Humayun R.H. Kabir

Free vibration analysis of thin arbitrarily laminated anisotropic plates using boundary-continuous displacement Fourier approach

A boundary continuous displacement based Fourier series solution to the boundary-value problem of free vibration of an arbitrarily laminated thin rectangular plate is presented. This powerful approach is employed to solve a system of three highly coupled partial differential equations arising from the Kirchhoff hypothesis as applied to an anisotropic laminate, with the SS2-type simply supported boundary conditions prescribed at all four edges. The accuracy of the computed eigenvalues (natural frequencies) is ascertained by studying the convergence characteristics of the lowest seven natural frequencies, and also by comparison with the computed degenerate FEM (finite element methods) results. Other important numerical results presented include variation of the response quantities of interest with geometric and material parameters, such as fiber orientation angle and longitudinal-to-transverse modulus ratio.

Thermal buckling response of shear flexible laminated anisotropic plates using a three-node isoparametric element

Abstract A recently developed three-node plate element with full integration scheme is extended to solve the problems of thermal buckling response of laminated composite square and skew plates. The plate element is based on the first-order shear deformation theory primarily suitable for moderately thick plates, but is also capable of handling thin plates without difficulty. The shear locking effects are nullified through use of shear correction terms in the transverse shear strain components. Thermal buckling temperatures including the critical one and mode shapes are numerically investigated. The present element shows excellent performance in the moderately thick to very thin plate regime. Results computed using the present element are generally in close agreement with the available analytical and higher order finite element methods solutions. The results computed using the present element are also compared with those due to QUAD4 element of NASTRAN (which supports only symmetric laminates) and SHELL99 element of ANSYS. Interesting results pertaining to the effects of shear deformation, lamination, skewness, etc., are discussed.

Effect of boundary constraint on the frequency response of moderately thick flat laminated panels

Abstract A boundary-discontinuous Fourier-type solution to the problem of free vibration of arbitrarily laminated moderately thick rectangular plates, subjected to general admissible boundary conditions, is presented. The YNS (Yang-Norris-Stavsky) theory incorporating first-order shear deformation, and rotary and in-plane inertias, is used in the formulation, resulting in a system of five highly coupled linear second-order partial differential equations (PDEs) with constant coefficients. For the purpose of illustration, the SSI and C4 boundary conditions are selected here because they provide the upper and lower bounds for elastically supported flat panels used in helicopter rotor blades, aircraft wings and other applications. Numerical results presented here include (i) comparison with the corresponding available FSDT-based boundary-continuous displacement-type Fourier and Galerkin solutions, and (ii) study of the effects of thickness, aspect ratio and fiber orientation angle on the normalized natural frequencies.

On boundary value problems of moderately thick shallow cylindrical panels with arbitrary laminations

An analytical (strong form) solution to the boundary value problem of moderately thick shallow cylindrical panels, with arbitrary laminations is presented. A double Fourier series approach is used to solve five second-order highly coupled partial differential equations that arise from the shallow shell formulation based on popular Donnell-Mushtari-Vlasov shell theory, and the first-order shear deformation-based through-thickness theory. An admissible boundary condition is considered to obtain numerical results that constitute the study of convergence of displacements, and moments; and spatial variations of them presented in the form of contour plotting for various parametric effects. These, hitherto unavailable, analytically obtained numerical results should serve as base-line solutions for future comparisons of popular approximate methods such as finite element, finite difference, Galerkin approach, Rayleigh-Ritz method, collocation method, least-squares method, and experimental results, for the case of arbitrary laminations.

On the Thermal Buckling Response of Shear-Flexible All-Edge Clamped Rectangular Plates:

Using an analytical approach, we investigate a thermal stability response for a rectangular plate with all-edge clamped boundary conditions. We consider the first-order shear deformation theory that utilizes shear flexible response, in order to obtain three highly coupled governing partial differential equations in three unknowns: one transverse displacement, and two independent rotations of the normal. The solution functions are assumed in the form of double Fourier series that satisfy the boundary conditions, as well as the partial differential equations. The results obtained from the analytical solution are compared with available finite element solutions. These analytically obtained results can be capitalized to check the accuracy of various approximate methods.

Frequency Response of a Three-Node Finite Element for Thick and Thin Plates

A shear-locking free isoparametric three-node triangular finite element is presented to study the frequency response of moderately thick and thin plates. Reissner/Mindlin theory that incorporates shear deformation effects is included into the element formulation. A shear correction term is introduced in transverse shear strain components to avoid the shear-locking phenomenon. The element is developed with a full integration scheme, hence, the element remains kinematically stable. Natural frequencies and mode shapes are obtained and compared with the available analytical and finite element solutions.

FREE VIBRATION RESPONSE OF CYLINDRICAL PANELS WITH HIGHER ORDER SHEAR DEFORMATION THEORY

Presented here is an analytical solution to the free vibration problem of an isotropic cylindrical panel with SS2-type simply supported boundary conditions based on Reddys third order shear deformation shell theory. Using the principle of virtual work, the Reddys shell theory generates five highly coupled partial differential equations in terms of three unknown displacements and two unknown rotations. The partial differential equations in conjunction with the prescribed boundary conditions are solved using displacement functions expressed in terms of double Fourier series expansion. Cylindrical panels with various aspect and thickness ratios are considered in the study of convergence behavior and parametric variation of the eigenvalues. The eigenvalues and mode shapes obtained in this study are compared with those obtained from the finite element software package ANSYS. The hitherto unavailable analytical solutions can be used as benchmarks for checking the accuracy of various approximate methods such as the Rayleigh–Ritz, finite element and finite difference methods.

ANALYTICAL SOLUTION OF DYNAMIC ANALYSIS OF SHEAR FLEXIBLE CLAMPED RECTANGULAR PLATES

A study on the suitability of a set of solution functions in an analytical solution approach for the dynamic response of a clamped, moderately-thick plate subjected to a dynamic load is presented. The solution methodology involves obtaining free vibration responses using a recently developed analytical solution approach based on the double Fourier series equations proposed by Kabir and Chaudhuri, and applying the modal superposition method in solving the dynamic response. The equilibrium equations for the plate are defined by a set of three highly-coupled partial differential equations that are based on the Mindlin theory. The solution functions are assumed in the form of double Fourier series that satisfy the boundary conditions. The dynamic load considered is a step impulse load applied at the center of the plate. Results include convergence characteristics of transverse displacement and moment and variation of these quantities with respect to aspect ratios. These analytical solutions are useful for testing the accuracy of numerical methods.

Double orthogonal set of solution functions for cross-ply laminated shear flexible cylindrical/doubly curved panels

An alternate set of solution functions in the form of orthogonal double Fourier series to analyse boundary-value problems of shear flexible general anti-symmetric cross-ply laminated cylinder/doubly curved panels is presented. Two types of boundary conditions are considered, simply supported, and all-edge clamped. The shell deformation characteristics are defined by five highly coupled partial differential equations in second order, according to the shell theory laid by Sanders. The solution functions that define the displacement behaviour of the shell presented in the form of double Fourier series are considered of mixed type, in contrast to the one prescribed earlier by Chaudhuri and Kabir. Efficiencies in the form of convergence and computational time are observed in the present method with improvement in the solutions. The numerical results, thus presented for various parametric effects can be considered as valuable tools to check the numerical solutions obtained from approximate methods.

Load Test on Reinforced Concrete Beams Strengthened by Carbon Fiber Laminated Plates - A Case Study

A load test was performed on reinforced concrete beams that were strengthened with carbon fibers. Seventy five thousands blocks were placed in four stages. A quality and management team worked continuously in the testing process. No creep effects were noticed with the use of carbon fiber plates, an interesting observation.