C.W. Lim

A higher order theory for vibration of shear deformable cylindrical shallow shells

Abstract The vibratory characteristics of thick cylindrical shallow shells of rȩctangular planform is investigated in this paper. A higher order shear deformation theory is proposed to analyse the effects of various shell geometries and boundary conditions on the vibration responses. The present higher order theory results in cubic expressions for the transverse shear distribution through shell thickness. The strain and kinetic integral energy expressions involving higher order shear deformation are derived in terms of Cartesian coordinates. An energy functional based on the principle of extremum energy is employed which yields a governing eigen-matrix equation. A set of orthogonally generated two-dimensional polynomial functions is adopted to approximate the deflections and rotations of the shells. The results, where possible, are compared with the available experimental and existing numerical results. Sets of new results for the hard simply supported and fully clamped cylindrical shallow shells are presented. Selected modes are illustrated in three-dimensional displacement plots.

Vibratory behaviour of shallow conical shells by a global Ritz formulation

Abstract The development of a new approach based on the energy principle is presented to study the free vibration of shallow conical shells. In the numerical procedure, a set of orthogonally generated and kinematically oriented shape functions, which satisfies the geometric boundary conditions at the outset, is proposed to overcome the mathematical complexity in expressing the geometry and variable surface curvature of these shells. To establish the accuracy of this method, convergence and comparison studies have been carried out. In this study, the effects of various shell parameters on the fundamental and higher mode frequencies are investigated. It is found that monotonic increases in the fundamental nondimensional frequency parameter occurs when the cone vertex angle or base subtended angle is increased independently for the cantilever conical shell. The fundamental frequency parameter also becomes higher for the fully clamped conical shell with higher panel-length to cone-length ( a / s ) ratio. A set of first published vibration mode shapes is also presented.

Vibration of pretwisted cantilever shallow conical shells

Abstract This paper presents a mathematical model to investigate the effects of initial twist on the vibratory characteristics of cantilever shallow conical shells. The energy functional is minimized according to the Ritz procedure to arrive at the governing eigenvalue equation. A set of orthogonally generated two-dimensional polynomials associated with a basic function, which accounts for the boundary expressions and constraints, is introduced to approximate the in-plane and transverse displacement amplitude functions. The complete procedure has been automated to compute the vibration frequencies and mode shapes for exemplary problems. In the numerical experiments, the convergence of eigenvalues is confirmed by increasing the degrees of polynomials employed in the admissible shape functions. To enhance the existing literature, a set of first known frequency parameters is presented. The paper highlights the important effects of angle of twist on the vibration frequencies and mode shapes of conical shells. The fundamental physical frequency ω decreases monotonically for a longer conical shell. The result shows that an increase in the angle of twist does not ensure higher torsional stiffness for a conical shell, which is in contradiction with previous observation for a pretwisted beam or plate. The symmetry of modes is absent when the angle of twist is non-zero.

A Ritz vibration analysis of doubly-curved rectangular shallow shells using a refined first-order theory

Integral expressions for strain and kinetic energies in vibration analysis of shear deformable doubly-curved shallow shells are presented. Although the formulation follows the first-order shear deformation theory, the consideration of Lame parameters for the transverse shear strain through shell thickness, which have been hitherto neglected by other researchers, shows linear distribution functions instead of constant values if the Lame parameters were dropped. Finite higher-order two-dimensional orthogonal polynomials (pb-2 functions) previously used in thin shell analyses [3–6] have been generalized from three to five degrees to account for the additional rotation fields of a moderately thick shell. These global shape functions satisfy the kinematic boundary conditions at the outset. The excellent performance and versatility of the computational methodology of the pb-2 Ritz method are illustrated in representative numerical simulations. The resultant matrix size is far smaller than other discretization methods and yet accurate solutions can be obtained. The validity of the results is verified through direct comparisons. The discrepancy of various formulations with and without the Lame parameters is presented and discussed.

Vibration of pretwisted cantilever trapezoidal symmetric laminates

SummaryA first known investigation on the vibratory characteristics of pretwisted composite symmetric laminates with trapezoidal planform is presented. A governing eigenvalue equation is derived based on the Ritz minimization procedure. This formulation shows that bending and stretching effects of these symmetric laminates are coupled by the presence of twisting curvature. For the solution method, a set of orthogonally generated shape functions is employed to approximate the transverse and in-plane displacements. These functions are generated through a proposed recurrence formula. During the orthogonalization process, a basic function is introduced to ensure the satisfaction of the essential boundary conditions. This proposed method is applied to determine the vibration response of the titled problem. The effects of angles of twist and laminated layers upon the vibration frequencies are examined. Convergence tests for selected examples are presented in which the accuracy of the results is established. It has been found that an increase in the angle of twist also strengthens the torsional stiffness of the laminated plates and therefore results in higher twisting frequencies. The present solutions, where possible, are verified by comparison with data of simplified examples that are available in the literature.

Vibration of perforated doubly-curved shallow shells with rounded corners

This study examines the natural frequency and vibratory characteristics of doubly-curved shallow shells having an outer super-elliptical periphery and an inner super-elliptical cutout. A superelliptical boundary in this context is defined as (2x/a)2n + (2y/b)2n = 1, where n = 1, 2, 3, …, ∞. This class of shells with rounded outer and inner corners has a great advantage over shells with a rectangular planform as stress concentration at the corners is greatly diffused. As a result, the high stress durability of such shells has a great potential for use in practical engineering applications, especially in aerospace, mechanical and marine structures. The doubly-curved shells investigated possess variable positive (spherical), zero (cylindrical) and negative (hyperbolic paraboloidal) Gaussian curvatures. A global energy approach is proposed to the study of such shell problems. The Ritz minimization procedure with a set of orthogonally generated two-dimensional polynomial functions is employed in the current formulation. This method is shown to yield better versatility, efficiency and less computational execution than the discretization methods.

A global continuum Ritz formulation for flexural vibration of pretwisted trapezoidal plates with one edge built in

Abstract The development of a continuum Ritz model for twisted plate vibration on the basis of minimization of the energy functional is presented. This method involves the use of a set of orthogonally generated pb -2 shape functions to approximate the transverse and in-plane deflections. In this paper, a two-dimensional recurrence formula is proposed to generate this set of functions. During the orthogonalization process, a basic function is introduced to each transverse and in-plane displacement to ensure the automatic satisfaction of the essential boundary conditions. The whole process can be easily implemented numerically and the accuracy of the method can be achieved by increasing the number of terms used in the polynomial functions. Following the above procedures, a governing eigenvalue equation is derived. This equation is solved to determine the vibration response of trapezoidal plates with initial twist. In order to verify the validity and reliability of this method, in addition to the convergence study, the present results are compared with the available values from the literature where possible.

Vibration of shallow conical shells with shear flexibility: A first-order theory

A formulation of shear deformation theory implemented numerically for the prediction of vibratory characteristics of shallow conical shell panels is presented. The derivation of thickness shear is assumed in a linear approximation. The Lame parameter for the transverse shear strain component, which has previously been neglected, is considered. This consideration accounts for the replacement of a term in transverse strain distribution through the shell thickness which results in linear transverse shear strain distribution in contrast to the constant distribution hitherto known to researchers in this field. The energy integral, which incorporates the shear deformation and rotary inertia, is minimized to derive the governing eigen-matrix equation. A set of benchmark frequency solutions is presented for two exemplary conical shells : the cantilever and the fully clamped shells. Some selected mode shapes in terms of mid-surface contour plots and three-dimensional meshes are also illustrated.

Flexural vibration of doubly-tapered cylindrical shallow shells

Abstract An approach based on the principle of energy is proposed to study the free flexural vibration of chordwise doubly-tapered cylindrical shallow shells. The variations in shell thickness are linear and symmetric. This topic is of practical interest, but one on which no previous work has been conducted. The analysis is performed using an efficient computational method based on the Ritz minimum energy approach. The in-plane and transverse displacement amplitude functions of the shells are approximated by sets of pb -2 shape functions with unknown coefficients. The pb -2 shape functions are basically a set of admissible functions composed of the product of a set of mathematically complete two-dimensional orthogonal polynomials and a boundary kinematically oriented basic function. The basic function is defined by the product of the equations of continuous piecewise boundary expressions of the shell planform each raised to an appropriate basic power corresponding to a free, simply supported or clamped edge, respectively. These pb -2 shape functions comply with the kinematic boundary conditions of the shells at the outset. Three classes of different shell configurations and boundary conditions are studied with selected mode shapes presented. A study on the ignoring of tangential inertia has shown little effect on the frequency response. However, the trend reveals a higher effect is evident as the shell curvature increases. The effects of symmetric thickness variation are reflected in the tabulated data, as well as the mode shape figures. Since no data for a doubly-tapered cylindrical shallow shell can be found in the open literature, the results presented in the current study can be used for future reference and comparison.

Effects of boundary constraints and thickness variations on the vibratory response of rectangular plates

Abstract The present study concentrates on the first known free flexural vibration of doubly-tapered rectangular plates subject to a variety of boundary constraints ranging from a cantilevered plate to a fully clamped plate. The Rayleigh-Ritz minimum total energy approach complemented by the global pb-2 shape functions as the admissible plate displacement amplitude functions is employed. The shape functions are, in principle, the product of a set of maethematically complete two-dimensional polynomials and a basic function. The basic function is formed from a product of the boundary piecewise geometric expressions of the plate each of which is raised to the power of 0, 1 or 2 corresponding to a free, simply supported or clamped edge, respectively. The shape functions satisfy the kinematic boundary conditions at the outset. This proposed computational model has a great advantage over the conventional finite element and the finite strip methods in terms of computation cost, numerical preparation and implementation, application versatility, and, in certain aspects, numerical accuracy. Comprehensive numerical results for six classes of plates with selected mode shapes are presented. These results may serve as a benchmark for future reference since no literature can be found for symmetric doubly-tapered plates as considered in this study.