Chaotic vibrations of beams on nonlinear elastic foundations subjected to reciprocating loads

Chaotic vibration of beams resting on a foundation with nonlinear stiffness is investigated in this paper. Cosine–cosine function is employed in modeling of the reciprocating load. The equation of motion is derived and solved to obtain corresponding Poincaré section in phase–space. Lyapunov exponent as a criterion for chaos indication is obtained. Dynamic behavior of the beam is examined in resonance condition. Homoclinic orbits are captured and their corresponding Melnikov's functions are established. A parametric study is then carried out and effects of linear and nonlinear parameters on the chaotic behavior of the system are studied.     

Bubble velocity jump discontinuity in polyacrylamide solutions: a photographic study

 It has been established that in the presence of elastic as well as surface tension forces, a jump can occur in the bubble velocity-bubble volume plot. It was proposed that this phenomenon was the result of an elastic instability at the bubble interface, which drastically changes the local boundary conditions. The origin of this change was mainly attributed to the magnitude of the elastic forces, which can extract surface active impurities at the gas-liquid interface. The purpose of this paper is to present photographic evidence of this hypothesis and to confirm past results. Received: 4 December 1998 Accepted: 17 February 1999     

Structural modeling of beams on elastic foundations with elasticity couplings

A new simplified structural model and its governing equations for beams on elastic foundations with elastic coupling are proposed. This modeling system is simple but appropriate for the initial structural design of large-scale submerged floating-beam structures moored by tension legs spaced at uniform interval along the beam. The model is actually for beam on discrete elastic supports rather than on continuous elastic foundations. Therefore, the governing equations are based on finite difference calculus and solutions for beams on discrete elastic supports with elasticity coupling are also proposed. To clarify the applicability limit of the proposed model, the equivalence between a beam on discrete elastic supports and that on continuous elastic foundation is investigated by comparisons of characteristic solutions.     

Vibration of beams with general boundary conditions due to a moving random load

Summary  The transverse vibrations of elastic homogeneous isotropic beams with general boundary conditions due to a moving random force with constant mean value are analyzed. The boundary conditions considered are: pinned–pinned, fixed–fixed, pinned–fixed, and fixed–free. Based on the Bernoulli beam theory, the problem is described by means of a partial differential equation. Closed-form solutions for the variance and the coefficient of variation of the beam deflection are obtained and compared for three types of force motion: accelerated, decelerated and uniform. The effects of beam damping and speed of the moving force on the dynamic response of beams are studied in detail. Received 3 December 2001; accepted for publication 30 April 2002     

弹性地基上Timoshenko梁的精确数值解

研究了弹性地基上Timoshenko梁的高精度有限元分析方法,利用控制微分方程的基本解建立了单元形函数,提出了弹性地基上Timoshenko梁分析的Trefftz单元。通过对引入的非节点自由度进行静力凝聚,得到的精确单元与常规单元具有相同的节点自由度。文中还讨论了有效降低计算过程中舍入误差的方法。算例结果表明,采用提出...     

On the application of the constant deflection-contour method in nonlinear vibrations of elastic plates

Summary  This paper concerns the application of the constant deflection-contour method to problems involving nonlinear vibrations. Two specific problems are considered: a clamped circular plate and an annular plate with free inner boundary. For the linear case, the results obtained offer excellent agreement with previous studies, indicating significant potential for the utilization of this method in different nonlinear cases. The analysis may be applied to other types of geometrical structures. Notwithstanding the fact that only a first-term approximation has been made for the deflection function, in conjunction with the Galerkin procedure, excellent agreement has been found. Additional analytical calculations could be made to improve accuracy, indicating that the method could prove particularly useful when employed with a symbolic manipulation package. Received 13 June 2001; accepted for publication 6 November 2001     

Instability of vibrations of a moving two-mass oscillator on a flexibly supported Timoshenko beam

Summary  Transversal vibrations of a uniformly moving two-mass oscillator on a Timoshenko beam of infinite length supported by a viscoelastic foundation are studied. By using integral transforms, the characteristic equation for the oscillator's vibrations is obtained. It is shown that the equation may have a root with a positive real part. The existence of such a root leads to the exponential increase of the amplitude of the oscillator vibrations, i.e. to instability. The reasons for the instability to occur are discussed. By employing the method of D-decomposition, the instability domains are found in the space of the system parameters. Received 30 October 2000; accepted for publication 28 March 2001     

Complete eigenfunction expansion form of the Green's function for elastic layered half-space

Summary  The complete eigenfunction expansion form of the Green's function for a 3-D elastic layered half-space in the frequency domain is derived in this paper. The expression of the Green's function presented here is an extension of that represented by the residue terms and the branch line integrals given by Lamb [1]. The present expression, however, clarifies the mathematical common frame between the residue terms and the branch line integrals with respect to the eigenfunctions and energy integrals. For the derivation, the concept of an energy integral for the improper eigenfunctions is newly developed. The improper eigenfunctions, which can be found in the wavenumbers for the branch cuts, are not in L ₂ space, so the definition of the energy integral requires some treatment. The energy integral is defined as the limit of the inner product of the improper eigenfunction and the definition function of the improper eigenfunction, for which the inner product remains finite. Via the definition of the energy integral, the kernel of the branch line integral is decomposed into the improper eigenfunction, and the complete eigenfunction expansion form of the Green's function is derived. The Green's function can thus be expressed by summation of normal modes, complex modes pointed out in [2], the integral of the improper eigenfunction and the residue at k=0 due to the singularity of the horizontal wavefunction. Received 3 May 2001; accepted for publication 23 August 2001     

Nonuniform warping torsion of orthotropic composite beams

Summary  The proposed work deals with the behaviour of orthotropic composite beams subjected to torsion, taking into consideration the effects due to nonuniform warping. We start from a one-dimensional approach based on the integral parameter method and consider any complex cross-sectional shape. The basic relations obtained lead to numerical computation based on the finite element method. As a special case of the general results obtained, an example of a sandwich beam is worked out in detail. Received 11 May 1999; accepted for publication 30 November 1999     

Generalized variational principles for thermopiezoelectricity

Summary  Based on the semi-inverse method of establishing variational principles proposed in [10], a family of variational principles (non Gurtin-type and not involving convolutions) for thermopiezoelectricity is deduced directly from the field equations and boundary/initial conditions. Present theory provides a more complete theoretical basis for the finite element applications and other direct variational methods such as Ritz's, Trefftz's and Kantorovitch's methods. Received 4 November 1999; accepted for publication 6 June 2000     

Thermal buckling and post-buckling of pinned–fixed Euler–Bernoulli beams on an elastic foundation

In this article, both thermal buckling and post-buckling of pinned–fixed beams resting on an elastic foundation are investigated. Based on the accurate geometrically non-linear theory for Euler–Bernoulli beams, considering both linear and non-linear elastic foundation effects, governing equations for large static deformations of the beam subjected to uniform temperature rise are derived. Due to the large deformation of the beam, the constraint forces of elastic foundation in both longitudinal and transverse directions are taken into account. The boundary value problem for the non-linear ordinary differential equations is solved effectively by using the shooting method. Characteristic curves of critical buckling temperature versus elastic foundation stiffness parameter corresponding to the first, the second, and the third buckling mode shapes are plotted. From the numerical results it can be found that the buckling load-elastic foundation stiffness curves have no intersection when the value of linear foundation stiffness parameter is less than 3000, which is different from the behaviors of symmetrically supported (pinned–pinned and fixed–fixed) beams. As we expect that the non-linear foundation stiffness parameter has no sharp influence on the critical buckling temperature and it has a slight effect on the post-buckling temperature compared with the linear one.     

A kinematic elastic nonshakedown theorem for implicit standard materials

Summary  Criteria for a priori recognition of the type of steady-state response induced by cyclic loads and prediction whether a structure will shakedown elastically or not, without the necessity of performing a step-by-step full analysis, have considerable importance. Melan and Koiter theorems provide criteria that guarantee whether elastic shakedown occurs or not under cyclic loads in case of perfect plasticity. However, there remain some aspects of the shakedown theory which deserve further study. One of these, concerned with more realistic nonassociative elastic–plastic constitutive material models, allowing for nonlinear kinematic and isotropic hardening suitable to describe the cyclic plastic behaviour of metallic materials, has strong motivation. Koiter's elastic nonshakedown theorem is reconsidered here, with the objective of extending it to the de Saxcé's implicit standard material class, which contains a wide class of nonassociative elastic–plastic material behaviours. Shakedown analysis is formulated by a kinematic approach based on the plastic accumulation mechanism concept due to Polizzotto. A sufficient condition for elastic nonshakedown and a distinct necessary condition are established. Then, an upper bound to the shakedown multiplier is evaluated. Received 15 February 2001; accepted for publication 18 October 2001     

The principle of correspondence between elastic and piezoelectric problems

Summary  A correspondence principle is established between elastic and piezoelectric problems for transversely isotropic materials, in such a way that the knowledge of an elastic solution yields fully coupled electro–elastic fields for the corresponding piezoelectric problem, provided the elastic solution is written in a certain form. The implementation of this principle is illustrated by constructing, in a routine way, several piezoelectric solutions involving crack and punch problems (one of them has not been solved previously). Received 12 Feburary 2002; accepted for publication 29 April 2002     

Analysis of 2D infinite anisotropic elastic strips and semi-strips

Summary  Using Stroh's formalism and the theory of analytic functions, simple and explicit solutions for a line dislocation in an infinite anisotropic elastic strip are obtained. The two boundaries of the strip are free of traction. The problem of a dislocation in an anisotropic elastic semi-infinite strip with traction-free boundaries is also studied. A set of singular integral equations governing the unknown functions is derived. When the medium is orthogonal anisotropic and the coordinate axes x ₁ x ₂ x ₃ are coincident with the material principal axes, all the eigenvalues of the material coefficient matrix are pure imaginary. Explicit expressions of the unknown functions are given for this case. The results obtained are valid not only for plane and anti-plane problems but also for coupled problems between in-plane and out-of-plane deformations. Received 30 October 2000; accepted for publication 28 March 2001     

A semi-infinite interfacial crack between two bonded dissimilar elastic strips

Summary  This paper is concerned with a semi-infinite interfacial crack between two bonded dissimilar elastic strips with equal thickness. Solutions for the complex stress intensity factor (SIF) and energy release rate (ERR) are obtained in closed form under in-plane deformations. During the procedure, the mixed boundary-value problem is reduced by means of the conformal mapping technique to the standard Riemann–Hilbert problem. In some limiting cases, the present solutions can cover the results found in literature. Received 21 February 2002; accepted for publication 2 July 2002 X.-F Wu's work was supported in part by the Milton E. Mohr Research Fellowship (2001, 2002) of the Engineering College at University of Nebraska-Lincoln.     

Material sensitivity analysis in homogenization of linear elastic composites

Summary  The main goal of the paper is to present theoretical aspects and the finite element method (FEM) implementation of the sensitivity analysis in homogenization of composite materials with linear elastic components, using effective modules approach. The deterministic sensitivity analysis of effective material properties is presented in a general form for an n-components periodic composite, and is illustrated by the examples of 1D as well as of 2D heterogeneous structures. The results of the sensitivity analysis presented in the paper confirm the usefulness of the homogenization method in computational analysis of composite materials the method may be applied to computational optimization of engineering composites, to the shape-sensitivity studies and, after some probabilistic extensions, to stochastic sensitivity analysis of random composites. Received 10 November 2000; accepted for publication 24 April 2001     

Generalized thermoelastic analysis of the stress-focusing effect in a solid cylinder under rotationally asymmetrical instantaneous thermal loading

Summary  This paper investigates the stress-focusing effect in an infinitely long cylinder under rotationally asymmetrical instantaneous thermal loading on the basis of the generalized thermoelastic Lord–Shulman (L-S) and Green–Lindsay (G-L) theories. Combined forms of the governing equations of both theories are given in a cylindrical coordinate system. The two-dimensional generalized thermoelastic problems are solved by numerical inversion of Laplace transform. Calculations have been performed to find distributions of thermal stresses on the basis of the L-S theory. Stress-focusing phenomena under different heating conditions are presented. The effects of thermomechanical coupling and relaxation time on the stress-focusing phenomena as well as the singularity of stresses are discussed. Received 15 November 2000; accepted for publication 15 November 2001     

Timoshenko curved beam bending solutions in terms of Euler-Bernoulli solutions

Summary  This paper presents the exact relationships between the deflections and stress resultants of Timoshenko curved beams and that of the corresponding Euler-Bernoulli curved beams. The curved beams considered are of rectangular cross sections and constant radius of curvature. They may have any combinations of classical boundary conditions, and are subjected to any loading distribution that acts normal to the curved beam centreline. These relationships allow engineering designers to directly obtain the bending solutions of Timoshenko curved beams from the familiar Euler-Bernoulli solutions without having to perform the more complicated shear deformation analysis. Accepted for publication 26 July 1996     

Morphological stability of epitaxial thin elastic films by van der Waals force

Summary  The morphological stability of epitaxial thin elastic films on a substrate by van der Waals force is discussed. It is found that only van der Waals force with negative Hamaker constant tends to stabilize the film, and the lower bound for the Hamaker constant is also obtained for the stability of thin film. The critical value of the undulation wavelength is found to be a function of both film thickness and external stress. The charateristic time-scale for surface mass diffusion scales to the fourth power to the wavelength of the perturbation. Received 4 December 2000; accepted for publication 31 July 2001     

Winkler-Pasternak弹性地基梁自由振动的二维弹性分析

基于二维线弹性理论,应用Hamilton原理,获得Winkler-Pasternak弹性地基梁自由振动的控制微分方程,应用微分求积法(DQM)数值研究了梁自由振动的无量纲频率特性。计算结果与已有的结果(Bernoulli-Euler梁和Timoshenko梁)比较表明,本文的分析方法对弹性地基长梁和短梁自由振动的研究都有效。最后考虑了几何参数对梁频率的影响,以及不同边界条件下地基系数对频率的影响和收敛性。