Bending of Composite Plates under Static and Dynamic Loading

Bespalova, E. I.; Kitaigorodskii, A. B.

doi:10.1007/s10778-005-0058-8

Cited by 12 publications

(10 citation statements)

References 4 publications

(2 reference statements)

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“…It should be noted that the numerical method proposed in [12,13] for shells with one plane of symmetry also applies to the case of loading under consideration. The importance of allowing for low orders of symmetry can be judged from [6,10], wherein it was established that not only quantitative but also qualitative effect of this factor on the natural vibrations of anisotropic bodies is significant.…”

Section: Introductionmentioning

confidence: 99%

Stability of Cylindrical Composite Shells under Torsion

2005

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The stability of fiber-reinforced cylindrical shells under torsion is analyzed in the case where the principal directions of elasticity in the layers do not coincide with the coordinate directions. The solution to the linearized equations of the technical theory of anisotropic shells is obtained in the form of trigonometric series. It is shown that for some reinforcement configurations the critical loads may depend on the direction of the torsional moment. It is also established that the minimum (in absolute value) eigenvalue does not always correspond to the critical load. This fact should be taken into account not only in the case of torsion but also in more complicated cases of loading Keywords: stability, torsion, cylindrical shell, composite with one plane of symmetry, critical loads Introduction. Problems of stability of cylindrical shells under torsion differ from those under loading of other types in both formulation and solution methods. It is obvious that the direction of the torsional moment is of no importance for isotropic and orthotropic shells. The mathematical solution should account for the fact that the absolute value of the critical load is also independent of the torsion direction, but its sign may be either plus or minus. The buckling of a shell is only possible under compressive loads of one sign such as axial load or pressure. This fact is also incorporated into the formulation of the eigenvalue problem. It is for these two types of loading that the most accurate solutions of stability problems have been obtained [2,4,8]. Less attention was given in the literature to the problem of stability under torsion [4,8,9]. Among the available analytic solutions, the most accurate seems to be Batdorf's solution to the stability problem for an isotropic cylindrical shell under torsion [8,9]. Its characteristic equation has the critical load parameter raised to the second power. If we determined its minimum value, it would become possible to calculate two critical values of the torsional moment differed only by sign. There are approximate analytic solutions for orthotropic shells [4]. For materials of lower order of symmetry, no satisfactory analytic solution has been found yet.In this paper, we solve the stability problem for cylindrical shells filament-wound in directions different from the coordinate axes. As in [9], the unknown functions are expanded into trigonometric series satisfying the hinged-support conditions. It should be noted that the numerical method proposed in [12,13] for shells with one plane of symmetry also applies to the case of loading under consideration. The importance of allowing for low orders of symmetry can be judged from [6, 10], wherein it was established that not only quantitative but also qualitative effect of this factor on the natural vibrations of anisotropic bodies is significant.1. Consider shells with an odd number of homogeneous anisotropic layers that have at each point only one plane of elastic symmetry parallel to the mid-surface. From the formal viewpoint, the...

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Section: Introductionmentioning

confidence: 99%

Stability of Cylindrical Composite Shells under Torsion

2005

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“…This paper continues the study [11], which was intended to analyze the stress-strain state of polygonal plates under static and dynamic loads. Here, we will analyze the dependence of the frequency properties of an L-shaped plate on boundary conditions and establish a relationship between the natural frequencies of the plate and its energy state.…”

mentioning

confidence: 87%

“…Such a numerical-and-analytic approach based on the generalized Kantorovich-Vlasov method in the form of the method of complete systems [1] is used here to solve stationary problems of bending of L-shaped plates. The configuration of plates will be described using the fictitious domain method and an analog of the force method of structural mechanics [4,6].This paper continues the study [11], which was intended to analyze the stress-strain state of polygonal plates under static and dynamic loads. Here, we will analyze the dependence of the frequency properties of an L-shaped plate on boundary conditions and establish a relationship between the natural frequencies of the plate and its energy state.…”

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confidence: 99%

Vibrations of polygonal plates with various boundary conditions

Bespalova

2007

Int Appl Mech

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The bending vibrations of polygonal (L-shaped) plates with different shapes and boundary conditions are studied. The natural frequencies are calculated using the inverse-iteration and Kantorovich-Vlasov methods. To take the configuration of the domain into account, the fictitious domain method and an analog of the force method of structural mechanics are used. Different trends in the dependence of the lowest natural frequency of an L-shaped plate on its geometry are illustrated for different boundary conditions. A correlation between the extreme values of the bending frequency and some relations for the energy characteristics of the plate is established Keywords: polygonal plate, bending vibrations, different boundary conditions, extreme frequency, energy characteristicsIntroduction. Polygonal plates attract considerable interest of experts in various fields. They are interesting to engineers as high-usage structural elements of building and architectural structures, to mechanicians as a possibility to examine the effect of domains of complicated configuration on the stress state, to mathematicians as an object of analysis of the singularity of solutions at corner points of different types, and to numerical analysts as an elementary example of nonconvex domains used in testing approximate methods [2,7,8,17,19]. Approaches to the solution of relevant problems are generally based on the ideas of the finite-difference and finite-element methods, which make it possible to discretely describe all complicating factors of the problem formulation (see, for example, [5,14]). A few problems with special boundary conditions such as hinged support can be solved by analytic methods based on Fourier series and transforms [8,17,19]. Intermediate between these alternative approaches are numerical-and-analytic methods. They considerably extend the range of solvable problems, keeping the continuum nature of the problem formulation [2,7,16]. Such a numerical-and-analytic approach based on the generalized Kantorovich-Vlasov method in the form of the method of complete systems [1] is used here to solve stationary problems of bending of L-shaped plates. The configuration of plates will be described using the fictitious domain method and an analog of the force method of structural mechanics [4,6].This paper continues the study [11], which was intended to analyze the stress-strain state of polygonal plates under static and dynamic loads. Here, we will analyze the dependence of the frequency properties of an L-shaped plate on boundary conditions and establish a relationship between the natural frequencies of the plate and its energy state.1. Problem Formulation and Solution Procedure. In our study, we will restrict ourselves to an isotropic plate of constant thickness [8]:

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“…The pseudo-force method for rod (beam) structures in problems with bilateral constraints and a known one-dimensional contact region was generalized in [11,12] to stationary problems for plate systems. To further generalize this method to contact problems for shells of revolution and rigid or elastic surfaces, it is necessary:…”

Section: Problem Formulation and Solution Techniquementioning

confidence: 99%

Contact Interaction Between a Laminated Shell of Revolution and a Rigid or Elastic Foundation

2005

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A method is proposed to solve the contact problem for laminated anisotropic shells of revolution. The method is based on a two-dimensional model that accounts for transverse shears and reduction. Also the method is based on the method of successive approximations, the generalized pseudo-force method, and a numerical-analytical method of solving boundary-value problems. The results obtained for a cylindrical shell of complex thickness structure are compared with those obtained in three-dimensional formulation Keywords: contact problem, anisotropic shell, laminated structure Introduction. In analyzing the stress-strain state of thin-walled structures on rigid or elastic foundations, it is necessary to describe the deformation of a shell and the mechanism of its interaction with the foundation [1-3, 6-10, 13].The model chosen to describe the deformation of a laminated shell should strike a reasonable balance between accuracy and implementability. The spatial theory of elasticity can adequately describe processes of interest, but may involve certain computational difficulties because of the high dimension of the corresponding problems. Approximate two-dimensional models of shells materially simplify the way the final result is obtained, but describe the contact interaction of bodies with different degrees of adequacy. For example, the classical theory of shells requires introducing concentrated forces at the contact boundary, which distort the contact pressure distribution. Allowing for transverse-shear strains eliminates the discontinuity of the shearing forces, but does not make the normal reactions at the boundary vanish. Additional tricks, such as an elastic layer between contacting surfaces, allow us to describe contact interaction more accurately, though describing the properties of such a layer involves difficulties [6,7]. A natural way to solve the contact problem for thin-walled shells is to allow for all kinds of transverse strains.In describing the interaction of contacting elements, the contact conditions are formulated as inequalities reflecting the nonnegativity of the constraint reactions in the contact region. In the absence of tangential forces, the geometrical contact condition is usually equality of displacements (strains) or curvatures of these elements [8].Expanding upon the studies [1, 2, 13], we propose here a method to solve static problems for laminated anisotropic shells of revolution contacting with rigid or elastic planes. The method is based on the nonclassical theory of shells and accounts for transverse-shear strains and reduction. To test the method, we will compare the results it produces with the exact solutions of some problems.

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Bending of Composite Plates under Static and Dynamic Loading

Cited by 12 publications

References 4 publications

Stability of Cylindrical Composite Shells under Torsion

Stability of Cylindrical Composite Shells under Torsion

Vibrations of polygonal plates with various boundary conditions

Contact Interaction Between a Laminated Shell of Revolution and a Rigid or Elastic Foundation

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