Buckling and vibration of polar orthotropic circular plate resting on Winkler foundation

https://doi.org/10.1016/j.jsv.2006.01.073Get rights and content

Abstract

Buckling and vibrational behavior of polar orthotropic circular plates of linearly varying thickness is presented on the basis of classical plate theory. The plate is resting on Winkler-type foundation. An approximate solution has been obtained by Ritz method, which employs basis functions based upon static deflection of polar orthotropic plates. The effect of elastic foundation and that of orthotropy on the natural frequency of plate has been illustrated for different values of taper parameter, flexibility parameter, in-plane force and nodal diameter. The critical buckling load for clamped and simply supported plates have been obtained. A comparison of results with those available in literature has been presented.

Introduction

The increasing use of orthotropic materials in modern aerospace structures has necessitated the study of vibrational characteristics of plate-type components fabricated by these materials. Circular and annular plates are extensively used as structural components for diaphragms and deckplates in launch vehicles. The consideration of thickness variation together with anisotropy not only reduces the size and weight of components, but also meets the desirability of high strength, corrosion resistance and high-temperature performance.

A considerable amount of work dealing with vibration of polar orthotropic circular/annular plates of uniform/non-uniform thickness is available in literature and few of them are reported in Refs. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. The problem of plates resting on an elastic foundation finds application in foundation engineering such as reinforced concrete pavements of high runways, foundation of deep wells, storage tanks and slabs of buildings(Szilard [12, p. 136]). Various models such as Winkler (Chonan [13], Gupta et al. [14] and Liew et al. [15]), Pasternak (Wang and Stephans [16]) and Vlasov (Bhattacharya [17]) have been proposed in the literature.

In various engineering applications, plates are often subjected to in-plane stressing due to compressive loads which may induce buckling, a phenomenon which is highly undesirable. This consideration is important for the design of structural components. Thus, the study of stability of plates assumes great significance. Keeping this in view Gupta et al. [18] analyzed the combined effect of axial force and elastic foundation of polar orthotropic annular plates using quintic splines technique.

Various numerical methods such as finite difference (Greenberg and stavsky [19]) and finite elements (Chen and Ren [20], Liu and Lee [21], Charbonneau [22]) require fine mesh size to obtain accurate results but are computationally expensive. Ritz method is one of the most popular methods for obtaining approximate solutions for frequencies and modes of vibration of elastic plates. It was applied by its inventor to study free vibration of a plate a century ago in 1909 [23]. Ritz method has the advantage of high accuracy and computational efficiency which greatly depends upon the choice of admissible functions. The method has also been used for thick plates [24] with different geometries. Bhat [25], [26] introduced the use of characteristic orthogonal polynomials in Rayleigh–Ritz method in the study of flexural vibration of rectangular and polygonal plates. Dickinson and Blasio [27] modified the set of orthogonal polynomials to study vibration and buckling of orthotropic rectangular plates. Liew et al. [28], [29] and Liew and Lam [30] proposed a set of two-dimensional plate functions generated by using Gram–Schmidt process in analyzing flexural vibration of triangular, rectangular and skew plates of uniform thickness in employing Rayleigh–Ritz method. Liew [31] applied the so-called pb-2 Ritz method to study free flexural vibrations of symmetrically laminated circular plates. Recently, Zhou et al. [32] used Ritz method for three-dimensional vibration analysis of thick rectangular plates while Zhou et al. [33] applied it to the study of circular and annular plates. Kang [34] used Ritz method for vibration analysis of circular and annular plates with nonlinear thickness variation. Kang et al. [35] applied Rayleigh–Ritz method for free vibration analysis of polar orthotropic circular plates.

In the present paper, the effect of elastic foundation on vibrations and buckling of polar orthotropic circular plates of linearly varying thickness with elastically restrained edge has been analyzed by Ritz method. A new type of basis functions based upon static deflection of polar orthotropic plates given by Lekhnitskii [36] have been chosen which leads to faster rate of convergence as compared to the choice of polynomial coordinate functions used in Ref. [37].

Section snippets

Analysis

Consider a thin circular plate of radius a, variable thickness h=h(r), resting on elastic foundation of modulus kf, elastically restrained against rotation by springs of stiffness kΦ and subjected to hydrostatic in-plane force N at the periphery. Let (r,θ) be the polar coordinates of any point on the neutral surface of the circular plate shown in Fig. 1.

The maximum kinetic energy of the plate is given byTmax=12ρω20a02πhW2rdθdr,where W is the transverse deflection, ρ the mass density and ω the

Method of solution: Ritz method

Ritz method requires that the functionalJ(W)=Umax-Tmax=120a02π[Dr{(2Wr2)2+2υθ2Wr2(1rWr+1r22Wθ2)}+Dθ(1rWr+1r22Wθ2)2+Dk{r(1rWθ)}2+kfW2+Nr(Wr)2+Nθ(1rWθ)2]rdθdr+12akφ02π(W(a,θ)r)2dθ-12ρω20a02πhW2rdθdrbe minimized.

Assuming the deflection function as W¯=cosnθi=0mAiWi(R)=cosnθi=0mAi(1+αiR4+βiR1+p)R2i+n,where Ai are undetermined coefficients, W¯=W/a and αi, βi are unknown constants to be determined from boundary conditions (Leissa [1, p. 14])KφdWi(1)dR=-(1-α)3[d2WidR2+υθ(1Rd

Numerical results

The frequency equation (14) is solved by hybrid secant method retaining the advantages of certainty of bisection and the speed of the secant method. The equation has been solved for various values of plate parameters such as taper α(=0.0, ±0.1, ±0.3), rigidity ratio Eθ/Er (=0.75, 1.0, 5.0, 10.0), foundation parameter Kf (=0.00 (0.01) 0.05) and nodal diameter n (=0, 1, 2) in presence of in-plane force parameter N¯ (=0, ±5, ±10). The Poisson's ratio νθ and shear modulus Dk0 has been fixed as 0.3

Discussion

Numerical results are presented in Table 1, Table 2, Table 3, Table 4 and Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12, Fig. 13, Fig. 14, Fig. 15 which show the effect of in-plane force parameter N¯ and foundation parameter Kf on natural frequencies of vibrations for different values of taper parameter α and rigidity ratio p2.

To choose the appropriate number of terms for the evaluation of frequency parameter Ω, a computer program was developed which

Conclusion

The paper presents results for natural frequencies and buckling loads for circular plates incorporating complicating effects such as polar orthotropy, thickness variation, elastic force and elastically restrained edge. Numerical results show that the presence of elastic foundation increases the frequency parameter for CL as well as SS plates. The critical buckling load parameter also gets increased due to elastic foundation whatever are other plate parameters. A desired frequency can be

References (40)

Cited by (48)

View all citing articles on Scopus
View full text