T.Y. Yang

Geometrically Nonlinear Finite Element Analysis of Imperfect Laminated Shells

Formulations and computational procedures are presented for the finite element analysis of laminated anisotropic composite thin shells including imperfections. The derivations of the nonlinear geometric element stiffness matrices were based on the total Lagrangian description. A 48 degree-of-freedom (d.o.f.) general curved shell element with arbitrary distribution of curvatures was used to model the shell middle- surface. Numerical results include the large deflection behavior of a variety of perfect plate and shell examples; buckling of a spherical shell with an axially symmetric im perfection ; and buckling of a cylindrical panel using measured initial transverse im perfections. A good comparison with existing results is obtained.

Static and Dynamic Formulation of a Symmetrically Laminated Beam Finite Element for a Microcomputer

A formulation, an efficient solution procedure, a microcomputer program, and a graphics routine for an anisotropic symmetrically laminated beam finite element in cluding the effect of shear deformation is introduced. The emphasis of the formulation and solution procedure is for simplicity, efficiency, and easy implementation on microcomputers. The element possesses six d.o.f.s at each of the two nodes: trans verse deflection and slope due to bending and shear, respectively, and a twisting angle and its derivative with respect to the beam axis. The formulation, solution procedure, and the program have been evaluated by performing a systematic choice of examples; whenever possible, the present solutions are compared with alternative existing solu tions.

Buckling, postbuckling, and nonlinear vibrations of imperfect plates

Formulations and computational procedures are presented for studying the geometrically nonlinear behavior, including buckling, postbuckling, and nonlinear vibrations of perfect and imperfect, isotropic and laminated thin plates. The finite-element method is used. The element used is a 48 degree-of-freedom thin flat plate rectangular element capable of modeling arbitrary imperfections. The incremental and total stiffness matrices for large displacement behavior are derived based on the total Lagrangian approach in conjunction with the Hamiltons principle. The geometric imperfections are treated by considering additional terms in the straindisplacement relations. Numerical results are presented for a variety of examples including: 1) postbuckling analysis of an isotropic, imperfect flat plate; 2) postbuckling analysis of a thin, cross-ply laminated imperfect plate; 3) free vibrations of an angle-ply laminated plate; 4) free-vibrations of an imperfect isotropic plate under the action of axial loads; 5) nonlinear free-vibrations of isotropic perfect flat plates; 6) nonlinear vibrations of axially loaded, isotropic perfect flat plates; and 7) nonlinear vibrations of an imperfect laminated plate. The results obtained in this study are compared with existing solutions and a good agreement is seen.

Free vibration of a pneumatic tire-wheel unit using a ring on an elastic foundation and a finite element model

Abstract Natural frequencies and mode shapes of a pneumatic tire without suspension are investigated using a 12-d.o.f., geometrically non-linear, doubly curved, thin shell finite element of revolution with laminate composite materials. The wheel is assumed to be free to move within its own plane. The results of the free vibration analysis indicate that only the radial modes of n = 1 are affected by the wheels freedom to move. To evaluate the finite element modeling, a simplified elastic ring-spring model is studied. The tire is modeled as a circular, elastic ring supported by distributed spring in both radial and circumferential directions. The wheel is modeled as a rigid mass to which the disturbed spring is attached. The two models are found to agree and complement each other. While the simplified ring-spring model is easy and practical to use to obtain preliminary results, the complex finite element model can give more detailed and accurate results for both free vibration and dynamic response analyses.

Natural frequencies and modes of rings that deviate from perfect axisymmetry

Given the experimental or theoretical natural frequencies and modes of an axisymmetric ring, an analytical method is presented that allows one to obtain the natural frequencies and modes when this ring is non-axisymmetric due to a mass or stiffness non-uniformity of the ring. A ring is the simplest model of an axisymmetric structure and treatment of its non-uniformities paves the way for treatment of more complicated structures such as tires, for example. The receptance method is employed to determine the natural frequencies and mode shapes of a circular ring which is deviating from axisymmetry due to a local mass of stiffness non-uniformity. The receptance of the ring is derived by utilizing the modal expansion method.

A sector finite element for dynamic analysis of thick plates

Abstract A 24 degree of freedom sector finite element is developed for the static and dynamic analysis of thick circular plates. The element formulation is based on Reissners thick plate theory. The convergence characteristic of the elements is first studied in a static example of an unsymmetrically loaded annular plate. The obvious advantageous effect of including the twist derivatives of deflection as degrees of freedom is shown. The elements are then used to analyze the natural frequencies of an annular plate with various ratios of inner to outer radius. The results are in good agreement with an alternative solution in which thick plate theory is used. The versatility of this finite element is finally demonstrated by performing free vibration analysis of an example of clamped sector plates with various thicknesses and different sectorial angles.

Free and Forced Nonlinear Dynamics of Composite Shell Structures

Natural frequencies and forced responses of thin laminated composite shells of general form are investigated using a high-order curved shell finite element. The effect of geo metric nonlinearity is included in the determination of the forced responses which are ob tained using Newmarks numerical integration scheme. A class of shell problems which are general with respect to geometry; boundary conditions; number, stacking sequence, and angle of orientation of the lamina; and load function can be solved using the present formulations. Numerical results and their comparisons with existing solutions are pre sented for cylindrical shells, shells of positive and negative Gaussian curvature, spherical caps, and shells of translation with different boundary conditions.

Free vibrations of a tire as a toroidal membrane

Abstract A tire is modeled as a toroidal membrane under internal pressure and mounted on a rim, to investigate its free vibration characteristics using a 12 d.o.f. rectangular membrane finite element. Such a modeling is valid if the tire is assumed to be incapable of supporting any weight in the absence of internal pressure. To verify the formulations of the membrane finite element, a flat rectangular membrane subject to in-plane loads and a circular cylindrical membrane under internal pressure are first analyzed. Analytical solutions for these cases are also derived. The analytical and numerical results are in good agreement. A toroidal membrane under internal pressure, assumed to model a low pressure tire, is studied next. Both the analytical derivation and the finite element solutions are presented. For the analytical solution the equations of motion yield a complicated differential equation for which an approximate solution is obtained by assuming that the parallel circle radius is constant as in the case of a bycycle wheel. The finite element solution successfully predicts the symmetrical and the twisting modes of vibration documented by other researchers, and is also in good agreement with the analytical results. The present formulations are useful to obtain a good first approximation of the free vibration response of a tire.

Natural frequencies and mode shapes of an automotive tire with interpretation and classification using 3-D computer graphics

Natural frequencies and mode shapes of a radial tire have been obtained by using an efficient, 12 degree of freedom, doubly curved thin shell finite element of revolution with smeared-out properties of laminate composite materials. The finite element formulation includes the geometrical non-linearities so that the prestressed state of the tire due to inflation is taken into account. While the basic formulation follows that of earlier work done at Purdue University, a general and efficient computational procedure and program have been developed, with a main feature being integration with computer graphics. Thus the complex tire geometry can be modeled more accurately and the free vibration mode shapes can be displayed graphically. This allows an interpretation and classification of mode shapes beyond the classical mode shapes of tires that have been presented in the literature. It allows further insight into the relationship between transverse and tangential motions beyond what has been conceived at the present state of the art of experimentation. Theoretical results are compared with experimental results obtained from modal analysis and good agreement is shown.

FREE VIBRATIONS OF FINITE ELEMENT PLATES SUBJECTED TO COMPLEX MIDDLE PLANE FORCE SYSTEMS

A finite element method is presented for free vibration analysis of thin plates subjected to complex force systems applied in the middle-plane of the plate. The equation of motion is characterized by the basic stiffness, consistent mass, and incremental stiffness matrices. The method is demonstrated by using both a conforming and a non-conforming rectangular plate element. Numerical examples of free vibration of plates under various inplane loadings for different aspect ratios and edge support conditions are illustrated. Particular attention is given to the influence of load intensity on the natural frequency of vibration and the extrapolated load corresponding to zero frequency which yields the static buckling load. It was found that in most cases the relationship between the square of the frequency and the load is not linear. The finite element solutions were also made for a sequence of gridwork refinements to show the convergence characteristics. Comparison of the natural frequencies (zero middle-plane load), the buckling loads and the frequency-load relationships with known analytical and numerical solutions indicates that the method gives good results even for relatively few elements.