Buckled Position Research Articles

Mechanical Behavior of Polyethylene Pipes under Strike-Slip Fault Movements.

The present paper addresses the mechanical behaviors and failure mechanisms of buried polyethylene (PE) pipes crossing active strike slip tectonic faults based on numerical simulation of the nonlinear response of the soil-pipeline system. The developed finite element (FE) model is first verified through comparing the simulation results with those from large-scale tests and good agreement between simulation and experimental measurements is obtained. The FE model is then applied to investigate the effects of fault crossing angle, pipe and soil properties on the mechanical behavior of PE pipe. The results indicate that the PE pipe crossing negative fault angles is primarily subjected to compression and bending, thus exhibits the phenomenon of buckling. With the increase of crossing angle, there is an increase of the axial strain and the maximum Mises stress in the buckled cross section, and a decrease of the distance between the buckling position and the fault plane. While for positive crossing angles, the PE pipe is mainly subjected to tension and relatively small bending. Increasing the crossing angle causes an increase in bending strain and a decrease in the axial strain. In addition, when the fault moving speed is slower, the axial strain and bending strain are larger, whereas the maximum Mises stress in the buckled cross section and the distance between the buckled position and the fault plane are reduced. Furthermore, the most severe deformation of the pipe is observed when it is buried in the sandy soil, followed by cohesive soil and loess soil.

Experiments on the dynamics of a cantilevered pipe conveying fluid and subjected to reverse annular flow

The dynamics of a pipe conveying fluid and counter-currently subjected to a partially confined external axial flow through a coaxial annular region over its upper portion has been investigated experimentally. Apart from its fundamental aspects, this flow configuration simulates one of the modi operandi of solution-mined caverns utilized for hydrocarbon storage. Beyond the instability, at sufficiently high flow rates, the flexible pipe impacts the rigid coaxial shorter outer tube. The effects of confinement length, i.e., the ratio of the confined length of the pipe to its total length, L′/L, and the ratio of external to internal flow velocities, Uo/Ui, on the onset of instability, as well as the dynamical behaviour of the system with impacting, have been investigated. It was found that for all L′/L ratios and a very small value of Uo/Ui, i.e., Uo/Ui=0.055, the system loses stability via flutter in the second mode. Increasing the confinement length ratio expedited the aforementioned flutter. At higher Uo/Ui ratios, i.e., Uo/Ui≥0.2, the pipe is subjected to a static deformation, followed by more complex behaviour at higher flow velocities. The post-instability impact displays a rich dynamical behaviour with both periodic and chaotic components. Depending on the various system parameters, mainly the Uo/Ui and L′/L ratios and the internal flow velocity, several types of impacting/sticking behaviour were observed. The pipe may undergo single-sided or two-sided impacting, oscillation over the buckled position and partial or complete rubbing or sticking on the outer rigid tube.

Snap-through buckling of fly ash cenosphere/epoxy syntactic foams under thermal environment

Experimental investigation on deflection behaviour of fly ash cenosphere/epoxy syntactic foam at room temperature and under thermal environment (three different heating conditions) is investigated. Influence of fly ash cenosphere volume fraction and nature of temperature variation on deflection behaviour of syntactic foam beam is discussed elaborately. Results reveal that the syntactic foam beam experience snap-through buckling under thermal environment and is reflected by two bifurcation points in temperature-deflection plot. It is observed that the time duration for which the foam beam stays in the first buckled position increases with increase in cenosphere content. Thermal environment induces compressive stresses in the samples causing such snap-through buckling. However, such phenomenon is not observed when mechanical compressive loads are applied under room temperature conditions. Temperature variation across the beam strongly influences snap-through buckling in syntactic foams in addition to volume fraction of filler content.

Thermal Postbuckling and Free Vibration of Extensible Microscale Beams Based on Modified Couple Stress Theory

AbstractThis paper presents a mathematical model and a computational approach for the thermal post-buckling and free vibration in the vicinity of the buckled equilibrium position of microbeams based on the modified couple stress Euler-Bernoulli beam theory and geometrically accurate relation. The governing equations for the whole analysis are established with a intrinsic material length scale parameter to capture the size effect. These equations, in conjunction with the corresponding boundary conditions, are decomposed into two two-point boundary value problems, which are solved using the shooting method. For static deformation, the geometric nonlinearity is involved and the size dependent postbuckling configuration is obtained as a function of temperature rise. For dynamic one, the small amplitude free vibration about the prebuckled position is developed following an assumed harmonic time mode, and the length scale and temperature rise dependent fundamental natural frequency is presented. Numerical computations are executed to illustrate the size dependency in the thermal postbuckling behaviors and fundamental frequency of the vibration around the buckled configuration of microbeams.

Nonlinear vibration of a post-buckled giant-magnetostrictive thin film laminated beam

One buckled simply supported giant magnetostrictive material (GMM) thin film laminated beam model subject to axial magnetic excitations is proposed. Themagnetostrictive force issimplified as a harmonic excitation. Applying the Hamilton principle and Galerkin approach the model is expressed as a one dimensional vibrating model with parametrical excitation. The multi-scales method is used to obtain the approximation of the solution when the vibration is around the buckled position and the amplitude of the oscillation is small. With the help of the undermined fundamental frequency method and normal form theory the Melnikov approach is improved to calculate more precise threshold value of the amplitude of excitation leading to chaos. The numerical simulation shows the performance of the new Melnikov approach is much better than the traditional way.

Vibration Analysis of a Postbuckled Microscale FG Beam Based on Modified Couple Stress Theory

On the basis of modified couple stress theory, the postbuckling behavior of the Euler-Bernoulli microscale FG beams is investigated by means of an exact solution method. The modified couple stress theory as a nonclassical continuum theory is capable of interpreting the size dependencies which become more significant at micro/nanoscales. The Von-Karman type nonlinear strain-displacement relationships are employed. The thermal effects are also incorporated into formulation. The governing equation of motion and the corresponding boundary conditions are derived using Hamilton’s principle. The material properties are assumed to be graded in the thickness direction according to the power-law distribution. A closed-form solution is obtained for the postbuckling deformation which is beyond the critical buckling load. To study the vibrations taking place in the vicinity of a buckled equilibrium position, the linear vibration problem is exactly solved around the first three buckled configurations. The natural frequencies of the lowest vibration modes around each of the first three buckled configurations are obtained. The influences of power-law exponent, boundary condition, length scale parameter, and thermal environment changes on the static deflection and free vibration frequencies are studied. A comparison is also made between the present results and those obtained via the classical beam theories.

Pseudo-nonlinear dynamic analysis of buckled pipes

In this study, the post-divergence behavior of fluid-conveying pipes supported at both ends is investigated using the nonlinear equations of motion. The governing equation exhibits a cubic nonlinearity arising from mid-plane stretching. Exact solutions for post-buckling configurations of pipes with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions are investigated. The pipe is stable at its original static equilibrium position until the flow velocity becomes high enough to cause a supercritical pitchfork bifurcation, and the pipe loses stability by static divergence. In the supercritical fluid velocity regime, the equilibrium configuration becomes unstable and bifurcates into multiple equilibrium positions. To investigate the vibrations that occur in the vicinity of a buckled equilibrium position, the pseudo-nonlinear vibration problem around the first buckled configuration is solved precisely using a new solution procedure. By solving the resulting eigenvalue problem, the natural frequencies and the associated mode shapes of the pipe are calculated. The dynamic stability of the post-buckling configurations obtained in this manner is investigated. The first buckled shape is a stable equilibrium position for all boundary conditions. The buckled configurations beyond the first buckling mode are unstable equilibrium positions. The natural frequencies of the lowest vibration modes around each of the first two buckled configurations are presented. Effects of the system parameters on pipe behavior as well as the possibility of a subcritical pitchfork bifurcation are also investigated. The results show that many internal resonances might be activated among the vibration modes around the same or different buckled configurations.

On the postbuckling and free vibrations of FG Timoshenko beams

The postbuckling behavior of functionally graded beams is investigated by means of an exact solution method. The Von-Karman type nonlinear strain–displacement relationships are employed. The effects of the transverse shear deformation and rotary inertia are also included based upon the Timoshenko beam theory. After writing the kinetic and potential energy functionals, the governing equations of motion including the axial, transverse deflections and also the cross sectional rotation are derived using the Hamilton’s principle. The material properties are assumed to be graded in the thickness direction according to the power-law distribution. Neglecting the inplane inertia, the three equations of motion are reduced to two nonlinear partial–integral–differential equations in terms of the transverse mid-plane deflection and the cross sectional rotation. FG beams are considered to have fixed–fixed, fixed–hinged, and hinged–hinged end conditions. A closed-form solution is achieved for the postbuckling deformation as a function of the exerted axial load which is beyond the critical buckling load. In order to study the vibrations taking place in the vicinity of a buckled equilibrium position, the linear vibration problem is exactly solved around the first buckled configuration of a hinged–hinged FG beam. This leads to a characteristic equation whose eigenvalues are the natural frequencies and the corresponding eigenvectors also determine the mode shapes. The influences of power-law exponent, some commonly used boundary conditions and beam geometrical parameters on the static deflection and free vibration frequencies are studied. A comparison of the present results with those obtained via Euler–Bernoulli beam theory clarifies the overestimation of the frequencies by the later one.

Cross-flow-induced instability and nonlinear dynamics of cylinder arrays with consideration of initial axial load

The instability and nonlinear dynamics of planar motions of a cylinder array subjected to cross-flow have been studied via a five-mode discretization of the governing partial differential equation, focusing on the effect of initial axial load externally imposed on the cylinder. Theoretical results based on a stability analysis have indicated that, with increasing initial axial load and flow velocity, the system may lose stability either via flutter or via buckling. The boundaries of these two forms of instability are predicted analytically. To explore the post-instability dynamics of the system, a Runge–Kutta scheme is used to solve the nonlinear governing equation of motion. Three typical behaviors, including limit cycle motions of the system, are obtained. It is shown that, for relatively low flow velocity, with increasing initial axial load, just beyond the pitchfork bifurcation the cylinder would settle in a buckled equilibrium position; and for high flow velocity, however, this phenomenon only occurs when the initial axial load becomes sufficiently large.

Nonlinear Dynamics of Supported Cylinder Subjected to Axial Flow

In this paper, the stability and nonlinear dynamics are studied for a slender flexible cylinder subjected to axial flow. A nonlinear model is presented, based on the corresponding linear equation of motion, for dynamics of the cylinder supported at both ends. The nonlinear terms considered here are only the additional axial force induced by the lateral motions of the cylinder. Using six-mode discretized equation, numerical simulations are carried out for the dynamical behavior of the cylinder to explain, with this relatively simple nonlinear model, the flutter instability found in experiment. The results of numerical analysis show that at certain value of flow velocity the system loses stability by divergence, and the new equilibrium (the buckled configuration) becomes unstable at higher flow leading to post-divergence flutter. As the flow velocity increases further, the quasiperiodic motion around the buckled position occurs, and this evolves into chaotic motions at higher flow.

LATERAL BUCKLING ANALYSIS OF ANGLED FRAMES WITH THIN-WALLED I-BEAMS

With the inclusion of warping deformations, lateral buckling behavior for angled frames with I-beams under the action of in-plan bending moments will be conducted in this paper. By adopting the joint equilibrium for the angled frame and the force-displacement relations for the members defined at the buckled position (rather than the initial position), the analytical solutions for buckling moments of an angled frame with simply supported ends are presented in closed form. The computed results indicate that for symmetric buckling, the buckling resistance to resist positive moments can be strengthened along with the increase of inclined angle of the member of an angled frame, but the buckling resistance against negative moments is reduced. On the other hand, for anti-symmetric buckled modes, the buckling strength to carry positive moments generally rises with the increase of inclined angles. However, an interesting phenomenon indicates that the buckling capacity to resist negative moments may reach its minimum at certain inclined angles. It means that the rotational restraints of torsion and bending at the joint of such a critical angled frame are situated in a minimum state to resist the anti-symmetric buckled modes.

Postbuckling and free vibrations of composite beams

Abstract An exact solution for the postbuckling configurations of composite beams is presented. The equations governing the axial and transverse deformations of a composite laminated beam accounting for the midplane stretching are derived. The inplane inertia and damping are neglected, and hence the two equations are reduced to a single nonlinear fourth-order partial–integral–differential equation governing the transverse deformations. We find out that the governing equation for the postbuckling of symmetric or asymmetric composite beams has the same form as that of beams made of an isotropic material. Composite beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions are considered. A closed-form solution for the postbuckling deformation is obtained as a function of the applied axial load, which is beyond the critical buckling load. To study the vibrations that take place in the vicinity of a buckled equilibrium position, we exactly solved the linear vibration problem around the first buckled configuration. Solving the resulting eigen-value problem results in the natural frequencies and their associated mode shapes. Both the static response represented by the postbuckling analysis and the dynamic response represented by the free vibration analysis in the postbuckling domain strongly depend on the lay-up of the laminate. Variations of the beam’s midspan rise and the fundamental natural frequency of the postbuckling domain vibrations with the applied axial load are presented for a variety of lay-up laminates. The ratio of the axial stiffness to the bending stiffness was found to be a crucial parameter in the analysis. This control parameter, through the selection of the appropriate lay-up, can be manipulated to help design and optimize the static and dynamic behavior of composite beams.

Experimental investigation of a buckled beam under high-frequency excitation

In this paper, we investigate theoretically and experimentally dynamics of a buckled beam under high-frequency excitation. It is theoretically predicted from linear analysis that the high-frequency excitation shifts the pitchfork bifurcation point and increases the buckling force. The shifting amount increases as the excitation amplitude or frequency increases. Namely, under the compressive force exceeding the buckling one, high-frequency excitation can stabilize the beam to the straight position. Some experiments are performed to investigate effects of the high-frequency excitation on the buckled beam. The dependency of the buckling force on the amounts of excitation amplitude and frequency is compared with theoretical results. The transient state is observed in which the beam is recovered from the buckled position to the straight position due to the excitation. Furthermore, the bifurcation diagrams are measured in the cases with and without high-frequency excitation. It is experimentally clarified that the high-frequency excitation changes the nonlinear property of the bifurcation from supercritical pitchfork bifurcation to subcritical pitchfork bifurcation and then the stable steady state of the beam exhibits hysteresis as the compressive force is reversed.

Buckling and Free Oscillations of Composite Microresonators

Free oscillations of piezoelectric, microelectromechanical resonators are considered in this effort. These resonators are modeled as clamped-clamped composite structures, with stepwise varying properties across the length of the resonator. The different features of the model development are discussed, and buckling in these resonators is studied. A nonlinear analysis conducted to study oscillations about a buckled position is presented. The results of the analysis are found to compare well with the experimental observations.

Nonlinear Random Response of Panels in an Elevated Thermal-Acoustic Environment

Sonic fatigue is generally considered as being one of the major design areas for the newest generation of high-speed flight vehicles. Efficient analysis methods for predicting nonlinear random response and fatigue life are urgently needed. This paper presents a finite element formulation for the prediction of nonlinear random response of thin isotropic/composite panels subjected simultaneously to high acoustic loads and elevated temperatures. Laminated plate theory and von Karman large displacement relations are used to derive the nonlinear equations of motion for an arbitrarily laminated composite panel subjected to combined acoustic and thermal loads. The nonlinear equations of motion in physical degrees of freedom are transformed to a set of coupled nonlinear equations in truncated modal coordinates, retaining fewer degrees of freedom. Numerical integration is employed to obtain the panel response to simulated Gaussian band-limited white noise. To validate the formulation, results are compared with existing linear and nonlinear solutions to assess the accuracy of nonlinear modal stiffness matrices and simulated random loads. Examples are given for an isotropic panel at various combinations of sound pressure levels and temperatures. Numerical results include rms values of maximum deflection and strain, time histories of deflection and strain response, probability distribution functions, power spectrum densities, and higher statistical moments. Numerical results predicted all three types of panel motions for a thermal buckled simply supported isotropic plate: linear random vibration about one of the buckled equilibrium position, snap-through motions between the two buckled positions, and nonlinear random response over both buckled positions.

Dynamic stability of a curved beam under sinusoidal loading

This paper is a study of snap-through properties of a non-linear dynamic buckling response to sinusoidal excitation of a clamped—clamped buckled beam. Using a simple formula, the highly non-linear motion of snap-through and its effects on the overall vibration response have been studied. The non-linear governing equation obtained here is solved using the Runge—Kutta (RK-4) numerical integration method. Critical parameters at the onset of the snap-through motion, which vary with different damping coefficients and linear circular frequencies of a flat beam, are studied and given in terms of the excitation level and response displacement. The relationships between static and dynamic responses at the start of the snap-through motion are also predicted. The analysis brings out various characteristic features of the phenomenon, i.e. (a) small oscillations about the buckled position, (b) chaotic motion of intermittent snap-through and (c) large oscillations of continuous snap-through motion crossing the two buckled positions. The non-linear dynamic instability behaviour of the beam, changing from the softening spring type to the hardening type, is due to the snap-through motion.

Non-linear free vibration analysis of composite plates with initial imperfections and in-plane loading

In this paper the non-linear response of layered composite plates with initial imperfections and subjected to in-plane preload is considered. The basic equations used in the analysis are those corresponding to an accurate shear deformation theory which employs parabolic shear strain variations across the thickness and requires no correction factors. The five governing non-linear equilibrium equations of the problem are reduced to a single non-linear ordinary differential equation (ODE) using a single mode approach in conjunction with the Galerkin method. Numerical results are obtained tor two problems: 1.(i) large amplitude vibration of imperfect plates and2.(ii) small oscillations in the vicinity of a static buckled position of an in-plane loaded two-layered (0 90) plate.