Some geometrically nonlinear problems of deformation of inelastic plates and shallow shells

This paper considers geometrically nonlinear problems of deformation of elastoplastic shallow shells and viscoelastoplastic plates where it is required to find kinematic loads for a given time interval such that a shell (plate) acquires prescribed residual deflections after these loads are applied and then removed. For some constraints, the correctness of the corresponding formulations (uniqueness of the solution and its continuous dependence on the problem data) is shown and iterative solution methods are justified.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 151–157, March–April, 2005.     

Modification of a generalized theory of shallow sandwich shells taking into account squeezing of the physically nonlinear filler

Zaporozh. University. Translated from Prikladnaya Mekhanika, Vol. 26, No. 12, pp. 39–45, December, 1990.     

Solvability of boundary value problems in the geometrically and physically nonlinear theory of shallow shells of Timoshenko type

This paper deals with the proof of the existence of solutions of a geometrically and physically nonlinear boundary value problem for shallow Timoshenko shells with the transverse shear strains taken into account. The shell edge is assumed to be partly fixed. It is proposed to study the problem by a variational method based on searching the points of minimum of the total energy functional for the shell-load system in the space of generalized displacements. We show that there exists a generalized solution of the problemon which the total energy functional attains its minimum on a weakly closed subset of the space of generalized displacements.     

Stress distribution in physically and geometrically nonlinear thin cylindrical shells with two holes

The elastoplastic state of thin cylindrical shells weakened by two circular holes is analyzed. The centers of the holes are on the directrix of the shell. The shells are made of an isotropic homogeneous material and subjected to internal pressure of given intensity. The distribution of stresses along the hole boundaries and over the zone where they concentrate (when the distance between the holes is small) is analyzed using approximate and numerical methods to solve doubly nonlinear boundary-value problems. The data obtained are compared with the solutions of the physically nonlinear (plastic strains taken into account) and geometrically nonlinear (finite deflections taken into account) problems and with the numerical solution of the linearly elastic problem. The stress-strain state near the two holes is analyzed depending on the distance between them and the nonlinearities accounted for __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 11, pp. 88–95, November 2005.     

Physically and geometrically nonlinear deformation of conical shells with an elliptic hole

The elastoplastic state of conical shells weakened by an elliptic hole and subjected to finite deflections is studied. The material of the shells is assumed to be isotropic and homogeneous; the load is constant internal pressure. The problem is formulated and a technique for numerical solution with allowance for physical and geometrical nonlinearities is proposed. The distribution of stresses, strains, and displacements along the hole boundary and in the zones of their concentration is studied. The solution obtained is compared with the solutions of the physically and geometrically nonlinear problems and a numerical solution of the linear elastic problem. The stress-strain state around an elliptic hole in a conical shell is analyzed considering both nonlinearities __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 69–77, February 2008.     

Physically and geometrically nonlinear deformation of thin-walled conical shells with a curvilinear hole

The elastoplastic state of thin conical shells with a curvilinear (circular) hole is analyzed assuming finite deflections. The distribution of stresses, strains, and displacements along the hole boundary and in the zone of their concentration are studied. The stress-strain state around a circular hole in shells subject to internal pressure of prescribed intensity is analyzed taking into account two nonlinear factors __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 73–79, April 2007.     

Investigation of the deformation and the stability of geometrically nonlinear beams

Summary The deformation and the stability of a geometrically nonlinear cantilever beam is investigated by using the concept of the deformation map, see Shilkrut [1,2]. The beam is loaded by two forces, a bending moment at its free end and a uniformly distributed load acting normally along the undeformed axis. The equilibrium differential equations for the above mentioned problem were derived by Reissner [3]. In order to create a deformation map, it is necessary to convert the boundary value problem into an initial value problem. Then the fourth order Runge-Kutta (R. K.) method can be used to solve numerically the system of equilibrium equations for the cantilever beam. The aim of this paper is to present new results for the above mentioned problem.

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Verformung und Stabilität von Balken unter Berücksichtigung der geometrischen Nichtlinearität

Übersicht Verformung und Stabilität eines Kragbalkens unter Berücksichtigung der geometrischen Nichtlinearität werden mit Hilfe des deformation map-Konzeptes (Shilkrut [1, 2]) untersucht. Der Balken wird an seinem freien Ende durch ein Biegemoment und zwei Einzelkräfte sowie durch eine gleichmäßig verteilte senkrechte Last entlang der unverformten Achse belastet. Die Gleichgewichtsbedingungen für dieses Problem wurden von Reissner [3] formuliert. Für die Definition einer deformation map ist die Umformung des Randwertproblems in ein Anfangswertproblem erforderlich. Das Runge-Kutta-(R. K.-) Verfahren vierter Ordnung kann dann zur numerischen Lösung des Systems der Gleichgewichts-bedingungen herangezogen werden. Ziel dieser Veröffentlichung ist es, neue Resultate für das erwähnte Problem vorzustellen.

     

Optimal design of reinforced cylindrical shells with corrosive wear taken into account

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 124–128, May–June, 1989.     

Analytical formulas of solutions of geometrically nonlinear equations of axisymmetric plates and shallow shells

Starting from the step-by-step iterative method, the analytical formulas of solutions of the geometrically nonlinear equations of the axisymmetric plates and shallow shells, have been obtained. The uniform convergence of the iterative method, is used to prove the convergence of the analytical formulas of the exact solutions of the equations.     

Analysis of the stability of shells of revolution with account for post-yield deformation

We propose a technique for the numerical analysis (using the finite-difference method) of the stability of ribbed shells of revolution with plastic deformations.S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of the Ukraine, Kiev. Translated from Priki. Mekh., Vol. 30, No. 6, pp. 65–72, 1994.     

On equations of the linear theory of shells with surface stresses taken into account

We construct equations of equilibrium and constitutive relations of linear theory of plates and shells with transverse shear strain taken into account, which are based on reducing the spatial elasticity relations with surface stresses taken into account to two-dimensional equations given on the shell median surface. We analyze the influence of surface elasticity moduli on the effective stiffness of plates and shells.     

Parametric vibrations of cylindrical shells subject to geometrically nonlinear deformation: multimode models

The nonlinear parametric vibrations of cylindrical shell are described by the Donnell–Mushtari–Vlasov equations. The motions are represented as a mode expansion. Discretization is performed using the Bubnov–Galerkin method. The describing-function method is used to study traveling waves and nonlinear normal modes in systems with and without dissipation