Vibrations and Stability of Shallow Ribbed Shells with a Rectangular Planform

A technique is developed to determine the critical stresses and natural frequencies of shallow ribbed shells with a rectangular planform and the corresponding buckling wave and natural modes. The effect of discrete rib arrangement on these characteristics is studied by numerical examples.     

Vibrations of ribbed shallow rectangular shells on an elastic foundation

A procedure to determine the natural frequencies and modes of ribbed shallow shells with rectangular planform on an elastic foundation is developed. The effect of the radius of curvature and the Winkler and Pasternak coefficients of subgrade reaction on the natural frequencies and modes of a shallow spherical shell with a square planform is analyzed. It is revealed that not only the natural frequencies but also the modes of the shell depend on the coefficients of subgrade reaction     

Vibrations of a complex-shaped panel

The free vibrations of flexible shallow shells with complex planform are studied. To analyze the natural frequencies and modes of linear vibrations, the R-function and Rayleigh–Ritz methods are used. A discrete model is obtained using the Bubnov–Galerkin method. The nonlinear vibrations are studied by combining the nonlinear normal mode method and the multiple-scales method. Skeleton curves of natural vibrations are drawn     

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     

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

The free vibrations of shallow orthotropic shells with variable thickness and rectangular planform are studied. The shear strains are taken into account. The spline approximation of unknown functions is used. The natural frequencies are calculated for different boundary conditions. The dependence of the natural frequencies on the curvature of the midsurface is examined. The natural frequencies of shells with constant and variable thickness are compared     

R-functions theory applied to investigation of nonlinear free vibrations of functionally graded shallow shells

Nonlinear free vibration of functionally graded shallow shells with complex planform is investigated using the R-functions method and variational Ritz method. The proposed method is developed in the framework of the first-order shear deformation shallow shell theory. Effect of transverse shear strains and rotary inertia is taken into account. The properties of functionally graded materials are assumed to be varying continuously through the thickness according to a power law distribution. The Rayleigh–Ritz procedure is applied to obtain the frequency equation. Admissible functions are constructed by the R-functions theory. To implement the proposed approach, the corresponding software has been developed. Comprehensive numerical results for three types of shallow shells with positive, zero and negative curvature with complex planform are presented in tabular and graphical forms. The convergence of the natural frequencies with increasing number of admissible functions has been checked out. Effect of volume fraction exponent, geometry of a shape and boundary conditions on the natural and nonlinear frequencies is brought out. For simply supported rectangular FG shallow shells, the results obtained are compared with those available in the literature. Comparison demonstrates a good accuracy of the approach proposed.     

Harmonic Vibrations of Shells with Cutouts Under Kinematic Conditions at Their Periphery

Problems on harmonic vibrations of shells with openings are considered for the case where displacements and angles of turn are specified at their periphery. A technique for determination of the natural frequencies and modes of vibrations is stated. The technique replaces the unknown peripheral forces and moments by a system of local statically equivalent loads. The intensity of these loads is found from the boundary conditions at the opening periphery. The solutions of the equations of shell theory for local loads are constructed by expanding the sought-for quantities into series in terms of the natural modes of the shell without openings. The performance of the approach developed is illustrated by calculating the natural frequencies and modes of shallow shells with a rectangular planform and various Gaussian curvatures     

Influence of discrete ribs on the vibrations of rectangular plates

A rectangular plate reinforced with longitudinal ribs is considered. The influence of the discrete ribs on the wavenumbers of harmonic waves propagating along the ribs is investigated. The effect of the stiffness parameters of the ribs on the natural frequencies and modes of a cross-ribbed plate is studied __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 2, pp. 65–78, February 2006.     

The Stress State of Stiffened Shallow Orthotropic Shells

An approach is proposed for stress analysis of elastic systems consisting of shallow shells having a rectangular planform and stiffened with rods in one direction. The shell curvature varying in the direction perpendicular to the ribs and piecewise-constant in another direction is taken into account. A system of ordinary differential equations and shell–rib conjugation conditions are derived after separation of variables for two simply supported opposite contours. A one-dimensional boundary-value problem is solved by a stable numerical method. The results of a stress–strain analysis of shipbuilding structural elements are presented as an example     

Influence of discretely arranged longitudinal ribs on the stability and vibrations of open circular cylindrical shells

Open cylindrical shells reinforced with quasiregular sets of longitudinal ribs and hinged at all edges are considered. The effect of the discrete arrangement and number of ribs on the critical stresses that cause instability under longitudinal compression and on the minimum natural frequencies of vibrations is examined. Numerical results are analyzed     

Natural Vibrations of Ribbed Cylindrical Shells with Low Shear Stiffness

A technique is proposed to determine the natural frequencies of rib-reinforced cylindrical shells with low shear stiffness. The equations of motion based on the Timoshenko model are used. The influence of the transverse shear moduli of the shell and ribs on the minimum natural frequencies and corresponding modes is studied by way of numerical examples. It is shown that the effect of the discrete ribs is different from that predicted by the classical theory of ribbed shells__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 4, pp. 66–74, April 2005.     

Natural Vibrations of a Cylindrical Shell with Circular Ribs Attached by Means of Discrete Elastic Elements

The vibrations of a cylindrical shell reinforced with circular ribs attached to it by means of discrete elastic elements are studied. The problem is solved by the finite-element method. The shell and ribs are modeled by a plane four-node finite element, which is a combination of a four-node plane stress element and a four-node flexural element. The effect of the stiffness of the elastic elements on the natural frequencies and modes is examined __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 10, pp. 108–113, October 2005.     

Multi-modal approach for large natural flexural vibrations of thermally stressed plates

By means of Berger's approximation, suitable for plates with immovable edges, the geometrically nonlinear problem is considerably simplified. Thermally loaded plates with polygonal planform under hard-hinged support conditions are considered, taking into account the effect of shear in transverse isotropy. The class of symmetric vibrations about the flat plate position is represented by a homogeneous and coupled set of Duffing-oscillators as a result of a multi-mode expansion. A unifying non-dimensional closed-form solution for the corresponding nonlinear natural vibration periods is given, which is independent of the special planform. The individual shape of the plate enters the transformation into real time through the linear natural frequencies, or, equivalently, through the linear eigen-values of an effectively prestressed membrane of the same planform.     

Effect of the stiffness of ribs on the vibrations and stability of open cylindrical shells

The effect of the stiffness of ribs on the minimum natural frequencies and critical stresses of axially compressed open cylindrical shells reinforced with a quasiregular set of longitudinal ribs is analyzed by way of numerical examples. It is shown that the earlier discovered phenomenon of abrupt decrease in the minimum frequencies is independent of rib stiffness for certain modes and a small number of ribs     

The natural vibrations of rectangular plates reinforced by orthogonal ribs

A technique is proposed to determine the fundamental frequencies of vibrations of rectangular plates strengthened by an orthogonal system of ribs. The necessity of allowing for the discrete arrangement of ribs is demonstrated by numerical examples. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 7, pp. 117–122, July, 2000.     

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.     

Natural Vibrations of a Cylindrical Shell Reinforced with Two-Layer Rings

The vibrations of a cylindrical shell reinforced with ring ribs attached to the shell by means of elastic elements are studied. The problem is solved by the finite-element method. The shell and ribs are modelled by a plane four-node finite element, which is a combination of a four-node plane stress element and a four-node flexural element. The effect of the stiffness of the elastic elements on the natural frequencies and modes is examined __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 8, pp. 105–110, August 2005.     

Free vibrations of flexible shallow shells of complex shape

A method is proposed for studying the free vibrations of flexible shallow shells with a complex planform. The method is based on variational and R-function methods. The R-function method allows constructing a system of basis functions in an analytic form. This makes it possible to reduce the Donnell-Mushtari-Vlasov equations to Duffing equations. The amplitude-frequency characteristics of shallow shells with a complex planform are given for different curvatures and boundary conditions. The results obtained are compared with published results for simply supported square shells to demonstrate the reliability and efficiency of the method __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 99–109, April 2007.     

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.     

Dispersion Curves for Harmonic Waves Propagating Along the Directrix of a Cylindrical Shell Reinforced with Rings

A method is proposed to study the influence of discrete ribs on the dispersion curves for harmonic waves propagating along the directrix of a simply supported cylindrical shell reinforced with rings. A numerical example is considered to demonstrate the necessity of accounting for this factor, the possibility of using a simplified formulation of the problem, and the necessity of accounting for the asymmetry of rib arrangement about the median surface of the shell