Optimization of reinforced cylindrical shells with nonuniform thickness

An optimum multilayer shell is designed whose stack of elementary layers has a nonuniform thickness. This optimization problem is solved numerically for the special cases of three-layer cylindrical shells with dynamic and static stability. The optimum variants of layer distribution in this model are compared with the optimum solutions in [1].Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 298–303, March–April, 1976.     

Refined linear theory of elastic anisotropic multilayer shells. Part 2

On the basis of the refined linear theory of elastic anisotropic multilayer shells of arbitrary shape derived in [1] it is established that a number of theorems of the linear theory of elasticity have analogues in the theory of multilayer anisotropic shells.For Part 1 see [1].Institute of Fluid Mechanics, Bucharest. Translated from Mekhanika Polimerov, No. 1, pp. 100–109, January–February, 1976.     

Stress-strain state in anisotropic multilayer shells with zones of nonideal contact between layers

Conclusions The theorem formulated here corresponds to the most general variational principle in the theory of elasticity. The equations and conditions derived from it constitute a complete system of relations necessary for defining and solving the problems which involve determining the stress-strain state in anisotropic multilayer shell structures. Assuming that some of the relations (2.2)–(2.9) are satisfied a priori, one can formulate other partial variational principles (Lagrange's, Reissner's, et al.).The result obtained here can be utilized for a correct derivation of two-dimensional equations for anisotropic multilayer shells of discrete structure, also as the starting point for devising approximate methods of solution of problems which involve determining the state of stress and strain in anisotropic multilayer shells.Translated from Mekhanika Kompozitnykh Materialov, No. 5, pp. 832–836, September–October, 1981.     

Comparative analysis of two algorithms for numerical solution of nonlinearstatic problems for multilayered anisotropic shells of revolution 2. Account of transverse compression

Two algorithms for numerical solution of static problems for multilayer anisotropic shells of revolution are discussed. The first algorithm is based on a differential approach using the method of discrete orthogonalization, and the second one—on the finite element method with linear local approximation in the meridional direction. It is assumed that the layers of the shell are made of linearly elastic, anisotropic materials. As the unknown functions, six displacements of the shell are chosen, which often simplifies the definition of static problems for multilayer shells. The calculation of a cross-ply cylindrical shell stretched in the axial direction is considered. It is shown that taking account of the transverse compression, anisotropy, and geometrical nonlinearity is important for the given class of problems.Tambov State Technical University, Tambov, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 4, pp. 435–446, May–June, 1999.     

Experimental-theoretical studies of the stability of orthotropic shells with a filler in axial compression

Conclusions 1. An analysis has been made of the solution to the problem of the stability of multilayer cylindrical shells having a filler and simple calculation formulas have been obtained for determining the critical forces.2. The stability of fiberglass-plastic shells with rubber-like fillers has been studied experimentally.3. Comparative experimental-theoretical studies of critical forces have been made, and the stability coefficients have been ascertained for the shell class under consideration.Translated from Mekhanika Polimerov, No. 3, pp. 485–489, May–June, 1978.     

Analysis of the effect of transverse shear strains on the stability of multilayer cylindrical shells under external pressure

The effect of transverse shear strains on the critical pressure is investigated using the results of the solution obtained for the problem of the stability "in the small" of elastic multilayer cylindrical shells of regular structure with alternating light and stiff layers. Attention is drawn to the need to estimate the state of stress of the shells in the critical-load zone with the object of studying the desirability of taking the shear effect into account in the stability calculations. The results obtained can be used in calculating the stability of shells made from resin-based composites (glass-reinforced plastics, graphite-reinforced plastics, etc.). The numerical calculations were carried out using a computer.Translated from Mekhanika Polimerov, No. 6, pp. 1066–1070, November–December, 1973.     

Design of anisotropic structures: Calculation,optimization, and tests

Conclusions A computational procedure based on a limiting state has been worked out which proposes the performance of tests of samples, small-scale models, and full-scale structures and a statistical analysis of the experimental data. Advantages of the method are shown in comparison with the computational method based on fracturing loads which pertain to estimating the accuracy of the computational scheme of structures and determining their safety coefficients. An optimality criterion is formulated within the framework of the method which is related to ensuring a maximum of the limiting (fracturing or critical) load with a constant mass of the structure.Applications of the method are given to estimation of the strength and stability of structures made out of polymeric and metal CM with the use of the Gol'denblat-Kopnov criterion and solutions based on linear shell theory. The conditions for realization of the optimality criterion of structures are determined. Recommendations are made for the rational reinforcement of structures made out of filamentary CM, and a procedure is proposed for the rational design of reinforced structures.As an example of the tests for stability of cylindrical shells made out of aluminum-magnesium alloy under external pressure, the legitimacy of the cause of the disagreement between the calculated and experimental critical loads, which consists of a discrepancy in the nature of wave formation at the instant of stability loss and initial imperfections, and the effectiveness of the computational method based on a limiting state are confirmed. Refined computational formulas for the critical loads of cylindrical shells made out of fiberglass, carbon fiber, and boraluminum under typical kinds of loading are proposed within the framework of the method and on the basis of an analysis of the results of tests, and a comparison is made of the effectiveness of the reinforcement of shells made out of aluminum-magnesium alloy and carbon fiber. The values of the safety coefficients of structures which guarantee their reliability are determined.Translated from Mekhanika Kompozitnykh Materialov, No. 2, pp. 262–271, March–April, 1980.     

Some problems of the theory and analysis of anisotropic plates and shells with low shear stiffness

Certain problems associated with the theory and analysis of plates and shells made of materials with low shear stiffness are discussed.Physicomechanical Institute, Academy of Sciences of the Ukrainian SSR, L'vov. Translated from Mekhanika Polimerov, No. 4, pp. 693–714, July–August, 1970.     

Application of the technical theory to sandwich shells with a solid polymeric core

The possibilities of using the technical theory for analyzing cylindrical sandwich shells with a core of low-modulus polymeric material are considered. It is shown to be necessary to make assumptions concerning the distribution of the deformations over the elements of the three-layer section and to take account of the shear strains in the core, the flexural rigidity in the longitudinal direction, and the Poisson ratio in determining the forces and moments. The theoretical conclusions have been experimentally confirmed by static tests on a model.All-Union Structural Engineering Correspondence Institute, Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 298–304, March–April, 1973.     

Certain problems in developing a theory and design methods for anisotropic shells and plates with finite stiffness in shear. A survey

A survey is presented here of research on theory and design methods for shells and plates made of anisotropic materials, such as laminate reinforced plastics [4–13], compliant in shear. The report is based mainly on studies made at the Western Science Center, Academy of Sciences of the Ukrainian SSR during the last five years (after the previous survey [8] has been published), with the introduction of generalized Timoshenko-type theories.Branch of Mathematical Physics, Institute of Mathematics, Academy of Sciences of the Ukrainian SSR, L'vov. Translated from Mekhanika Polimerov, No. 2, pp. 269–284, March–April, 1975.     

Iterative method of designing uniform and laminated shells of revolution made of highly elastic materials

Conclusion An iterative method was proposed for numerical solution of problems concerning the axisymmetric deformation of uniform and laminated shells of revolution made of highly elastic materials. Calculations performed for rubber, rubber-fiber, and rubber-layered shells demonstrate the reliability and effectiveness of the method.Translated from Mekhanika Kompozitnykh Materialov, No. 1, pp. 109–116, January–February, 1990.     

Analysis of plates of polymeric material subjected to local loads

An approximate method of analyzing plates of polymeric material under local loading is described. The state of stress and strain and the stability of the plate are investigated.Moscow Institute of Electronic Machine Building. Translated from Mekhanika Polimerov, No. 3, pp. 430–434, May–June, 1976.     

Shear effects during bending of oriented glass-reinforced plastics

An oriented glass-reinforced plastic may be treated as a regular multilayer material. Given large number of layers we can replace the discrete model with a continuous medium whose elastic constants are determined by the properties of the components. Using this approach and assuming that the reinforcing and the resin obey Hooke's law, we examine the problem of the bending of oriented glass-reinforced plastics and by means of two examples demonstrate the importance of shear effects in the bending of beams. The experimental results are in good agreement with theory.Mekhanika Polimerov, Vol 1, No. 2, pp. 38–46, 1965     

Second-approximation shear theory for shallow layered shells and plates

Analysis of a second-approximation refined shear model for shallow layered composite shells and plates with a substantially inhomogeneous structure over the thickness is presented. The tangential displacements and corresponding normal stresses are expressed in the form of a polynomial of the fith degree in the transverse coordinate and contain squared rigidity characteristics. In this way, the accuracy of results and practical coincidence with the 3D solutions is ensured. Based on the refined model, a theory of shallow layered shells is developed. A system of resolving equations of sixteenth power together with appropriate boundary conditions was obtained and solved analytically. It is shown that the area of application of the formed model is extended as compared with the model of the first approximation. The model proposed allows us to examine the stress-strain state of layered composite structures of substantially different thickness and physical-mechanical characteristics of the layers, including the possibility of simulating relatively large shear deformations of rigid layers separated by a low-modulus thin interlayer pliable to transverse shear.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Ukrainian Transport University, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 3, pp. 363–370, May–June, 1998.     

Problems in dynamics of viscoelastic multilayer anisotropic shells and plates. Part 3: Optimization of vibration immunity characteristics of composite cylindrical shell

Conclusions Equations have been proposed which describe steady-state vibrations of multilayer shells made of composite materials. On the basis of these equations some simplest modes of deformation have been analyzed. The problem of optimizing the vibration immunity has been solved for a cylindrical shell serving as a component of a simple mechanical system.Translated from Mekhanika Kompozitnykh Materialov, No. 2, pp. 258–262, March–April, 1982.     

Experimental investigation of the anisotropy of strength of bolted joints of laminated composites upon variation of geometric and design parameters

Conclusion In the present work we investigated the anisotropy of the characteristics of tensile (buckling) strength and elasticity of plates made of KMU-4L in a bolted joint in a wide range of parameters of the material (stacking, number of layers), geometric parameters of the joint, parameters of the fastening element and of design parameters. An analysis of the results showed that the strength of the bolted joint under static loading can be increased for thin plates with h = 1–2 mm by local reinforcement of the zone of the hole by layers with = ±45 °, and also by increasing the moment of preliminary tightening; for plates with h = 3–4 mm the only feasible measure is increasing the moment of preliminary tightening to 20–40 Nm. All the results of the experiment were graphically generalized in the form of surfaces of maximal stress; this made it possible to predict the load bearing capacity of a joint for intermediate values of the parameters by calculating the safety factors.Translated from Mekhanika Kompozitnykh Materialov, No. 2, pp. 268–272, March–April, 1990.     

Local buckling of compressed elements of layered viscoelastic material

The problem of local surface buckling of compressed elements of layered material is solved using the equations of the theory of multilayer plates [1, 2, 5]. The local buckling deformations are described by means of the solutions of differential-difference equations damping into the interior of the medium. The effect of compressive forces, buckling wavelength, and the elastic and viscoelastic constants of the material on the behavior of the deformations and on the thickness of the layer to which significant buckling extends is investigated. Numerical estimates are presented for the buckling parameters of compressed laminated-plastic elements.Moscow Power Engineering Institute. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 816–821, September–October, 1968.     

Numerical modeling of nonlinear deformation and buckling of composite plate-shell structures under pulsed loading

Nonlinear three-dimensional problems of dynamic deformation, buckling, and posteritical behavior of composite shell structures under pulsed loads are analyzed. The structure is assumed to be made of rigidly joined plates and shells of revolution along the lines coinciding with the coordinate directions of the joined elements. Individual structural elements can be made of both composite and conventional isotropic materials. The kinematic model of deformation of the structural elements is based on Timoshenko-type hypotheses. This approach is oriented to the calculation of nonstationary deformation processes in composite structures under small deformations but large displacements and rotation angles, and is implemented in the context of a simplified version of the geometrically nonlinear theory of shells. The physical relations in the composite structural elements are based on the theory of effective moduli for individual layers or for the package as a whole, whereas in the metallic elements this is done in the framework of the theory of plastic flow. The equations of motion of a composite shell structure are derived based on the principle of virtual displacements with some additional conditions allowing for the joint operation of structural elements. To solve the initial boundary-value problem formulated, an efficient numerical method is developed based on the finite-difference discretization of variational equations of motion in space variables and an explicit second-order time-integration scheme. The permissible time-integration step is determined using Neumann's spectral criterion. The above method is especially efficient in calculating thin-walled shells, as well as in the case of local loads acting on the structural element, when the discretization grid has to be condensed in the zones of rapidly changing solutions in space variables. The results of analyzing the nonstationary deformation processes and critical loads are presented for composite and isotropic cylindrical shells reinforced with a set of discrete ribs in the case of pulsed axial compression and external pressure.Scientific Research Institute of Mechanics, Lobachevskii Nizhegorodsk State University, N. Novgorod, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 757–776, November–December, 1999.     

The theory of orthotropic layered cylindrical shells

The author examines orthotropic layered cylindrical shells for which the moduli of elasticity of the load-carrying layers substantially exceed the shear modulus between layers. This class of structure includes, in particular, shells made of orthotropic glass-reinforced plastic. In this case the classical theory based on the Kirchhoff-Love hypotheses requires refinement; the corresponding equations obtained as a result of approximating the distribution of shear stresses or displacements over the thickness of the shell by a certain known function are presented in [4, 7, 8]. In this paper equations that make it possible to construct the stress distribution over the shell thickness are obtained within the framework of the engineering theory on the basis of the hypothesis of the incompressibility of a normal element.Mekhanika Polimerov, Vol. 4, No. 1, pp. 136–144, 1968     

Optimum control of the forced motions of systems with continuously distributed parameters : PMM vol. 38, n 6, 1974, pp. 1056–1062

We propose one of the possible versions of the optimum control of the forced motions of elastic systems of the type of rods, plates, and shells. We apply the procedure developed to elementary problems on the transition of a freely-supported rod or plate from an initial state φ, ψ to the rest state in the least possible time T in the presence of a constraint on the forcing load. We use the elementary results of theory of the l-problem of moments of Krein [1–3].