Bending tests to estimate the axial force in tie-rods

This paper presents a static method for the axial load identification of prismatic structural elements, with known geometric and elastic properties, which can be idealized as simply supported beams constrained by two end rotational springs. To this aim, the beam is subjected to an additional, transversal static force and the flexural displacements are measured at three given cross sections. Numerical and experimental tests are developed to validate the analytical procedure. In principle, use can be made of the proposed algorithm to evaluate both the axial force and the flexural stiffness coefficients of the end constraints. In fact, very good agreement is obtained between estimated and measured values of the axial force. Vice versa, the end stiffness identification gives reliable results for low values of the axial force only whereas, for all other cases, scattered and unreliable results are obtained.     

考虑轴力二阶效应悬臂梁的支承及裂纹参数识别

建立了轴向压力作用下悬臂裂纹梁边界支承和裂纹损伤程度识别方法．首先，将悬臂梁边界非完整支承等效为竖向和扭转弹簧、梁中开裂纹等效为内部扭转弹簧，利用Laplace变换，得到了边界弹性支承、考虑轴向压力二阶效应、具有任意裂纹数目Euler-Bernoulli悬臂梁弯曲挠度的解析解．其次，提出了边界弹性支承弹簧柔度和裂纹等效扭转弹簧柔度的识别方法．最后，通过数值试验，考察了轴向压力，裂纹深度以及测量误差等对识别结果的影响，说明了本文考虑轴向压力二阶效应的悬臂梁边界支承弹簧柔度及裂纹等效扭转弹簧柔度识别方法的适用性和可靠性，结果表明：相比于应变测量误差，挠度测量误差对裂纹损伤程度识别结果影响更加敏感，且轴向压力对裂纹损伤程度识别影响较小，因此，应严格控制挠度的测量误差．同时，边界支承扭转弹簧柔度的识别误差大于其竖向弹簧柔度识别误差．这些结果为实际工程中边界非完整支承悬臂裂纹梁的参数识别提供了指导．     

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.     

Nonlinear forced vibration analysis of postbuckled beams

A numerical solution methodology is proposed herein to investigate the nonlinear forced vibrations of Euler–Bernoulli beams with different boundary conditions around the buckled configurations. By introducing a set of differential and integral matrix operators, the nonlinear integro-differential equation that governs the buckling of beams is discretized and then solved using the pseudo-arc-length method. The discretized governing equation of free vibration around the buckled configurations is also solved as an eigenvalue problem after imposing the boundary conditions and some complicated matrix manipulations. To study forced and nonlinear vibrations that take place around a buckled configuration, a Galerkin-based numerical method is applied to reduce the partial integro-differential equation into a time-varying ordinary differential equation of Duffing type. The Duffing equation is then discretized using time differential matrix operators, which are defined based on the derivatives of a periodic base function. Finally, for any given magnitude of axial load, the pseudo -arc-length method is used to obtain the nonlinear frequencies of buckled beams. The effects of axial load on the free vibration, nonlinear, and forced vibrations of beams in both prebuckling and postbuckling domains for the lowest three vibration modes are analyzed. This study shows that the nonlinear response of beams subjected to periodic excitation is complex in the postbuckling domain. For example, the type of boundary conditions significantly affects the nonlinear response of the postbuckled beams.     

Nonlinear dynamic analysis of Timoshenko beams by BEM. Part I: Theory and numerical implementation

In this two-part contribution, a boundary element method is developed for the nonlinear dynamic analysis of beams of arbitrary doubly symmetric simply or multiply connected constant cross section, undergoing moderate large displacements and small deformations under general boundary conditions, taking into account the effects of shear deformation and rotary inertia. Part I is devoted to the theoretical developments and their numerical implementation and Part II discusses analytical and numerical results obtained from both analytical or numerical research efforts from the literature and the proposed method. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse loading and bending moments in both directions as well as to axial loading. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, to the axial displacement and to two stress functions and solved using the Analog Equation Method, a BEM based method. Application of the boundary element technique yields a nonlinear coupled system of equations of motion. The solution of this system is accomplished iteratively by employing the average acceleration method in combination with the modified Newton–Raphson method. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. The proposed model takes into account the coupling effects of bending and shear deformations along the member, as well as the shear forces along the span induced by the applied axial loading.

     

A shearable and thickness stretchable finite strain beam model for soft structures

Soft materials and structures have recently attracted lots of research interests as they provide paramount potential applications in diverse fields including soft robotics, wearable devices, stretchable electronics and biomedical engineering. In a previous work, an Euler–Bernoulli finite strain beam model with thickness stretching effect was proposed for soft thin structures subject to stiff constraint in the width direction. By extending that model to account for the transverse shear effect, a Timoshenko-type finite strain beam model within the plane-strain context is developed in the present work. With some kinematic hypotheses, the finite deformation of the beam is analyzed, constitutive equations are deduced from the theory of finite elasticity, and by employing the standard variational method, the equilibrium equations and associated boundary conditions are derived. In the limit of infinitesimal strain, the new model degenerates to the classical extensible and shearable elastica model. The corresponding incremental equilibrium equations and associated boundary conditions are also obtained. Based on the new beam model, analytical solutions are given for simple deformation modes, including uniaxial tension, simple shear, pure bending, and buckling under an axial load. Furthermore, numerical solution procedures and results are presented for cantilevered beams and simply supported beams with immovable ends. The results are also compared with the previously developed finite strain Euler–Bernoulli beam model to demonstrate the significance of transverse shear effect for soft beams with a small length-to-thickness ratio. The developed beam model will contribute to the design and analysis of soft robots and soft devices.     

Free vibration analysis of functionally graded material beams based on Levinson beam theory

Free vibration response of functionally graded material (FGM) beams is studied based on the Levinson beam theory (LBT). Equations of motion of an FGM beam are derived by directly integrating the stress-form equations of elasticity along the beam depth with the inertial resultant forces related to the included coupling and higherorder shear strain. Assuming harmonic response, governing equations of the free vibration of the FGM beam are reduced to a standard system of second-order ordinary differential equations associated with boundary conditions in terms of shape functions related to axial and transverse displacements and the rotational angle. By a shooting method to solve the two-point boundary value problem of the three coupled ordinary differential equations, free vibration response of thick FGM beams is obtained numerically. Particularly, for a beam with simply supported edges, the natural frequency of an FGM Levinson beam is analytically derived in terms of the natural frequency of a corresponding homogenous Euler-Bernoulli beam. As the material properties are assumed to vary through the depth according to the power-law functions, the numerical results of frequencies are presented to examine the effects of the material gradient parameter, the length-to-depth ratio, and the boundary conditions on the vibration response.     

Vibration and stability of an axially moving viscoelastic beam with hybrid supports

Vibration and stability are investigated for an axially moving beam constrained by simple supports with torsion springs. A scheme is proposed to derive natural frequencies and modal functions from given boundary conditions of an elastic beam moving at a constant speed. For a beam constituted by the Kelvin model, effects of viscoelasticity on the free vibration are analyzed via the method of multiple scales and demonstrated via numerical simulations. When the axial speed is characterized as a simple harmonic variation about the constant mean speed, the instability conditions are presented for axially accelerating viscoelastic beams in parametric resonance. Numerical examples show the effects of the constraint stiffness, the mean axial speed, and the viscoelasticity.     

饱和多孔弹性Timoshenko悬臂梁的拟静力弯曲

在经典单相Timoshenko梁变形和孔隙流体仅沿多孔梁轴向运动的假定下,基于不可压饱和多孔介质的三维理论,本文首先建立了横观各向同性饱和多孔弹性Timoshenko悬臂梁拟静力弯曲的一维数学模型,并给出了相应的边界条件。其次,利用Laplace变换及其数值逆变换,分析了端部不同渗透条件下,饱和多孔弹性Timoshenko悬臂梁在端部梯载荷作用下的拟静力响应,给出了饱和多孔Timoshenko悬臂梁弯曲时挠度、弯矩以及孔隙流体压力等效力偶等随时间的响应曲线,并与饱和多孔Euler-Bernoulli悬臂梁的响应进行了比较,考察了梁长细比对弯曲的影响。数值结果表明：固相骨架与孔隙流体的相互作用具有粘性效应,梁弯曲的拟静态挠度具有蠕变行为,端部渗透条件对梁的弯曲响应有显著的影响,并且,饱和多孔弹性Timoshenko悬臂梁的拟静态响应亦存在Mandel-Cryer现象。     

Subharmonic and ultra-subharmonic response of nonlinear elastic beams subjected to harmonic excitation

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求解弹性梁的普遍化方法

介绍了一种求解弹性梁的新方法．该方法利用奇异函数与拉普拉斯变换相结合的方法导出弹性梁弯曲变形的普遍表达式，并利用边界条件确定约束力，对具有任意支承形式、受力状况和阶梯形状的静定或超静定梁具有普适性．     

Non-linear flexural–torsional dynamic analysis of beams of arbitrary cross section by BEM

In this paper, a boundary element method is developed for the non-linear flexural–torsional dynamic analysis of beams of arbitrary, simply or multiply connected, constant cross section, undergoing moderately large deflections and twisting rotations under general boundary conditions, taking into account the effects of rotary and warping inertia. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse loading in both directions as well as to twisting and/or axial loading. Four boundary value problems are formulated with respect to the transverse displacements, to the axial displacement and to the angle of twist and solved using the Analog Equation Method, a BEM based method. Application of the boundary element technique leads to a system of non-linear coupled Differential–Algebraic Equations (DAE) of motion, which is solved iteratively using the Petzold–Gear Backward Differentiation Formula (BDF), a linear multistep method for differential equations coupled to algebraic equations. The geometric, inertia, torsion and warping constants are evaluated employing the Boundary Element Method. The proposed model takes into account, both the Wagner's coefficients and the shortening effect. Numerical examples are worked out to illustrate the efficiency, wherever possible the accuracy, the range of applications of the developed method as well as the influence of the non-linear effects to the response of the beam.     

轴向力作用任意不连续梁自由振动分析的广义函数解

摘要：首先运用分布理论建立了轴向力作用下含多个不连续点的欧拉梁的自由振动的统一微分方程。不连续点的影响由广义函数（Dirac delta函数）引入梁的振动方程。微分方程运用Laplace变换方法求解;与传统方法不同的是,本文方法适用于含任意类型的不连续点和多种不连续点组合情况的梁,求得的模态函数为整个不连续梁的一般解。由于模态函数的统一化以及连续条件的退化,特征值的求解得到了极大的简化。最后,以轴向力作用下多跨梁—弹簧质量块系统模型为例验证了本文方法的正确性与有效性。     

THE STABILITY OF AN AXIALLY ACCELERATING BEAM ON SIMPLE SUPPORTS WITH TORSION SPRINGS

The axially moving beams on simple supports with torsion springs are studied. The general modal functions of the axially moving beam with constant speed have been obtained from the supporting conditions. The contribution of the spring stiffness to the natural frequencies has been numerically investigated. Transverse stability is also studied for axially moving beams on simple supports with torsion springs. The method of multiple scales is applied to the partialdifferential equation governing the transverse parametric vibration. The stability boundary is derived from the solvability condition. Instability occurs if the axial speed fluctuation frequency is close to the sum of any two natural frequencies or is two fold natural frequency of the unperturbed system. It can be concluded that the spring stiffness makes both the natural frequencies and the instability regions smaller in the axial speed fluctuation frequency-amplitude plane for given mean axial speed and bending stiffness of the beam.     

A realistic estimation of the effective breadth of ribbed plates

In this paper a realistic estimation of the effective breadth of a stiffened plate is presented. For the estimation of the effective breadth the adopted model contrary to the models used previously takes into account the resulting inplane forces and deformations of the plate as well as the axial forces and deformations of the beam, due to combined response of the system. The analysis consists in isolating the beams from the plate by sections parallel to the lower outer surface of the plate. The forces at the interface, which produce lateral deflection and inplane deformation to the plate and lateral deflection and axial deformation to the beam, are established using continuity conditions at the interface. The solution of the arising plate and beam problems, which are nonlinearly coupled, is achieved using the analog equation method. After the solution of the plate––beams system is achieved, the distribution of the axial stresses across the plate, resulting from both the bending and the inplane action of the plate, is obtained. Integrating this distribution across the plate the values of the effective breadth are obtained. The influence of these values from the beam stiffness and their variation along the longitudinal direction of the plate are shown as compared with those obtained from various codes through numerical examples with great practical interest.     

Euler–Bernoulli beams with multiple singularities in the flexural stiffness

Euler–Bernoulli beams under static loads in presence of discontinuities in the curvature and in the slope functions are the object of this study. Both types of discontinuities are modelled as singularities, superimposed to a uniform flexural stiffness, by making use of distributions such as unit step and Dirac's delta functions. A non-trivial generalisation to multiple different singularities of an integration procedure recently proposed by the authors for a single singularity is presented in this paper. The proposed integration procedure leads to closed form solutions, dependent on boundary conditions only, which do not require enforcement of continuity conditions along the beam span. It is however shown how, from the solution of the clamped-clamped beam, by considering suitable singularities at boundaries in the flexural stiffness model, responses concerning several boundary conditions can be recovered. Furthermore, solutions in terms of deflection of the beam are obtained for imposed displacements at boundaries providing the so called shape functions. The above mentioned shape functions can be adopted to insert beams with singularities as frame elements in a finite element discretisation of a frame structure. Explicit expressions of the element stiffness matrix are provided for beam elements with multiple singularities and the reduction of degrees of freedom with respect to classical finite element procedures is shown. Extension of the proposed procedure to beams with axial displacement and vertical deflection discontinuities is also presented.     

Exact buckling solution for two-layer Timoshenko beams with interlayer slip

This paper deals with the buckling behavior of two-layer shear-deformable beams with partial interaction. The Timoshenko kinematic hypotheses are considered for both layers and the shear connection (no uplift is permitted) is represented by a continuous relationship between the interface shear flow and the corresponding slip. A set of differential equations is obtained from a general 3D bifurcation analysis, using the above assumptions. Original closed-form analytical solutions of the buckling load and mode of the composite beam under axial compression are derived for various boundary conditions. The new expressions of the critical loads are shown to be consistent with the ones corresponding to the Euler–Bernoulli beam theory, when transverse shear stiffnesses go to infinity. The proposed analytical formulae are validated using 2D finite element computations. Parametric analyses are performed, especially including the limiting cases of perfect bond and no bond. The effect of shear flexibility is particularly emphasized.     

Elasticity solutions for plane anisotropic functionally graded beams

This paper considers the plane stress problem of generally anisotropic beams with elastic compliance parameters being arbitrary functions of the thickness coordinate. Firstly, the partial differential equation, which is satisfied by the Airy stress function for the plane problem of anisotropic functionally graded materials and involves the effect of body force, is derived. Secondly, a unified method is developed to obtain the stress function. The analytical expressions of axial force, bending moment, shear force and displacements are then deduced through integration. Thirdly, the stress function is employed to solve problems of anisotropic functionally graded plane beams, with the integral constants completely determined from boundary conditions. A series of elasticity solutions are thus obtained, including the solution for beams under tension and pure bending, the solution for cantilever beams subjected to shear force applied at the free end, the solution for cantilever beams or simply supported beams subjected to uniform load, the solution for fixed–fixed beams subjected to uniform load, and the one for beams subjected to body force, etc. These solutions can be easily degenerated into the elasticity solutions for homogeneous beams. Some of them are absolutely new to literature, and some coincide with the available solutions. It is also found that there are certain errors in several available solutions. A numerical example is finally presented to show the effect of material inhomogeneity on the elastic field in a functionally graded anisotropic cantilever beam.     

The influence of the axial force on the vibration of the Euler–Bernoulli beam with an arbitrary number of cracks

The exact closed-form solution for the vibration modes and the eigen-value equation of the Euler–Bernoulli beam-column in the presence of an arbitrary number of concentrated open cracks is proposed. The solution is provided explicitly as functions of four integration constants only, to be determined by the standard boundary conditions. The enforcement of the boundary conditions leads the exact evaluation of the vibration frequencies as well as the buckling load of the beam-column and the corresponding eigen-modes. Furthermore, the presented solution allows a comprehensive evaluation of the influence of the axial load on the modal parameters of the beam. The cracks, which are not subjected to the closing phenomenon, are modelled as a sequence of Dirac’s delta generalised functions in the flexural stiffness. The eigen-mode governing equation is formulated over the entire domain of the beam without enforcement of any further continuity condition. The influence of the axial load on the vibration modes of beam-columns with different number and position of cracks, under different boundary conditions, has been analysed by means of the proposed closed-form expressions. The presented parametric analysis highlights some abrupt changes of the eigen-modes and the corresponding frequencies.     

CLASSICAL AND HOMOGENIZED EXPRESSIONS FOR BUCKLING SOLUTIONS OF FUNCTIONALLY GRADED MATERIAL LEVINSON BEAMS

The relationship between the critical buckling loads of functionally graded material(FGM) Levinson beams(LBs) and those of the corresponding homogeneous Euler-Bernoulli beams(HEBBs) is investigated. Properties of the beam are assumed to vary continuously in the depth direction. The governing equations of the FGM beam are derived based on the Levinson beam theory, in which a quadratic variation of the transverse shear strain through the depth is included.By eliminating the axial displacement as well as the rotational angle in the governing equations,an ordinary differential equation in terms of the deflection of the FGM LBs is derived, the form of which is the same as that of HEBBs except for the definition of the load parameter. By solving the eigenvalue problem of ordinary differential equations under different boundary conditions clamped(C), simply-supported(S), roller(R) and free(F) edges combined, a uniform analytical formulation of buckling loads of FGM LBs with S-S, C-C, C-F, C-R and S-R edges is presented for those of HEBBs with the same boundary conditions. For the C-S beam the above-mentioned equation does not hold. Instead, a transcendental equation is derived to find the critical buckling load for the FGM LB which is similar to that for HEBB with the same ends. The significance of this work lies in that the solution of the critical buckling load of a FGM LB can be reduced to that of the HEBB and calculation of three constants whose values only depend upon the throughthe-depth gradient of the material properties and the geometry of the beam. So, a homogeneous and classical expression for the buckling solution of FGM LBs is accomplished.