Nonlinear Dynamic Buckling Research Articles

Nonlinear thermal vibration and dynamic buckling of eccentrically stiffened sandwich-FGM cylindrical shells containing fluid

The governing equations for analysing thermal vibration and dynamic buckling of eccentrically stiffened sandwich functionally graded cylindrical shells full filled with fluid and surrounded by elastic foundations in thermal environment are derived by using the classical shell theory, the geometrical nonlinearity in von Karman-Donnell sense, the smeared stiffener technique and Pasternak’s foundation model. Solutions of the problem are established according to the Galerkin’s method and Runge–Kutta method. The effects of fluid pressure, stiffeners, geometrical ratios, temperature and elastic foundation on the dynamic responses of shells are investigated.

Nonlinear vibration and dynamic buckling analyses of sandwich functionally graded porous plate with graphene platelet reinforcement resting on Winkler–Pasternak elastic foundation

The nonlinear vibration and the dynamic buckling of a graphene platelet reinforced sandwich functionally graded porous (GPL-SFGP) plate are thoroughly investigated in this paper. The investigated GPL-SFGP plate consists of two metal face layers and a functionally graded porous core with graphene platelet reinforcement. The effects of the Winkler–Pasternak elastic foundation, thermal environment and damping are incorporated. The open-cell metal foam model is implemented to model the mechanical properties of the porous core. Axial compressive stress is applied on the GPL-SFGP plate by exerting various compressive loading speeds at one edge of the plate. Grounded on the classical plate theory, both motion and geometric compatibility equations of the plate are deduced by introducing the Von Kármán strain-displacement relationship and stress function. Both the Galerkin and the fourth-order Runge–Kutta methods are implemented to solve the governing equation of the dynamic system. Meticulously designed numerical experiments are conducted to identify the critical influential factors of the dynamic stability of the GPL-SFGP plate. The influences of loading speed, damping ratio, temperature variation, initial imperfection, elastic foundation parameters, porosity, GPL weight fraction and the dimensions of the GPL on the overall dynamic stability of the GPL-SFGP plate are evidently demonstrated.

Nonlinear Dynamic Buckling of Fixed Shallow Arches under an Arbitrary Step Radial Point Load

AbstractThis paper investigates the nonlinear elastic in-plane dynamic buckling of a fixed shallow circular arch that was subjected to an arbitrarily located step radial point load, which has not b...

Nonlinear dynamic buckling of the imperfect orthotropic E-FGM circular cylindrical shells subjected to the longitudinal constant velocity

In this study, an analytical approach on the nonlinear dynamic buckling of the orthotropic circular cylindrical shells made of exponential law functionally graded material (E-FGM) subjected to the longitudinal constant velocity is investigated with the incorporation of mercurial damping effect. The material properties are assumed to vary gradually in the thickness direction according to an exponential distribution function of the volume fraction of constituent materials. Theoretical formulations are derived based on improved Donnell shell theory (DST) and accounting for von-Kármán strain–displacement relation, initial imperfection and damping effect. By applying Galerkin method and Airy's stress function, the obtained nonlinear differential equations are solved numerically by the fourth-order Runge–Kutta method. The nonlinear dynamic stability of the orthotropic FG cylindrical shell is assessed based on Budiansky–Roth criterion. Additionally, a parametric study is conducted to demonstrate the effects of various velocities, initial imperfections, damping ratios, inhomogeneous parameters on nonlinear dynamic buckling behavior of an imperfect orthotropic FG cylindrical shell. Comparing results with those in other publications validates the proposed method.

Nonlinear dynamic stability analysis of Euler–Bernoulli beam–columns with damping effects under thermal environment

In this study, a unified nonlinear dynamic buckling analysis for Euler–Bernoulli beam–columns subjected to constant loading rates is proposed with the incorporation of mercurial damping effects under thermal environment. Two generalized methods are developed which are competent to incorporate various beam geometries, material properties, boundary conditions, compression rates, and especially, the damping and thermal effects. The Galerkin–Force method is developed by implementing Galerkin method into force equilibrium equations. Then for solving differential equations, different buckled shape functions were introduced into force equilibrium equations in nonlinear dynamic buckling analysis. On the other hand, regarding the developed energy method, the governing partial differential equation for dynamic buckling of beams is also derived by meticulously implementing Hamilton’s principles into Lagrange’s equations. Consequently, the dynamic buckling analysis with damping effects under thermal environment can be adequately formulated as ordinary differential equations. The validity and accuracy of the results obtained by the two proposed methods are rigorously verified by the finite element method. Furthermore, comprehensive investigations on the structural dynamic buckling behavior in the presence of damping effects under thermal environment are conducted.

Nonlinear size-dependent dynamic buckling analysis of embedded micro cylindrical shells reinforced with agglomerated CNTs using strain gradient theory

This article presents an analysis on the nonlinear dynamic buckling of a micro cylindrical shell reinforced by carbon nanotubes (CNTs) with considering agglomeration effects. The structure is surrounded by an elastic medium and is subjected to magnetic field and harmonic mechanical load. Mindlin theory is employed to model the structure and the modified strain gradient theory (SGT) is also used to capture the size effect. Mori–Tanaka approach is used to estimate the equivalent material properties of the nanocomposite cylindrical shell and consider the CNTs agglomeration effect. The motion equations are derived using Hamilton’s principle and the differential quadrature method (DQM) along with Bolotin method is employed to solve them for obtaining the dynamic stability region. The effect of different parameters including magnetic field, CNTs volume percent and agglomeration effect, boundary conditions, elastic medium, size effect and length to thickness ratio on the dynamic buckling behavior of the system is studied. The results indicate that considering elastic medium, magnetic field and size effects, increases the stiffness of the structure and so, the dynamic instability region occurs at higher frequency while considering the CNTs agglomeration effect has inverse effect. Also, by increasing the CNTs volume percent and considering the clamped–clamped boundary condition, the dynamic behavior of the system improves.

Nonlinear dynamic characteristics and stability of composite orthotropic plate on elastic foundation under thermal environment

An analytical computational scheme for nonlinear dynamic characteristics and stability of an eccentrically composite orthotropic plate on Winkler-Pasternak elastic foundation subjected to different axial velocities is proposed with the incorporation of mercurial damping effects under thermal environment. Incorporating the classical plate theory and Von-Kármán strain-displacement relation, the nonlinear compatibility equation is derived. The Galerkin method and Airy’s stress function are implemented to establish the nonlinear dynamic buckling equation accommodating the thermal and damping effects. Then the developed nonlinear differential equations are solved numerically by the fourth-order Runge-Kutta method. The characteristics of natural frequency, linear and nonlinear vibration, frequency-amplitude curve and nonlinear dynamic responses are investigated by the developed approach with validations by other literatures. The nonlinear dynamic buckling loads are determined by using Budiansky-Roth criterion. Additionally, various effects of velocity, damping ratio, temperature change, buckling mode, initial imperfection and foundation parameter on nonlinear dynamic buckling of the orthotropic plate are discussed.

Effect of forcing frequency on nonlinear dynamic pulse buckling of imperfect rectangular plates with different boundary conditions

The present study was aimed to investigate the influence of forcing frequency on nonlinear dynamic pulse buckling of imperfect rectangular plates with six different boundary conditions. The Galerkin's approximate method on the basis of polynomial and trigonometric mode shape functions is used to reduce the governing nonlinear partial differential equations to ordinary nonlinear differential equations. Moreover a numerical study of these governing equations is accomplished by Runge Kutta integration methods. The convergence of the polynomial and trigonometric mod shape functions are investigated to compute the dynamic response of plate. The effects of frequency of impulse loading and boundary conditions on the deflection histories of plate are studied. The dynamic response of plate subjected to impulsive loading with different forcing frequency is compared to results obtained by exponential impulsive loading. The results show that, by increasing the forcing frequency of impulsive loading, the maximum displacement of plate increases and converge with lower values to response of plate subjected to exponential impulse. Moreover, different boundary conditions and various pulse functions have significant influence on the dynamic response of the plate.

Dynamic snap-through buckling of CNT reinforced composite sandwich spherical caps

This paper investigates the nonlinear dynamic buckling response of axisymmetric carbon nanotubes reinforced composite sandwich spherical caps subjected to suddenly applied pressure. To solve this problem, a finite element approach is adopted using an axisymmetric shear flexible shell element, free from spurious constraints causing shear and membrane locking especially for the case of thin shells. The geometric nonlinearity is accounted for in the formulation using von Karman’s strain-displacement relations. Newmark’s integration technique along with the modified Newton-Raphson iteration scheme is employed to solve the nonlinear governing equations. The critical dynamic pressure value is considered as the pressure load beyond which the maximum average displacement response shows instant growth in the time history of the shell structure. The obtained results are validated against the available analytical solutions for the case of isotropic spherical caps. A detailed parametric study is carried out to bring out the effects of shell geometry parameter, volume fraction of the CNT, core-to-face sheet thickness, temperature, boundary conditions and different types of pulse on the dynamic snap-through characteristics of spherical shells.

Nonlinear dynamical analyses of eccentrically stiffened functionally graded toroidal shell segments surrounded by elastic foundation in thermal environment

Abstract In this study, the nonlinear vibration and dynamic buckling of eccentrically stiffened functionally graded toroidal shell segments surrounded by an elastic medium in thermal environment are presented. The governing equations of motion of eccentrically stiffened functionally graded toroidal shell segments are derived based on the classical shell theory with the geometrical nonlinear in von Karman-Donnell sense and the smeared stiffeners technique. Furthermore, the dynamical characteristics of shells as natural frequencies, nonlinear frequency–amplitude relation, nonlinear dynamic responses and the nonlinear dynamic critical buckling loads evaluated by Budiansky-Roth criterion are considered. The effects of characteristics of functionally graded materials, geometrical ratios, elastic foundation, pre-loaded axial compression and temperature on the dynamical behavior of shells are investigated.

A new approach for nonlinear dynamic buckling of S-FGM toroidal shell segments with axial and circumferential stiffeners

This paper presents an analytical approach to study the nonlinear dynamic buckling of sigmoid functionally graded material (S-FGM) toroidal shell segments surrounded by elastic foundations subjected to axial compression. Shells are reinforced by closely spaced rings and stringers. The nonlinear dynamic buckling analysis is based on the classical shell theory with von Kármán–Donnell-type of geometrical nonlinearity. By using the Galerkin method, the closed-form expression of the deflection–time curves is obtained. Numerical solutions are presented in tabular and graphical forms to investigate the deflection–time curves of FGM shells surrounded by elastic foundations. Finally, the influences of geometrical parameters, volume fraction index, elastic foundations, and the effectiveness of stiffeners on the nonlinear dynamic buckling of the S-FGM toroidal shell segments are discussed. The present results are compared with the available data for a special case.

Nonlinear thermal dynamic buckling of FGM beams

Abstract Based on the nonlinear dynamic analysis, dynamic buckling and imperfection sensitivity of the FGM Timoshenko beam subjected to sudden uniform temperature rise are studied. Initial geometric imperfection of the beam is also taken into account. It is assumed that during deformation the beam is resting over a conventional three-parameter elastic foundation with softening/hardening cubic non-linearity. The analysis is performed with considering temperature dependency assumption of each thermo-mechanical property of the FGM beam. The governing nonlinear dynamic equations are derived based on the generalized Hamilton principle. In the spatial approximation of the problem, a set of ordinary differential equations in time is obtained by the conventional multi-term Ritz method. These equations are converted into a set of algebraic equations by utilizing the Newmark family of time approximation scheme. The obtained non-linear algebraic equations are solved via the well known Newton–Raphson iterative scheme. The Budiansky–Roth criterion is used to detect the unbounded motion type of dynamic buckling. Results reveal that for beams with stable post-buckling equilibrium path, no dynamic buckling occurs according to the Budiansky–Roth criterion. However, dynamic buckling may occur for the FGM beams resting on sufficiently stiff softening elastic foundation due to their unstable post-buckling equilibrium paths.

Nonlinear dynamic buckling of imperfect rectangular plates with different boundary conditions subjected to various pulse functions using the Galerkin method

In this paper, the nonlinear dynamic pulse buckling of imperfect rectangular plate subjected to sinusoidal, exponential, damping and rectangular pulse functions with six different boundary conditions is investigated. In order to solve the large deformation equations of plate, Galerkin method together with trigonometric mode shape functions is applied. Also, the nonlinear coupled time integration of the governing equation of plate is a solved employing fourth-order Runge–Kutta method. The effects of boundary conditions, pulse functions, initial imperfection, force pulse amplitude and geometrical parameters of the shock spectrum of a plate and deflection histories of plate for impulsive, dynamic and quasi-periodic force loads are studied. In this study, the effects of boundary conditions, pulse functions, initial imperfection, force pulse amplitude and geometric parameter in nonlinear dynamic response are investigated. According to the results for impulsive loading, the displacement response reaches its peak after shock duration and in the dynamic and quasi-static pulse loading, the maximum response of plate occurs during and before shock duration. Moreover, with increasing the loading amplitude, length of the plate and initial imperfection, the maximum displacement of plate increases. Different boundary conditions and various pulse functions have significant influence on the dynamic response of the plate. By increasing the restriction in supports, the resistance ability against deformation and stability of plate increases.

Semi-analytical approach for analyzing the nonlinear dynamic torsional buckling of stiffened functionally graded material circular cylindrical shells surrounded by an elastic medium

In this study, we propose a semi-analytical method for analyzing the nonlinear dynamic behavior of functionally graded material (FGM) thin circular cylindrical shells surrounded by an elastic medium under a time-dependent torsional load. The shell is stiffened by a closely spaced orthogonal FGM stiffeners system, which is attached to the inner surface of the shell. The elastic medium is assumed to follow the two-parameter elastic foundation model proposed by Pasternak. The fundamental relations and governing equations for stiffened cylindrical shells are derived using the Donnell shell theory with von Karman geometrical nonlinearity and the smeared stiffeners technique. A deflection function with three terms is employed to consider the nonlinear buckling shape in a more correct manner. The Galerkin method is used. The critical dynamic torsional load is found by using the fourth order Runge–Kutta method and Budiansky–Roth criterion. Based on computations, we investigated the effects of the stiffener, foundations, material, and dimensional parameters on the dynamic buckling behavior of the shell.

An analytical approach to analyze nonlinear dynamic response of eccentrically stiffened functionally graded circular cylindrical shells subjected to time dependent axial compression and external pressure. Part 2: Numerical results and discussion

Based on the classical thin shell theory with the geometrical nonlinearity in von Karman-Donnell sense, the smeared stiffener technique, Galerkin method and an approximate three-term solution of deflection taking into account the nonlinear buckling shape is chosen, the governing nonlinear dynamic equations of eccentrically stiffened functionally graded circular cylindrical shells subjected to time dependent axial compression and external pressure is established in part 1. In this study, the nonlinear dynamic responses are obtained by fourth order Runge-Kutta method and the nonlinear dynamic buckling behavior of stiffened functionally graded shells under linear-time loading is determined by according to Budiansky-Roth criterion. Numerical results are investigated to reveal effects of stiffener, input factors on the vibration and nonlinear dynamic buckling loads of stiffened functionally graded circular cylindrical shells.

Closed form solutions for predicting static and dynamic buckling behaviors of a drillstring in a horizontal well

The nonlinear static and dynamic buckling (snaking motion) analysis of a rotating drilling string laterally constrained in a horizontal well is presented, through a group of fourth-order nonlinear partial differential equations . The analytical approximate solutions to the static and dynamic buckling are obtained via combining Newton linearization with Harmonic Balance Method, and Galerkin's method, respectively. On the basis of the analytical approximate solutions, static post-buckling deformation, critical dynamic buckling load, and two different kinds of snaking motions (i.e. the pipe moves up and down around its static buckling configuration; the pipe moves from one side of the wellbore to the other side) are investigated. Accuracy of the approximate solutions is verified by comparing with numerical solutions. Theoretical results are useful for practical design applications related to calculation of buckling loads and selection of bottom hole-assembly (BHA) elements and pipe rotational speeds . What's more, the solving procedures of accurate analytical approximate solutions to the snaking motions yield rapid convergence with respect to exact numeric solutions. The present results are valid for small as well as large oscillation amplitudes. By combining Newton linearization with Harmonic Balance Method and Galerkin's method, the analytical approximate solutions are explicitly established for static and dynamic buckling of a horizontal drilling string, respectively. • Nonlinear static and dynamic buckling of drillpipe in a horizontal well is studied. • Analytical approximate solutions are obtained by Improved HB and Galerkin method. • The present solution procedures are brief. • Accuracy of approximate solution is verified by comparing with numerical solution.

Nonlinear static and dynamic buckling of eccentrically stiffened functionally graded cylindrical shells under axial compression surrounded by an elastic foundation

This paper presents an analytical approach to investigate the nonlinear buckling of imperfect eccentrically stiffened functionally graded thin circular cylindrical shells subjected to axial compression and surrounded by an elastic foundation. Based on the classical thin shell theory with the geometrical nonlinearity in von Karman-Donnell sense, initial geometrical imperfection, the smeared stiffeners technique and Pasternak’s two-parameter elastic foundation, the governing equations of eccentrically stiffened functionally graded cylindrical shells are derived. The functionally graded cylindrical shells are reinforced by homogeneous ring and stringer stiffener system on internal and (or) external surface. The resulting equations are solved by the Galerkin method to obtain the explicit expression of static critical buckling load, post-buckling load-deflection curve and nonlinear dynamic motion equation. The nonlinear dynamic responses are found by using fourth order Runge-Kutta method. The dynamic critical buckling loads of shells are considered for step loading of infinite duration and linear-time compression. The obtained results show the effects of foundation, stiffeners and input factors on the nonlinear buckling behavior of these structures.

Nonlinear dynamic analysis of eccentrically stiffened functionally graded circular cylindrical thin shells under external pressure and surrounded by an elastic medium

A semi-analytical approach eccentrically stiffened functionally graded circular cylindrical shells surrounded by an elastic medium subjected to external pressure is presented The elastic medium is assumed as two-parameter elastic foundation model proposed by Pasternak. Based on the classical thin shell theory with the geometrical nonlinearity in von Karman–Donnell sense, the smeared stiffeners technique and Galerkin method, this paper deals the nonlinear dynamic problem. The approximate three-term solution of deflection shape is chosen and the frequency–amplitude relation of nonlinear vibration is obtained in explicit form. The nonlinear dynamic responses are analyzed by using fourth order Runge–Kutta method and the nonlinear dynamic buckling behavior of stiffened functionally graded shells is investigated according to Budiansky–Roth criterion. Results are given to evaluate effects of stiffener, elastic foundation and input factors on the frequency–amplitude curves, natural frequencies, nonlinear responses and nonlinear dynamic buckling loads of functionally graded cylindrical shells.

Multiple unstable equilibrium branches and non-linear dynamic buckling of shallow arches

An arch under a suddenly-applied load will oscillate about its equilibrium position. If the suddenly-applied load is sufficiently large, the oscillation may reach a position on the unstable equilibrium branch of the arch, triggering its dynamic buckling. In many cases, arches are supported by other structural members or by elastic foundations which provide elastic types of rotational restraints to the ends of the arch. When the rotational end restraints of an arch are not equal to each other, the in-plane non-linear equilibrium path of the arch may have multiple unstable branches, which will influence the dynamic buckling of the arch significantly. This paper investigates effects of multiple unstable equilibrium branches on the non-linear in-plane dynamic buckling of a shallow circular arch under a suddenly-applied central concentrated load. Two methods based on the energy approach are used to derive the dynamic buckling loads. It is found that the number and magnitude of dynamic buckling loads are influenced significantly by the number of unstable equilibrium branches, by the stiffness of the unequal rotational end restraints, and by the included angle and slenderness ratio of the arch.

Nonlinear static and dynamic buckling analysis of imperfect eccentrically stiffened functionally graded circular cylindrical thin shells under axial compression

An analytical approach is presented to investigate the nonlinear static and dynamic buckling of imperfect eccentrically stiffened functionally graded thin circular cylindrical shells subjected to axial compression. Based on the classical thin shell theory with the geometrical nonlinearity in von Karman–Donnell sense, initial geometrical imperfection and the smeared stiffeners technique, the governing equations of motion of eccentrically stiffened functionally graded circular cylindrical shells are derived. The functionally graded cylindrical shells with simply supported edges are reinforced by ring and stringer stiffeners system on internal and (or) external surface. The resulting equations are solved by the Galerkin procedure to obtain the explicit expression of static critical buckling load, post-buckling load–deflection curve and nonlinear dynamic motion equation. The nonlinear dynamic responses are found by using fourth-order Runge–Kutta method. The dynamic critical buckling loads of shells under step loading of infinite duration are found corresponding to the load value of sudden jump in the average deflection and those of shells under linear-time compression are investigated according to Budiansky–Roth criterion. The obtained results show the effects of stiffeners and input factors on the static and dynamic buckling behavior of these structures.