Dynamic Buckling Analysis Research Articles

Dynamic visco-elastic buckling analysis of an in vitro airway model: prediction of circumferential and axial buckling mode numbers

The purpose of the research reported here was to elucidate the mechanism of formation of the various lobes observed in asthmatic airways by both theoretical and experimental analysis employing an in vitro airway model. The rationale is that the elucidated mechanism will facilitate the development of new diagnostic methods and treatment regimens for asthma. Lobe formation was analyzed on the basis of an assumption of cross-sectional buckling of the airway. Here, we propose a dynamic visco-elastic buckling model analysis of the airway for the prediction of circumferential and axial buckling mode numbers. The calculated circumferential buckling mode numbers were in reasonably good agreement with those measured in the dynamic buckling experiment using the in vitro airway model. The calculated axial buckling mode numbers were in qualitative agreement with those observed in the experiment. The non-dimensional parameters related to the remodeling and the consequent pathologies occurring in asthmatic airways were also shown, and the influence of changes in the non-dimensional parameters on the circumferential and axial buckling mode numbers was also calculated. The circumferential and axial buckling mode numbers decreased due to thickening and stiffening of the basement membrane. Thickening of the tissues surrounding the basement membrane and the increase in the mucus secreted in the airway lumen were modeled as an increase in the added mass on the basement membrane. The results of calculation showed that the circumferential and axial buckling mode numbers increased because of the thickening of the surrounding tissues and the increase in mucus secretion. We suggest that it may be possible to diagnose the severity of asthma by using the results of the calculation of the changes in the buckling mode numbers caused by the changes in the strength of the remodeling. The physiological reality of the in vitro airway model reported here is discussed using the non-dimensional parameters.

Nonlinear static and dynamic buckling analysis of functionally graded shallow spherical shells including temperature effects

This paper presents an analytical approach to investigate the nonlinear static and dynamic unsymmetrical responses of functionally graded shallow spherical shells under external pressure incorporating the effects of temperature. Governing equations for thin FGM spherical shells are derived by using the classical shell theory taking into account von Karman–Donnell geometrical nonlinearity. Approximate solutions are assumed and Galerkin procedure is applied to determine explicit expressions of static critical buckling loads of the shells. For the dynamical response, motion equations are numerically solved by using Runge–Kutta method and the criterion suggested by Budiansky–Roth. A detailed analysis is carried out to show the effects of material and geometrical parameters, boundary conditions and temperature on the stability and dynamical characteristics of FGM shallow spherical shells.

Buckling analysis and optimal structural design of supercavitating vehicles using finite element technology

The supercavitating vehicle is an underwater vehicle that is surrounded almost completely by a supercavity to reduce hydrodynamic drag substantially. Since the cruise speed of the vehicle is much higher than that of conventional submarines, the drag force is huge and a buckling may occur. The buckling phenomenon is analyzed in this study through static and dynamic approaches. Critical buckling load and pressure as well as buckling mode shapes are calculated using static buckling analysis and a stability map is obtained from dynamic buckling analysis. When the finite element method (FEM) is used for the buckling analysis, the solver requires a linear static solver and an eigenvalue solver. In this study, these two solvers are integrated and a consolidated buckling analysis module is constructed. Furthermore, Particle Swarm Optimization (PSO) algorithm is combined in the buckling analysis module to perform a design optimization computation of a simplified supercavitating vehicle. The simplified configuration includes cylindrical shell structure with three stiffeners. The target for the design optimization process is to minimize total weight while maintaining the given structure buckling-free.

Dynamic Stability Analysis of Shells Using a Finite Element Based Reduction Method

AbstractA finite element formulation of a reduction method for dynamic stability analysis of imperfection‐sensitive shell structures is presented. The reduction method makes use of a perturbation approach, initially developed for static buckling and later extended to dynamic buckling analysis. The single mode dynamic buckling analysis and its extension to parametric excitation analysis are described. The approach is available within a general purpose finite element code. Characteristic results for the parametric excitation analysis of a composite cylindrical shell are shown. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Nonlinear thermomechanical dynamic buckling analysis of imperfect viscoelastic composite/sandwich shells by a double-superposition global–local theory and various constitutive models

Almost no dynamic buckling analysis has been performed so far for the sandwich/multilayer viscoelastic shells. Even the vibration analyses of the mentioned shells have been restricted to the harmonic loads ignoring the transverse stresses and their continuity at the mutual interfaces of the layers, and the transverse flexibility of the shell. In the present paper, a high-order double-superposition global–local theory inherently suitable for nonlinear analyses is proposed and employed for nonlinear dynamic buckling and postbuckling analyses of imperfect viscoelastic composite/sandwich cylindrical shells subjected to thermomechanical loads. Depending on the nature of the applied loads, both complex modulus and hierarchical constitutive models are used for the viscoelastic materials. Results reveal that as the time duration of the suddenly applied loads decreases beyond the first natural period of the shell, the dynamic buckling load becomes much higher than the static buckling load, especially for the rectangular load–time histories. Furthermore, the relaxation behavior of the viscoelastic material may decrease the dynamic buckling load.

Dynamic buckling analysis of composite cylindrical shells using a finite element based perturbation method

In this paper a finite element formulation of a reduction method for dynamic buckling analysis of imperfection-sensitive shell structures is presented. The reduction method makes use of a perturbation approach, initially developed for static buckling and later extended to dynamic buckling analysis. The implementation of a single-mode dynamic buckling analysis in a general purpose finite element code is described. The effectiveness of the approach is illustrated by application to the dynamic buckling of composite cylindrical shells under axial and radial step loads. Results of the reduction method are compared with results available in the literature. The results are also compared with full model finite element explicit dynamic analysis, and a reasonable agreement is obtained.

A nonlinear double-superposition global–local theory for dynamic buckling of imperfect viscoelastic composite/sandwich plates: A hierarchical constitutive model

Abstract Nonlinear dynamic thermo-mechanical buckling and postbuckling analyses of imperfect viscoelastic composite laminated/sandwich plates are performed by a proposed theory that takes into account all the interlaminar kinematic and transverse stress continuity conditions, for the first time. Even the dynamic buckling analysis of the multi-layered/sandwich plates employing the hierarchical constitutive model has not been performed before. The proposed theory is a double-superposition high-order global–local theory that is calibrated based on the nonlinear strain–displacement expressions for the thermoelastic loadings taking into account the structural damping. The buckling loads are determined based on a criterion previously published by the author. Various complex sensitivity analyses evaluating effects of the relaxation parameters, rate of the loading, sudden heating, and pre-stress on thermo-mechanical buckling of the viscoelatic multi-layered/sandwich plates are performed. Results show that the viscoelastic behavior may decrease the buckling load. Sudden dynamic buckling loads are higher due to the reflected stress waves.

A double-superposition global–local theory for vibration and dynamic buckling analyses of viscoelastic composite/sandwich plates: a complex modulus approach

A higher-order global–local theory is proposed based on the double-superposition concept for free vibration and dynamic buckling analyses of viscoelastic composite/sandwich plates subjected to thermomechanical loads. In contrast to all theories proposed so far for analysis of the viscoelastic plates, the continuity conditions of the transverse shear and normal stresses at the layer interfaces and the nonzero traction conditions at the top and bottom surfaces of the sandwich plates are satisfied. Another novelty is that these conditions may be satisfied for viscoelastic plates with temperature-dependent material properties and nonlinear behaviors subjected to thermomechanical loads. Furthermore, transverse flexibility is also taken into account. Some dynamic buckling/wrinkling analyses of the viscoelastic plates are performed in the present paper, for the first time. Comparisons made between results of the paper and results reported by well-known references confirm the accuracy and the efficiency of the proposed theory and the relevant solution algorithm.

Rate-dependent constitutive model of poly(ethylene terephthalate) for dynamic analysis

Uniaxial tensile testing at strain rates ranging from 10^(-3) to 10^(-1) s^(-1) was carried out to study the rate-dependent mechanical behavior for poly(ethylene terephthalate) (PET) used in the packaging industry. The experimental results show that a rate-dependent plastic behavior exists for PET material. The value of the yield strength was found to increase with the increasing strain rate. A new constitutive model based on the improved Cowper-Symonds rate-dependent constitutive model is proposed to describe the mechanical behavior of PET material in the strain rate ranging from 10^(-3) to 10^(-1) s^(-1), providing more accurate material data for the subsequent simulation analysis of drop test and dynamic buckling. The predictions obtained using the proposed model are compared with experimental results of the improved Cowper-Symonds model. The simulating results of the proposed model agree well with the experimental data. For a low strain rate, the predictions of this model are more precise than those obtained using the improved Cowper-Symonds model. This confirms that the new constitutive model is suitable for describing the mechanical behavior of PET material at a low strain rate and modeling impact problem.

Exact solution and dynamic buckling analysis of a beam-column system having the elliptic type loading

This paper presents a closed form solution to the dynamic stability problem of a beam-column system with hinged ends loaded by an axial periodically time-varying compressive force of an elliptic type, i.e., a 1cn2(τ, k 2) + a 2sn2(τ, k 2) + a 3dn2(τ, k 2). The solution to the governing equation is obtained in the form of Fourier sine series. The resulting ordinary differential equation is solved analytically. Finding the exact analytical solutions to the dynamic buckling problems is difficult. However, the availability of exact solutions can provide adequate understanding for the physical characteristics of the system. In this study, the frequency-response characteristics of the system, the effects of the static load, the driving forces, and the frequency ratio on the critical buckling load are also investigated.

Non-linear dynamic thermo-mechanical buckling analysis of the imperfect laminated and sandwich cylindrical shells based on a global-local theory inherently suitable for non-linear analyses

The available accurate shell theories satisfy the interlaminar transverse stress continuity conditions based on linear strain–displacement relations. Furthermore, in majority of these theories, either influence of the transverse normal stress and strain or the transverse flexibility of the shell has been ignored. These effects remarkably influence the non-linear behavior of the shells especially in the postbuckling region. Furthermore, majority of the buckling analyses performed so far for the laminated composite and sandwich shells have been restricted to linear, static analysis of the perfect shells. Moreover, almost all the available shell theories have employed the Love–Timoshenko assumption, which may lead to remarkable errors for thick and relatively thick shells. In the present paper, a novel three-dimensional high-order global-local theory that satisfies all the kinematic and the interlaminar stress continuity conditions at the layer interfaces is developed for imperfect cylindrical shells subjected to thermo-mechanical loads.In comparison with the layerwise, mixed, and available global-local theories, the present theory has the advantages of: (1) suitability for non-linear analyses, (2) higher accuracy due to satisfying the complete interlaminar kinematic and transverse stress continuity conditions, considering the transverse flexibility, and releasing the Love–Timoshenko assumption, (3) less required computational time due to using the global-local technique and matrix formulations, and (4) capability of investigating the local phenomena. To enhance the accuracy of the results, compatible Hermitian quadrilateral elements are employed. The buckling loads are determined based on a criterion previously published by the author.

ENERGY APPROACH FOR DYNAMIC BUCKLING OF AN UNDAMPED ARCH MODEL UNDER STEP LOADING WITH INFINITE DURATION

Performing a dynamic buckling analysis of structures is more difficult than carrying out its static buckling analysis counterpart. Some structures have a nonlinear primary equilibrium path including limit points and an unstable equilibrium path. They may also have bifurcation points at which equilibrium bifurcates from the primary equilibrium path to an unstable secondary equilibrium path. When such a structure is subjected to a load that is applied suddenly, the oscillation of the structure may reach the unstable primary or secondary equilibrium path and the structure experiences an escaping-motion type of buckling. For these structures, complete solutions of the equations of motion are usually not needed for a dynamic buckling analysis, and what is really sought are the critical states for buckling. Nonlinear dynamic buckling of an undamped two degree-of-freedom arch model is investigated herein using an energy approach. The conditions for the upper and lower dynamic buckling loads are presented. The merit of the energy approach for dynamic buckling is that it allows the dynamic buckling load to be determined without the need to solve the equations of motion. The solutions are compared with those obtained by an equation of motion approach.

Energy approach for dynamic buckling of shallow fixed arches under step loading with infinite duration

Shallow fixed arches have a nonlinear primary equilibrium path with limit points and an unstable postbuckling equilibrium path, and they may also have bifurcation points at which equilibrium bifurcates from the nonlinear primary path to an unstable secondary equilibrium path. When a shallow fixed arch is subjected to a central step load, the load imparts kinetic energy to the arch and causes the arch to oscillate. When the load is sufficiently large, the oscillation of the arch may reach its unstable equilibrium path and the arch experiences an escaping-motion type of dynamic buckling. Nonlinear dynamic buckling of a two degree-of-freedom arch model is used to establish energy criteria for dynamic buckling of the conservative systems that have unstable primary and/or secondary equilibrium paths and then the energy criteria are applied to the dynamic buckling analysis of shallow fixed arches. The energy approach allows the dynamic buckling load to be determined without needing to solve the equations of motion.

Dynamic buckling of columns by biaxial moments and uniform end torque

A new concept of uniform torque is proposed for the dynamic torsional buckling analysis. A dynamic biaxial moments and torque buckling theory is presented for analysis in structural dynamics. Second-order effects of the axial force, biaxial moments and torque are considered. The consistent natural boundary moments and forces are derived to ensure the symmetry of the dynamic stiffness matrix in fulfilling the requirement of the reciprocal theorem and conservation of energy. The exact dynamic stiffness matrix is obtained using power series expansion. The derivatives of the analytical dynamic stiffness matrix with respect to different loading and geometric parameters are derived explicitly for sensitivity and continuation analyses. Generally distributed axial force can be analyzed without difficulty. It is pointed out that non-uniform sections may not be handled by power series due to the convergent problem. Global pictures for all kinds of linear dynamic buckling are given for the first time. The methodology is based on finite element formulation and therefore it can easily be extended to analyze structural frames.

Dynamic Visco-elastic Buckling Analysis for Airway Model

In order to clarify the mechanism by which the lung airway narrows during an asthma attack, dynamic buckling analysis of the wall was conducted. The wall was modeled using a visco-elastic thin-walled circular cylinder of the Voigt model for the planestress state. A governing equation for dynamic buckling was derived, and in the equation, the contraction of smooth muscle was replaced by uniform inward transmural pressure. The non-dimensional parameters for the buckling wave number n were nondimensional retardation time τ, non-dimensional increasing velocity of inward transmural pressure β, thickness radius ratio α2, radius length ratio η, density ratio ζ, and Poisson's ratio ν. The validity of the theoretical model was confirmed by comparing the calculated wave number with that obtained from the experiment, in which a silicone rubber tube blended with silicone potting gel was used as the in vitro airway model. In addition, the wave number n increased with β. It was necessary to consider the damping effect of the tube model or the airway wall, and n increased by 1.5 to 2 due to the additional mass effect of surrounding tissues of the basement membrane in the airway wall.

Non-linear dynamic thermo-mechanical buckling analysis of the imperfect sandwich plates based on a generalized three-dimensional high-order global–local plate theory

Abstract The available plate theories either have not considered the interlaminar stress continuity condition or have been calibrated based on linear strain–displacement relations. Moreover, almost all buckling analyses performed so far employing the global–local plate theories, were restricted to linear, static buckling analyses of the perfect plates, neglecting the transverse normal strain and stress. Researches available in literature for dynamic buckling analyses of the sandwich plates are very rare. In the present paper, a generalized high-order global–local theory that satisfies all the kinematic and transverse stress continuity conditions at the interfaces of the layers, is proposed to investigate dynamic buckling of imperfect sandwich plates subjected to thermo-mechanical loads. In comparison to the layerwise, mixed, and available global–local theories, the present theory has the advantages of: (1) less required computational time due to using the global–local technique and matrix formulations, (2) higher accuracy due to satisfying the complete interlaminar kinematic and transverse stress continuity conditions and considering the transverse flexibility, (3) suitability for non-linear analyses, (4) capability of investigating the local phenomena, such as the wrinkling. To enhance the accuracy of the results, compatible Hermitian quadrilateral elements are employed. The buckling loads are determined based on a criterion previously published by the author.

Vibration and dynamic buckling control of imperfect hybrid FGM plates with temperature-dependent material properties subjected to thermo-electro-mechanical loading conditions

Dynamic buckling analysis of FGM plates has not been accomplished so far. In the present paper, vibration and dynamic buckling of FGM rectangular plates with surface-bonded or embedded piezoelectric sensors and actuators subjected to thermo-electro-mechanical loading conditions are investigated. A finite element formulation based on a higher-order shear deformation theory is developed. Both initial geometric imperfections of the plate and temperature-dependency of the material properties are taken into account. Dynamic buckling of plates already pre-stressed by other forms of loading conditions is assumed to occur under suddenly applied thermal or mechanical loads. A nine-node second order Lagrangian element, an efficient numerical algorithm for solving the resulted highly non-linear governing equations, and an instability criterion already proposed by the author are employed. A simple negative velocity feedback control is used to actively control the dynamic response of the plate. Results show that generally, initial geometric imperfections lead to an increased fundamental bending natural frequency and decreased buckling loads. Furthermore, buckling mitigation due to utilizing integrated piezoelectric sensors and actuators is mainly achieved in extremely high gain values. Therefore, the piezoelectricity effect on the buckling load is small in applicable voltages. It is also noticed that the temperature-dependency and initial geometric imperfections remarkably affect the buckling loads.

Analysis of Airway Narrowing Based on Dynamic Buckling Model

In order to elucidate the mechanism of airway narrowing in asthma, dynamic buckling analysis was performed for basement membrane. The basement membrane was treated as two-dimensional circular thin-walled viscoelastic shell of Voigt model and contraction of smooth muscle was replaced by inward transmural pressure applied to the basement membrane. Governing equations for the dynamic buckling was derived and wave number of buckling mode obtained were compared with numerical simulation results by non-linear FEM and with experimental results. The non-dimensional parameters are thickness radius ratio, non-dimensional increasing speed of inward transmural pressure, non-dimensional retardation time and Poisson ratio. Wave number of buckling mode is determined as the number having maximum amplification from initial imperfection mode. The wave number of buckling mode increases with increase in non-dimensional increasing speed of inward transmural pressure and it decreases with increase in non-dimensional retardation time. It has been known that wave number of buckling mode decreases due to thickening and hardening of basement membrane caused by the remodeling in asthma. This fact is originated from decrease in non-dimensional increasing speed of inward transmural pressure due to the remodeling. In the dynamic buckling analysis the damping effect of the basement membrane is necessary to take into consideration, because the primary component of the basement membrane is collagen having damping characteristics.

Dynamic buckling of flat and curved sandwich panels with transversely compressible core

In the present study, the effect of the transverse compressibility of the core on the transient dynamic response of structural sandwich panels under rapid loading conditions is investigated. The analysis is based on a higher-order sandwich shell theory in an effective multilayer formulation. The model is based on the standard Kirchhoff–Love hypothesis for the face sheets whereas a first/second order power series expansion is employed for the core. Consistent equations of motion and boundary conditions are derived by means of Hamilton’s principle. An analytical solution is obtained by an extended Galerkin procedure. The theory is applied to the dynamic buckling and postbuckling analyses of plane and curved sandwich panels subjected to rapidly applied tangential and transverse loads. It is observed that the transverse compressibility of the core can have distinct effects on both, the frequency and the amplitudes of the resulting free oscillations. Due to interactions between the overall oscillation and a local oscillation in the presence of a face wrinkling instability mode, the dynamic response can be chaotic, resulting in oscillating face wrinkling instability modes with unpredictable amplitudes.

Dynamic buckling of a 2-DOF imperfect system with symmetric imperfections

Non-linear dynamic buckling of a two-degree-of freedom (2-DOF) imperfect planar system with symmetric imperfections under a step load of infinite duration (autonomous system) is thoroughly discussed using energy and geometric considerations. This system under the same load applied statically exhibits (prior to limit point) an unstable symmetric bifurcation lying on the non-linear primary equilibrium path. With the aid of the total energy-balance equation of the system and the particular geometry (due to symmetric imperfections) of the plane curve corresponding to the zero level total potential energy “surface” exact dynamic buckling loads are obtained without solving the non-linear initial-value problem. The efficiency and the reliability of the technique proposed herein is demonstrated with the aid of various dynamic buckling analyses which are compared with numerical simulation using the Verner–Runge–Kutta scheme, the accuracy of which is checked via the energy-balance equation.