Investigation of the Nonlinear Behavior of Beams*

ABSTRACT A system of equations that describes the nonlinear behavior of teams is presented. The differential equations are solved by using the Galerkin method, and a system of nonlinear algebric equations is obtained. A method to deal with discontinuities in structural properties and load distribution along a beam is presented. The derivation includes general expressions for gravity-type loads. Two methods of calculating the resultant moment along the beam are described. One method incorporates differentiation of the expressions for displacements, while the second method is based on integration of loads along the beam. The numerical model is used in order to investigate displacements and moment resultants along a cantilevered beam. Different problems that are associated with nonlinear behavior are presented and discussed. Good agreement is obtained between theoretical and experimental results.     

Experimental investigation of transient flexural waves in beams with discontinuities of cross section

There has been little experimental work on flexural wave propagation in general and on flexural wave propagation in beams with discontinuities of cross section in particular. Experimental data are obtained for various test beams subjected to eccentric longitudinal impact. The bending strain versus time results are presented for several positions along a uniform beam and finite beams (of circular cross section) with discontinuities of cross section. Bending strain histories are recorded at several positions before and after the discontinuity. The effect of reflections on the propagated flexural wave is illustrated. The dispersion of the traveling flexural wave caused several alternating peaks within the duration of the original positive input pulse. The importance of investigating discontinuities of cross section in structures subjected to impact loading is clearly manifested.     

旋转SMA纤维混杂复合材料薄壁梁的自由振动

研究具有SMA主动纤维的旋转复合材料单闭室薄壁截面梁的耦合自由振动问题.基于Hamilton原理并结合SMA纤维复合材料薄壁梁的二维截面内力(矩)与位移(转角)关系方程,导出旋转单闭室截面薄壁复合材料梁的1D耦合自由振动分析模型.该模型还考虑薄壁梁调矩角和预锥角的作用.采用Galerkin法求解振动模型,获得梁耦合振动...     

Dynamic stability of pre-twisted beams with non-constant spin rates under axial random forces

This paper investigates the dynamic stability of a pre-twisted cantilever beam spinning along its longitudinal axis with a periodically varying speed and acted upon by an axial random force at the free end. The spin rate of the beam is characterized as a small periodic perturbation superimposed on a constant speed, and the axial force is assumed as the sum of a static force and a weakly stationary random process with a zero mean. Both the periodically varying spin rate and the axial random force may lead to parametric instability of the beam. In this work, the finite element method is applied first to get rid of the dependence on the spatial coordinate. The method of stochastic averaging is then adopted to obtain Ito’s equations for the system response under different resonant frequency combinations. Finally, the first-moment and the second-moment stability conditions of the beam are derived explicitly. Numerical results are presented for a simple harmonic speed perturbation and a Gaussian white noise axial force.     

Development, modeling and deflection analysis of hybrid micro actuator with integrated thermal and piezoelectric actuation

Micro actuators are irreplaceable part of motion control in minimized systems. The current study presents an analytical model for a new Hybrid Thermo Piezoelectric micro actuator based on the combination of piezoelectric and thermal actuation mechanisms. The micro actuator structure is a double PZT cantilever beam consisting of two arms with different lengths. The presented micro actuator uses the structure of electrothermal micro actuator in which polysilicon material is replaced by PZT. Also the voltage and poling directions are considered in the lengthwise of PZT beams. As a result, the piezoelectric actuation mechanism is based on d ₃₃ strain coefficient. The tip deflection of micro actuator is obtained using Timoshenko beam theory. Analytical results are compared with FEM results along with other reported results in the literature. The effects of geometrical parameters and PZT material constants on actuator tip deflection are studied to provide an efficient optimization of HTP micro actuator.     

轴向扩散下简支饱和多孔弹性梁的大挠度分析

基于多孔介质理论和弹性梁的大挠度理论,并考虑轴向变形,在孔隙流体仅沿轴向扩散的假设下,建立了微观不可压饱和多孔弹性梁大挠度弯曲变形的一维非线性数学模型.在此基础上,忽略饱和多孔弹性梁的轴向应变,并利用Galerkin截断法,研究了两端可渗透的简支饱和多孔弹性梁在突加横向均布载荷作用下的拟静态弯曲,给出了饱和多孔梁弯曲时挠度、弯矩和轴力以及孔隙流体压力等效力偶等沿轴线的分布曲线.揭示了大挠度非线性和小挠度线性模型的结果差异,指出大挠度非线性模型的结果小于相应小挠度线性模型的结果,并且这种差异随着载荷的增大而增大.计算表明:当无量纲载荷参数q>5时,应该采用大挠度非线性数学模型进行研究.     

斜拉桥主梁模型承载性能试验研究

通过对斜拉桥主梁模型在不同工况及施加横向预应力方案的电测试验,得出了主梁的应力分布规律。试验结果表明,箱梁的撑杆和横向预应力的设计,对箱梁顶板的应力状态有很大影响。施加斜索力后,箱梁顶板的横向拉应力均小于1MPa,除横向预应力作用区域外,箱梁顶板的纵向应力为均匀分布的压应力,若在所有撑杆截面施加横向预应力,则撑杆只出现压应力。文中的试验数据可为优化结构、不同设计方案的选择提供理论根据。同时设计了对模型施加预应力的方法,解决了模型预应力的模拟问题。     

An experimental study of bending impact waves in beams

An experimental study of the propagation of flexural waves in an elastic beam of circular cross section subjected to an approximate-step-function bending moment is given. The test beam was a low-carbon-steel bar 2 in. diam × 30-ft long and was suspended in a vertical position by a pin located near the upper end. The step moment was applied at the upper pinned end of the bar by an arrangement of two high-pressure, nitrogen-operated cylinders. The strains were measured with strain gages located at eight stations along the bar. The experimental results are compared with results obtained from a solution of the “Timoshenko” beam by W. Flügge and E. E. Zajac for a semi-infinite pinned-end beam subjected to a suddenly applied bending moment at the pinned end. The experimental results are correlated with the predictions of the theoretical solution.     

Shape Monitoring of a Beam Structure from Measured Strain or Curvature

This paper presents results of numerical simulations and an experimental investigation of a method to determine shape of a beam from curvature and/or strain measurements. The purpose of this work was to develop an efficient and accurate method that can be used in real time shape monitoring of beam type structures with possible extension to more complex systems. A method based upon solving a set of continuity equations is presented. Numerical simulations were implemented to minimize the number of sensors and to determine the most beneficial sensor locations and sensor/model configuration to capture the shape in a timely and effective manner. Simulations showed that dividing the beam into segments (elements) and placing sensors at the Gauss point locations of each segment gave only 0.14% systematic error while using three elements and two Gauss points per element. An experiment was designed using an aluminum beam combined with a data acquisition system and a shape reconstruction algorithm. The real-time reconstruction of shape from curvature data was accomplished using strain gages for the curvature estimates. The results were compared to a technique based on position only data and point cloud image data. Overall, consistent results were obtained. The percent difference between the experimental and photogrammetry results fluctuated from 1.4 to 3.5% when various magnitudes of concentrated loads were applied to the beam. This methodology may be useful in real-time shape control and shape modification systems with potential applications in structural health monitoring and damage detection.     

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.     

Large deflection of curved elastic beams made of Ludwick type material

The large deflection of an axially extensible curved beam with a rectangular cross-section is investigated. The elastic beam is assumed to satisfy the Euler-Bernoulli postulation and be made of the Ludwick type material. Through reasonably simplified integration, the strain and curvature of the axis of the beam are presented in implicit formulations. The governing equations involving both geometric and material nonlinearities of the curved beam are derived and solved by the shooting method. When the initial curvature of the beam is zero, the curved beam is degenerated into a straight beam,and the predicted results obtained by the present model are consistent with those in the open literature. Numerical examples are further given for curved cantilever and simply supported beams, and the couplings between elongation and bending are found for the curved beams.     

Vibration based damage detection in a uniform strength beam using genetic algorithm

Cantilever steel beams of uniform strength are having various industrial applications. In particular when it is used as leaf spring it undergoes very large deflection in comparison to beam of uniform cross section. The damage occurs in these beams mainly due to fatigue loading. Early detection of damage in such type of beams is very essential to avoid a major failure or accident. In this paper, firstly formulation of an objective function for the genetic search optimization procedure along with the residual force method are presented for the identification of macroscopic structural damage in an uniform strength beam. Two cases have been investigated here. In the first case the width is varied keeping the strength of beam uniform throughout and in the second case both width and depth are varied to represent a special case of uniform strength beam. The developed model require experimentally determined data as input and detect the location and extent of the damage in the beam. Here, experimental data are simulated numerically by using finite element models of structures with inclusion of random noise on the vibration characteristics. It has been shown that the damage may be identified for the said problems with a good accuracy.     

An efficient procedure to find shape functions and stiffness matrices of nonprismatic Euler–Bernoulli and Timoshenko beam elements

Nonprismatic beam modeling is an important issue in structural engineering, not only for versatile applicability the tapered beams do have in engineering structures, but also for their unique potential to simulate different kinds of material or geometrical variations such as crack appearing or spreading of plasticity along the beam. In this paper, a new procedure is proposed to find the exact shape functions and stiffness matrices of nonprismatic beam elements for the Euler–Bernoulli and Timoshenko formulations. The variations dealt with here include both tapering and abrupt jumps in section parameters along the beam element. The proposed procedure has found a simple structure, due to two special approaches: The separation of rigid body motions, which do not store strain energy, from other strain states, which store strain energy, and finding strain interpolating functions rather than the shape functions which suffer complex representation. Strain interpolating functions involve low-order polynomials and can suitably track the variations along the beam element. The proposed procedure is implemented to model nonprismatic Euler–Bernoulli and Timoshenko beam elements, and is verified by different numerical examples.     

Analysis,modelling, and optimization of laminated glasses as plane beam

A simply supported glass/polyvinyl butyral (PVB)/glass beam is modelled by plane finite elements. The distribution of strain and stress through the beam thickness and along its axis is obtained as a result of linear finite element analysis. It shows that the bending stress in the glass layers is determinant for the load-bearing capability of laminated glasses, but the shear in the PVB-interlayer plays an important role for glass-layer interaction. A mathematical model of triplex glass beam is derived, consisting of a bending curvature differential equation and a differential equation of PVB-interlayer shear interaction. The derived equations are solved analytically with boundary conditions of simply supported beam under uniform transverse load. A parametric study of the derived mathematical model is carried out. The model is utilized for lightweight structure optimization of layer thicknesses. The results of the optimization show that laminated glasses could be superior to monolithic glasses.     

Geometric nonlinear formulation for curved beams with varying curvature

Based on exact Green strain of spatial curved beam, the nonlinear strain-displacement relation for plane curved beam with varying curvature is derived. Instead of using the previous straight beam elements, curved beam elements are used to approximate the curved beam with varying curvature. Based on virtual work principle, rigid-flexible coupling dynamic equations are obtained. Physical experiments were carried out to capture the large overall motion and the strain of curved beam to verify the present rigid-flexible coupling formulation for curved beam based on curved beam element. Numerical results obtained from simulations were compared with those results from the physical experiments. In order to illustrate the effectiveness of the curved beam element methodology, the simulation results of present curved beam elements are compared with those obtained by previous straight beam elements. The dynamic behavior of a slider-crank mechanism with an initially curved elastic connecting rod is investigated. The advantage of employing generalized-α method is pointed out and the special nonlinear dynamic characteristics of the curved beam are concluded.     

Kinematically exact curved and twisted strain-based beam

The paper presents a formulation of the geometrically exact three-dimensional beam theory where the shape functions of three-dimensional rotations are obtained from strains by the analytical solution of kinematic equations. In general it is very demanding to obtain rotations from known rotational strains. In the paper we limit our studies to the constant strain field along the element. The relation between the total three-dimensional rotations and the rotational strains is complicated even when a constant strain field is assumed. The analytical solution for the rotation matrix is for constant rotational strains expressed by the matrix exponential. Despite the analytical relationship between rotations and rotational strains, the governing equations of the beam are in general too demanding to be solved analytically. A finite-element strain-based formulation is presented in which numerical integration in governing equations and their variations is completely omitted and replaced by analytical integrals. Some interesting connections between quantities and non-linear expressions of the beam are revealed. These relations can also serve as useful guidelines in the development of new finite elements, especially in the choice of suitable shape functions.     

The micro-connection of the second-order moiré fringes and its order determination,and the determination of the boundary strains

The micro-connection method for determining the centre lines of second-order moiré fringes presented in this paper can enhance the accuracy in measuring strain fields, and generally can determine the absolute order of the second-order moiré fringes. The strain data obtained from curved beam experiment are in good accordance with theory. The relationship between the second-order moiré fringe and the strains at the specific points of the specimen is derived. Hence a reciprocate shift method is presented for determining strain distributions in the non-overlapping region of the shifted moiré patterns, (usually in the region of specimen where no second-order moiré fringes occur is the boundary region).     

Geometrically Nonlinear Cantilever under Stochastic Loading Vectors

This paper deals with the static and dynamic response of acantilever beam affected by Gaussian and non-Gaussian vectorprocesses. The load is modeled by three different correlationstructures based on the second-order Markov process, and thenon-Gaussian random features are described by translationprocesses. Both the linear and geometrically nonlinear behaviorare described using the finite element approach, and the beammechanical characteristics (natural frequencies and damping) arevaried to investigate their influence on the response. Finally,the Monte Carlo approach is used to estimate the responsestatistics. The transverse displacement and the bending moment atthe beam joints are used to summarize the obtained results interms of skewness and kurtosis coefficients, and probabilitydensity functions. The study presented in this paper providesinformation on the influence of the load nature and the structuralbehavior on the response random features.     

Polychromatic X-ray method for residual-stress measurements in a subsurface layer

The classical sin² method with characteristic X-rays is widely used to measure residual stress nondestructively in the steel members of a structure or a machine. With this method it is, however, difficult to measure the three-dimensional stress distribution with a steep gradient that occurs along the depth direction in a subsurface layer of the material after surface treatment such as grinding or cold rolling. This paper presents a new polychromatic X-ray method for residual-stress measurements in a subsurface layer. The relationship between the diffracted beam peak of the polychromatic X-ray and the strain along the depth direction in a subsurface layer was obtained by theoretical analysis. It was modeled by numerical simulation to obtain probable values of the parameters, and these were used along with experimental X-ray data to derive an experimental value for the stress gradient. This was compared with the values predicted from plate bending theory.Paper was presented at the 1994 SEM Spring Conference on Experimental Mechanics held in Baltimore, MD on June 6–9.