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.     

Finite element formulation for dynamics of delaminated composite beams with piezoelectric actuators

As a part of an effort to develop a model-supported method for detection of delaminations in composite beams with the use of time responses to external excitations, a finite element formulation for dynamics of a composite beam with delamination and attached piezoelectric actuators is developed. In this formulation account is taken of transverse shear deformation and nonlinear through-thickness variation of the longitudinal displacement. Parameters that characterize the delamination are incorporated into the formulation that makes the finite element model convenient for use in conjunction with damage identification (not discussed in the present paper). Computational predictions of frequencies show good agreement with experimental results.     

Optimal elastic design of beams

According to the principle of minimum complementary energy a mathematical statement of optimal strength design problem for elastic beams is formulated in this research, which is an extremum problem of functionals with equality and inequality constraints. Further the application of the Lagrangian multiplier method yields the necessary conditions for extrema. A set of relations that must be satisfied for the optimal solution follows afterwards. This set of relations can be used to verify the optimality of a uniform strength design or any feasible elastic design. An iterative numerical method to find the optimal solution when the uniform strength design is not optimal is also presented in this paper.Project supported by the Science and Technics Fund of the Chinese National Educational Committee.     

Analysis of micropolar elastic beams

In this paper, a linear theory for the analysis of beams based on the micropolar continuum mechanics is developed. Power series expansions for the axial displacement and micro-rotation fields are assumed. The governing equations are derived by integrating the momentum and moment of momentum equations in the micropolar continuum theory. Body couples and couple stresses can be supported in this theory. After some simplifications, this theory can be reduced to the well-known Timoshenko and Euler–Bernoulli beam theories. The nature of flexural and longitudinal waves in the infinite length micropolar beam has been investigated. This theory predicts the existence of micro-rotational waves which are not present in any of the known beam theories based on the classical continuum mechanics. Also, the deformation of a cantilever beam with transverse concentrated tip loading has been studied. The pattern of deflection of the beam is similar to the classical beam theories, but couple stress and micro-rotation show an oscillatory behavior along the beam for various loadings.     

Finite elastic straining

Summary Finite elastic straining is analysed with all quantities referred consistently to the deformed body taken as the function domain. The straining-displacement of a typical point is relative to a set of axes imbedded in the body at one arbitrary point and rotating in fixed space with that neighborhood if necessary in a particular problem. The resulting |plane stress' equations have precisely the same form as in the classical theory but relate to |true' quantities in the deformed body.The solution of a circular hole in a deformed sheet under simple tension is given and checks closely with experiment on rubber. Cauchy strains of order 65% and local rotation of order 30° are found to occur at the hole boundary.The solution of a deformed quadrantal cantilever is given. Cauchy strains of several hundred percent and local rotation of order 90° occur.Any boundary value problem already solved for the classical infinitesimal strains theory can be applied directly as a finite strains solution for the deformed body.Notation x, y, z, r, , z Cartesian and polar co-ordinates respectively - , Normal and shear true stresses respectively - , Normal and shear true strains respectively - r Position vector - Airy stress function - S Simple tensile stress applied to sheet - a Radius of circular hole in deformed sheet - a, b Inner and outer radii of quadrantal cantilever - u Straining-displacement vector - u, v Straining-displacement scalar components - E, True Young's modulus and Poisson's ratio respectively - c ₁, c ₂ Local unit vectors in principal normal strains directions - i, j Cartesian axes constant unit vectors - Stress dyadic or tensor - First stress invariant - I Idemfactor or spherical tensor - P Shear load per unit thickness applied to quadrantal cantilever - A, B, D, N, H, K, L Arbitrary constants of integration     

An integral approach for large deflection cantilever beams

A new integral approach is proposed to solve the large deflection cantilever beam problems. By using the moment integral treatment, this approach can be applied to problems of complex load and varying beam properties. This versatile approach generally requires only simple numerical techniques thus is easy for application. Treatment for typical loading and beam property conditions are presented to demonstrate the capability of this approach.     

Exact deflection solutions of beams with shear piezoelectric actuators

Exact deflection models of beams with n actuators of shear piezoelectric are developed analytically. To formulate the models, the first-order and higher-order beam theories are used. The exact solutions are obtained with the aid of the state-space approach and Jordan canonical form. A case study is presented to evaluate the performance of the authors’ previously reported models. Through a demonstrative example, a comparative study of the first-order and higher-order beams with two shear piezoelectric actuators is attained. It is shown that the first-order beam cannot predict the beam behavior when compared with the results of the higher-order beam. Further applications of the solutions are presented by investigating the effects of actuators lengths and locations on the deflected shapes of beams with two piezoelectric actuators. Some interesting deflection curves are presented. For example, the deflection curve of a H–H beam resembles saw teeth that rotate clockwise about the central location with the increase of actuators lengths. The presented exact solutions can be used in the design process to obtain detailed deformation information of beams with various boundary conditions. Moreover, the presented analysis can be readily used to perform precise shape control of beams with n actuators of shear piezoelectric.     

Finite twist of composite beams

Summary A pretwisted composite beam with straight centroidal axis, subjected to axial tension and torsion is proved to achieve a better structural efficiency if the reinforcing fibers are located along helicoidal generators in the pretwisted configuration, than in the Hodges' problem where those fibers are parallel to the beam axis.

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Sommario Si trova che una trave svergolata in materiale composito, con asse baricentrico diritto, soggetto a trazione e torsione, raggiunge una maggiore efficienza strutturale se le fibre di rinforzo sono dispose già inizialmente secondo linee generatrici elicoidali piuttosto che in direzione assiale come supposto in un precedente lavoro di Hodges.

     

Asymptotic analysis of layered elastic beams

We consider an elastic beam formed by three layers, fixed at one end and loaded at the free end. We call adherents the upper and lower layers ${\Omega }_{+}^{?}$ and ${\Omega }_{?}^{?}$ and an adhesive layer ${\Omega }_m^{?}$. We denote by $?h_{\pm ,m}$ the thickness of each layer and we suppose that the stiffness of the adhesive layer is ${?}^2$, with respect to that of the adherents. By an asymptotic analysis we obtain the zeroth order limit problem and the form of the second order displacements. To cite this article: M. Serpilli, C. R. Mecanique 333 (2005).     

BIPARAMETRIC PERTURBATION SOLUTIONS OF LARGE DEFLECTION PROBLEM OF CANTILEVER BEAMS

The large deflection problem of cantilever beams was studied by means of the biparametric perturbation method and the first order derivative substitution from pseudolinear analysis approach. This kind of substitution can transform the basic equation, an integral differential equation into nonlinear algebraic ones, thus simplify computational process. Compared with present results, it indicates that the large deflection problem solved by using pseudolinear analysis can lead to simple and precise results.     

弹性地基梁损伤诊断研究

利用试验得到的振动参数评估结构的破损情况，是当前结构工程学科十分活跃的领域。由于弹性地基梁的振动模态受地基和梁两方面因素的影响，其损伤诊断问题变得十分复杂。本文通过对两靖自由弹性地基梁的灵敏性分析发现弹性地基梁的前两阶自由模态主要与地基有关，利用这一特性构造了两级识别的方法，并引入优化领域寻优能力极强的遗传算法进行识别，找到了令人满意的答案。     

Bending of beams on three-parameter elastic foundation

The bending of a Timoshenko beam resting on a Kerr-type three-parameter elastic foundation is introduced, its governing differential equations are formulated and analytically solved, and the solutions are discussed and applied to particular problems. Parametric analyses of elastically supported beams of infinite and finite length are carried out and comparisons are made between one, two or three-parameter foundation models and more accurate 2D finite element models. In order to estimate the necessary soil parameters, an analytical procedure based on the modified Vlasov model is proposed. The presented solutions and applications show the superiority of the Kerr-type foundation model compared to one or two-parameter models.     

Ⅰ型裂纹对薄壁偏心柱弹性挠度的影响

本文重点研究了裂纹对薄壁截面偏心受压柱弹性挠度的影响,分析模型假设偏心柱两墙铰支;截面绕强轴发生单向弯曲变形：I型裂纹位于受力最为严重的中间截面的受拉翼缘上,首先简单介绍了Rayleigh-Ritz变分法求得的弹性挠度三角函数级数解,该解的优点是可以在裂纹截面满足变形协调条件,适应于裂纹在薄壁截面上的沿弯矩面外方向扩展的情况,然后针对工字形,箱形两种双轴对称薄壁截面柱,具体地分析了在不同荷载,偏心距和长细比条件下裂纹长度对偏心柱挠度的影响规律和范围,理论分析和数值结果表明,在其它条件不变的情况下,柱的长细比越小或偏心距越大,裂纹引起的挠度增量越明显。,     

Asymptotic theories of elastic beams and rods

Summary Through the asymptotic approach to elastic theory of beam-like structural elements, various categories of problems are distinguished with different first approximation formulations, showing different degrees of non-linearity.

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Sommario Per la teoria elastica di elementi strutturali tipo trave, il metodo asintotico porta a individuare varie categorie di problemi, retti da diverse formulazioni con diversi gradi di non-linearità, qui considerate in forma di prima approssimazione. Il metodo, oltre a fornire giustificazione di formulazioni classiche, offre la possibilità di approfondimenti sistematici.

     

空间桁架结构大位移问题的有限元分析方法

本文基于总体拉格朗日坐标描述法,采用Ｋｉｒｃｈｏｆｆ应力张量和Ｇｒｅｅｎ应变张量定义,导出了严格意义下的杆单元增量列式,计算表明本文方法可以有效用于空间桁架结构大位移问题分析。     

An integral equation approach to finite deflection of elastic plates

The present paper concerns an approximate integral equation approach to finite deflection of elastic plates with aribitrary plan form. With the combined help of the Berger equation governing non-linear bending and a weighted residual technique for boundary-value problem, a boundary integral equation is formulated for immovable edge conditions. We here clarify the formulation and show that the calculation can be performed, with a slight modification, through a procedure similar to that conventionally employed in the linear bending analysis. Availability of the derived integral equation and the solution scheme is shown by way of simple numerical examples.     

A new technique for large deflection analysis of non-prismatic cantilever beams

This paper studies the very large deflection behavior of prismatic and non-prismatic cantilever beams subjected to various types of loadings. The formulation is based on representing the angle of rotation of the beam by a polynomial on the position variable along the deflected beam axis. The coefficients of the polynomial are obtained by minimizing the integral of the residual error of the governing differential equation and by applying the beam’s boundary conditions. Several numerical examples are presented covering prismatic and non-prismatic cantilever beams subjected to uniform, non-uniform distributed loads and tip concentrated loadings in vertical and horizontal directions. The loads considered in this study are restricted to the non-follower type loads. Cases with different loadings and geometries are compared with MSC/NASTRAN computer package. However, for some very large deflection case, the MSC/NASTRAN failed to predict the deflected shape due to divergence problems.     

Simply supported elastic beams under parametric excitation

In this paper, the nonlinear characteristics of the parametric resonance of simply supported elastic beams are investigated. Considering a geometrically exact formulation for unsharable and inextensible elastic beams subject to support motions, the integral-partial-differential equation of motion is obtained. The third-order perturbation of the equation of motion is then determined in a form amenable to an asymptotic treatment. The method of multiple scales is used to obtain the equations that describe the modulation of the amplitude and phase of parametric-resonance motions. The stability and bifurcations of the system are investigated considering, in particular, the frequency-response function. Furthermore, experimental results are shown to confirm the theoretically predicted stability and bifurcations.     

Finite element system of elastic structure

The article deals with the problems of controllability, observability and stabilizability of an elastic-structural system treated by the finite element method. The results obtained here agree with that obtained in distributed parameter-system model, nevertheless, they are more convenient than those in carrying out the computation with a computer, at the same time the method appears much easier that the conventional one. In section one, the system's controllability and observability are studied and some conditions which are easier to be justified by computer are given. In section two, the problem of stabilizing an elastic object by the use of linear feedback is fully discussed. As the attained results there show that, so far as an elastic-structural system is concerned, it is possible to assign arbitrary frequencies of vibration only by the use of displacement feedback, however, it is impossible to stabilize the system while the system is completely controllable. While the velocity feedback can stabilize the system, but its ability is limited. The case of rigid body motion involved in the system equation has also been discussed. In section three, the control of a straight beam is treated by the finite element method. The whole system of a beam can be decomposed into four irrelevant subsystems of tension-compression, torsion, bending in two directions, their controllability and observability are also analyzed respectively. The controllability and observability of segment-shaped beam are discussed in the end.