Predicting buckling loads from vibration data

There are analytical methods for predicting the buckling loads of columns with the boundaries ideally fixed, i.e., simply supported or built-in, or partially fixed. Vibration-test results may furnish a practical method of measuring the fixity. In this investigation a beam, that may or may not be loaded as a column, is assumed to have a torsional spring at each end such that a zero torsional stiffness corresponds to a simply supported end and an infinite torsional stiffness corresponds to a built-in end. From a Rayleigh-Ritz analysis, the buckling load and the fundamental frequency of the beam are each computed as a function of the torsional stiffness. This procedure leads to a one-to-one nondimensional relationship between the buckling load and the natural frequency. From these calculations, it is seen that regardless of the degree of clamping of one end relative to the other end, all that is needed to predict the buckling load within a 15-percent range is a knowledge of the theoretical buckling load of the simply supported column; the theoretical fundamental frequency of the simply supported beam; and the experimental fundamental frequency. Experimental results are presented to support the theory.     

Dynamic analysis to infinite beam under a moving line load with uniform velocity

I.IntroductionAninfinitebeamonelasticfoundationnotonlycanbelookedonasdynamicmodelforaSuspensionbridgeoratensiondiagonalbridgel']butalsocanbeusedindynamicsanalysistoarailtrack,therefore,dynamicresponseofinfinitebeamundermovillgloadhasbecomethefocusofdiscussioninpastseveraldecades.TheproblemalsohasbeeninvestigatedbyTimoshenko12j,Frybal'],Steelel'],Lee15],etal.Wefind,however,thatallofthestudiesonthissubjectonlydiscussedmovingpointloadproblem.Actually,eitherautomobileloadoratrainloadisalinedist…     

Bending of a poroelastic beam with lateral diffusion

Bending an elastic beam leads to a complicated 3D stress distribution, but the shear and transverse stresses are so small in a slender beam that a good approximation is obtained by assuming purely uniaxial stress. In this paper, we demonstrate that the same is true for a saturated poroelastic beam. Previous studies of poroelastic beams have shown that, to satisfy the Beltrami–Michell compatibility conditions, it is necessary to introduce either a normal transverse stress or shear stresses in addition to the bending stress. The problem is further complicated if lateral diffusion is permitted. In this study, a fully coupled finite element analysis (FEA) incorporating the lateral diffusion effect is presented. Results predicted by the “exact” numerical solution, including load relaxation, pore pressure, stresses and strains, are compared to an approximate analytical solution that incorporates the assumptions of simple beam theory. The applicability of the approximate beam-bending solution is investigated by comparing it to FEA simulations of beams with various aspect ratios. For “beams” with large width-to-height ratios, the Poisson effect causes vertical deflections that cannot be neglected. It is suggested that a theory of plate bending is needed in the case of poroelastic media with large width-to-height ratios. Nevertheless, use of the approximate solution yields very small errors over the range of width-to-height ratios (viz., 1–4) explored with FEA.     

The effect of transverse normal strain in contact of an orthotropic beam pressed against a circular surface

The contact problem of a straight orthotropic beam pressed onto a rigid circular surface is considered using beam theories that account for transverse shear and transverse normal deformations. The circular nature of the rigid surface emphasizes the difference between Euler Bernoulli theory behavior, where point loads develop at the edge of contact, and the higher order theories that predict non-singular pressure distributions. While Timoshenko beam theory is the simplest theory that addresses this behavior, the prediction of a maximum value of pressure at the edge of contact contradicts the elasticity theory result that contact pressure must drop to zero. Transverse normal strain is therefore introduced, both to study this fundamental discrepancy and to include an important effect in many contact problems. To investigate this effect, higher order beam theories that account for both constant and linear transverse normal strain through the beam thickness are derived using the principle of virtual work. The resulting orthotropic beam theories depend on the bending stiffness (EI), shear stiffness (GA), axial stiffness (EA₁) and transverse normal stiffness (EA₂), which are independent stiffness parameters that can differ by orders of magnitude. The above mentioned contact problem is then solved analytically for these theories, along with the Timoshenko beam model which assumes zero transverse normal strain. The results for different orthotropic materials show that inclusion of transverse normal deformation has a significant effect on the contact pressure solution. Furthermore, the solution using higher order beam theories encompasses the two extremes of a Hertz-like contact pressure when the half contact length is smaller than the thickness of the beam, and the Timoshenko beam theory case when the half contact length is much larger than the thickness. Concerning the behavior of the pressure at the edge of contact, adherence to the boundary conditions required by the principle of virtual work, shows that while the pressure does tend to zero, it does not become zero unless artificially enforced. In this regard the solution for the case of linear strain is better than that for constant strain. All beam solutions are validated with plane elasticity solutions obtained using the commercial finite element software ABAQUS.     

Experimental and numerical investigations of the responses of a cantilever beam possibly contacting a deformable and dissipative obstacle under harmonic excitation

In this paper, the dynamics of a cantilever beam subjected to harmonic excitations and to the contact of an obstacle is studied with the help of experimental and numerical investigations. The steel flexible structure is excited close to the free end with a shaker and may come into contact with a deformable and dissipative obstacle. A technique for modeling contact phenomena using piece-wise linear dynamics is applied. A finite-dimensional modal model is developed through a Galerkin projection. Concentrated masses, dampers and forces are considered in the equations of motion in such a way that the boundary conditions are those of a cantilever beam. Numerical studies are conducted by assuming finite-time contact duration to investigate the frequency response of the impacted beam for different driving frequencies. Experimental results have been extrapolated through a displacement laser sensor and a load cell. The comparison between numerical and experimental results show many qualitative and quantitative similarities.The novelty of this paper can be synthetized in (a) the development of experimental results that are in good agreement with the numerical implementation of the introduced model; (b) the development of a comprehensive contact model of the beam with an unilateral, deformable and dissipative obstacle located close to the tip; (c) the possibility of accounting for higher modes for the cantilever beam problem, and hence of analyzing how the response varies when moving the excitation (and/or the obstacle) along the beam, and of investigating the effect of the linearly elastic deformability of the built‐in end of the beam; (d) an easy and intuitive solution to the problem of accounting for spatially singular masses, dampers, springs and forces in the motion equations; (e) the possibility of accounting for finite gap and duration of the contact between beam and obstacle.     

梁长对移动荷载下梁响应影响研究

本文研究了梁跨长对移动荷载下梁稳态响应的影响.在以往的研究中,通常采用Galerkin方法或者有限元方法计算不同长度的梁的动力响应.当梁长的增加不再明显改变梁的动力响应时,可认为梁此时的长度可以代替无穷长时的情况.采用上述方法的研究表明,当梁长大于10 m时,就可以用有限长梁近似无限长梁的响应.本文通过求解有限长梁和无...     

Closed-form solution of a shear deformable,extensional ring in contact between two rigid surfaces

Contact of a circular ring with a flat, rigid ground is considered using curved beam theory and analytical methods. Applications include tires, springs, and stiffeners, among others. The governing differential equations are derived using the principle of virtual work and the formulation includes deformations due to bending, transverse shear and circumferential extension. The three associated stiffness quantities, EI, GA and EA, respectively, remain as independent parameters in the differential equations. This allows the special cases such as an inextensible Timoshenko beam (EI and GA) or an extensible Euler beam (EI and EA) to be obtained directly by the appropriate limits. The effect of these three stiffness parameters on the contact pressure solution is studied, which shows how those fundamental parameters can be selected for the purpose of the application. Although the formulation is for small displacement theory, both radial and circumferential distributed loads are considered, which allows the pressure in the deformed state to be vertical rather than radial, which is shown to be important. Closed form expressions for all force and displacement quantities are obtained in terms of the angular location of the edge of contact, which must be determined numerically. Extensibility complicates the analytical expressions within the contact region, and a series solution is proposed in this case. A two-term asymptotic expression for the stiffness of the ring is determined analytically. Finally, all solutions are validated using the commercial finite element software ABAQUS, with attention to non-linear behavior and the range of validity of these solutions.     

Static analysis of an infinite beam resting on a tensionless Pasternak foundation

This paper addresses the static response of an infinite beam supported on a unilateral (tensionless) two-parameter Pasternak foundation and subjected to complex transverse loads, including self weight. The transfer displacement function method (TDFM) is employed to determine the initially unknown lengths that remain in contact. In contrast to a Winkler Foundation System (WFS), the lift-off points in a PFS (Pasternak Foundation System) are not necessarily at zero displacement but may be determined sequentially through considering the compatibility conditions at the junctions of contact and non-contact segments. After the response of the whole system including the beam and foundation is expressed through the displacement constants of the initial segment, the contact problem is reduced to two nonlinear algebraic equations with two unknowns. The foundation reactions and the internal actions of the beam may also be determined from the displacement response of the system. Two simple cases are solved to illustrate the influence of the foundation stiffness factors and finally, a third example of a beam with several contact segments is presented to demonstrate the application of the TDFM.     

高速荷载下多孔饱和地基的动力响应

研究高速荷载作用下梁与多孔饱和半空间的动力响应。由Fourier变换求解多孔饱和固体的动力基本方程，根据梁与半空间的接触条件得出多孔饱和半空间上梁的垂直位移的表达式。文中的数值算例考虑了荷载移动速度对梁的动力位移的影响，并与相应的弹性半空间问题作了对比。从算例中可以发现荷载移动速度对动力位移有很大的影响，当移动速度与半空间的表面波速相近时，地面会当产生很大的振动，同时还发现当速度大于介质的剪切波速时，多孔饱和半空间上梁的动力响应与弹性半空间上梁的动力响应有很大的差别。     

FREE VIBRATION ANALYSIS OF SIMPLY SUPPORTED BEAM WITH BREATHING CRACK USING PERTURBATION METHOD

In this paper, a new analytical method for vibration analysis of a cracked simply supported beam is investigated. By considering a nonlinear model for the fatigue crack, the governing equation of motion of the cracked beam is solved using perturbation method. The solution of the governing equation reveals the superhaxmonics of the fundamental frequency due to the nonlinear effects in the dynamic response of the cracked beam. Furthermore, considering such a solution, an explicit expression is also derived for the system damping changes due to the changes in the crack parameters, geometric dimensions and mechanical properties of the cracked beam. The results show that an increase in the crack severity and approaching the crack location to the middle of the beam increase the system damping. In order to validate the results, changes in the fundamental frequency ratios against the fatigue crack severities are compared with those of experimental results available in the literature. Also, a comparison is made between the free response of the cracked beam with a given crack depth and location obtained by the proposed analytical solution and that of the numerical method. The results of the proposed method agree with the experimental and numerical results.     

隧道开挖影响下地层-基础体系的接触力学响应分析

针对隧道正交下穿既有结构施工力学响应的预测问题, 建立了考虑多体接触作用的隧道施工扰动下地层?基础体系力学响应解析预测方法. 该方法将地层视为均匀各向同性的线弹性体, 通过引入接触理论考虑地层与基础间的接触作用, 并提出“隧道开挖与基础作用换序求解”的新解析思路,确定了最终状态接触压力, 解决了隧道开挖及多体接触耦合作用下接触压力难以确定的问题, 进而依据弹性力学解的叠加性获得目标问题的解析解答. 通过对比该解析解与ABAQUS数值解, 发现两者吻合良好. 基于本方法开展参数分析, 研究了地层参数、隧道埋深、隧道边界径向位移以及外荷载集度对地表竖向附加位移、接触压力和基础内力分布的影响规律. 结果表明: 本方法可准确预测地层?基础体系的接触力学响应, 实现了地层与基础间接触力学行为的量化描述; 地层杨氏模量和泊松比对地层?基础体系力学响应的影响分别侧重于变形和受力, 而隧道埋深和隧道边界径向位移变化对受力变形均有较大影响; 地层位移受隧道开挖扰动与多体接触效应的耦合作用, 且接触影响范围局限在接触区域附近; 隧道开挖使接触压力产生“中间释放、端部集中”的重分布现象, 并由此造成基础内力的大幅增长. 当开挖扰动剧烈时, 甚至产生竖向位移不连续的脱空接触现象. 研究成果对城市浅埋隧道施工影响下地层?基础体系力学响应预测具有重要的理论意义和应用价值.      

Analytical estimates of the contact zone area for a pressurized flat-oval cylindrical shell placed between two parallel rigid plates

The paper presents an analytical estimate for an area of contact for a thin-walled noncircular cylindrical shell placed between two parallel rigid plates with an initial gap and then pressurized hydrostatically up to the contact appearance. The Euler–Bernoulli beam is used to model the shell deformation under the plane strain assumption. Such a simplification allows one to obtain the simplest closed-form estimate for the contact zone area. The first approximation is obtained neglecting the deformation of the curvilinear segments of the flat oval shell cross-section while the solution for the curved beam loaded by the homogeneous pressure is considered as a second approximation. The accuracy of the proposed analytical solutions as well as their usability in the preliminary design of thin-walled elements of various cooling systems is validated by the results of both numerical simulations and experimental tests.     

Response of an infinite beam resting on a tensionless elastic foundation subjected to arbitrarily complex transverse loads

This paper presents an investigation into the static response of an infinite beam supported on a unilateral (tensionless) elastic foundation and subjected to arbitrary complex loading, including self-weight. A new numerical method is developed to determine the initially unknown lengths that remain in contact. Based on the continuity conditions at the junctions of contact and non-contact segments, the response of the whole beam may be expressed through the displacement constants of the initial segment, reducing the contact problem to two nonlinear algebraic equations with two unknowns. The technique has been named the transfer displacement function method (TDFM). Comparison with the exact results of a particular limiting case shows the expected complete agreement. Finally, an example of a beam with several contact segments is presented and verified by the application of equilibrium conditions.     

Transient responses of beams and plates subject to travelling load. Miscellaneous results

The steady-state response of a free and infinite Timoshenko beam is specified analytically in terms of non-dimensional displacements and stresses. The beam is supposed loaded by a travelling concentrated force or a moving step load. By a validated explicit numerical calculation, it is shown how a load travelling on a beam at constant velocity, from defined time and abscissa, generates a response which evolves towards the steady-state solution for a part, and towards a quantified transient solution for another part. Asymptotic values are given for the transient displacements and stresses according to the time and the speed of the loading. The solution is also found for a plate subject to a pressure, which spreads respecting the cylindrical symmetry. It is possible to identify in the response a part which follows the pressure front, and which is comparable with the steady-state response of a beam, and another transient part, which generates displacements and stresses with a much less oscillating character. An asymptotic solution is also presented for the plate.The whole series of the results makes it possible to better understand qualitatively the beginning of the transient response of a beam or of a plate to a moving load, and also makes it possible to estimate the stresses and displacements without needing specialised numerical codes.     

移动载荷作用下沥青路面稳态响应与计算

为了研究移动载荷作用下沥青路面稳态响应,将沥青路面看作为作用在Kelvin黏弹性地基上不仅具有弹性而且具有黏滞性的无限长梁,建立了梁的稳态响应数学模型. 找到了稳态响应与瞬态响应的关系,由瞬态响应解析解得到了稳态响应近似解析解,并进行了实例计算.从而解决了稳态数学模型难以求解和计算的问题. 计算结果表明,依所给的道路参数和车辆载荷,只要响应位置距初始位置一定距离,移动载荷作用下沥青路面的瞬态响应即为稳态响应.     

Smooth static and dynamic indentation of a cantilever beam

The static and dynamic indentation of structural elements such as beams and plates continue to be intriguing problems, especially for scenarios where large area contacts are expected to occur. Standard methods of indentation analyses use a beam theory solution to obtain an overall load–displacement relationship and then a Hertzian contact solution to calculate local stresses under the indenter. However, these techniques are only applicable in a fairly limited class of problems: the stress distribution in the contact region will differ significantly from a Hertzian one when the contact length exceeds the thickness of the beam. The indentation models developed herein are improvements over existing GLOBAL/LOCAL models for static and dynamic indentation of cantilever beams. Maximum contact stresses, beam displacements, and contact force time histories are obtained and compared with the predictions of current static and dynamic indentation models. The validity of the solutions presented herein is further assessed by comparing the results obtained to the predictions of modified beam theory solutions.     

Tensionless contact of a finite beam: Concentrated load inside and outside the contact zone

For a finite beam with a nonzero gap distance, an asymmetric concentrated load can be either inside or outside of the contact zone. A new governing equation is given for the case of a concentrated load outside the contact zone. By numerically solving the left-side and right-side contact lengths of the beam, a criterion is established to determine whether the concentrated load is inside or outside the contact zone. A more general approach on the tensionless contact of a beam is thus presented.     

考虑轴力二阶效应的损伤梁弯曲摄动解

将损伤梁等效为阶梯型变刚度Euler-Bernoulli梁,利用Heaviside广义函数,给出了阶梯型变刚度梁抗弯刚度的统一表达式．在此基础上,考虑轴向压力二阶效应,并以损伤为摄动参数,得到了均布横向载荷作用下,简支损伤梁弯曲挠度的一阶和二阶摄动解析解,并数值分析了摄动解析解的精度和损伤梁的弯曲变形特性,结果表明：随着轴向压力和刚度损伤参数的增加,挠度一阶和二阶摄动解析解误差增加,挠度二阶摄动解析解误差通常小于其一阶摄动解析解误差,且二阶摄动解的误差很小,满足工程应用的精度．同时,损伤梁的挠度和转角分布与完整梁的挠度和转角分布差异较大,在刚度变化位置处损伤梁转角斜率存在突变．这些结果可为轴力作用下Euler-Bernoulli梁损伤识别提供理论支撑．     

General solution for dynamical problem of infinite beam on an elastic foundation

Based on the theory of Euler-Bernoulli beam and Winkler assumption for elasticfoundation,a mathematical model is presented.By using Fourier transformation for spacevariable,Laplace transformation for time variable and convolution theorem for theirinverse transformations,a general solution for dynamical problem of infinite beam on anelastic foundation is obtained.Finally,the cases of free vibration,impulsive response andmoving load are also discussed.     

固定接触界面法向静弹性刚度

基于Hertz接触理论推导了两个微凸体之间互相作用的法向接触静弹性刚度.根据修正后的一个微接触点的平截面积尺寸分布,给出了界面的总法向接触静弹性条件刚度、总条件载荷的解析解.将法向静弹性刚度的解析解嵌入到有限元软件中,获得整机的理论模态.通过实验对解析解进行了定量验证.以机床结合部为研究对象,在理论振型与实验振型一致的...