Anand V. Singh

Free in-plane vibration of isotropic non-rectangular plates

Abstract A numerical study for the free in-plane vibration of non-rectangular plate is presented in this paper. The plate is defined by four curved boundaries using eight points and natural co-ordinates to map the geometry. The displacement nodes are introduced at the boundary of the plate as well as inside the perimeter. Each displacement node has two degrees of freedom and much higher order polynomials than the ones used for the geometric representation are used to represent the displacement fields. Using the energy functional, the matrix equation is obtained and solved for natural frequencies and associated mode shapes. Numerical results are calculated first for rectangular plates with three sets of boundary conditions and then compared with the data published in the literature by other researchers. Additional results are also presented and discussed in this paper for the in-plane vibration of rhombic and circular–annular–sector plates.

On the finite displacement analysis of quadrangular plates

Abstract In this paper, a numerical method for the linear and geometrically non-linear static analysis of thin plates is presented. The method begins with the elasticity equations pertaining to strain components, stresses, displacement components, strain energy and work due to externally applied loads. The plate geometry is defined by a quadrangular boundary with four straight edges and the natural coordinates in conjunction with the Cartesian coordinates are used to map the geometry. The matrix equation of equilibrium is derived using the work–energy principle with the displacement fields expressed by algebraic polynomials, the coefficients of which are then manipulated to satisfy the kinematic boundary conditions. To validate the results from the present method, square plates having all sides fully fixed and all sides simply supported without in-plane movement are analysed. Comparison is made for the uniformly loaded square plate with the results obtained by Levy who solved the non-linear plate bending problem using the Th.von Ka ’ rma ’ n ’ s equations. Rhombic plates are examined and numerical results corresponding to these cases are presented in this paper. Very good comparison of the results regarding deflection and bending stresses with other sources available in the literature is found.

Nonlinear Forced Vibrations of Laminated Piezoelectric Plates

A numerical approach is presented for linear and geometrically nonlinear forced vibrations of laminated composite plates with piezoelectric materials. The displacement fields are defined generally by high degree polynomials and the convergence of the results is achieved by increasing the degrees of polynomials. The nonlinearity is retained with the in-plane strain components only and the transverse shear strains are kept linear. The electric potential is approximated layerwise along the thickness direction of the piezoelectric layers. In-plane electric fields at the top and bottom surfaces of each piezoelectric sublayer are defined by the same shape functions as those used for displacement fields. The equation of motion is obtained by the Hamilton’s principle and solved by the Newmark’s method along with the Newton–Raphson iterative technique. Numerical procedure presented herein is validated by successfully comparing the present results with the data published in the literature. Additional numerical examples are presented for forced vibration of piezoelectric sandwich simply supported plates with either a homogeneous material or laminated composite as core. Both linear and nonlinear responses are examined for mechanical load only, electrical load only, and the combined mechanical and electrical loads. Displacement time histories with uniformly distributed load on the plate surface, electric volts applied on the top and bottom surfaces of the piezoelectric plates, and mechanical and electrical loads applied together are presented in this paper. The nonlinearity due to large deformations is seen to produce stiffening effects, which reduces the amplitude of vibrations and increases the frequency. On the contrary, antisymmetric electric loading on the nonlinear response of piezoelectric sandwich plates shows increased amplitude of vibrations.

Transient Vibration Analysis of Open Circular Cylindrical Shells

A numerical method based on the Rayleigh-Ritz method has been presented for the forced vibration of open cylindrical shells. The equations are derived from the three-dimensional strain-displacement relations in the cylindrical coordinate system. The middle surface of the shell represents the geometry, which is defined by an angle that subtends the curved edges, the length, and the thickness. The displacement fields are generated with a predefined set of grid points on the middle surface using considerably high-order polynomials. Each grid point has five degrees of freedom, viz., three translational components along the cylindrical coordinates and two rotational components of the normal to the middle surface. Then the strain and kinetic energy expressions are obtained in terms of these displacement fields. The differential equation governing the vibration characteristics of the shell is expressed in terms of the mass, stiffness, and the load consistent with the prescribed displacement fields. The transient response of the shell with and without damping is sought by transforming the equation of motion to the state-space model and then the state-space differential equations are solved using the Runge-Kutta algorithm.

Linear and Nonlinear Dynamic Responses of Various Shaped Laminated Composite Plates

A unified approach to study the forced linear and geometrically nonlinear elastic vibrations of fiber-reinforced laminated composite plates subjected to uniform load on the entire plate as well as on a localized area is presented in this paper. To accommodate different shapes of the plate, the analytical procedure has two parts. The first part deals with the geometry which is interpolated by relatively low-order polynomials. In the second part, the displacement based p-type method is briefly presented where the displacement fields are defined by significantly higher-order polynomials than those used for the geometry. Simply supported square, rhombic, and annular circular sector plates are modeled. The equation of motion is obtained by the Hamiltons principle and solved by beta-m method along with the Newton-Raphson iterative scheme. Numerical procedure presented herein is validated successfully by comparing present results with the previously published data, convergence study, and fast Fourier transforms of the linear and nonlinear transient responses. The geometric nonlinearity is seen to cause stiffening of the plates and in turn significantly lowers the values of displacements and stresses. Also as expected, the frequencies are increased for the nonlinear cases.

Geometrically non-linear dynamic analysis of laminated shells using Bézier functions

Abstract In this paper a geometrically non-linear theory for the analysis of open deep laminated shell panels is presented. Parabolic variation of the transverse shear stresses through the thickness of the shell and the effects of rotary inertia are included in the formulation. Linear and non-linear stiffness matrices are derived using orthogonal curvilinear coordinate system for a general doubly curved deep laminated shell. The solution of the equation of motion is based on the Ritz method. Therefore, the Bezier surface patches are used as the admissible displacement fields to represent the shells middle surface displacement and rotation components and the resulting equation of motion is solved to obtain the transient response, using Beta-m time integration and Newton-Raphson iterations. A very good convergence of the responses is observed by using only the fifth order Bezier surface patches. The dynamic responses of cross-ply laminated circular and non-circular cylindrical panels pinned at the straight edges under a central point load are studied. Effect of the eccentricity on the dynamic response of non-circular cylindrical panels having the same plan form as the circular cylindrical panel is also examined.

On asymmetric vibrations of layered orthotropic shells of revolution

Abstract A simple and computationally efficient numerical approach is employed in this study for free vibrations of complicated shaped layered shells of revolution. Each layer of the shell is assumed to be constructed from an orthotropic material. The effects of shear deformation and rotary inertia have been included in the formulation, where stiffness and consistent mass matrices are evaluated using reduced integration technique. The approach yields accurate results for shells, if the mid-surfaces are represented by a series of mutually tangential circular arcs. Numerical examples include the natural frequencies and corresponding mode shapes of three-layered freely supported circular cylindrical shells, clamped shallow spherical shells and a cylindrical shell closed by a torispherical head.

A Numerical Free Vibration Analysis of Annular Elliptic Plates

A numerical method is presented in this paper for the free vibration analysis of circular and elliptical first order shear deformable plates. In this method, the ellipse is mapped into a circle and then the circular geometry of the plate is mapped using parabolic interpolation function of natural coordinates and eight nodal points of prescribed coordinates. The displacement fields are defined by a set of relatively very high order interpolation functions and for the displacement degrees of freedom a set of nodal points are defined separate from those of the geometric interpolation. Numerical results for the fully clamped elliptical plate are obtained and compared with the available data from the literature. Additional results for the simply 1‐supported complete elliptical plate and the annular elliptical plates subjected to various boundary conditions are presented and discussed.

Stresses in Welded Pad Reinforced Nozzle in Spherical and Ellipsoidal Pressure Vessel

When a piping system is attached to spherical or ellipsoidal pressure vessel heads, localized stresses of immensely high magnitudes are developed in the vicinity of the shell-pipe juncture due to pressure loads and/or thrust and moment loads. This stress problem is generally overcome by welding a pad to the main shell around the nozzle. This reinforcement alters the stress distribution pattern as well as changes the flexibilities of the nozzle-shell juncture substantially. Analytical investigations using the thin shell theory and experimental testing of this problem have been carried out in the past[1–5].