Y.P. Tseng

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.

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.

Bending analysis of bimodular laminates using a higher-order finite strip method

Abstract The higher-order plate theory associated with the finite strip method is developed to analyze bimodular composite laminates in this paper. The transverse shear deformation can be effectively evaluated and the shear correction coefficients are not required. The maximum deflections and neutral surface locations of sinusoidally loaded rectangular bimodulus plates are determined. The convergence and accuracy of the proposed formulation is well demonstrated by several numerical examples. The efficiencies of computational cost and memory storage are also noted.

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.

Stability of laminated plates using finite strip method based on a higher-order plate theory

A higher-order shear deformable plate finite strip element is developed and employed to investigate the critical buckling load of composite laminated plates. The warping of cross-section and transverse shear deformation can be accurately predicted by the higher-order plate theory. Meanwhile, fewer degrees of freedom in the finite strip method than those in the finite element method are required. The good convergence characteristics of the finite strip approach and the relative efficiency of particular economization schemes are then demonstrated through several numerical results.

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

A refined finite strip method using higher-order plate theory

Abstract The higher-order plate theory is adopted in the finite strip method to analyse orthotropic laminated plates in this paper. Several examples with the existing elasticity solutions are illustrated to validate the accuracy and efficiency of the present formulation. Although fewer degrees of freedom are required, the present model yields the same or even better displacement and flexural stress results than the higher-order plate element. The through-thickness distribution of transverse shear stress is also properly predicted through the stress equilibrium equation.

Free Vibration Analysis of Bimodulus Laminated Plates Using Higher-Order Plate Element

Abstract A C° isoparametric higher‐order plate element is developed to analyze the free vibration of bimodulus laminated plates. The equations of motion for the higher‐order plate theory are also derived variationally. The natural frequencies and neutral surface locations are determined for benchmark problems. The numerical results are compared to available analytical solutions, and excellent agreement is observed. Obviously, the present formulation is more accurate than the first‐order theory.