 此外在优化过程中肋条位置改变时只需要重新计算位移转换矩阵, 又避免了网格重构.      

弹性地基双层梁理论下的混凝土路面力学分析

为建立混凝土路面结构受力分析计算模型,以Winkler弹性地基梁模型为基础,推导出了弹性地基双层梁理论的表达式;给定边界条件,利用MATLAB软件获得了无限长弹性地基梁在集中力作用下的挠度表达式。将混凝土路面结构简化为弹性地基上的双层梁,当车辆荷载作用于混凝土路面时,在集中载荷的作用下,建立了面层与基层的微分平衡方程。应用广义“初参数”法,得到了双层梁位移和应力的解析解。通过算例,对面层及基层的变形和应力进行了分析,结果表明:增大面层、基层的轴惯性矩和地基的弹性常数,可以有效地减少面层和基层的变形量,降低最大应力数值,但抗弯刚度对基层和面层的弯矩受力影响不大。最后将结果与ANSYS分析结果进行了比较,佐证了解的可靠性,研究结果可为混凝土路面结构设计提供依据。     

非均匀Winkler弹性地基上变厚度矩形板自由振动的DTM求解

针对非均匀Winkler弹性地基上变厚度矩形板的自由振动问题，通过一种有效的数值求解方法——微分变换法（DTM），研究其无量纲固有频率特性。已知变厚度矩形板对边为简支边界条件，其他两边的边界条件为简支、固定或自由任意组合。采用DTM将非均匀Winkler弹性地基上变厚度矩形板无量纲化的自由振动控制微分方程及其边界条件变换为等价的代数方程，得到含有无量纲固有频率的特征方程。数值结果退化为均匀Winker弹性地基上矩形板以及变厚度矩形板的情形，并与已有文献采用的不同求解方法进行比较，结果表明，DTM具有非常高的精度和很强的适用性。最后，在不同边界条件下分析地基变化参数、厚度变化参数和长宽比对矩形板无量纲固有频率的影响，并给出了非均匀Winkler弹性地基上对边简支对边固定变厚度矩形板的前六阶振型。     

基于无网格法的非均匀弹性地基上变厚度加筋板弯曲与固有频率分析

采用基于移动最小二乘近似的无网格方法并结合一阶剪切变形理论,分析了非均匀弹性地基上变厚度加筋板的弯曲和固有频率问题.首先,用节点对变厚度板和筋条分别进行离散,导出变厚度板和筋条的势能;其次,利用筋条与变厚度板之间的位移协调条件将筋条的节点参数转换为板的节点参数,再将两者的势能进行叠加得到变厚度加筋板的总势能,并根据能量法得到其动能;最后,利用最小势能原理及Hamilton原理分别得到弯曲控制方程与振动控制方程.由于采用的方法不能直接施加位移边界,故采用完全转换法处理位移边界.本文先计算变厚度板的弯曲及非均匀弹性地基板的固有频率,与文献对比验证方法的有效性;然后对非均匀弹性地基上变厚度加筋板弯曲与 自由振动进行了计算,并将计算结果与有限元结果进行了对比.结果表明,本文方法计算所得结果与文献解及有限元结果之间的误差均小于5％,验证了该方法在计算非均匀弹性地基上变厚度加筋板弯曲与固有频率问题的有效性.     

Natural vibration of a sandwich beam on an elastic foundation

The natural vibration of an elastic sandwich beam on an elastic foundation is studied. Bernoulli’s hypotheses are used to describe the kinematics of the face layers. The core layer is assumed to be stiff and compressible. The foundation reaction is described by Winkler’s model. The system of equilibrium equations is derived, and its exact solution for displacements is found. Numerical results are presented for a sandwich beam on an elastic foundation of low, medium, or high stiffness __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 57–63, May 2006.     

Structural modeling of beams on elastic foundations with elasticity couplings

A new simplified structural model and its governing equations for beams on elastic foundations with elastic coupling are proposed. This modeling system is simple but appropriate for the initial structural design of large-scale submerged floating-beam structures moored by tension legs spaced at uniform interval along the beam. The model is actually for beam on discrete elastic supports rather than on continuous elastic foundations. Therefore, the governing equations are based on finite difference calculus and solutions for beams on discrete elastic supports with elasticity coupling are also proposed. To clarify the applicability limit of the proposed model, the equivalence between a beam on discrete elastic supports and that on continuous elastic foundation is investigated by comparisons of characteristic solutions.