Solution of contact problems on the basis of a refined theory of plates and shells

Solutions of contact mixed boundary-value problems for a plate and for a cylindrical shell are given. These solutions are obtained with the use of equations for shells constructed by expanding solutions of elasticity theory equations with respect to the Legendre polynomials. Results of numerical simulations of the stress state in the vicinity of points with changing conditions on the frontal faces of the shell are presented. The results obtained are compared with analytical solutions of elasticity theory problems and with solutions obtained on the basis of the classical equations of the shell theory. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 169–176, September–October, 2008.     

Determination of the axisymmetric geometrically nonlinear thermoviscoelastoplastic stress-strain state of shells of revolution

A method is developed for determining the axisymmetric thermoviscoelastoplastic stress-strain state of shells subjected to bending and torsion. The problem is solved in a geometrically nonlinear formulation with allowance for transverse shear. The geometrically nonlinear deformation of an annular plate, the thermoviscoelastoplastic deformation of a cylindrical shell, and the limiting state of a corrugated shell are studied as examples. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 12, pp. 40–48, December, 1999.     

Interaction of an elastic shell with two-dimensional flows of an ideal fluid

In [1, 2], Kiselev and Rapoport investigated the flow of a jet over an elastic plate and shell. In the present paper, the problem of two-sided flow past an elastic shell is investigated in the exact nonlinear formulation. At a sufficiently high rigidity and small curvature of the shell in undeformed state it is shown that the problem has a unique solution, and a method is proposed for finding it. Some results of calculations are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 139–143, September–October, 1981.     

Experimental Investigation of the Natural Vibrations of Shells of Revolution with a Liquid and Added Masses

It is established that the effect of rigidly attached masses on the natural frequencies is considerably greater than that of the liquid filling the shell. It is shown that the resonant vibrations of the shell can be excited by a wind load on a thin plate rigidly attached to the shell with an added mass     

Non-symmetrical large deformation of a shallow thin conical shell

I.IntroductionItisimportanttoresearchnon-symmetricalquestionsofshallowcollicalshellsintheoryoronapplication.Asonekindofpressurevessel'sparts,shallowconicalshellsareverycommonlyusedillellgineerillgpractice,becausethedifficultyofmanul\lctul.illgthemis'small.AlthotlghwelookupmanyChineseandtbreignperiodicalswhicharctlblctobefound,wehavenotyeth'ulldarticlesanddocumentsfornon-symmetricalandnolllinearquestionsofshitllowconictllshells.Oval'rccelltyears,ProlbssorWangXinzhiandhiscolleagueshavedonealot…     

Flow around a flexible cylindrical shell by a plane stream of an ideal liquid

The flow by a plane stream of an ideal liquid around a cylindrical shell of zero flexural stiffness (a soft cylindrical shell), or a gas bubble on the boundary of which forces of tension act, was studied in [1–6]. The flow around an elastic plate in a linear formulation was considered in [7, 8]. We consider the flow, around a flexible cylindrical shell which possesses a flexural stiffness and at the same time admits large displacements, by a plane system of an ideal incompressible liquid. An application of methods of the theory of functions of a complex variable leads to an effective solution of the problem. The shape of the shell, the forces in it, the forces acting on the shell, and the field of velocities of the flow of the liquid are determined.     

Natural vibrations of a cantilever thin elastic orthotropic cylindrical shell

The natural vibrations of a cantilever thin elastic orthotropic circular cylindrical shell are studied. Dispersion equations for the determination of possible natural frequencies of cantilever closed shells and open shells with Navier hinged boundary conditions at the longitudinal edges are derived from the classical dynamic theory of orthotropic cylindrical shells. It is proved that there are asymptotic relationships between these problems and the problems for a cantilever orthotropic strip plate and for a cantilever rectangular plate and the eigenvalue problem for a semi-infinite closed orthotropic cylindrical shell with free end and for the same but open shell with Navier hinged boundary conditions at the longitudinal edges. A procedure to identify types of vibrations is presented. Orthotropic cylindrical shells with different radii and lengths are used as an example to find approximate values of the dimensionless natural frequency and damping factor for vibration modes __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 68–91, May 2008.     

Corotational nonlinear analyses of laminated shell structures using a 4-node quadrilateral flat shell element with drilling stiffness

A new 4-node quadrilateral flat shell element is developed for geometrically nonlinear analyses of thin and moderately thick laminated shell structures. The fiat shell element is constructed by combining a quadrilateral area co- ordinate method （QAC） based membrane element AGQ6- II, and a Timoshenko beam function （TBF） method based shear deformable plate bending element ARS-Q12. In order to model folded plates and connect with beam elements, the drilling stiffness is added to the element stiffness matrix based on the mixed variational principle. The transverse shear rigidity matrix, based on the first-order shear deformation theory （FSDT）, for the laminated composite plate is evaluated using the transverse equilibrium conditions, while the shear correction factors are not needed. The conventional TBF methods are also modified to efficiently calculate the element stiffness for laminate. The new shell element is extended to large deflection and post-buckling analyses of isotropic and laminated composite shells based on the element independent corotational formulation. Numerical re- sults show that the present shell element has an excellent numerical performance for the test examples, and is applicable to stiffened plates.     

On solution of problems of dynamics of plates and shells with local structure heterogeneities

A general approach to the solution of nonstationary problems of the theory of shells with local structure heterogeneities is proposed. The approach is based on representation of solutions for the systems of differential motion equations of shell theory in the form of expansions in terms of free forms of vibrations (FFV). Two implementations of the approach, involving FFV expansions of certain members and FFV expansions of the system as a whole, are considered. The capabilities of the approach are illustrated by results for specific problems. In the first implementation, we determine the dynamic stress-strain state of a square plate with three rod supports. A round plate connected to a coaxial circular cylindrical shell is considered as an example of FFV expansion of the system as a whole. Special Design and Engineering Office, S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 10, pp. 46–53, October, 1999     

Interaction of a spherical shockwave and an elastic circular cylindrical shell

The paper studies the interaction of a spherical shock wave with an elastic circular cylindrical shell immersed in an infinite acoustic medium. The shell is assumed infinitely long. The wave source is quite close to the shell, causing deformation of just a small portion of the shell, which makes it possible to represent the solution by a double Fourier series. The method allows the exact determination of the hydrodynamic forces acting on the shell and analysis of its stress state. Some characteristic features of the stress state are described for different distances to the wave source. Formulas are proposed for establishing the safety conditions of the shell.Translated from Prikladnaya Mekhanika, Vol. 40, No. 9, pp. 94–104, September 2004.     

Nonlinear stability of a shallow unsymmetrical heated orthotropic sandwich shell of double curvature with orthotropic core

Equations which govern the behavior of an elastic unsymmetrical, orthotropic sandwich shell of double curvature with orthotropic core having different elastic characteristics under uniform heating are derived. The face sheet may be of unequal thickness of different materials. However, a restriction that the radii of curvature of the shell elements be large compared with the overall thickness of the sandwich shell is imposed. The variational procedure has been used to obtain the five equations which govern the behavior of the heated orthotropic sandwich shell for the stability. In case of symmetry the equations resemble with those of Grigolyuk. Finally, the numerical results of a square or a rectangular simply supported curved plate section of a cylindrical shell under thermal loading have been computed and compared with other known results. The graphs have been drawn to show the effects of different sandwich material for immovable and movable edge conditions.     

THICKNESS-SHEAR VIBRATION OF CIRCULAR CRYSTAL PLATE IN CYLINDRICAL SHELL AS PRESSURE SENSOR

Based on the theory for small fields superposed on relatively larger fields in an electroelastic body, & theoretical analysis is performed on a circular plate thickness-shear crystal resonator sealed in a circular cylindrical shell for pressure measurement. A simple expression is obtained for pressure induced frequency shifts in the resonator, which is examined for design optimization. Numerical results show that the frequency shifts depend linearly on the pressure, and that a pressure sensor with a softer outer shell or a smaller thickness ratio of the crystal plate to the outer shell has higher sensitivity.     

Stability of an Elastic System Consisting of a Ribbed Cylindrical Shell and Noncrossing Longitudinal Plates

The paper presents a method for stability analysis of an elastic system consisting of a ribbed cylindrical shell and noncrossing plates. The influence of the width and thickness of the plate on the buckling stresses and modes is analyzed     

Stresses in shallow axisymmetric shells using a casting procedure

Normal and radial displacements of a shallow thin sheil of revolution are related by theoretical considerations. Radial displacements are calculated from slope measurements on the generatrices of the initial shell surface (before loading) and of the final shell surface (after loading). Membrane and bending stresses over the entire shell surface are then computed from the measured slope values and the calculated radial displacements. A measuring technique is developed which is especially suitable for shallow shells enclosed in small spaces. It consists mainly of making a cast of the initial and final shell surface using an epoxy platic. The disklike cast is cut along its diameter. The slope values are measured optically along the line which is formed by the intersection of the cutting plane with the deflection surface. The method is applied to a shallow thin shell of revolution in contact with a concentric piston and to a clamped circular plate, both experiencing large axisymmetric deflection.     

曲厚壳单元在斜板拱桥分析中的应用

介绍了利用曲厚壳单元对斜板拱桥进行空间受力分析,并将曲厚壳单元所得结果与采用实体单元分析的结果和模型实验所得结果进行了比较,证明利用曲厚壳单元来分析斜板拱桥是行之有效的。     

Non-linear vibrations of imperfect free-edge circular plates and shells

Large-amplitude, geometrically non-linear vibrations of free-edge circular plates with geometric imperfections are addressed in this work. The dynamic analog of the von Kármán equations for thin plates, with a stress-free initial deflection, is used to derive the imperfect plate equations of motion. An expansion onto the eigenmode basis of the perfect plate allows discretization of the equations of motion. The associated non-linear coupling coefficients for the imperfect plate with an arbitrary shape are analytically expressed as functions of the cubic coefficients of a perfect plate. The convergence of the numerical solutions are systematically addressed by comparisons with other models obtained for specific imperfections, showing that the method is accurate to handle shallow shells, which can be viewed as imperfect plate. Finally, comparisons with a real shell are shown, showing good agreement on eigenfrequencies and mode shapes. Frequency-response curves in the non-linear range are compared in a very peculiar regime displayed by the shell with a 1:1:2 internal resonance. An important improvement is obtained compared to a perfect spherical shell model, however some discrepancies subsist and are discussed.     

On the influence of a transverse magnetic field on the vibration spectra of shallow shells

In the design of electric machines, devices, and plasma generator bearing constructions, it is sometimes necessary to study the influence of magnetic fields on the vibration frequency spectra of thin-walled elements. The main equations of magnetoelastic vibrations of plates and shells are given in [1], where the influence of the magnetic field on the fundamental frequencies and vibration shapes is also studied. When studying the higher frequencies and vibration modes of plates and shells, it is very efficient to use Bolotin’s asymptotic method [2–4]. A survey of studies of its applications to problems of elastic system vibrations and stability can be found in [5, 6]. Bolotin’s asymptotic method was used to obtain estimates for the density of natural frequencies of shallow shell vibrations [3] and to study the influence of the membrane stressed state on the distribution of frequencies of cylindrical and spherical shells vibrations [7, 8]. In a similar way, the influence of the longitudinal magnetic field on the distribution of plate and shell vibration frequencies was studied [9, 10]. It was shown that there is a decrease in the vibration frequencies of cylindrical shells under the action of a longitudinal magnetic field, and the accumulation point of the natural frequencies moves towards the region of lower frequencies [10]. In the present paper, we study the influence of a transverse magnetic field on the distribution of natural frequencies of shallow cylindrical and spherical shells, obtain asymptotic estimates for the density of natural frequencies of shell vibrations, and compare the obtained results with the empirical numerical results.     

The Stress-Strain State of a Composite Spherical Shell with Two Polar Openings

The stress–strain state of a deep spherical composite shell is analyzed. In the general case, the shell is weakened by openings located at the poles. The variational difference method is applied to solve the problem. Calculations are carried out for an orthotropic shell of low shear stiffness.     

Influence of the boundary conditions on the stress state of a flexible cylindrical shell of variable stiffness in a magnetic field

The effect of the boundary conditions on the stress state of a circular cylindrical shell of variable thickness (stiffness) is analyzed using a geometrically nonlinear problem statement. The cylindrical shell is subject to a magnetic field, external electric current, and nonstationary mechanical load. Numerical results are presented and analyzed     

Motion of the detonation products of a point-ignited charge in a cylindrical shell

A numerical solution is obtained for the two-dimensional nonsteady problem of the motion of detonation products from a cylindrical high-explosive charge enclosed in a shell with the initiation of detonation at a central point in the end of the charge. The detonation products propagate in vacuum. The strength of the shell is not considered. A three-term equation of state is used for the detonation products.