Dynamic stability of a porous cylindrical shell

This paper is devoted to a closed cylindrical shell made of a porous-cellular material. The mechanical properties vary continuously on the thickness of a shell. The mechanical model of porosity is as described as presented by Magnucki, Stasiewicz. A shell is simply supported on edges. On the ground of assumed displacement functions the deformation of shell is defined. The displacement field of any cross section and linear geometrical and physical relationships are assumed in cylindrical coordinate system. The components of deformation and stress state were found. Using the Hamilton's principle the system of differential equations of dynamic stability is obtained. The forms of unknown functions are assumed and the system of a differential equations is reduced to a simple ordinary equation of dynamic stability of shell (Mathieu's equation). The derived equation are used for solving a problem of dynamic stability of porous-cellular shell with intensity of load directed in generators of shell. The critical loads are derived for a family of porous shells. The unstable space of family porous shells is found. The influence a coefficient of porosity on the stability regions in Figures is presented. The results obtained for porous shell are compared to a homogeneous isotropic cylindrical shell. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A free boundary problem and stability for the rectangular plate

In the recent paper [13] we have answered the question of stability for the linear circular plate which is being axially compressed by a force greater than the critical value and contacts a plane obstacle. In this case there are radially symmetric solutions and the contact region is a disk of a smaller radius. This simplified the determination of the critical parameter values for which the plane jumps to another state. For the rectangular plate continuation has to be applied to the variational inequality in order to determine the contact region and evalute the stability criterion. A numerical method is developed for a discretization of the problem and is used to compute the critical load both in the simply supported and the clamped case.     

Interaction of multiple cracks in a rectangular plate

This paper is concerned with interaction of multiple cracks in a finite plate by using the hybrid displacement discontinuity method (a boundary element method). Detail solutions of the stress intensity factors (SIFs) of the multiple-crack problems in a rectangular plate are given, which can reveal the effect of geometric parameters of the cracked body on the SIFs. The numerical results reported here illustrate that the boundary element method is simple, yet accurate for calculating the SIFs of multiple crack problems in a finite plate.     

Analysis and modeling of a flexible rectangular cantilever plate

Flexible plate structures with large deflection and rotation are commonly used structures in engineering. How to analyze and solve the cantilever plate with large deflection and rotation is still an unsolved problem. In this paper, a general nonlinear flexible rectangular cantilever plate considering large deflection and rotation angle is modeled, solved and analyzed. Hamilton's principle is applied to obtain the nonlinear differential dynamic equations and boundary conditions by introducing a coordinate transformation between the Cartesian coordinate system and the deformed local coordinate system. Stress function relating to in-plane force resultants and shear forces is given for the first time for complex coupling equations caused by coordinate transformation. The nonlinear equations and the solving method are validated by experiments. Then, harmonic balance method is adopted to get the nonlinear frequency-response curves, which shows strong hardening spring characteristic of this system. Runge–Kutta methods are used to reveal complex nonlinear behaviors such as 5 super-harmonic resonance, bifurcations and chaos for general nonlinear flexible rectangular cantilever plate.     

An efficient rectangular plate element

A new 12-parameter rectangular plate element is presented by use of the double set parameter method. The error in the energy norm is of orderO(h ² ), one order higher than the commonly used Adini nonconforming element.     

Vibration of a viscoelastic rectangular orthotropic plate on a viscoelastic foundation

The integrodifferential equation describing the vibration of a viscoelastic rectangular orthotropic plate resting on a viscoelastic foundation with two foundation moduli is investigated. The solution of the starting equation is obtained by the method of straight lines [1] in conjunction with the averaging method [2]. By setting c₁=0 or c₂=0 or their combinations in the solutions obtained we obtain various particular cases that are generalized in the present investigation.Institute of Cybernetics and Computer Center, Academy of Sciences of the Uzbek SSR, Tashkent. Institute of Mechanics and Earthquake Resistance of Structures, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, No. 6, pp. 1087–1094, November–December, 1970.     

Elastic stability of all edges simply supported,stepped and stiffened rectangular plate under uniaxial loading

In this paper using finite difference method the lower bound buckling load for simply supported stepped and stiffened rectangular plate with uniformly distributed compressive forces in one direction is discussed. The plate is divided into 400 rectangular meshes. The partial derivatives are approximated using central difference formula. Three hundred and sixty-one equations are formed and using MATLAB the least eigenvalue is obtained. The buckling coefficients are calculated for different types of stepped plates and the results are presented in tables and graphs for ready use by designers. Two separate tables are presented in comparison with the exact buckling factor for stepped rectangular plate by Xiang, Wei and Wang. The results give a close agreement.     

Elastic stability of all edges simply supported,stepped and stiffened rectangular plate under Biaxial loading

In this paper using finite difference method the lower bound buckling load for simply supported (a) stepped and stiffened rectangular thin plate (b) linear and non-linear variation of thickness (c) uniformly distributed compressive forces in both directions (d) uniformly distributed compressive force in y direction and non-uniform distribution of compressive force in x-direction is discussed. The thin plate is divided into 900 rectangular meshes. The partial derivatives are approximated using central difference formula. Eight hundred and forty one equations are formed and using the program developed and the least eigenvalue is obtained. The buckling coefficients are calculated for different types of stepped and non prismatic plates and the results are presented in tables and graphs for ready use by designers. Buckling factors for some cases are presented in the form of three separate tables and compared with the values obtained by Xiang, Wei and Wang. The results are in close agreement.     

Delamination buckling of a rectangular orthotropic composite plate containing a band crack

The delamination buckling problem for a rectangular plate made of an orthotropic composite material is studied. The plate contains a band crack whose faces have an initial infinitesimal imperfection. The subsequent development of this imperfection due to an external compressive load acting along the crack is studied through the use of the three-dimensional geometrically nonlinear field equations of elasticity theory for anisotropic bodies. A criterion of initial imperfection is used in determining the critical forces. The corresponding boundary-value problems are solved by employing the boundary-form perturbation technique and the FEM. Numerical results for the critical force are presented.     

Bending of a rectangular orthotropic glass-reinforced plastic plate under a transverse load

The Bubnov-Galerkin method is used to solve the problem of the bending of a clamped flexible orthotropic plate. An equation relating the deflection and the load is obtained. An experiment on a plate made of polyester plastic reinforced with satin-weave glass fabric is described. It is shown that within certain limits the deflections at the center of the plate and the combined stresses along the clamped edges are in good agreement with the theoretical values. The limits of applicability of the first-approximation formulas are investigated.All-Union Scientific-Research Institute of Railroad Transport, Ural Division, Sverlovsk. Translated from Mekhanika Polimerov, No. 4, pp. 674–679, July–August, 1969.     

Exact control of vibrations of a rectangular plate by boundary forces

The problem of control of vibrations of a right-angled plate is considered with restrictions posed on the absolute value of the control forces. The purpose of control is to bring the plate into the rest state in finite time.     

Unsteady MHD flow of a non-Newtonian fluid on a porous plate

This paper deals with the study of the MHD flow of non-Newtonian fluid on a porous plate. Two exact solutions for non-torsionally generated unsteady hydromagnetic flow of an electrically conducting second order incompressible fluid bounded by an infinite non-conducting porous plate subjected to a uniform suction or blowing have been analyzed. The governing partial differential equation for the flow has been established. The mathematical analysis is presented for the hydromagnetic boundary layer flow neglecting the induced magnetic field. The effect of presence of the material constants of the second order fluid on the velocity field is discussed.     

Cylindrical bending of an elastic rectangular sandwich plate on a deformable foundation

Cylindrical bending of an elastic rectangular sandwich plate having a rigid filler and resting on an elastic foundation is considered. To describe the kinematics of the plate, asymmetric across its thickness, the hypotheses of broken normal are assumed. The reaction of the foundation is described by the Winkler model. A system of equilibrium equations is derived, and its exact solution is obtained in terms of displacements. A numerical analysis of the solution is presented.     

Perturbation analysis of a modified second grade fluid over a porous plate

A modified second grade non-Newtonian fluid model is considered. The model is a combination of power-law and second grade fluids in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The flow of this fluid is considered over a porous plate. Equations of motion in dimensionless form are derived. When the power-law effects are small compared to second grade effects, a regular perturbation problem arises which is solved. The validity criterion for the solution is derived. When second grade effects are small compared to power-law effects, or when both effects are small, the problem becomes a boundary layer problem for which the solutions are also obtained. Perturbation solutions are contrasted with the numerical solutions. For the regular perturbation problem of small power-law effects, an excellent match is observed between the solutions if the validity criterion is met. For the boundary layer solution of vanishing second grade effects however, the agreement with the numerical data is not good. When both effects are considered small, the boundary layer solution leads to the same solution given in the case of a regular perturbation problem.     

Boundary collocation method for a cracked rectangular plate with double external tension

In this article, the boundary collocation method is employed to investigate the problems of a central crack in a rectangular plate which applied double external tension on the outer boundary under the assumption that the dimensions of the plate are much larger than that of the crack. A set of stress functions has also been proposed based on the theoretical analysis which satisfies the condition that there is no external force on the crack surfaces. It is only necessary to consider the condition on the external boundary. Using boundary collocation method, the linear algebra equations at collocation points are obtained. The least squares method is used to obtain the solution of the equations, so that the unknown coefficients can be obtained. According to the expression of the stress intensity factor at crack tip, we can obtain the numerical results of stress intensity factor. Numerical experiments show that the results coincide with the exact solution of the infinite plate. In particular, this case of the double external tension applied on the outer boundary is seldom studied by boundary collocation method.     

Hydromagnetic effects on the three-dimensional flow past a porous plate

Hydromagnetic effects on the three-dimensional flow of an electrically conducting viscous incompressible fluid past a porous plate with periodic suction has been analysed. The uniform flow is subjected to a transversely applied magnetic field. The mathematical analysis is presented for the hydromagnetic boundary layer flow neglecting the induced magnetic field. Approximate solutions for the components of velocity field and the skin-frictions due to them are obtained and discussed with the help of a graph and tables.