Analytical Solutions For Buckling Research Articles

The analytical solution for buckling of curved sandwich beams with a transversely flexible core subjected to uniform load

In this paper, linear buckling analysis of a curved sandwich beam with a flexible core is investigated. Derivation of equations for face sheets is accomplished via the classical theory of curved beam, whereas for the flexible core, the elasticity equations in polar coordinates are implemented. Employing the von-Karman type geometrical non-linearity in strain-displacement relations, nonlinear governing equations are resulted. Linear pre-buckling analysis is performed neglecting the rotation effects in pre-buckling state. Stability equations are concluded based on the adjacent equilibrium criterion. Considering the movable simply supported type of boundary conditions, suitable trigonometric solutions are adopted which satisfy the assumed edge conditions. The critical uniform load of the beam is obtained as a closed-form expression. Numerical results cover the effects of various parameters on the critical buckling load of the curved beam. It is shown that, face thickness, core thickness, core module, fiber angle of faces, stacking sequence of faces and openin angle of the beam all affect greatly on the buckling pressure of the beam and its buckled shape.

Application of the VIM to thermal buckling of composite beams on an elastic foundation

AbstractIn this paper, thermal buckling analysis of composite laminated beams resting on a two-parameter elastic foundation is carried out by means of the variational iteration method (VIM). The solution procedure is based on the first-order shear deformation beam theory of Timoshenko combined with the von Karman type of geometrical non-linearity. Both edges of the beam are assumed to be axially restrained while may possess various out-of-plane boundary conditions. The elastic foundation is modelled via the two-parameter Pasternak medium which takes into account the vertical interaction of the elements. The governing equilibrium equations are obtained by means of the virtual displacement principle. Stability equations are then obtained by means of the adjacent equilibrium criterion. These equations are uncoupled to extract individual equations in terms of lateral displacement. The resulted differential equations are solved via the VIM. The study examines the VIM in obtaining the closed-form analytical solutions for thermal buckling of composite laminated beams. Furthermore, it is shown that for some special materials with negative thermal expansion coefficient, buckling may occur under cooling rather than heating or it may not occur at all.

Buckling of Functionally Graded Cylindrical Shells Under Combined Thermal and Compressive Loads

This article focuses on analytical solutions for bifurcation buckling of FGM cylindrical shells under thermal and compressive loads. A new solution methodology is established based on Hamilton's principle. The fundamental problem is subsequently transformed into the solutions of symplectic eigenvalues and eigenvectors, respectively. Then, by applying a unidirectional Galerkin method, imperfection sensitivity of an imperfect FGM cylindrical shell is discussed in detail. The solutions reveal that boundary conditions, volume fraction exponent, FGM properties, and temperature rise distribution significantly influence the buckling behavior. Critical stresses are reduced greatly due to the existence of initial geometric imperfections.

Effects of in-plane loads on vibration of laminated thick rectangular plates resting on elastic foundation: An exact analytical approach

Abstract In this article, an exact analytical solution for vibration and buckling of symmetrically laminated thick rectangular plates on elastic foundation subjected to different types of in-plane loading is presented. It is assumed that the laminated rectangular plate is symmetric and composed of transversely isotropic layers. Based on the third order shear deformation plate theory, three coupled partial differential governing equations of motion of the plate are obtained. Doing some mathematical manipulations, these equations are converted into a sixth order and a second order decoupled partial differential equations. By applying different types of in-plane loading and using the Levy solution, vibration and buckling of symmetric laminated rectangular plate resting on two-parameter elastic foundation is solved analytically. The accurate natural frequencies of the laminated rectangular plates with six different boundary conditions are presented for several thickness–length ratios, some aspect ratios, different elastic foundation parameters and various in-plane loading ratios. The results show that the in-plane loading can increase or decrease the natural frequency of laminated rectangular plate depending on the boundary condition and the type of loading. Finally, the mode shape contour plots are depicted for a laminated rectangular plate with various boundary conditions and different in-plane loadings.

Accurate symplectic space solutions for thermal buckling of functionally graded cylindrical shells

This paper derives new analytical solutions for thermal bifurcation buckling of cylindrical shells made of functionally graded materials (FGMs) with temperature-dependent material properties. The Donnell’s shell theory is adopted and a symplectic solution methodology is established through the Hamiltonian variational principle. The fundamental buckling problem is then converted into the solving for the symplectic eigenvalues and eigenvectors. The solutions reveal that boundary conditions and temperature-dependent FGM properties have significant influence on thermal buckling behavior. It is also concluded that temperature field conditions cannot be neglected for FGCSs being rich in thermal sensitive compositions.

Analytical solutions for bending, vibration, and buckling of FGM plates using a couple stress-based third-order theory

Analytical solutions of a general third-order plate theory that accounts for the power-law distribution of two materials through thickness and microstructure-dependent size effects are presented. The Navier solution technique is adopted to derive analytical solutions to simply supported rectangular plates. The modulus of elasticity and the mass density are assumed to vary only through thickness of plate and a single material length scale parameter of a modified couple stress theory captures the microstructure-dependent size effects. Examples of bending, buckling, and vibration problems are presented to show effects of the power-law distribution of two materials and the microstructure-dependent size parameter.

Finite element analysis on thermal upheaval buckling of submarine burial pipelines with initial imperfection

In-service hydrocarbons must be transported at high temperature and high pressure to ease the flow and prevent the solidification of the wax fraction. The pipeline containing hot oil will expand longitudinally due to the rise in temperature. If such expansion is resisted, for example by frictional effects over a kilometer or so of pipeline, compressive axial stress will be built up in the pipe-wall. The compressive forces are often so large that they induce vertical buckling of buried pipelines, which can jeopardize the structural integrity of the pipeline. A typical initial imperfection named continuous support mode of submarine pipeline was studied. Based on this type of initial imperfection, the analytical solution of vertical thermal buckling was introduced and an elastic-plasticity finite element analysis (FEA) was developed. Both the analytical and the finite element methodology were applied to analyze a practice in Bohai Gulf, China. The analyzing results show that upheaval buckling is most likely to build up from the initial imperfection of the pipeline and the buckling temperature depends on the amplitude of initial imperfection. With the same amplitude of initial imperfection, the triggering temperature difference of upheaval buckling increases with covered depth of the pipeline, the soil strength and the friction between the pipeline and subsoil.

Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory

In this paper, static bending and buckling of a functionally graded (FG) nanobeam are examined based on the nonlocal Timoshenko and Euler–Bernoulli beam theory. This non-classical (nonlocal) nanobeam model incorporates the length scale parameter (nonlocal parameter) which can capture the small scale effect. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The governing equations and the related boundary conditions are derived using the principal of the minimum total potential energy. The Navier-type solution is developed for simply-supported boundary conditions, and exact formulas are proposed for the deflections and the buckling load. The effects of nonlocal parameter, aspect ratio, various material compositions on the static and stability responses of the FG nanobeam are discussed. Some illustrative examples are also presented to verify the present formulation and solutions. Good agreement is observed. The results show that the new nonlocal beam model produces larger deflection and smaller buckling load than the classical (local) beam model.

A new analytical solution for lateral-torsional buckling of arches under axial uniform compression

A circular arch with a uniform open thin-walled cross-section that is subjected to a radial load distributed uniformly around the arch axis may suddenly displace laterally and twist out of its plane of loading by buckling in a lateral-torsional mode. The classical solution for the elastic lateral-torsional buckling load of such an arch is based on the assumption that the uniform radial load produces a uniform axial compressive force in a circular arch without in-plane bending. This assumption is approximately correct for deep arches. However, the uniform radial load may produce substantial bending actions in shallow arches, and so the assumption does not hold for shallow arches. Hence, it is doubtful whether the classical solution can correctly predict the lateral-torsional buckling load of shallow circular arches under a uniform radial load. The prebuckling behavior of a pin-ended circular arch under a uniform radial load is investigated in this paper. It is found that circular arches under a uniform radial load are subjected to combined axial compressive and bending actions, and that the bending action in shallow arches is substantial. It is also ascertained that the axial compressive force in shallow arches is much smaller than the nominal axial compressive force used in the classical buckling analysis. This paper derives a new unified analytical solution for the lateral-torsional buckling load of pin-ended arches by accounting for the combined bending and axial compressive actions. Comparisons with finite element results show that the new solution can accurately predict the lateral-torsional buckling load of both shallow and deep circular arches under a uniform radial load, while the classical solution predicts incorrect lateral-torsional buckling loads for shallow arches. The effects of the in-plane boundary conditions on the lateral-torsional buckling load are also investigated. The analytical solution for the lateral-torsional buckling load accounting for the effects of in-plane fixed boundary conditions is also derived. It is found that the in-plane fixed boundary condition greatly increases the lateral-torsional buckling load of shallow arches. In addition, the effects of the height of application of the load on the lateral-torsional buckling are studied, and the analytical solution for the buckling load including the effects of the height of the load application is derived. The agreements of these analytical solutions with finite element results are very good.

Unified analytical solution for dynamic elastic buckling of beams for various boundary conditions and loading rates

The problem of dynamic elastic buckling of Euler–Bernoulli beams subjected to axial loads is studied analytically. The dynamic axial loading is accomplished by a constant displacement rate of one end of the beam with respect to the other. The axial loading rates are considered slow enough to obviate the need to account for axial wave propagation effects. The dynamics of the beam is formulated in the modal domain considering the lowest static buckling mode. The dynamic stability of the beam is investigated as a response problem assuming an initial deformed geometry in terms of an eccentricity favouring the mode shape. The governing partial differential equation is condensed into a unified ordinary differential equation for various boundary conditions using a single dimensionless parameter in terms of beam geometry, material properties, loading rate, and appropriate coefficients corresponding to different boundary conditions. The results obtained for the dynamic response of beams for various values of the dimensionless parameter and initial eccentricity suggest that the solutions can be combined into a single unified analytical expression for the dynamic buckling load. It is shown that the corresponding dynamic response curves can also be collapsed into a single curve using the dimensionless peak load and the associated time parameter. The accuracy of the unified analytical expression for dynamic buckling is verified against exact solution of the ordinary differential equation considering three problems in terms of dimensional quantities for different boundary conditions and loading rates. Further the validity of the single mode dynamic buckling formulation is examined by comparing the results obtained for various boundary conditions with the numerical results from the dynamic response of a large number of degree of freedom finite element model of the beam without any restriction on the deformation mode shape. This unified solution has potential application in optimum design for dynamic collapse of truss type structures subjected to dynamic loads.

A VARIATIONAL PRINCIPLE APPROACH FOR BUCKLING OF CARBON NANOTUBES BASED ON NONLOCAL TIMOSHENKO BEAM MODELS

Based on the nonlocal elastic theory and variational principle, new Timoshenko beam models and analytical solutions for buckling of carbon nanotubes considering nanoscale size effect and shear deformation are established. New equilibrium equations and higher-order boundary conditions are derived and the buckling behavior of carbon nanotubes is numerically investigated. The numerical solutions confirm that nanotube stiffness is enhanced by nanoscale size effect and reduced by shear deformation. It is also concluded that nanotubes with different boundary conditions show varying sensitivity to changes in nanoscale and dimension. Comparison with molecular dynamics simulation results verifies the accuracy and reliability of this new analytical nonlocal Timoshenko beam model.

Analytical Solution of Elastic-Plastic Buckling of a Square Steel Tube Column under Axial Compressive Load and Bending Moment

Based on Ježek method of computing the elastic-plastic buckling of the member under the axial compressive load and the bending moment, the analytical expressions of calculating the ultimate load of buckling about the neutral axis with the moment of inertia for a square steel tube column are derived. By degenerated into the analytical expressions of the rectangular column and compared with the values of the finite element analysis (FEA) method, it shows that the analytical method in this paper is valid, which provides a new method of theoretical study for the elastic-plastic buckling of the member.

An analytical solution for buckling of moderately thick functionally graded sector and annular sector plates

In this article, an analytical solution for buckling of moderately thick functionally graded (FG) sectorial plates is presented. It is assumed that the material properties of the FG plate vary through the thickness of the plate as a power function. The stability equations are derived according to the Mindlin plate theory. By introducing four new functions, the stability equations are decoupled. The decoupled stability equations are solved analytically for both sector and annular sector plates with two simply supported radial edges. Satisfying the edges conditions along the circular edges of the plate, an eigenvalue problem for finding the critical buckling load is obtained. Solving the eigenvalue problem, the numerical results for the critical buckling load and mode shapes are obtained for both sector and annular sector plates. Finally, the effects of boundary conditions, volume fraction, inner to outer radius ratio (annularity) and plate thickness are studied. The results for critical buckling load of functionally graded sectorial plates are reported for the first time and can be used as benchmark.

Stability analysis of functionally graded rectangular plates under nonlinearly varying in-plane loading resting on elastic foundation

In this research work, an exact analytical solution for buckling of functionally graded rectangular plates subjected to non-uniformly distributed in-plane loading acting on two opposite simply supported edges is developed. It is assumed that the plate rests on two-parameter elastic foundation and its material properties vary through the thickness of the plate as a power function. The neutral surface position for such plate is determined, and the classical plate theory based on exact neutral surface position is employed to derive the governing stability equations. Considering Levy-type solution, the buckling equation reduces to an ordinary differential equation with variable coefficients. An exact analytical solution is obtained for this equation in the form of power series using the method of Frobenius. By considering sufficient terms in power series, the critical buckling load of functionally graded plate with different boundary conditions is determined. The accuracy of presented results is verified by appropriate convergence study, and the results are checked with those available in related literature. Furthermore, the effects of power of functionally graded material, aspect ratio, foundation stiffness coefficients and in-plane loading configuration together with different combinations of boundary conditions on the critical buckling load of functionally graded rectangular thin plate are studied.

Lateral buckling of non-trenched high temperature pipelines with pipelay imperfections

An analysis method for the buckling process of a pipe section with a random pipelay imperfection is proposed. Four basic lateral modes, acquired by finite-element (FE) eigenvalue buckling analysis, are combined to provide the needed grid configurations for describing a real pipelay imperfection and an arc-length algorithm is used to analyze the snap-through process of the shell-element-grid model under nonlinear frictional boundary conditions. This paper also presents evaluation methods for the lateral buckling of two types of pipe-in-pipe systems that are used in the offshore oil and gas industry. For evaluating the buckling and postbuckling of compliant pipe-in-pipe systems FE analyses were carried out to judge the occurrence of the system buckling and furthermore to check postbuckling stresses induced in the buckles. The calculated results of the modified Riks algorithm indicate that only when high temperature would not trigger an abrupt short-wavelength buckle and when no yielding has been induced in the unavoidable long-wavelength buckles, the thermal stability and safety of compliant pipe-in-pipe systems can be proved. In the non-compliant pipe-in-pipe systems, firstly small-amplitude buckles of the carrier pipe may occur in the annulus between carrier pipe and casing pipe and the contact forces between the spacers and the casing pipe may drive the buckle of the pipe-in-pipe systems on the seabed. Based on the classical analytical solution of pipe buckling, four potential buckling modes corresponding to finite-element models are developed to evaluate the stability and the postbuckling strength of such pipe-in-pipe systems.

Analytical Solution of Elastic-Plastic Buckling of an H-Shaped Steel Column under Axial Compressive Load and Bending Moment

Based on Ježek method of computing the elastic-plastic buckling of the member under the axial compressive load and the bending moment, the analytical expressions of calculating the ultimate load of buckling about the neutral axis with the maximum moment of inertia for an H-shaped column are derived. By degenerated into the analytical expressions of the rectangular column and compared with the values of the finite element analysis (FEA) method, it shows that the analytical method in this paper is valid, which provides a new method of theoretical study for the elastic-plastic buckling of the member.

Analytical solution for buckling of asymmetrically delaminated Reissner’s elastic columns including transverse shear

The exact analytical solution of buckling in delaminated columns is presented. In order to investigate analytically the influence of axial and shear strains on buckling loads the geometrically exact beam theory is employed with no simplification of the governing equations. The critical forces are then obtained by the linearized stability theory. In the paper, we limit the studies to linear elastic columns with a single delamination, but with arbitrary longitudinal and vertical asymmetry of delamination and arbitrary boundary conditions. The studies of quantitative and qualitative influence of transverse shear are shown in detail and extensive results for buckling loads with respect to delamination length, thickness and longitudinal position are presented.

Analytical solutions for buckling of simply supported rectangular plates due to non-linearly distributed in-plane bending stresses

Rigorous analytical solutions are obtained for the plane stress problem of a rectangular plate subjected to non-linearly distributed bending loads on two opposite edges. They are then used in a Galerkin type solution to obtain the corresponding convergent buckling loads. It is shown that the critical bending moment depends significantly on the actual edge load distribution and further the number of nodal lines of the buckled configuration can also be different from that corresponding to a linear antisymmetric distribution of the bending stresses. Results are tabulated for future use while judging approximate numerical solutions.

Analytical solutions for buckling of rectangular plates under non-uniform biaxial compression or uniaxial compression with in-plane lateral restraint

The objective of this work is to examine the effect of a non-uniform distribution of the applied edge loads on their net critical value with reference to the title problem. This forms a sequel to an earlier work on uniaxial compression of plates without any in-plane restraint at lateral edges to suppress the Poisson effect. A rigorous superposition approach is employed for plane stress analysis of the loaded plate, and the resulting non-uniform in-plane stress field is fully accounted for in the subsequent stability analysis which is based on Galerkin's approach with a complete set of admissible functions. Pertinent results are presented to highlight the influence of various non-uniform distributions as well as an in-plane restraint at the lateral edges.

Natural frequencies of thin-walled curved beams

New equations of motion governing dynamic behavior of thin-walled curved beams were investigated based on the study of Kang and Yoo [Thin-walled curved beams, I: formulation of nonlinear equations, J. Eng. Mech., ASCE 120 (10) (1994) 2072–2101; Thin-walled curved beams, II: analytical solutions for buckling of arches, J. Eng. Mech., ASCE 120 (10) (1994) 2102–2125]. Explicit numerical expressions were derived to predict the complex dynamic behavior of the thin-walled curve beams. Stiffness and mass matrix of the curved beam element for finite element analyses were formulated to allow explicit evaluation of the dynamic behavior. The formulations were conducted using not only typical six degree of freedom but also additional warping degree of freedom at each node. This paper proved the validity and the convergence of the presented formulation using the curved beam element. The paper included comparisons of the natural frequencies of the thin-walled curved beams from the finite element formulations with those from other researchers’ study. The presented formulation and the new equations are sufficiently accurate for use in study and evaluation of curved architectural structures and bridges.