Free vibrations of rectangular orthotropic shallow shells with varying thickness

The paper proposes a numerical-analytic approach to studying the free vibrations of orthotropic shallow shells with double curvature and rectangular planform. The approach is based on the spline-approximation of unknown functions. Calculations are carried out for different types of boundary conditions. The influence of the mid-surface curvature and variable thickness on the behavior of dynamic characteristics is studied __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 102–115, June 2007.     

Free vibrations of shallow orthotropic shells with variable thickness and rectangular planform

The free vibrations of shallow doubly curved orthotropic shells with rectangular planform and varying thickness is solved using a refined formulation and the spline-approximation method. Various boundary conditions are considered. The effect of the curvature of the mid-surface on the spectrum of natural frequencies is examined. The natural frequencies and modes of orthotropic shells of constant and varying thickness are compared and analyzed     

Solution describing the natural vibrations of rectangular shallow shells with varying thickness

Natural vibrations of shallow cylindrical shells with rectangular plan and varying thickness are studied using a spline-approximation method developed previously. Computation is carried out for different types of boundary conditions. The effect of the curvature of the midsurface on the natural frequencies is examined. The natural frequencies of shells with constant and varying thickness are compared __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 89–98, April 2007.     

Stress state of nonthin orthotropic shells with varying thickness and rectangular planform

The stress-strain state of a shallow orthotropic shell with rectangular planform and thickness varying in two coordinate directions is studied. A refined problem formulation is used. Different boundary conditions are considered. A numerical analytic approach based on the spline approximation and discrete orthogonalization is developed. The stress-strain state of shallow orthotropic shells whose thickness is varied keeping its mass constant is studied Translated from Prikladnaya Mekhanika, Vol. 44, No. 8, pp. 91–102, August 2008.     

Stress-strain analysis of closed nonthin orthotropic conical shells of varying thickness

The approach developed to solve two-dimensional static problems for nonthin conical shells of varying thickness is used to examine the effect of the geometrical parameters on the stress-strain state of shells. The approach is based on spline-approximation and a stable numerical method of solving one-dimensional problems __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 46–58, June 2008.     

Natural vibrations of corrugated orthotropic shells of revolution

The torsional and longitudinal–flexural vibrations of corrugated orthotropic shells are investigated. Relations, including the equations of motion in forces and moments and Hooke’s relations, are obtained using the Kirchhoff–Love hypotheses. The influence of the geometric parameters of the shell (corrugation amplitude and length) on the eigenfrequencies and natural vibrations modes is studied for fixed-end shells. It is found that during torsional vibrations, increasing the corrugation amplitude and increasing the number of corrugations leads to a decrease in the resonant frequencies. In the case of torsional and longitudinal–flexural vibrations, the influence of the corrugation amplitude on the natural vibration modes is investigated.     

Nonlinear vibrations of orthotropic shallow shells of revolution

A set of nonlinearly coupled algebraic and differential eigenvalue equations of nonlinear axisymmetric free vibration of orthotropic shallow thin spherical and conical shells are formulated.following an assumed time-mode approach suggested in this paper. Analytic solutions are presented and an asymptotic relation for the amplitude-frequency response of the shells is derived. The effects of geometrical and material parameters on vibrations of the shells are investigated.     

The R-function method used to solve nonlinear bending problems for orthotropic shallow shells on an elastic foundation

The paper proposes a method to solve geometrically nonlinear bending problems for thin orthotropic shallow shells and plates interacting with a Winkler–Pasternak foundation under transverse loading. This method is based on Ritz’s variational method and the R-function method. The developed algorithm and software are used to solve a number of test problems and to study complex-shaped shells. The effect of the shape of shells, the boundary conditions, the stiffness of the foundation, and the load distribution on the behavior of isotropic and orthotropic shells undergoing geometrically nonlinear bending is studied     

Characteristic orthogonal polynomials in the study of transverse vibrations of nonhomogeneous rectangular orthotropic plates of bilinearly varying thickness

Effect of nonhomogeneity on the vibrational characteristics of thin orthotropic rectangular plates of bilinearly varying thickness has been studied using boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method. The thickness variation is taken as the Cartesian product of linear variations along two concurrent edges of the plate. The orthogonal polynomials in two variables are generated using the Gram-Schmidt process. The nonhomogeneity of the plate material is assumed to arise due to linear variations in Young’s moduli, shear modulus and density of the plate with the in-plane coordinates. Numerical results have been computed for four different combinations of clamped, simply supported and free edges. Effect of thickness variation together with varying values of aspect ratio and nonhomogeneity on the natural frequencies is illustrated for the first three modes of vibration. Three dimensional mode shapes have been presented. Comparison has been made with the known results.     

Forced vibrations of orthotropic shells: nonclassical boundary-value problems

The forced vibrations of a cylindrical orthotropic shell are studied. Two types of boundary conditions on the outer surface are examined considering that the displacement vector prescribed on the inner surface varies harmonically with time. Asymptotic solutions of associated dynamic equations of three-dimensional elasticity are found. Amplitudes of forced vibrations are determined and conditions under which resonance occurs are established. Boundary-layer functions are defined. The rate of their decrease with distance from the ends inside the shell is determined. A procedure of joining solutions for the internal boundary-layer problem is outlined in the case for the, if clamping boundary conditions are prescribed at the ends     

Natural vibrations of orthotropic,ribbed, cylindrical shells

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 27, No. 10, pp. 83–90, October, 1991.     

Nonlinear vibrations of rectangular plates with linearly varying thickness

Simplified nonlinear governing differential equations proposed by Berger for static cases and extended by Nash and Modeer for dynamic cases are used to analyse the title problem. Steady-state harmonic oscillations are assumed and the time variable is eliminated by a Kantorovich averaging method. The enclosure or comparison theorem of Collatz is then applied to the reduced equations to obtain the upper and lower bounds for the fundamental nonlinear frequency of simply-supported rectangular plates with linearly varying thickness. The fundamental eigenvalues are given for several taper and aspect ratios.Nomenclature a, b dimensions of plates - A _i series coefficients - D Eh ³/12(1– ²) flexural rigidity - D ₀ Eh ₀ ³ /12(1– ²) - E Young's modulus - h thickness, h ₀(1+x) - h ₀ thickness parameter - N _x , N _y stress resultants in the X and Y directions - N (N _x +N _y )/(1+) - P ₁, P ₂, ... parameters - Q ₁, Q ₂, ... parameters - R[X, (A/h ₀)²] bounding function - t time - u, v in-plane displacements - lateral deflections of plate - X=x/a dimensionless co-ordinate - x, y rectangular co-ordinates - y _n (X) series related to - thickness taper ratio - parameter in the neighbourhood of - error-function associated with differential equation - eigenvalue relating to frequency - Poisson's ra-tio - plate material specific weight - (X) function related to plate deflection - (X) admissible functions - circular frequency