Nonlinear Dynamic Buckling Research Articles

Nonlinear dynamic buckling of pinned–fixed shallow arches under a sudden central concentrated load

The structural behavior of a shallow arch is highly nonlinear, and so when the amplitude of the oscillation of the arch produced by a suddenly-applied load is sufficiently large, the oscillation of the arch may reach a position on its unstable equilibrium paths that leads the arch to buckle dynamically. This paper uses an energy method to investigate the nonlinear elastic dynamic in-plane buckling of a pinned–fixed shallow circular arch under a central concentrated load that is applied suddenly and with an infinite duration. The principle of conservation of energy is used to establish the criterion for dynamic buckling of the arch, and the analytical solution for the dynamic buckling load is derived. Two methods are proposed to determine the dynamic buckling load. It is shown that under a suddenly-applied central load, a shallow pinned–fixed arch with a high modified slenderness (which is defined in the paper) has a lower dynamic buckling load and an upper dynamic buckling load, while an arch with a low modified slenderness has a unique dynamic buckling load.

Nonlinear Dynamic Buckling for Truncated Sandwich Shallow Conical Shell Structure Subjected to Pulse Impact

Nonlinear dynamic buckling for sandwich shallow conical shell structure under uniform triangular pulse is investigated. Based on the Reissner’s assumption and Hamiton’s principle, the nonlinear dynamic governing equation of sandwich shallow spherical shells is derived. The corresponding nonlinear dynamic response equations are obtained by Galerkin method and solved by Runge-Kutta method. Budiansky-Roth criterion expressed by displacements of rigid center is employed to determine the critical impact bucking load. The effects of geometric parameters and physical parameters on impact buckling are discussed.

The nonlinear dynamic buckling response of functionally graded truncated conical shells

In this study, the nonlinear dynamic buckling of functionally graded (FG) truncated conical shells subjected to axial compressive load varying as a linear function of time is investigated. The material properties of the FG truncated shell are assumed to vary continuously through the thickness of the shell. The nonlinear pre-buckling deformations of the FG truncated conical shell are taken into account. The fundamental relations and modified Donnell type nonlinear dynamic stability and compatibility equations of the FG truncated conical shell are derived and solved by using the Superposition principle, Galerkin and Runge–Kutta methods. The values of the dimensionless nonlinear critical time parameter have been found numerically. Finally, carrying out some computations, the effects of the compositional profiles, the variation of the truncated conical shell geometric parameters and the axial loading speed on the dimensionless linear and nonlinear critical time parameters have been studied. Comparing the results of this study with those in the literature validates the present analysis.

Nonlinear dynamic buckling of shallow circular arches under a sudden uniform radial load

The structural behavior of a shallow arch is highly nonlinear, and so when the amplitude of the oscillation of the arch produced by a suddenly applied load is sufficiently large, the oscillation of the arch may reach a position at its primary unstable equilibrium path or secondary bifurcation unstable equilibrium path, leading the arch to buckle dynamically. This paper presents an analytical study of the nonlinear dynamic in-plane buckling of a shallow circular arch under a uniform radial load that is applied suddenly and with an infinite duration. The principle of conservation of energy is used to establish the criterion for dynamic buckling of the arch, and the analytical solution for the dynamic buckling load is derived. It is shown that under a suddenly applied uniform radial load, a shallow pinned–fixed arch has a unique possible dynamic buckling load, while shallow pinned–pinned and fixed–fixed arches may have two possible dynamic buckling loads: a lower dynamic buckling load and an upper dynamic buckling load. The dynamic buckling loads of a shallow arch under a suddenly applied uniform radial load with infinite duration are found to be lower than their static counterparts, and to increase with an increase of the arch included angle and slenderness. The effect of static preloading on the dynamic buckling of an arch is also investigated. It is found that the pre-applied static load decreases the dynamic buckling load of the arch, but increases the sum of the pre-applied load and the dynamic buckling load.

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.

Nonlinear dynamical analysis of eccentrically stiffened functionally graded cylindrical panels

Based on the classical shell theory with the geometrical nonlinearity in von Karman–Donnell sense and the smeared stiffeners technique, the governing equations of motion of eccentrically stiffened functionally graded cylindrical panels with geometrically imperfections are derived in this paper. The characteristics of free vibration and nonlinear responses are investigated. The nonlinear dynamic buckling of cylindrical panel acted on by axial loading is considered. The nonlinear dynamic critical buckling loads are found according to the criterion suggested by Budiansky–Roth. Some numerical results are given and compared with the ones of other authors.

Nonlinear Dynamic Buckling of FGM Shallow Conical Shells under Triangular Pulse Impact Loads

In this paper, nonlinear dynamic buckling of FGM shallow conical shells under the action of triangular pulse impact loads are investigated. The nonlinear dynamic governing equation of symmetrically FGM shallow conical shells is built. Using Galerkin method, the nonlinear dynamic governing equation is solved, and the nonlinear dynamic response equation of symmetrically FGM shallow conical shells is obtained. The Runge-Kutta method is introduced to numerically solve the nonlinear dynamic response equation and the impact response curve is achieved. Budiansky-Roth motion criterion expressed by the displacement of the peak of the shell is employed to determine the critical impact buckling load. The influences of geometric parameters and gradient constants on impact buckling are discussed as well.

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.

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.

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.

Nonlinear dynamic buckling of functionally graded cylindrical shells subjected to time-dependent axial load

Abstract This paper is presented to solve the nonlinear dynamic buckling problem of a new type of composite cylindrical shells, made of ceram/metal functionally graded materials. The material properties vary smoothly through the shell thickness according to a power law distribution of the volume fraction of the constituent materials. The dynamic axial load is set in a linear increase form with regard to time. By taking the temperature-dependent material properties into account, the effect of environmental temperature rise is included. The nonlinear dynamic equilibrium equation of the shell was obtained by applying an energy method, and was then solved using the four-order Runge–Kutta method. The critical condition was eventually determined using B-R dynamic buckling criterion. Numerical results show the dynamic buckling load is higher than its static counterpart. Meanwhile, various effects of the inhomogeneous parameter, loading speed, dimension parameter, environmental temperature rise and initial geometrical imperfection on nonlinear dynamic buckling are discussed.

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.

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Nonlinear dynamic response and buckling of laminated cylindrical shells with axial shallow groove based on a semi-analytical method

The effect of axial shallow groove on the nonlinear dynamic response and buckling of laminated cylindrical shells subjected to radial compression loading was investigated. Based on the first-order shear deformation theory (FSDT), the nonlinear dynamic equations involving the transverse shear deformation and initial geometric imperfections were derived with the Hamilton philosophy. The axial shallow groove of the laminated composite cylindrical shell was treated as the initial geometric imperfections in the dynamic equations. A semi-analytical method of expanding displacements and loads along the circumferential direction and employing the finite difference method along the axial direction and in the time domain is used to solve the governing equations and obtain the dynamic response of the laminated shell. The B-R criterion was employed to determine the critical loads of dynamic buckling of the shell. The effects of the parameters of the shallow groove on the dynamic response and buckling were discussed in this paper and the results show that the axial shallow grooves greatly affect the dynamic response and buckling.

Nonlinear Dynamic Thermal Buckling of Functionally Graded Spherical Caps

isotropic/orthotropic/functionally graded material spherical shells suddenly exposed to thermal environment is rather meager in the literature and such studies are important to the structural designers. In the present work, the nonlinear dynamic thermal buckling of functionally graded spherical caps is investigated using a threenoded shear flexible axisymmetric curved shell element based on field-consistencyprinciple[6].Geometricnonlinearityisassumedin thepresentstudy,usingvonKarman’sstrain-displacement relations. In addition, the formulation includes in-plane and rotary inertia effects. The material properties are graded in the thickness direction according to the power-law distribution in terms of volume fractions of the constituents of the material. The nonlinear governing equations derived are solved employing Newmark’s numerical integration method in conjunction with the modified Newton– Raphson iteration scheme. The critical dynamic buckling temperature difference is taken as the temperature difference between the shell surfaces corresponding to a sudden jump in the maximum average displacement in the time history [2,12]. Numerical results arepresented that considerdifferent values ofgeometrical parameter and power-law index.

On the nonlinear axisymmetric dynamic buckling behavior of clamped functionally graded spherical caps

Here, the dynamic thermal buckling behavior of functionally graded spherical caps is studied considering geometric nonlinearity based on von Karman's assumptions. The formulation is based on first-order shear deformation theory and it includes the in-plane and rotary inertia effects. The material properties are graded in the thickness direction according to the power-law distribution in terms of volume fractions of the material constituents. The effective material properties are evaluated using homogenization method. The governing equations obtained using finite element approach are solved employing the Newmark's integration technique coupled with a modified Newton–Raphson iteration scheme. The pressure load corresponding to a sudden jump in the maximum average displacement in the time history of the shell structure is taken as the dynamic buckling load. The present model is validated against the available isotropic case. A detailed numerical study is carried out to highlight the influences of shell geometries, power law index of functional graded material and boundary conditions on the dynamic buckling load of shallow spherical shells.

Dynamic buckling of a simple geometrically imperfect frame using Catastrophe Theory

This paper deals with nonlinear static and dynamic buckling of a geometrically imperfect two-bar frame due to initially crooked bars, which is subjected to an eccentrically applied load at its joint. The analysis is facilitated by considering the frame (being a continuous system) as one degree-of-freedom (1-DOF) system with generalized coordinate unknown the column axial force and then by employing catastrophe theory. Through a local analysis via Taylor's expansion of the nonlinear equilibrium equation of the frame, one can classify the total potential energy (TPE) function of the frame to the canonical form of the corresponding TPE function of the seven elementary Thom's catastrophes. Using energy criteria static catastrophes are extended to the corresponding dynamic catastrophes of undamped frames under step loading (autonomous systems) by conveniently determining the dynamic singularity and bifurcational sets. Numerical examples associated with static and dynamic fold catastrophes demonstrate the efficiency and reliability of the present approach.

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.