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Double-beam systems have found many applications in civil and mechanical engineering. Extensive works of research have been conducted on the investigation of double-beam systems in which some have studied the vibration characteristics of general double-beam systems without considering the effect of axial load, whereas others investigated the vibration and buckling of axially loaded double-beam systems with classical boundary conditions. A few previous studies have considered a general axially loaded double-beam system, but none of them have provided a general solution to calculate the critical buckling load of the system with arbitrary boundary conditions. In the present study, free vibration and buckling of an axially loaded double-beam system, which was composed of two nonidentical beams with generalized boundary conditions, were analytically studied by using the Euler–Bernoulli beam theory. A semianalytical method was proposed to calculate the natural frequencies and mode shapes of the system, and an explicit formula was derived to determine the critical buckling load of the system. Parametric investigations were performed to study the influences of the connecting stiffness of the interlayer and axial loads on the dynamic characteristics and critical buckling load of the system. Results show that the proposed method can be used for free vibration and buckling analyses of generalized double-beam systems subjected to axial loads. [ABSTRACT FROM AUTHOR]

An accurate nonlinear hygrothermal-electro-elastic (HTEE) buckling analysis of piezoelectric fiber-reinforced composite cylindrical shells subjected to the coupled loading effects of axial compression and hydrostatic pressure was established by considering the nonuniform prebuckling effect. Nonlinear governing equations were derived based on higher-order shear deformation theory and Novozhilov's nonlinear shell theory. Accurate critical buckling stresses and pressures and explicit buckling modes for both axisymmetric and nonaxisymmetric buckling were obtained by the Galerkin method. A comparison between the new prediction and existing results is presented and excellent agreement is reported. A comprehensive parametric study of geometric parameters, end conditions, distribution patterns, and hygrothermal-electric multiphysical fields on the buckling behavior of HTEE composite cylindrical shell is also analyzed and discussed. [ABSTRACT FROM AUTHOR]

• The stability of sandwich nanobeams are interconnected via Kerr-type foundation is investigated. • The three couple equations are extracted using the Eringen's nonlocal elasticity theory. • The buckling loads are determined using the generalized differential quadrature method. • Results are discussed and the importance of some geometric and material parameters is learned. In this study, the endurable buckling load of the elastically connected parallel sandwich nano-beams with varying cross-sections is assessed. In this regard, a layered beam system consisting of two parallel axially loaded tapered sandwich nanobeams that are interconnected via a Kerr-type three-parameter elastic foundation is considered. The geometric properties are assumed to be changed exponentially along the length of the beam element. The governing equilibrium equations of the system are described by a set of three coupled homogeneous differential equations, which originates in the context of Eringen's non-local elasticity theory and Euler beam model. Then, the numerical differential quadrature technique is used to estimate the endurable axial critical loads. Eventually, a comprehensive parameterization research is performed to investigate the sensitivity of linear buckling resistance to tapering ratio, nonlocal parameter, stiffness of elastic connections, volume fraction exponent, and thickness ratio. The research work of the present study is novel, and the attained numerical outcomes can be used as benchmarks for future researches in this field. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive analysis of a nonlinear buckling of a cylinder with a crack with the new regression equation. For this purpose, the effective material properties and geometry parameters are considered. For geometry properties, the length, thickness and diameter for cylinder, the length, location and angle of crack are considered. For material properties, the modulus of elasticity, yield stress and plastic properties are considered. Then using the response surface method, a regression equation for predicting the cylinder's critical buckling load is proposed. To validate the results, some experimental tests are presented and the results show that there is good agreement between the proposed equation and experimental tests. Then the effects of considering parameters on the critical buckling load of the cylinder are investigated. The results show that in low values for thickness the yield stress has a more negligible effect, but for large thickness values, the yield stress has more effect on critical buckling load. Furthermore, the critical crack occurs when the crack locates perpendicular to the direction of the loading. Also, the crack's location and the cylinder's boundary condition are effective parameters on critical buckling load. Because of the interaction effect of material and geometry properties, the material and geometry parameters should be considered simultaneously to predict the critical buckling load. [ABSTRACT FROM AUTHOR]

The sensitivity of critical buckling load and critical stress concerning different geometrical and physical parameters of Euler{Bernoulli nanobeams with defects is studied. Eringen's nonlocal theory of elasticity is used for the determination of critical buckling load for stepped nanobeams subjected to axial loads for different support conditions. An analytical approach to study the impact of discontinuities and boundary conditions on the critical buckling load and critical stress of nanobeams has been developed. Critical buckling loads of stepped nanobeams are defined under the condition that the nanoelements are weakened with stable crack-like defects. Simply supported, clamped and cantilever nanobeams with steps and cracks are investigated in this article. The presented results are compared with the other available results and are found to be in a close agreement. [ABSTRACT FROM AUTHOR]

• Equations of a hyperelastic beam under time-varying axial loading are derived. • It falls into a barreling state with an axial load lower than the critical load. • Its transverse vibration is analyzed by Galerkin and harmonic balance methods. • The material parameter can change the behavior of its transverse vibration. Equations of motion of a hyperelastic beam under time-varying axial loading are derived via the extended Hamilton's principle in this work, where the transverse vibration is coupled with the longitudinal vibration, and nonlinear vibrations of the beam in the subcritical buckling regime are investigated. Complex nonlinear boundary conditions of the beam are determined under some geometric constraints. The critical buckling load is first determined through linear bifurcation analysis. Effects of material and geometric parameters on the forced longitudinal vibration of the beam are numerically investigated. Steady harmonic shapes of the beam at different times under harmonic axial loading are determined. The beam is in the barreling deformation state even when the axial load is not in excess of the critical buckling load. The governing equation for the nonlinear transverse vibration of the beam is obtained by decoupling its equations of motion. Natural frequencies of the free linearized transverse vibration of the beam are studied. By applying the eigenfunction expansion method, the governing equation for the nonlinear transverse vibration of the beam transforms to a series of strongly nonlinear ordinary differential equations (ODEs). Two-to-one internal resonance of the beam is studied by the numerical integration method and its phase-plane portraits are obtained. The harmonic balance method and pseudo arc-length method are used to determine steady-state periodic solutions of the beam from the strongly nonlinear ODEs, and amplitude-frequency responses of the beam are determined. Effects of the external mean axial load, excitation amplitude, and damping coefficient on the amplitude-frequency response of the beam are numerically investigated. Combined effects of the external excitation amplitude and frequency on response amplitudes are also investigated. [ABSTRACT FROM AUTHOR]

This study aims to analyze the effect of thermomechanical coupling damage caused by laser irradiation on a Carbon Fiber Reinforced Polymer (CFRP) cylindrical shell with axial compression using finite element method (FEM). The quasi-static thermomechanical coupling damage process of a typical CFRP structure was studied intensively. First, a thermomechanical coupling model was established, then the influence of the size of the laser spot, the thickness of the shell and the different exposure location of the laser spot on the critical buckling load (CBL) were investigated. The results show that the value of CBL decreases near quadratically as the shell thickness decreases. The value of CBL decreases nonlinearly with increase of the spot size. The CBL decreases and then increases until the laser reaches the midpoint of cylinder axis as the laser moves along the axial direction of the cylinder; the CBL decreases dramatically as the laser deviates from the axis. [ABSTRACT FROM AUTHOR]

Pile foundations used in liquefiable deposits often have large slenderness ratios, and these piles are particularly vulnerable to earthquake-induced soil liquefaction and occur buckling instability under axial load alone as soils around the pile liquefy. However, insight into the precise impact of inertial load on the buckling behavior of piles in liquefiable deposits is still lacking, and the current methods used to calculate the critical buckling load of the pile in liquefiable ground usually ignore inertial load. In light of the above, two shaking table tests were carried out to clarify the impact of inertial load on the buckling behavior of piles partly embedded in saturated sands. Moreover, a novel calculation method considering inertial load was presented to accurately estimate the critical buckling load of the pile in liquefiable deposits. The feasibility of the presented method had been verified using the results from a centrifuge experiment and an analytical method along with a finite element method. Furthermore, parametric analysis was conducted to study the effect of soil type, peak ground acceleration (PGA), pile diameter, and unsupported length ratio on the critical buckling load of the pile in liquefiable soils. Finally, comparing the calculated results of critical buckling load between the method presented in this paper and the method in the design code confirmed the limitation of the current design code. These combined analyses demonstrate that the critical buckling load of the pile decreases due to the action of inertial load, resulting in the pile being more susceptible to buckling failure. Therefore, designers have to consider the impact of inertial load during the seismic design of pile foundations in liquefiable soils. • Two shake-table tests are carried out to study the effect of inertial load on piles in liquefiable soils. • A novel method for calculating the critical buckling load of piles considering inertial loads is presented. • The key parameters influencing the buckling instability of piles are analyzed. [ABSTRACT FROM AUTHOR]

This paper focuses on the dynamic buckling of thin cylindrical shells with arbitrary axisymmetric thickness variation under time dependent axial pressure. Based on the derivation of stability and compatibility equations of variable thickness cylindrical shells under dynamic external pressure by Aksogan and Sofiyev, the corresponding stability and compatibility equations of thin cylindrical shells with arbitrary axisymmetric thickness variation under dynamic axial pressure are obtained and expressed in nondimensional form. Combining the small parameter perturbation method, Fourier series expansion and the Sachenkov–Baktieva method, analytical formulas of the critical buckling load of thin cylindrical shells with arbitrary axisymmetric thickness variation under axial pressure that varies as a power function of time are obtained. Two cases of thickness variation are introduced to research the critical dynamic buckling load with the present formulas. Effects of thickness variation parameters and loading speed of dynamic axial pressure on the critical buckling load are discussed. The method is also applied to determine the critical dynamic buckling load of thin cylindrical shells with a classical cosine form thickness variation. Results revel that the thickness variation and pressure parameters play a major role in dictating the buckling capacity of thin cylindrical shells under dynamic axial pressure. [ABSTRACT FROM AUTHOR]

Buckling loads of partial composite columns under distributed axial loads is investigated in this paper for the first time. The interlayer interaction corresponding to the level of interfacial bonding imperfection in the layered heavy composite columns is formulated in the model by a shear slip/stiffness modulus. Governing differential equations and boundary equations are derived and represented in a general dimensionless form. A semi-analytical solution is applied to the governing buckling equations of the presented model using power-series technique to extract critical loads of partial composite columns. Five different classical end types are considered namely clamped–clamped (C-C), clamped-pinned (C-P), clamped-sliding (C-S), clamped-free (C-F) and pinned–pinned (P-P). Also, for two extreme cases of non-composite/zero-interaction and full-composite/perfectly-bonded layered columns, exact closed-form characteristic buckling equations are introduced. A convergence study is conducted to ensure stability of the applied power-series solution. It is demonstrated that the obtained buckling loads for partial composite columns with different end conditions approach those obtained from the exact closed-from solution for the full-composite extreme case when the interfacial shear modulus approaches infinity. Effect of imperfect bonding between the column layers and slip on critical buckling loads is investigated. It is shown that a more realistic model based on the partial composite interaction hypothesis predicts critical loads that are less than those based on idealized composite columns with perfect interfacial bonding. [ABSTRACT FROM AUTHOR]

• A general nonlinear double-beam model resting on an elastic foundation is newly established. • Double-beam systems have multivalued critical buckling loads, and the carrying capacity of targeted beam can be greatly strengthened through the distribution of the two axial loads. • A simple criterion is proposed to quickly estimate whether or not the axial loads would trigger the buckling of the system under various symmetric/asymmetric boundaries. • In-phase and out-of-phase postbuckling paths and their associated stabilities of symmetric/asymmetric systems are investigated in detail, and the snap-through buckling phenomenon is detected. • Strengthening the system stiffness may even trigger the out-of-phase buckling more easily for asymmetric systems. Double-beam systems composed of two interconnected parallel beams have become basic mechanical components of modern engineering structures. Focusing on the asymmetric features in boundary conditions and other structural parameters, this paper studies the buckling and postbuckling behaviors of a double-beam system supported on an elastic foundation. The nonlinear governing differential equations of the system are derived by using the Hamilton's principle, and discretized via the differential quadrature method. Then the pseudo-arc length algorithm is adopted to overcome difficulties of numerical continuation around bifurcation points. Compared with single beam systems, double-beam systems exhibit some peculiarities like multivalued critical buckling loads, in-phase and out-of-phase postbuckling paths. Based on the proposed procedures, these buckling and postbuckling behaviors are investigated in detail. For symmetric boundary cases of the two sub-beams, the multivalued critical buckling loads of the system compose a cluster of critical curves that intersect at a point determined only by the critical loads of single beams. Hence a simple criterion based on single beam critical loads is proposed to quickly estimate whether or not the axial loads would trigger the buckling of the system. However for asymmetric boundary cases, the cluster of critical curves locate in a region that depends on the intrinsic double-beam characteristics, and the criterion is revised accordingly. Also for asymmetric systems, some unique features of the out-of-phase postbuckling paths are detected. It is found that their stabilities are quite different from those of symmetric systems, and that strengthening the system stiffness may even trigger the out-of-phase buckling more easily. Graphical abstract [Display omitted]. [ABSTRACT FROM AUTHOR]

This paper presents an analytical solution for calculating the critical buckling load of simply supported cellular columns when they buckle about the major axis. The solution takes into account the influence of web shear deformation on the buckling of cellular columns and is derived using the stationary principle of potential energy. The formula derived for calculating the critical buckling load is validated using finite element analysis results. It is shown from the present analytical solution that the web shear deformation can significantly reduce the buckling resistance of cellular columns. The influence of the shear deformation on the critical buckling load increases with the cross-section area of the tee section and the radius of circular holes but decreases with the length and the web thickness of the cellular column. [ABSTRACT FROM AUTHOR]

In this study, a two-stage optimization framework is proposed for cylindrical or flat stiffened panels under uniform or non-uniform axial compression, which are extensively used in the aerospace industry. In the first stage, traditional sizing optimization is performed. Based on the buckling or collapse-like deformed shape evaluated for the optimized design, the panel can be divided in sub-regions each of which shows characteristic deformations along axial and circumferential directions. Layout optimization is then performed using a stiffener spacing distribution function to represent the location of each stiffener. A layout coefficient is assigned to each sub-region and the overall layout of the panel is optimized. Three test problems are solved in order to demonstrate the validity of the proposed optimization framework: remarkably, the load-carrying capacity improves by 17.4 %, 66.2 % and 102.2 % with respect to the initial design. [ABSTRACT FROM AUTHOR]

Highlights: [•] Elastically connected parallel beams under compressive axial loading. [•] Kerr type elastic layer. [•] Explicit expressions of natural frequencies and the associated amplitude ratio. [•] Effects of compressive axial load on the responses of the double beam system. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the mechanical buckling load of an eccentrically stiffened truncated conical shells made of functionally graded materials and subjected to axial compressive load and external uniform pressure by analytical method. Shells are reinforced by stringers and rings. The change of spacing between stringers in the meridional direction is taken into account. Material properties of shell are graded in the thickness direction according to a volume fraction power-law distribution. The equilibrium and linearized stability equations for stiffened shells are derived based on the classical shell theory and smeared stiffeners technique. The resulting equations which they are the couple set of three variable coefficient partial differential equations in terms of displacement components are investigated by Galerkin method and the closed-form expression for determining the buckling load is obtained. The effects of stiffeners, material and dimensional parameters are analyzed in detail. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Based on Reddy's third-order shear deformation theory, an exact closed-form solution is proposed to describe linear buckling of transversely isotropic laminated rectangular plates under either mono- or bi-axial compressive in-plane loads. To this end, the coupled governing equations are exactly converted to two sets of uncoupled equations for in-plane and transverse deformations of symmetric laminated plates. The new uncoupled equations are analytically solved by applying both Navier and Lévy-type solution methods. The validity and high accuracy of the current exact solution are evaluated by comparing the present results with their counterparts reported in literature. [ABSTRACT FROM AUTHOR]

Buckling instability of slender piles may occur based on the effect of excessive axial loads acting on a pile with diminishing effective stress and shear strength owing to surrounding soil liquefaction. However, several guidelines for pile design often provide insufficient consideration of buckling failure, and it is significant to study the buckling failure of piles in liquefiable sites for the design and inspection of piles. In this paper, an efficient method of determining the critical buckling load of piles considering the material and geometric nonlinearities of piles based on the excess pore pressure distribution is proposed based on the beam-on nonlinear Winkler foundation (BNWF) model. Then, the method is verified and validated using results from centrifuge tests. Next, plots of the main effects and analysis of variance (ANOVA) were used for the sensitivity analysis of pile failure modes under axial loading. The initial imperfections and flexural rigidity of the pile have significant effects on the pile failure modes, and the pile is susceptible to buckling failure with low initial imperfections and flexural rigidity. Furthermore, the effects of the flexural rigidity of the pile, unsupported length ratio, initial imperfections, and degree of soil layer liquefaction on the critical buckling load of the pile in liquefiable soil are examined. The axial load-carrying capacity of the pile decreases with increasing initial imperfection amplitude, decreasing flexural rigidity of the pile, decreasing unsupported length ratio and severer degree of soil layer liquefaction. Finally, an estimating formula for predicting the critical buckling load of piles in liquefiable soil and an application example are presented. • A reliable approach is proposed to estimate the buckling load of piles during soil liquefaction. • Several key influencing factors on failure modes and buckling load of pile are discussed. • A simplified estimation method to predict the critical buckling load during the soil liquefaction is presented. [ABSTRACT FROM AUTHOR]

This work investigated buckling resistance of an X80 steel pipe containing a corrosion defect under axial compressive loading by finite element analysis. A 3-dimensional numerical model was developed to determine parametric effects on buckling behavior of the corroded pipeline. These included dimension of the corrosion defect (i.e., length, depth and width), pipeline geometry (i.e., outer diameter and wall thickness), internal pressure and mechanical properties of the steel (i.e., yielding strength, tensile strength and strain hardening exponent). The critical buckling load of the corroded pipeline is primarily affected by the depth and width of the corrosion defect, while the defect length is the least important. As the defect depth and width increase, the critical buckling load decreases, but the effect of corrosion width is not apparent when the dimensionless corrosion width w π D (w is width of corrosion defect and D is the pipe outer diameter) is greater than 0.5. The threshold defect length affecting the critical axial load of the pipeline is L D t = 4.86 (t is pipe wall thickness). When L D t > 4.86 , the influence of the defect length is ignorable. Pipeline containing a long corrosion defect feature of double buckling wave peaks, and a short corrosion defect contains a single buckling wave peak. The critical buckling load of the corroded pipeline subjected to the axial compression load increases with increased pipe outer diameter and wall thickness, but decreases with internal pressure. The increased yield and tensile strengths of the steel can improve the capacity to prevent local buckling of the corroded pipeline. The strain hardening exponent has a limited effect on the critical buckling load, which shows a significant decrease when the operating pressure is up to the critical internal pressure, resulting in yielding of the pipeline. • Developed a finite element model for buckling analysis of corroded X80 steel pipeline. • Determined the effects of corrosion dimension and pipeline geometry on buckling. • Determined the critical buckling load as a function of corrosion depth, width and length. • Determined the effects of pipe geometry and steel properties on critical buckling load. [ABSTRACT FROM AUTHOR]

This paper carries out the nonlinear stability of nanocomposite multilayer organic solar cell (NMOSC) subjected to axial compressive loads. The model of organic solar cell is assumed to consist five layers: Al, P3HT:PCBM, PEDOT:PSS, IOT and Glass. Based on the classical plate theory, the basic equations are established taking into account the effect of elastic foundations and initial imperfection. The approximation solutions are selected based on the boundary conditions of the four edges of NMOSC. The equation which indicates the relationship between axial compressive loads and deflection amplitude of NMOSC is obtained by using the Galerkin method. Bees Algorithm is applied to maximize the value of critical buckling load with nine variables including the thickness of five layers, the length and the width of NMOSC and two stiffness coefficients of elastic foundations. The numerical results show the effect of geometrical and material parameters, initial imperfection and elastic foundations on the nonlinear static stability and the critical buckling load of NMOSC. Optimal values of nine geometrical parameters of NMOSC are also determined. • The nonlinear stability of nanocomposite multilayer organic solar cell is investigated. • The classical plate theory is used to establish basic equations. • The effect of elastic foundations and initial imperfection are taken into account. • Effects of material and geometrical properties, elastic foundations and initial imperfection are discussed. • Bees Algorithm is used to determine the maximum value of critical buckling load depending on nine variables. [ABSTRACT FROM AUTHOR]

This study is devoted to the modeling and simulation of uncertainties in the constitutive elastic properties of material constituting a circular column under axial compression. To this aim, a probabilistic model dedicated to the construction of positive-definite random elasticity matrices was first used, involving two stochastic parameters: the mean value and a dispersion parameter. In order to compute the nonlinear effects between load and lateral deflection for the buckling problem of the column, a finite element framework combining a Newton-Raphson solver was developed. The finite element tool was validated by comparing the as-obtained critical buckling loads with those from Euler's formula at zero-fluctuation of the elasticity matrix. Three levels of fluctuations of material uncertainties were then propagated through the validated finite element tool using the probabilistic method as a stochastic solver. Results showed that uncertain material properties considerably influenced the buckling behavior of columns under axial loading. The coefficient of variation of a critical buckling load over 500 realizations were 15.477%, 26.713% and 41.555% when applying dispersion parameters of 0.3, 0.5 and 0.7, respectively. The 95% confidence intervals of column buckling response were finally given. The methodology of modeling presented in this paper is a potential candidate for accounting material uncertainties with some instabilities of structural elements under compression. [ABSTRACT FROM AUTHOR]