Exact Buckling Research Articles

BUCKLING OF NANO-RINGS/ARCHES BASED ON NONLOCAL ELASTICITY

This paper is concerned with the bifurcation buckling of nano-rings and nano-arches where the allowance for small scale effect is catered for by using Eringen's nonlocal theory of elasticity. Exact buckling solutions for nano-rings and nano-arches under uniform radial pressure are derived and the influence of small scale effect on the buckling pressures and mode shapes is investigated. The new results presented will be useful to engineers who are designing nano-rings and nano-arches to be used in MEMS and NEMS devices.

Buckling Loads of Two-Layer Composite Columns with Interlayer Slip and Stochastic Material Properties

This paper presents an efficient stochastic buckling model for studying the structural reliability of layered composite columns with interlayer slip between the layers and random material and loading parameters. The model is based on the exact buckling model, response surface method, and Monte Carlo simulations. The probability of failure is investigated for a different number of random variables, sample points, and various degrees of response surfaces. The results show that the probability of failure is considerably affected by the type (deterministic or probabilistic) of the loading and its distribution.

An analytical model for the buckling of plates under mixed boundary conditions

An analytical approach for the buckling analysis of rectangular plates under mixed boundary conditions is presented. In order to solve the partial differential equation governing the problem at hand a method of separation of variables is here adopted, by introducing the displacement field as a result of the scalar product of two vectors which combine prescribed and unknown scalar functions. By following this strategy, exact buckling solutions for a wide class of problems, in which mixed boundary conditions can be assigned relaxing some usual constraints, are determined, and buckling load of plates, where biaxial tensile and compressive loads are applied in presence of piecewise clamped and partially supported sides, obtained analytically. Several cases of engineering interest are finally analyzed, and comparisons of the theoretical outcomes with literature data and Finite Element-based numerical results are also shown, in order to highlight the effectiveness of the proposed strategy.

Elastic stability of all edges simply supported, stepped and stiffened rectangular plate under uniaxial loading

In this paper using finite difference method the lower bound buckling load for simply supported stepped and stiffened rectangular plate with uniformly distributed compressive forces in one direction is discussed. The plate is divided into 400 rectangular meshes. The partial derivatives are approximated using central difference formula. Three hundred and sixty-one equations are formed and using MATLAB the least eigenvalue is obtained. The buckling coefficients are calculated for different types of stepped plates and the results are presented in tables and graphs for ready use by designers. Two separate tables are presented in comparison with the exact buckling factor for stepped rectangular plate by Xiang, Wei and Wang. The results give a close agreement.

Inelastic buckling load of a locally weakened reinforced concrete column

This paper presents a mathematical model for investigating the buckling of geometrically perfect, materially and kinematically non-linear, locally weakened reinforced concrete columns. The present analytical model is based on the linearized stability theory and is capable of predicting exact buckling loads. Based on a systematic parametric analysis, the effects of the weakness location, its magnitude, the number of weaknesses and the material non-linearity of concrete and reinforcing steel on buckling loads have been investigated. The results are given in terms of graphs showing in detail that the material non-linearity of concrete and reinforcing steel and the local weaknesses have a significant influence on the buckling load.

Exact buckling solution for two-layer Timoshenko beams with interlayer slip

This paper deals with the buckling behavior of two-layer shear-deformable beams with partial interaction. The Timoshenko kinematic hypotheses are considered for both layers and the shear connection (no uplift is permitted) is represented by a continuous relationship between the interface shear flow and the corresponding slip. A set of differential equations is obtained from a general 3D bifurcation analysis, using the above assumptions. Original closed-form analytical solutions of the buckling load and mode of the composite beam under axial compression are derived for various boundary conditions. The new expressions of the critical loads are shown to be consistent with the ones corresponding to the Euler–Bernoulli beam theory, when transverse shear stiffnesses go to infinity. The proposed analytical formulae are validated using 2D finite element computations. Parametric analyses are performed, especially including the limiting cases of perfect bond and no bond. The effect of shear flexibility is particularly emphasized.

Fundamental Buckling of Annular Plates with Elastically Restrained Guided Edges Against Translation

This study deals with the exact buckling solutions of annular plates with an elastically restrained guided edge against translation. The classical plate theory is used to derive the governing differential equation for annular plate with elastically restrained guided edge against translation. The buckling mode may not be axisymmetric as previously assumed. In certain cases, an asymmetric mode would yield a lower buckling load. This is due to switching of mode. This work presents the critical buckling load parameters for axisymmetric and asymmetric buckling modes. Extensive data is tabulated so that pertinent conclusions can be arrived at on the influence of translational restraints, Poisson's ratio and other boundary conditions on the buckling of uniform isotropic annular plates. The numerical results obtained, are in good agreement with the previously published data. In this paper the characteristic equations are exact, therefore the results can be calculated to any accuracy. Comparison of studies demonstrates the accuracy and stability of this work.

Buckling mosaic of a circular plate on a partial elastic foundation

Buckling of plates resting on an elastic foundation is important in numerous practical applications such as concrete pavements and foundations on soil. The buckling of circular plates has been reported by several authors (Yu 1957, Galletly 1959, Kline and Hancock 1965, Wolkowisky 1969, Gupta et al. 1991, Gupta and Ansari 1998, Wang 2005, Gupta et al. 2006). All of the above sources considered a circular plate resting on a full elastic foundation. The present Note studies the buckling of a circular plate on a partial concentric elastic foundation which is important for centrally supported or stiffened plates. We shall investigate whether there are any other asymmetric buckling modes aside from those reported by Wang (2005). Our solutions will also be exact, such that no numerical errors are incurred. Exact buckling criteria also serve as standards for checking results from approximate or numerical methods (Wang et al. 2005).

Nonlinear inelastic analysis of steel-concrete composite beam-columns using the stability functions

In this study, a flexibility-based finite element method considering geometric and material nonlinearities is developed for analyzing steel-concrete frame structures. The stability functions obtained from the exact buckling solution of the beam-column subjected to end moments are used to accurately capture the second-order effects. The proposed method uses the force interpolation functions, including a moment magnification due to the axial force and lateral displacement. Thus, only one element per a physical member can account for the interaction between the bending moment and the axial force in a rational way. The proposed method applies the Newton method based on the load control and uses the secant stiffness method, which is computationally both efficient and stable. According to the evaluation result of this study, the proposed method consistently well predicts the nonlinear inelastic behavior of steel-concrete composite frames and gives good efficiency.

Exact buckling analysis of composite elastic columns including multiple delamination and transverse shear

The exact analytical solution of buckling in beams with multiple delaminations 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. The parametric studies are designed so that they give us fundamental understanding of the effects of the delamination number, length and position on the buckling load. The effect of shear is found to be of substantial importance.

Exact solution of the multi-cracked Euler–Bernoulli column

The use of distributions (generalized functions) is a powerful tool to treat singularities in structural mechanics and, besides providing a mathematical modelling, their capability of leading to closed form exact solutions is shown in this paper. In particular, the problem of stability of the uniform Euler–Bernoulli column in presence of multiple concentrated cracks, subjected to an axial compression load, under general boundary conditions is tackled. Concentrated cracks are modelled by means of Dirac’s delta distributions. An integration procedure of the fourth order differential governing equation, which is not allowed by the classical distribution theory, is proposed. The exact buckling mode solution of the column, as functions of four integration constants, and the corresponding exact buckling load equation for any number, position and intensity of the cracks are presented. As an example a parametric study of the multi-cracked simply supported and clamped–clamped Euler–Bernoulli columns is presented.

Exact static analysis of partially composite beams and beam-columns

The ordinary differential equations and general solutions for the deflection and internal actions and, especially, the pertaining consistent boundary conditions for partially composite Euler–Bernoulli beams and beam-columns are presented. Static loading conditions, including transverse and axial loading and first- and second-order analyses are considered. The theoretical procedure is applicable to general loading and boundary conditions for uniform composite beams and beam-columns with interlayer slip. Further, the exact closed form characteristic equations and their associated exact buckling length coefficients for composite columns with interlayer slip are derived for the four Euler boundary conditions. It is shown that these coefficients are the same as those for ordinary fully composite (solid) columns, except for the Euler clamped-pinned case. For the clamped-pinned case, the difference between the exact buckling length coefficient and the corresponding value for solid columns is less than 1.8% depending on the so-called composite action parameter and relative bending stiffness parameter. Correspondingly, the maximum deviation between the exact and approximate buckling load is at most 2.5%. These small differences can in most practical cases be neglected. Also, the maximum theoretical range for the relative bending stiffness for partially composite beams and beam-columns is derived. An effective bending stiffness, valuable in the determination of the critical buckling load for partially composite members, is derived. This effective bending stiffness is also suitable for analysing approximate deflections and internal actions or stresses in composite beams with flexible shear connection. The beam-column analysis is applied to a specific case. The difference in the approaches to the first- and second-order analysis is illustrated and the results clearly show the magnification in the actions and displacements due to the second-order effect. The magnification of the internal axial forces is different from magnifications obtained for the other internal actions, since only that portion of an internal axial force that is induced by bending is magnified by the second-order effect.

Simplified buckling analysis of skeletal structures

A simplified approach is proposed for buckling analysis of skeletal structures, which employs a ‘rotational spring analogy’ for the formulation of the geometric stiffness matrix. The benefit of this analogy is that it offers an intuitive framework, which is based on the common notions of linear structural analysis. Assuming that the structural deflections prior to buckling are negligible, a linear eigenvalue problem, utilising the geometric and material stiffness matrices, can be easily formulated and solved for the critical buckling loads. This can be further simplified using an assumed mode, where the rotational spring analogy is shown to provide considerable computational benefits and significant insight into the buckling of various forms of skeletal structure. In this context, the use of different assumed modes can be conceived as a process of probing the structure to establish the most likely mode for buckling and the corresponding critical load. It is also shown that the approximation inherent in the assumed mode approach together with the discrete form of the rotational spring analogy can be significantly improved through modal combinations and increasing the number of elements, respectively, where convergence to the exact buckling solution is demonstrated. Several illustrative examples are provided in this paper, which highlight the simplicity of the proposed approach, its application using a linear structural analysis tool and its ability to shed significant light on important issues in buckling analysis of skeletal structures.

Buckling of intermediate ring supported cylindrical shells under axial compression

This paper is concerned with the elastic buckling of axially compressed, circular cylindrical shells with intermediate ring supports. The simple Timoshenko thin shell theory and the more sophisticated Flügge thin shell theory have been adopted in the modeling of the cylindrical shells. We used these two representative theories to examine the sensitivity of the buckling solutions to the different degree of approximations made in shell theories. By dividing the shell into segments at the locations of the ring supports, the state-space technique is employed to derive the solutions for each shell segment and the domain decomposition method utilized to impose the equilibrium and compatibility conditions at the interfaces of the shell segments. First-known exact buckling factors are obtained for cylindrical shells of one and multiple intermediate ring supports and various combinations of boundary conditions. Comparison studies are carried out against benchmark solutions and independent numerical results from ANSYS and p-Ritz analyses. The influence of the locations of the ring supports on the buckling behaviour of the shells is examined.

Shell Theory Accuracy with Regard to Initial Postbuckling Behavior of Cylindrical Shell

are the normal strains in the prebuckling state. The buckling anal- ysis for the nondebonded imperfect beams is then similar to that presented earlier for the free-vibration problem. In the case of a generally debonded beam, an exact buckling solution cannot be obtained and approximate treatments should be employed. How- ever, if the beam is with through-length delamination at the kth interface, for example, an exact solution can be obtained, just as it can for the nondebonded imperfect beam, but with K (k)

Stability of Nonuniform Cracked Bars Under Arbitrarily Distributed Axial Loading

This study is concerned with the stability analysis of nonuniform bars with arbitrary number of cracks and with arbitrary distribution of flexural stiffness or arbitrarily distributed axial loading. The homogeneous solutions of the governing differential equation for buckling of a nonuniform uncracked bar are derived for several important cases. A model of massless rotational spring is adopted to describe the local flexibility induced by cracks. Then a new approach that combines the exact buckling solution of a nonuniform uncracked bar, the model of massless rotational spring, and the transfer matrix method is presented for the title problem. The main advantage of the proposed method is that the eigenvalue equation for buckling of a nonuniform bar with an arbitrary number of cracks, arbitrary distribution of flexural stiffness, or arbitrarily distributed axial loading can be conveniently determined from a second-order determinant. The decrease in the determinant order as compared with other methods leads to significant savings in the computational effort. A numerical example is given to illustrate the reliability of the proposed approach through comparisons with numerical solutions and to study the effect of cracks on the stability of a nonuniform bar. Nomenclature ai = depth of the ith crack Ci =fl exibility of the ith rotational spring Ci1, Ci2 = constants of integration 1F2(a1; a2; a3; a4) =h ypergeometric series f () =fl exibility function hi = height of cross section at xi Jν(x) = first Bessel function of the vth order Ki(x) =fl exural stiffness at x in the ith

Exact Buckling Analysis of Constrained Stepped Columns

An analytical technique, called the Effective Stiffness Method (ESM), is developed for static buckling analysis of one-dimensional continua. Based on a distributed transfer function formulation, the ESM delivers exact buckling solutions for constrained stepped columns, eccentrically loaded columns, and columns with initial deflection. The method also obtains highly accurate buckling solutions for columns with arbitrary non-uniform cross-section. The ESM treats different column configurations in a systematic manner, and is numerically efficient. Unlike conventional analytical methods, the proposed method does not use series of continuous functions. The capabilities and advantages of the ESM are demonstrated in numerical examples.

Exact Solutions for Buckling of Multispan Rectangular Plates

This paper presents exact solutions for buckling of multispan rectangular plates having two opposite edges simply supported and the other two edges being either free, simply supported, or clamped. The Levy solution procedure is employed to develop an analytical approach for buckling analysis of multispan plates. The Levy solution for each span is derived and the continuity along the interface of two spans is ensured through the implementation of the essential and natural boundary conditions at the interface. Extensive buckling factors, most of which are first-known exact solutions, are given in tabular and design chart forms for two- and three-unequal-span square plates subjected to uniaxial in-plane load in the x or y directions and biaxial in-plane load. The influence of the span ratios and plate boundary conditions on the buckling factors is discussed. Buckling factors are also obtained for two-, three-, and four-equal-span rectangular plates with various edge support conditions. The exact buckling solutions presented in this paper are of benchmark values for such plates.

Classes of exact solutions for buckling of multi-step non-uniform columns with an arbitrary number of cracks subjected to concentrated and distributed axial loads

The governing differential equation for buckling of a multi-step non-uniform column subjected to combined concentrated and distributed axial loads, each step of which has an arbitrary number of cracks with or without spring supports, is expressed in the form of bending moment. A model of massless rotational spring is adopted to describe the local flexibility induced by cracks in the column. In this paper, the distribution of flexural stiffness of a non-uniform column is arbitrary, and the distribution of axial forces acting on the column is expressed as a functional relation with the distribution of flexural stiffness and vice versa. The governing equation for buckling of a one-step non-uniform column is reduced to a differential equation of the second-order without the first-order derivative by means of functional transformation. Then, this kind of differential equation is reduced to Bessel equations and other solvable equations for six important cases. The exact buckling solutions of one-step non-uniform columns are thus found. Then a new approach that combines the exact buckling solution of a one-step column and the transfer matrix method is presented to establish the eigenvalue equation for buckling of a multi-step non-uniform column with spring supports. The main advantage of the proposed method is that the eigenvalue equation for buckling of a non-uniform column with an arbitrary number of cracks, any kind of two end supports and various spring supports at intermediate points can be conveniently determined from a second order determinant. Due to the decrease in the determinant’s order as compared with previously developed procedures the computational time required by the present method can be reduced significantly. A numerical example is given to examine the accuracy of the proposed method and to investigate the effect of cracks on buckling of a multi-step non-uniform column.

EXACT BUCKLING AND VIBRATION SOLUTIONS FOR STEPPED RECTANGULAR PLATES

This paper is concerned with the determination of exact buckling loads and vibration frequencies of multi-stepped rectangular plates based on the classical thin (Kirchhoff) plate theory. The plate is assumed to have two opposite edges simply supported while the other two edges can take any combination of free, simply supported and clamped conditions. The proposed analytical method for solution involves the Levy method and the state-space technique. By using this analytical method, exact buckling and vibration solutions are obtained for rectangular plates having one- and two-step thickness variations. These exact solutions are extremely useful as benchmark values for researchers developing numerical techniques and software for analyzing non-uniform thickness plates.