Figure - available from: Mathematical Problems in Engineering
This content is subject to copyright. Terms and conditions apply.
Critical buckling loads versus thickness ratio of a four-lobed cross section cylindrical shell with variable thickness, (l=4,h=0.02).
Source publication
The static buckling of a cylindrical shell of a four-lobed cross section of variable thickness subjected to non-uniform circumferentially compressive loads is investigated based on the thin-shell theory. Modal displacements of the shell can be described by trigonometric functions, and Fourier's approach is used to separate the variables. The govern...
Similar publications
Lateral torsional buckling may occur in an unrestrained beam. A beam is considered to be
unrestrained when its compression flange is free to laterally rotate and laterally displace.
In this paper, the finite element analysis is used to investigate the lateral torsional buckling behavior
of I-beam with and without web opening. The analysis considers...
The paper presents an analysis of the stress state in a sandwich open conical shells during of stability loss. The shells under consideration consist of a lightweight core layer and two face-layers which are load-carrying. The thickness of those faces is assumed to be equal, and the thickness of the core is about 80% of the whole shell thickness. T...
In this work, we have determined the mean square relative displacement, elastic constant, anharmonic effective potential, correlated function, local force constant, and other thermodynamic parameters of diamond-type structured crystals under high-pressure up to 14 GPa. The parameters are calculated through theoretical interatomic Morse potential pa...
The stability and post-buckling response of simple perfect and imperfect systems made of nonlinear elastic material are thoroughly discussed. The perfect systems correspond to the three types of bifurcation points, i.e. stable symmetric, unstable symmetric and asymmetric. Material-dependent stability conditions are properly established. It is found...
A nondestructive, inexpensive method of measuring textural quality of raw apples is described. Termed the “sphere” test, it measures the compression of an apple produced by a fixed load on a standard diameter sphere. It is compared to a method for measuring dynamic elastic property. Both are evaluated in terms of the shearing measurements and senso...
Citations
... The Fourier series are widely used in engineering [1,2]. In the mechanics of solids and structures, Fourier series are frequently adopted for finding numerical and analytic solutions [3][4][5]. ...
... The means of the summability methods adopted in this work are briefly described in Section 3. Extensive numerical examples on both a multiphase composite and a material with voids are provided in Section 4: in these examples, partial sums, iterated Fejér partial sums, and Riesz means for the local stress are compared with solutions provided by accurate FEM analyses. 2 Mathematical Problems in Engineering ...
In the past, Eshelby’s method for a single inclusion in an infinite medium has been extended to periodic heterogeneous materials for the evaluation of global properties. In this work, the extended Eshelby’s method is used for the evaluation of local fields such as strain and stress in periodic heterogeneous materials characterized by unit cells containing multiple fibres or voids with different geometric and mechanical properties. The proposed method provides Fourier coefficients that are used to construct partial sums of three-dimensional trigonometric Fourier series of local fields. These partial sums exhibit unwanted effects such as the Gibbs phenomenon. In order to attenuate these effects, the behaviour of iterated Fejér partial sums and means of the Riesz summability method is investigated. Extensive numerical examples on both a multiphase composite and a material with voids are provided: in the examples, partial sums, iterated Fejér partial sums, and Riesz means for the local stress are compared with FEM solutions. The numerical comparison shows Riesz means perform better than partial sums and iterated Fejér partial sums and are effective in approximating elastic local fields in periodic heterogeneous solids.
... To make the data easily analyzed, we introduce the nondimensional coordinate = / and make 1 = / , 2 = ℎ/ . Thus, the governing differential equations (13) and (9b) can be, respectively, replaced by the following forms: The displacement equations can be obtained by (6a)-(6c), which arẽ ...
... According to the displacement differential equation of given beam and the Saint-Venant torsional equation [12,13], by (21), we have ...
The classical shell theory (CST) without considering the shear deformation has been commonly used in the calculation of shells structures recently. However, the impact of theory of plates and shells subjected to the shear deformation on the calculation is increasingly pronounced along with the wide use of composite laminated structures. In this paper, based on first-order shear deformation theory (FSDT) of cylindrical shells, the displacement control differential equation of moderately thick cylindrical shells has been obtained, so has been the edge force at longitudinal of the shells. Meanwhile, a group of unit force is introduced to deduce the displacement of edge beam under the action of edge force. A join condition of moderately thick cylindrical ribbed shells is established according to the continuity of displacement as well. Most notably, the displacement analytical solution of bending problems of moderately thick cylindrical ribbed shells is obtained, which has profound theoretical significance for further improving the analytical solution of moderately thick cylindrical shells.
... For example, Greenberg and Stavsky (1995) analyzed the buckling response of composite cylindrical shells subject to circumferentially non-uniform axial loads. Another study was carried out by Ahmed (2009). Hao et al. (2012) developed a two-stage optimization framework with adaptive sampling including relatively high-fidelity surrogate models of stiffened panels under non-uniform compression. ...
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.
This paper analytically studies the buckling of a cylindrical shell having varying thickness under non-uniform axial compressive loads for the first time, which widely exists in engineering practice. A novel quadratic perturbation technique is developed to establish general buckling load formulas for the shell. This method overcomes the difficulties of traditional energy methods in solving high order determinants and deriving direct expressions for buckling loads when shell thickness and axial load are unknown. Applying presented formulas, various shell thicknesses and axial loads are analyzed, and a series of new results for buckling loads are obtained and validated. Even for classical cosine thickness variation under uniform axial compression, we also give general conclusions compared with Koiter’s results by the energy method. The effects of thickness variations and load distribution parameters on buckling loads are analyzed in detail. The presented study in this paper fills the gap and establishes a foundation of buckling analysis for non-uniformly loading cylindrical shells with variable thickness. Certainly, the established formulas are general and available for buckling resistance capacity evaluation for the shells under all circumstances involving thickness variations or/and non-uniform axial compressive loads.
Stability of thin-walled structures is investigated on the basis of the geometrically nonlinear theory of shells. This allows us to monitor a series of shell deformation processes under different load changes. The local and general loss of stability can be determined by observing changes in the shape of the shell curved surface before and after critical loads. The shell material can be isotropic or orthotropic, but linear-elastic. A mathematical model of deformation of a shell is the functional of total potential energy of deformation of the shell.
Two methods are applied to minimize the total energy functional of deformation of the shell. One is based on the method of L-BFGS with discrete approximation of the unknown functions by NURBS-surfaces. This enables taking into account various forms of fixing of the shell contour and the complex shape of this circuit. The other employs the Ritz method and the best parameter continuation method in the continuous approximation of the unknown displacement functions and the angles of rotation of the normal. This technique allows us to find the upper and lower critical loads and the bifurcation point. A combination of these techniques makes it possible to study both the subcritical and supercritical behavior of the structure and to identify the local and global buckling of the shell and their relationship.
On the curve of equilibrium states versus “load q – deflection W” one can see all moments of buckling of the shell caused by “swatting” of some part of it. Thus, after each buckling a significant change in the shape of the curved surface is observed. For illustration purposes the changes in the shell shape at subcritical and supercritical stages are plotted against its three-dimensional surface. After the general loss of stability the shell deforms under loading without any significant change in its shape, i.e. it behaves as a plate.
In recent years there are more and more structures made of composite materials, especially in the form of thin-walled shells, being applied in various fields of technology. When using composite materials such as concrete or fiberglass, reinforcing elements are often placed along the axes of the curvilinear coordinate system of the shell, and in this case, the structure can be considered as orthotropic.
There are a lot of papers on the calculation of orthotropic shells, but they do not adequately investigate a number of important factors that influence the stress-strain state of the shell and its stability. In particular, the calculation of reinforced shells does not take into account such factors as in-plane shear, shear and torsional stiffness of ribs, etc.
The paper presents the mathematical model of deformation of thin orthotropic shells of revolution, based on the model of Timoshenko – Reissner. The model takes into account the design of reinforcement with the shear and torsional stiffness of the ribs, geometric nonlinearity and also the irregular shape of the shell. Possibility of application of methods and algorithms which are used in the study of isotropic shells is shown. The presented model investigates the stress-strain state and stability of thin orthotropic reinforced shells of revolution more adequate.
