Bifurcation of equilibrium of flexible bars subject to eccentric loads

The multiple equilibrium shapes, i. e., bifurcation behavior of the flexible bars subject to axial eccentric force are analyzed by the large deflection theory. Three types of elastica of equilibrium shape are discussed, and the method for determining the various classes of bifurcation of equilibrium due to given loads is presented.     

Postbuckling behavior of variable-arc-length elastica connected with a rotational spring joint including the effect of configurational force

This paper aims to study the behavior of a variable-arc-length (VAL) elastica subjected to the end loading, where a rotational spring joint is placed within the span length of the elastica. One end of the elastica is rested on the pinned support while the other end is placed into the sleeve support. The length of the elastica can be fed into the system through sleeve support by the end thrust where the effect of configurational force has been considered. A rotational spring joint is located within the span length of the elastica. From the equilibrium equations, moment–curvature expression, geometric relations, and boundary conditions, the closed-form solution in terms of elliptic integral of the first and second kinds can be demonstrated. The results obtained from elliptic integral method are validated with those from the shooting method and they are in excellent agreement. In order to interpret the behavior of the elastica, load–deflection curves and equilibrium shapes are established. Interesting features of the results are demonstrated. Particularly, when the stiffness of the spring joint becomes zero, the secondary buckling and the multiple equilibrium shapes can be captured in which the stable equilibrium shapes can be evaluated by using the vibration analysis. For a low value of the stiffness of the spring joint, the elastica has a possibility to exhibit the hardening behavior. When the stiffness of the spring joint becomes large, the elastica shows the softening behavior and its shape is identical to a single portion of VAL elastica.     

Snapping of an elastica under various loading mechanisms

This paper studies the snap-through buckling of a hinged elastica subject to a midpoint force. The focus is placed on how different load models affect the deformation and snapping load. Three different load models are considered. In the first model, the point force is fixed onto a midpoint of the elastica. In the second model, the point force is fixed on a central line in space. In the third model, the external force is applied through a rigid bar, which is allowed to slide along the central line in space. When the loaded elastica deforms symmetrically, as in the case when the magnitude of the point force is small, there is no difference between these three load models. However, when the elastica deforms unsymmetrically, the three load models produce different results. Vibration method is used to determine the stability of the equilibrium configurations. It is found that the elastica may snap via either a sub-critical pitchfork bifurcation or a limit-point bifurcation. If the elastica snaps via a sub-critical pitchfork bifurcation, the pre-snapping deformation is symmetric. If the elastica snaps via a limit-point bifurcation, on the other hand, the pre-snapping deformation is unsymmetric. For a specified initial shape of the elastica, different load models predict different snapping loads and pre-snapping deformations. The theoretical predictions are confirmed by experimental observations.     

Bending of a cantilever rectangular plate loaded discontinuously

The cantilever rectangular plates discussed previously are all loaded continuously. For example, the load may be either a uniform or a concentrated load at the free ledge of the plate. Now we go a step further to deal with the case of a discontinuously loaded rectangular cantilever plate. The problem to be solved will involve a concentrated load at the center of the plate, as shown in Fig. 1. The method of solution used is the same as before.     

Complete solution of the stability problem for elastica of Euler's column

The stability of postcritical equilibrium forms of a simply supported column loaded with an axial force is analyzed. Investigating the sign of the second variation of the column's total energy, we obtain the Sturm-Liouville boundary-value problem, which is solved numerically. The stability conditions are formulated in terms of eigenvalues of the problem. The complete solution to the column plane elastica is given. The ranges of the compressive force corresponding to stable equilibrium configurations of the column are established.     

Investigation of the Effect of Axial Loads on the Transverse Vibrations of a Vertical Cantilever with Variable Parameters

The method of influence function is applied to the solution of the boundary-value problem on the free transverse vibrations of a vertical cantilever and a bar subjected to axial loads. To demonstrate the capabilities of the method, a cantilever with the free end under two types of loading — point forces (conservative and follower) and a load distributed along the length (dead load) — is analyzed. A characteristic equation in the general form, which does not depend on the cantilever shape and on the type of axial load, is given. The Cauchy influence function depends on the cantilever shape and the type of axial load. As an example, a tapered cantilever subjected to conservative and follower forces and an elastically supported bar under the dead load are considered in detail. The characteristic equation derived allows one to evaluate the natural frequencies and the Euler critical loads. It is shown that the calculated natural frequencies and critical forces are in a good agreement with the exact values when several terms are retained in the characteristic series. The high accuracy of the method is also confirmed     

STATIC STUDY OF CANTILEVER BEAM STICTION UNDER ELECTROSTATIC FORCE INFLUENCE

I. INTRODUCTION For capacitor-like microelectromechanical systems (MEMS) structure[1??6], the voltage betweenthe structure and substrate causes attractive force. The sources of the voltage can be an arti?ciallymounted device[2 , 7??9] or the temporar…     

Furthest reach of a uniform cantilevered elastica

A uniform elastic cantilever is subjected to a uniformly distributed load or a concentrated load at its tip. The angle of the fixed end with the horizontal is varied until the maximum horizontal distance (projection) from the fixed end to the horizontal location of the tip is attained. The beam is modeled as an inextensible elastica, and numerical results are obtained with the use of a shooting method. For the optimal solution (furthest reach), the tip is below the level of the fixed end. Experiments are conducted to verify the analysis for a heavy cantilever (i.e., only subjected to its self-weight).     

Elastica of a variable-arc-length circular curved beam subjected to an end follower force

This paper presents postbuckling behaviors of a variable-arc-length (VAL) circular curved beam subjected to an end follower force. One end of the VAL circular curved beam is hinged while the other end is supported by a frictionless slot, which is fixed horizontally and vertically but is allowed to rotate corresponding to loading direction. When the VAL circular curved beam is deformed, the total arc-length of the circular curved beam varies. Two approaches have been applied for the solution of this problem. The first approach is an elliptic integrals method based on elastica theory, which yields the exact closed-form solution in terms of the first and second kinds of elliptic integrals. For validation of the results, the shooting method is employed for a numerical solution by developing the set of nonlinear governing differential equations together with boundary conditions, and then integrating them by using the fourth-order Runge–Kutta algorithm. The results from both approaches are in very good agreement. From the results, it is found that the VAL circular curved beam subjected to an end follower force can be deformed in many mode shapes. For the first and third modes, the beam exhibits both stable and unstable configurations, whereas for the second mode only an unstable configuration exists. The influences of initial curvature on the critical load and the deformed configurations are highlighted.     

Deformations of a clamped–clamped elastica inside a circular channel with clearance

In this paper, we study the planar deformations of an elastica inside a circular channel with clearance. One end of the elastica is fully clamped, while the other end is partially clamped in the lateral direction and is subject to a pushing force longitudinally. In the experiment we first observe various deformation patterns after pushing the elastica through the partial clamp. Both symmetric and asymmetric deformations are recorded. Special attention is focused on the contact conditions between the elastica and the circular channel. In order to analyze the elastica deformation theoretically, we first divide the elastica into several elementary sub-domains depending on the contact condition between the elastica and the circular channel. In each sub-domain the elastica is either loaded only at the ends or in full contact with the outer wall. Armed with these basic equilibrium analyses, we proceed to calculate and classify the loaded elastica into several deformation patterns. Finally, we present the load-deflection curves, both theoretically and experimentally, which relate the longitudinal forces at both ends to the elastica length increase inside the channel. The branching phenomena predicted theoretically agree fairly well quantitatively with the experimental measurements.     

Cantilever beam equilibrium configurations

This article investigates a method for obtaining all equilibrium configurations of a cantilever beam subjected to an end load with a constant angle of inclination. The formulation is based on plane finite-strain beam theory in the elastic domain. An example of a cantilever beam subjected to a horizontal pressure force is discussed in detail.     

Flexural-torsional bifurcations of a cantilever beam under potential and circulatory forces I: Non-linear model and stability analysis

The stability of a cantilever elastic beam with rectangular cross-section under the action of a follower tangential force and a bending conservative couple at the free end is analyzed. The beam is herein modeled as a non-linear Cosserat rod model. Non-linear, partial integro-differential equations of motion are derived expanded up to cubic terms in the transversal displacement and torsional angle of the beam. The linear stability of the trivial equilibrium is studied, revealing the existence of buckling, flutter and double-zero critical points. Interaction between conservative and non-conservative loads with respect to the stability problem is discussed. The critical spectral properties are derived and the corresponding critical eigenspace is evaluated.     

Large deflections of cantilever beams of non-linear elastic material under a combined loading

Large deflection of cantilever beams made of Ludwick type material subjected to a combined loading consisting of a uniformly distributed load and one vertical concentrated load at the free end was investigated. Governing equation was derived by using the shearing force formulation instead of the bending moment formulation because in the case of large deflected member, the shearing force formulation possesses some computational advantages over the bending moment formulation. Since the problem involves both geometrical and material non-linearities, the governing equation is complicated non-linear differential equation, which would in general require numerical solutions to determine the large deflection for a given loading. Numerical solution was obtained by using Butcher's fifth order Runge-Kutta method and are presented in a tabulated form.     

ANALYSIS OF SHEAR LAG EFFECT OF TWIN-CELL BOX GIRDERS

针对单箱双室箱梁,考虑各翼板间剪力滞翘曲的差异,并结合全截面轴力自平衡条件,定义了箱梁各翼板的剪滞翘曲位移函数. 利用最小势能原理,建立了双室箱梁考虑剪力滞效应的控制微分方程. 对一典型的单箱双室简支箱梁,利用空间板壳数值方法和本文解析解方法,研究了满跨均布载荷和跨中集中力作用下截面的剪力滞分布规律. 结果表明,本文提出的剪力滞翘曲位移模式能够反映双室箱梁各翼板间剪力滞翘曲的差异,本文解析解与有限元数值解吻合良好. 双室箱梁中腹板部位顶、底板处的剪力滞效应与边腹板部位有一定差异,对算例结构,中腹板部位的顶、底板应力小于边腹板部位的应力.     

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.     

POLYNOMIAL SOLUTIONS TO PIEZOELECTRIC BEAMS (Ⅱ)——ANALYTICAL SOLUTIONS TO TYPICAL PROBLEMS

IntroductionThis paper is a continuation of Ref.[1],in which a series of orthotropic piezoelectricplane problems was solved and the corresponding exact solutions were obtained with the trial-and-error method,on the basis of the general solution expressed …     

POLYNOMIAL SOLUTIONS TO PIEZOELECTRIC BEAMS(Ⅱ)--ANALYTICAL SOLUTIONS TO TYPICAL PROBLEMS

For the orthotropic piezoelectric plane problem, a series of piezoelectric beams is solved and the corresponding analytical solutions are obtained with the trialand-error method on the basis of the general solution in the case of three distinct eigenvalues, in which all displacements, electrical potential, stresses and electrical displacements are expressed by three displacement functions in terms of harmonic polynomials. These problems are cantilever beam with cross force and point charge at free end, cantilever beam and simply-supported beam subjected to uniform loads on the upper and lower surfaces, and cantilever beam subjected to linear electrical potential.     

Analytical solutions for plane problem of functionally graded magnetoelectric cantilever beam

In this paper, an exact analytical solution is presented for a transversely isotropic functionally graded magneto-electro-elastic (FGMEE) cantilever beam, which is subjected to a uniform load on its upper surface, as well as the concentrated force and moment at the free end. This solution can be applied for any form of gradient distribution. For the basic equations of plane problem, all the partial differential equations governing the stress field, electric, and magnetic potentials are derived. Then, the expressions of Airy stress, electric, and magnetic potential functions are assumed as quadratic polynomials of the longitudinal coordinate. Based on all the boundary conditions, the exact expressions of the three functions can be determined. As numerical examples, the material parameters are set as exponential and linear distributions in the thickness direction. The effects of the material parameters on the mechanical, electric, and magnetic fields of the cantilever beam are analyzed in detail.     

The equilibrium state of a structure subject to frictional contact

A structure in frictional contact subject to static loads has not, in general, a unique static equilibrium state. This is because the state, displacements and contact forces, depend on the load history of the structure.In cases where the exact load history is not known it would be of interest to find a state that is in some sense likely and define this as the equilibrium state. In this paper, it is assumed that the state with the smallest potential energy is the most likely one. The implication of this definition of likely state is analysed and shows that the resulting problem basically can be seen as a generalization of the frictionless contact problem to structures where no frictionless state is possible, i.e. structures where non-zero friction forces are necessary to satisfy force equilibrium.The results of several numerical experiments are given. The structures in the experiments are trusses and structures modelled by the finite element method. Both a sequential quadratic programming method and an enumeration method are used to solve the likely-state problem.     

Remarks on formulating an adhesion problem using Euler’s elastica (draft)

Three formulations for the problem of an elastica adhering to a rigid surface are discussed and compared. These include stationary principles, the surface integral of Eshelby’s energy-momentum tensor, and the material (configurational) force balance. The configuration at static equilibrium is predicted in closed form for a pair of structures that arise in nano- and microscale applications.