Parametric resonances in the two-to-one resonant beams on elastic foundation

The nonlinear vibration of the inextensional beam on the elastic foundation under parametric resonance and two-to-one internal resonance is investigated. Considering the inextensional condition and the second-order moment of the subgrade reaction, the extended Hamilton principle is applied to derive the motion equation of the beam on elastic foundation. Then the multimodal discretization and the method of multiple scales are used to obtain the modulation equations. The nonlinear response is examined by means of the frequency- and force–response curves. It is shown that the two-mode solution is born when the single-mode solution undergoes the pitchfork bifurcation. The shooting method and numerical simulations are applied to investigate the dynamic solutions. Particular attention is placed on the effects of the cut-off frequency on the nonlinear response.     

Exact solution and stability of postbuckling configurations of beams

We present an exact solution for the postbuckling configurations of beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions. We take into account the geometric nonlinearity arising from midplane stretching, and as a result, the governing equation exhibits a cubic nonlinearity. We solve the nonlinear buckling problem and obtain a closed-form solution for the postbuckling configurations in terms of the applied axial load. The critical buckling loads and their associated mode shapes, which are the only outcome of solving the linear buckling problem, are obtained as a byproduct. We investigate the dynamic stability of the obtained postbuckling configurations and find out that the first buckled shape is a stable equilibrium position for all boundary conditions. However, we find out that buckled configurations beyond the first buckling mode are unstable equilibrium positions. We present the natural frequencies of the lowest vibration modes around each of the first three buckled configurations. The results show that many internal resonances might be activated among the vibration modes around the same as well as different buckled configurations. We present preliminary results of the dynamic response of a fixed–fixed beam in the case of a one-to-one internal resonance between the first vibration mode around the first buckled configuration and the first vibration mode around the second buckled configuration.     

非线性弹性基础上矩形板热后屈曲分析

给出非线性弹性基础上矩形板在均匀和非均匀（抛物型）热分布作用下的后屈曲分析。采用摄动——Ｇａｌｅｒｋｉｎ混合法给出完善和非完善矩形板热屈曲载荷和热后屈曲平衡路径。数值计算结果表明，非线性弹性基础上矩形板具有不稳定的热后屈曲平衡路径，且对初始几何缺陷是敏感的     

Stability and optimization of a clamped beam elastically restrained against translation on one end resting on Winkler foundation

In this study, stability and bimodal optimization of clamped beam elastically restrained against translation on one end subjected to a constant axially load are analyzed. The beam is positioned on elastic Winkler type foundation. The Euler method of adjacent equilibrium configuration is used in deriving the nonlinear governing equations. The critical load parameters, axial force and stiffness of foundation, are obtained for beam with the unit cross-sectional area.The shape of the beam stable against buckling that has minimal volume is determined by using Pontryagin’s maximum principle. The optimality conditions for the case of bimodal optimization are derived. The cross-sectional area for optimally designed beam is found from the solution of a nonlinear boundary value problem. New numerical results are obtained. A first integral (Hamiltonian) is used to monitor accuracy of integration. It is shown that there is the saving in material for the same buckling force.     

Postbuckling of internal pressure loaded FGM cylindrical shells surrounded by an elastic medium

This paper presents a study on the postbuckling response of a functionally graded cylindrical shell of finite length embedded in a large outer elastic medium and subjected to internal pressure in thermal environments. The surrounding elastic medium is modeled as a tensionless Pasternak foundation that reacts in compression only. The postbuckling analysis is based on a higher order shear deformation shell theory with von Kármán–Donnell-type of kinematic nonlinearity. The thermal effects due to heat conduction are also included and the material properties of functionally graded materials (FGMs) are assumed to be temperature-dependent. The nonlinear prebuckling deformations and the initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine the postbuckling response of the shells and an iterative scheme is developed to obtain numerical results without using any assumption on the shape of the contact region between the shell and the elastic medium. Numerical solutions are presented in tabular and graphical forms to study the postbuckling behavior of FGM shells surrounded by an elastic medium of tensionless elastic foundation of the Pasternak-type, from which results for conventional elastic foundations are obtained as comparators. The results reveal that the unilateral constraint has a significant effect on the postbuckling response of shells subjected to internal pressure in thermal environments when the foundation stiffness is sufficiently large.     

Transient elastic wave propagation in an infinite Timoshenko beam on viscoelastic foundation

This paper presents an effective numerical method for solving elastic wave propagation problems in an infinite Timoshenko beam on viscoelastic foundation in time domain. In order to use the finite element method to model the local complicated material properties of the infinite beam as well as foundation, two artificial boundaries are needed in the infinite system so as to truncate the infinite beam into a finite beam. This treatment requires an appropriate boundary condition derived and applied on the corresponding truncated boundaries. For this purpose, the time-dependent equilibrium equation of motion for beam is changed into a linear ordinary differential equation by using the operator splitting and the residual radiation methods. Simultaneously, an artificial parameter is employed in the derivation. As a result, the high-order accurate artificial boundary condition, which is local in time, is obtained by solving the ordinary differential equation. The numerical examples given in this paper demonstrate that the proposed method is of high accuracy in dealing with elastic wave propagation problems in an infinite foundation beam.     

Analytical solution to a mixed boundary value elastic problem of a roller-guided panel of laminated composite

This study presents an analytical solution to elastic field in a roller-guided panel of symmetric cross-ply laminated composite material. The mixed boundary value two-dimensional plane stress elasticity problem is formulated in terms of a single displacement potential function. This reduces the problem to the solution of a single fourth order partial differential equation of equilibrium as the other equilibrium equation is satisfied automatically. The solution is obtained in terms of an infinite Fourier series. To present some numerical results, a panel of glass/epoxy laminated composite is considered and different components of stress and displacement at different sections of the panel are presented graphically. To justify the present analytical solution, it is compared with the finite element solution obtained by using the commercial software ANSYS. It is found that the two solutions agree well with each other. This ensures that the formulation developed in this study based on the displacement potential approach can be used to obtain analytical solution of an elastic field in structural elements of laminated composite under any mode of boundary conditions prescribed in terms of either stress, displacement or any combination of these.     

On the Vibrations of a Simply Supported Square Plate on a Weakly Nonlinear Elastic Foundation

In this paper an initial-boundary value problem for a weakly nonlinear plate equation with a quadratic nonlinearity will be studied. This initial-boundary value problem can be regarded as a simple model describing free oscillations of a simply supported square plate on an elastic foundation. It is assumed that the foundation has a different behavior for compression and for expansion. An approximation for the solution of the initial-boundary value problem will be constructed using a two-timescales perturbation method. The existence and uniqueness of the solution of the problem will be proved. Also the asymptotic validity of the constructed approximations will be shown on long timescales. For specific parameter values, it turns out that complicated internal resonances occur.     

On the determination of missing boundary data for solids with nonlinear material behaviors,using displacement fields measured on a part of their boundaries

The paper is devoted to the derivation of a numerical method for expanding available mechanical fields (stress vector and displacements) on a part of the boundary of a solid into its interior and up to unreachable parts of its boundary (with possibly internal surfaces). This expansion enables various identification or inverse problems to be solved in mechanics. The method is based on the solution of a nonlinear elliptic Cauchy problem because the mechanical behavior of the solid is considered as nonlinear (hyperelastic or elastoplastic medium). Advantage is taken of the assumption of convexity of the potentials used for modeling the constitutive equation, encompassing previous work by the authors for linear elastic solids, in order to derive an appropriate error functional. Two illustrations are given in order to evaluate the overall efficiency of the proposed method within the framework of small strains and isothermal transformation.     

Deformation of a three-layer elastoplastic beam on an elastic foundation

We study the thermal force bending of an elastoplastic three-layer beam with a rigid filler; the beam is connected with an elastic foundation. The broken normal hypothesis is adopted to describe the kinematics of a packet nonsymmetric with respect to the thickness. The foundation reaction is described by Winkler’s model. The system of equilibrium equations and its exact solution in displacements are obtained and numerical results for a three-layer metal-polymeric beam are presented.     

NUMERICAL ANALYSIS OF THE LARGE DEFLECTION OF AN ELASTIC-PLASTIC BEAM

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Postbuckling and mode jumping analysis of composite laminates using an asymptotically correct, geometrically non-linear theory

An analytic method is presented in this paper to study the postbuckling and mode jumping behavior of bi-axially compressed composite laminates. The governing partial differential equations (PDEs) are derived rigorously from an asymptotically correct, geometrically non-linear theory. A novel and relatively simpler solution approach is developed to solve the two coupled fourth-order PDEs, namely, the compatibility equation and the dynamic governing equation. The generalized Galerkin method is used to solve boundary value problems corresponding to antisymmetric angle-ply and cross-ply composite plates, respectively. The variety of possible modal interactions is expressed in an explicit and concise form by transforming the coupled non-linear governing equations into a system of non-linear ordinary differential equations (ODEs).

The comparison between the present method and the finite element analysis (FEA) shows a pretty good match in their numerical results in the primary postbuckling region. While the FEA may lose its convergence when solution comes close to the secondary bifurcation point, the analytic approach has the capability of exploring deeply into the post-secondary buckling realm and capture the mode jumping phenomenon for various combinations of plate configurations and in-plane boundary conditions. Free vibration along the stable primary postbuckling and the jumped equilibrium paths are also studied.     

A quasilinearization approach to the solution of elastic beams on nonlinear foundations

In this paper, the solution of a beam on nonlinear elastic foundation whose deflection satisfies the nonlinear boundary value problem (1, 2), is studied by means of the theory of quasilinearization. The problem is formulated in Section 2 where conditions for the existence and uniqueness of the solution are stated. In Section 3, the idea of quasilinearization is introduced and the positivity of an associated linear differential operator is investigated. In Section 4 the usual version of quasilinearization, i.e. The Newton-Raphson-Kantorovich sequence, is presented and conditions under which this sequence is monotonically convergent, are established. In Section 5, an alternative successive approximation scheme whose derivation relies on ideas of quasilinearization, is presented. Finally, an example is solved by numerical procedures based in the methods discussed in previous sections.     

Slowing down of the growth of a crack of variable width under the influence of a temperature field

This paper touches upon changes in the temperature field near the ends of a crack of variable width under the action of an inhomogeneous stress field. The solution of the boundaryvalue problem on the equilibrium of the crack with partially contacting edges (there is adhesion of the edges on a certain part of the contact zone and slippage on another one) under the action of the outer inhomogeneous stress field, temperature field, and efforts on the contacting surfaces of the crack of variable width comparable with displacements in an elastic state is reduced to the problem of linear conjugation of analytic functions.     

From wrinkling to global buckling of a ring on a curved substrate

We present a combined analytical approach and numerical study on the stability of a ring bound to an annular elastic substrate, which contains a circular cavity. The system is loaded by depressurizing the inner cavity. The ring is modeled as an Euler–Bernoulli beam and its equilibrium equations are derived from the mechanical energy which takes into account both stretching and bending contributions. The curvature of the substrate is considered explicitly to model the work done by its reaction force on the ring. We distinguish two different instabilities: periodic wrinkling of the ring or global buckling of the structure. Our model provides an expression for the critical pressure, as well as a phase diagram that rationalizes the transition between instability modes. Towards assessing the role of curvature, we compare our results for the critical stress and the wrinkling wavelength to their planar counterparts. We show that the critical stress is insensitive to the curvature of the substrate, while the wavelength is only affected due to the permissible discrete values of the azimuthal wavenumber imposed by the geometry of the problem. Throughout, we contrast our analytical predictions against finite element simulations.     

The exact solution for the general bending problems of conical shells on the elastic foundation

The general bending problem of conical shells on the elastic foundation (Winkler Medium) is not solved. In this paper, the displacement solution method for this problem is presented. From the governing differential equations in displacement form of conical shell and by introducing a displacement function U(s,θ), the differential equations are changed into an eight-order soluble partial differential equation about the displacement function U(s,θ) in which the coefficients are variable. At the same time, the expressions of the displacement and internal force components of the shell are also given by the displacement function U(s θ). As special cases of this paper, the displacement function introduced by V.S. Vlasov in circular cylindrical shell^[5], the basic equation of the cylindrical shell on the elastic foundation and that of the circular plates on the elastic foundation are directly derived.Under the arbitrary loads and boundary conditions, the general bending problem of the conical shell on the elastic foundation is reduced to find the displacement function U(s,θ).The general solution of the eight-order differential equation is obtained in series form. For the symmetric bending deformation of the conical shell on the elastic foundation, which has been widely usedinpractice,the detailed numerical results and boundary influence coefficients for edge loads have been obtained. These results have important meaning in analysis of conical shell combination construction on the elastic foundation,and provide a valuable judgement for the numerical solution accuracy of some of the same type of the existing problem.     

Response of an infinite beam resting on a tensionless elastic foundation subjected to arbitrarily complex transverse loads

This paper presents an investigation into the static response of an infinite beam supported on a unilateral (tensionless) elastic foundation and subjected to arbitrary complex loading, including self-weight. A new numerical method is developed to determine the initially unknown lengths that remain in contact. Based on the continuity conditions at the junctions of contact and non-contact segments, the response of the whole beam may be expressed through the displacement constants of the initial segment, reducing the contact problem to two nonlinear algebraic equations with two unknowns. The technique has been named the transfer displacement function method (TDFM). Comparison with the exact results of a particular limiting case shows the expected complete agreement. Finally, an example of a beam with several contact segments is presented and verified by the application of equilibrium conditions.     

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.     

Buckling of a twisted and compressed rod

We consider the problem of determining the stability boundary for an elastic rod under thrust and torsion. The constitutive equations of the rod are such that both shear of the cross-section and compressibility of the rod axis are considered. The stability boundary is determined from the bifurcation points of a single nonlinear second order differential equation that is obtained by using the first integrals of the equilibrium equations. The type of bifurcation is determined for parameter values. It is shown that the bifurcating branch is the branch with minimal energy. Finally, by using the first integral, the solution for one specific dependent variable is expressed in terms of elliptic integrals. The solution pertaining to the complete set of equilibrium equations is obtained by numerical integration.