C.S. Huang

In-plane vibration of laminated curved beams with variable curvature by dynamic stiffness analysis

Based on the Timoshenko-type curved beam theory, the free vibration of composite laminated beams of variable curvature is studied. The effects of shear deformation and rotary inertia are both considered. By incorporating the dynamic stiffness method and the series solution, an analytical solution is developed. The arch is decomposed into as many elements as needed for the accuracy of solution. In each element, a series solution is formulated in terms of polynomials, the coefficients of which are related to each other through recurrence formulas. As a result, in order to obtain an accurate solution, one does not need a lot of terms in series solution and in Taylor expansion series for the variable coefficients in the governing equations. Compared to the previous literature, the accuracy and efficiency of the present formulation are validated. This methodology can be applied to laminated curved beams of complex arch structures without any difficulty.

Comprehensive exact solutions for free vibrations of thick annular sectorial plates with simply supported radial edges

Abstract The first known exact solutions are derived for the free vibrations of thick (Mindlin) annular sectorial plates having simply supported radial edges and arbitrary conditions along the circular edge. The general solutions to the Mindlin differential equations of motion contain non-integer order ordinary and modified Bessel functions of the first and second kinds, and six arbitrary constants of integration. Frequency determinant equations are derived for thick annular sectorial plates with circular edges having all nine possible combinations of clamped, simply supported, or free boundary conditions. Extensive amounts of nondimensional frequency parameters are presented for thickness ratio ( h a ) values of ≅ 0,0.1, and 0.2; radii ( b a ) values within the range of 0.1–0.7; and sector angle values of 180° ⩽ α ⩽ 360° for which, in the range of α > 180°, no previously published results are known to exist. Frequency results obtained for thick annular sectorial plates are compared to those determined for classically thin ( h a ≅ 0 ) ones. The frequencies for 360° annular sectorial plates (i.e. annular plates having a hinged crack) are compared with those for complete circular annular plates. The exact solutions presented herein are useful to researchers for determining the correctness of approximate numerical procedures and software packages for thick plate vibration analyses.

Stress singularities at angular corners in first-order shear deformation plate theory

Abstract The first-known Williams-type singularities caused by homogeneous boundary conditions in the first-order shear deformation plate theory (FSDPT) are thoroughly examined. An eigenfunction expansion method is used to solve the three equilibrium equations in terms of displacement components. Asymptotic solutions for both moment singularity and shear-force singularity are developed. The characteristic equations for moment singularity and shear-force singularity and the corresponding corner functions due to ten different combinations of boundary conditions are explicated in this study. The validity of the present solution is confirmed by comparing with the singularities in the exact solution for free vibrations of Mindlin sector plates with simply supported radial edges, and with the singularities in the three-dimensional elasticity solution for a completely free wedge. The singularity orders of moments and shear forces caused by various boundary conditions are also thoroughly discussed. The singularity orders of moments and shear forces are compared according to FSDPT and classic plate theory.

Out-of-plane dynamic analysis of beams with arbitrarily varying curvature and cross-section by dynamic stiffness matrix method

Abstract The first known dynamic stiffness matrix for noncircular curved beams with variable cross-section is developed, with which an exact solution of the out-of-plane free vibration of this type of beam is derived. By using the Laplace transform technique and the developed dynamic stiffness matrix and equivalent nodal force vector, the highly accurate dynamic responses, including the stress resultants, of the curved beams subjected to various types of loading can be easily obtained. The dynamic stiffness matrix and equivalent nodal force vector are derived based on the general series solution of the differential equations for the out-of-plane motion of the curved beams with arbitrary shapes and cross sections. The validity of the present solution for free vibration is demonstrated through comparison with published data. The accuracy of the present solution for transient response is also confirmed through comparison with the modal superposition solution for a simply-supported circular beam subjected to a moving load. With the proposed solution, both the free vibration and forced vibration of non-uniform parabolic curved beams with various ratios of rise to span are carried out. Nondimensional frequency parameters for the first five modes are presented in graphic form over a range of rise-to-span ratios (0.05≤h/l≤0.75) with different variations of the cross-section. Dynamic responses of the fixed–fixed parabolic curved beam subjected to a rectangular pulse are also presented for different rise-to-span ratios.

An exact solution for in-plane vibrations of an arch having variable curvature and cross section

Abstract An exact solution for in-plane vibration of arches with variable curvature as well as cross section has been developed using the famous Frobenius method combined with the dynamic stiffness method. The effects of shear deformation and rotary inertia are taken into account. A convergent solution is always guaranteed without numerical difficulties. An important by-product of this series solution is that the first known dynamic stiffness matrix for an arch with variable curvature and variable cross section is also explicitly formulated. Some new numerical results are given for non-dimensional frequencies of parabolic arches with a certain type of variation of cross section along the arch that is often used in practical structures. Extensive and accurate (six significant figure ) non-dimensional frequency tables and graphic charts are presented for a series of parabolic arches showing the effects of rise to span length, slenderness ratio, and variation of cross section.

An analytical solution for in-plane free vibration and stability of loaded elliptic arches

Abstract In-plane free vibration and stability analyses of elliptic arches subjected to a uniformly distributed vertical static loading are performed here. A variational principle is applied to derive the governing equations for free vibration and stability of preloaded arches, considering the effect of the extensibility of the arch centerline but neglecting the effect of shear deformation. Particular attention is given to present a general procedure for combining series solutions with stiffness matrixes to construct an analytical solution for free vibration and stability of loaded arches with varying curvature. The correctness of the proposed solution is verified through a convergence study on the vibration frequencies of a loaded circular arch and by comparing the results with published data. The solution is further applied to investigate the behaviors of clamped or fixed–free elliptic arches.

An accurate solution for the responses of circular curved beams subjected to a moving load

In this paper, an accurate and effective solution for a circular curved beam subjected to a moving load is proposed, which incorporates the dynamic stiffness matrix into the Laplace transform technique. In the Laplace domain, the dynamic stiffness matrix and equivalent nodal force vector for a moving load are explicitly formulated based on the general closed-form solution of the differential equations for a circular curved beam subjected to a moving load. A comparison with the modal superposition solution for the case of a simply supported curved beam confirms the high accuracy and applicability of the proposed solution. The internal reactions at any desired location can easily be obtained with high accuracy using the proposed solution, while a large number of elements are usually required for using the finite element method. Furthermore, the jump behaviour of the shear force due to passage of the load is clearly described by the present solution without the Gibbs phenomenon, which cannot be achieved by the modal superposition solution. Finally, the present solution is employed to study the dynamic behaviour of circular curved beams subjected to a moving load considering the effects of the loading characteristics, including the moving speed and excitation frequency, and the effects of the characteristics of curved beams such as the radius of curvature, number of spans, opening angles and damping. The impact factors for displacement and internal reactions are presented. Copyright

Influence of corner stress singularities and nonstructural mass on the vibrations of cantilevered skewed trapezoidal plates

Abstract This work examines the combined influences of re-entrant corner stress singularities and nonstructural mass on the natural frequencies of cantilevered, skewed trapezoidal plates. Using the Ritz method in conjunction with classical thin plate theory, the vibratory transverse displacements are assumed as mathematically complete polynomials and admissible corner functions, which account for the singular bending stress behavior at the re-entrant corner. Detailed numerical studies show that the convergence of upper-bound frequencies of skewed trapezoids with nonstructural mass is enhanced when the oft-used Ritz trial space of polynomials is augmented by admissible corner functions. To close an apparent void in the plate vibration data base, accurate nondimensional frequencies are calculated for thin, isotropic trapezoidal plates (including parallelogram and triangular ones as special cases). An extensive amount of frequency data is reported which summarize the combined effects of geometrical parameters such as skew angle and chord ratio, and of dynamical system parameters such as mass ratio, and the nonstructural mass position. The results obtained by the present method are compared with those obtained by alternative theoretical approaches.