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Trigonometric Functions (trigonometric + function)
Selected AbstractsPost-buckling analysis of imperfect laminates using finite strips based on a higher-order plate theoryINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 15 2003G. P. Zou Abstract A finite strip procedure has been developed for the post-buckling analysis of composite laminates when subjected to progressive end shortening. The finite strips are developed based on a higher-order shear deformation plate theory and there are nine variables at each nodal line. Initial imperfection expressed in the form of suitable trigonometric function is allowed. Examples including isotropic plates and laminates with arbitrary lay-up arrangement are presented. Numerical results for laminates with and without initial imperfection are used to illustrate the effect of imperfection. Copyright © 2003 John Wiley & Sons, Ltd. [source] Stress analyses of laminates under cylindrical bendingINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 1 2008Tarun Kant Abstract A semi-analytical approach for evaluation of stresses and displacements in composite and sandwich laminates under cylindrical bending subjected to transverse load has been developed in this paper. Two dimensional (2D) partial differential equations (PDEs) of such a laminate are obtained by imposing plane-strain conditions of elasticity. The fundamental dependent variables are so selected in this formulation that they satisfy the continuity of displacements and transverse interlaminar stresses at the laminate interface through the thickness. The set of governing PDEs are transformed into a set of coupled first-order ordinary differential equations (ODEs) in thickness direction by assuming suitable global orthogonal trigonometric functions for the fundamental variables satisfying the boundary conditions. These ODEs are numerically integrated by a specially formulated ODE integrator algorithm involving transformation of a two-point boundary value problem (BVP) into a set of initial value problems (IVPs). Numerical studies on both composite and sandwich laminates for various aspect ratios are performed and presented. Accuracy of the present approach is demonstrated by comparing the results with the available elasticity solution. It is seen that the present results are in excellent agreement with the elasticity solutions. Some new results for sandwich laminates and for uniform loading condition are presented for future reference. Copyright © 2006 John Wiley & Sons, Ltd. [source] Coupling of mapped wave infinite elements and plane wave basis finite elements for the Helmholtz equation in exterior domainsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2003Rie Sugimoto Abstract The theory for coupling of mapped wave infinite elements and special wave finite elements for the solution of the Helmholtz equation in unbounded domains is presented. Mapped wave infinite elements can be applied to boundaries of arbitrary shape for exterior wave problems without truncation of the domain. Special wave finite elements allow an element to contain many wavelengths rather than having many finite elements per wavelength like conventional finite elements. Both types of elements include trigonometric functions to describe wave behaviour in their shape functions. However the wave directions between nodes on the finite element/infinite element interface can be incompatible. This is because the directions are normally globally constant within a special finite element but are usually radial from the ,pole' within a mapped wave infinite element. Therefore forcing the waves associated with nodes on the interface to be strictly radial is necessary to eliminate this internode incompatibility. The coupling of these elements was tested for a Hankel source problem and plane wave scattering by a cylinder and good accuracy was achieved. This paper deals with unconjugated infinite elements and is restricted to two-dimensional problems. Copyright © 2003 John Wiley & Sons, Ltd. [source] Hydroelastic vibrations of flexible rectangular tanks partially filled with liquidINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2007Ding Zhou Abstract In this paper, the three-dimensional vibratory characteristics of flexible rectangular tanks partially filled with liquid are studied. The surface waves of the liquid are taken into account in the analysis. Both the bulging modes of the tank-wall vibration and the sloshing modes of the liquid oscillation are investigated. The vibrating modes of the liquid,tank system are divided into four distinct categories: double symmetric modes (SS); antisymmetric,symmetric modes (AS); symmetric,antisymmetric modes (SA) and double antisymmetric modes (AA). Each of these categories is separately investigated. The velocity potential of the liquid is analytically deduced by using a combination of the superposition method and the method of separation of variables. According to the liquid,tank interface conditions and the orthogonality of trigonometric functions, the coefficients in the solution of liquid velocity potential are expressed in the integral forms including the tank,wall dynamic deflection. A set of reasonable static beam functions is constructed as the admissible functions of the tank-wall vibration. The eigenfrequency equation of the liquid,tank system is derived by using a combination of the Rayleigh,Ritz method and the Galerkin method. Convergence study demonstrates the high accuracy and small computational cost of the proposed approach. Finally, some numerical results are presented for the first time. Copyright © 2006 John Wiley & Sons, Ltd. [source] An enriched meshless method for non-linear fracture mechanicsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2004B. N. Rao Abstract This paper presents an enriched meshless method for fracture analysis of cracks in homogeneous, isotropic, non-linear-elastic, two-dimensional solids, subject to mode-I loading conditions. The method involves an element-free Galerkin formulation and two new enriched basis functions (Types I and II) to capture the Hutchinson,Rice,Rosengren singularity field in non-linear fracture mechanics. The Type I enriched basis function can be viewed as a generalized enriched basis function, which degenerates to the linear-elastic basis function when the material hardening exponent is unity. The Type II enriched basis function entails further improvements of the Type I basis function by adding trigonometric functions. Four numerical examples are presented to illustrate the proposed method. The boundary layer analysis indicates that the crack-tip field predicted by using the proposed basis functions matches with the theoretical solution very well in the whole region considered, whether for the near-tip asymptotic field or for the far-tip elastic field. Numerical analyses of standard fracture specimens by the proposed meshless method also yield accurate estimates of the J -integral for the applied load intensities and material properties considered. Also, the crack-mouth opening displacement evaluated by the proposed meshless method is in good agreement with finite element results. Furthermore, the meshless results show excellent agreement with the experimental measurements, indicating that the new basis functions are also capable of capturing elastic,plastic deformations at a stress concentration effectively. Copyright © 2003 John Wiley & Sons, Ltd. [source] On the differentiation of the Rodrigues formula and its significance for the vector-like parameterization of Reissner,Simo beam theoryINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2002M. Ritto-Corrêa Abstract In this paper we present a systematic way of differentiating, up to the second directional derivative, (i) the Rodrigues formula and (ii) the spin-rotation vector variation relationship. To achieve this goal, several trigonometric functions are grouped into a family of scalar quantities, which can be expressed in terms of a single power series. These results are then applied to the vector-like parameterization of Reissner,Simo beam theory, enabling a straightforward derivation and leading to a clearer formulation. In particular, and in contrast with previous formulations, a relatively compact and obviously symmetric form of the tangent operator is obtained. The paper also discusses several relevant issues concerning a beam finite element implementation and concludes with the presentation of a few selected illustrative examples. Copyright © 2002 John Wiley & Sons, Ltd. [source] | ||||||||||