LI Jing-yu,HU Yu-ren

(School of Naval Architecture,Ocean and Civil Engineering,Shanghai Jiao Tong University,Shanghai 200030,China)

1 Introduction

Detailed reviews of vibration of various elastic systems were discussed by Lynn and Kumbasar[1].The crack ratio was also paid a critical attention in the study.The analyses of vibration of a cracked rectangular plate were investigated by Stahl and Keer[2],who have solved the eigen problems of simply supported plate by using homogeneous Fredholm integral equation.Maruyama and Ichinomiya[3]obtained experimentally the actual frequencies for rectangular plates with straight narrow slits,and the effects of lengths,locations and inclination angles of slits on the natural frequency and the mode shape were discussed.Hirano and Okazaki[4],Neku and Solecki[5]have analyzed the internal crack problem by means of finite Fourier transformation.The discontinuities of the displacement and the slope of the crack were expanded into Fourier series,then the characteristic equation in form of an infinite determinant was obtained.Yuan and Dickson[6]and Lee and Lim[7]used the Rayleigh-Ritz approach for the study of the free vibration of the system.Further the results of the finite element method for dynam-ic analysis of a thin rectangular plate with a through crack under bending,twisting and shearing have been formulated by Qian[8].Krawezuk[9-10],Ostachowiez[11]and Liew[12]employed the decomposition method to determine the vibration frequencies of cracked plates.They assumed the cracked plate domain to be an assemblage of small sub domains with the appropriate functions formed and led to a governing Eigen value equation.Ramamurti and Neogy[13]have applied the generalized Rayleigh-Ritz method to determine the natural frequencies of cracked cantilevered plates.The results show that the natural frequency decreases with increasing crack length at the same plate.All the above references have made the conclusion that the natural frequency and the vibration amplitude are functions of the crack location and length.

In recent years,significant efforts have been published in the area of non-destructive damage evaluation for damage identification in structures[14-22].These methods are based on the fact that local damages usually cause decrease in the structures stiffness,which produces the change in vibration characteristic such as natural frequencies,mode shapes and curvature mode shapes.When the changes of the vibration characteristics are examined,the location and magnitude of the structural damage can be identified.Among these vibration characteristics,the natural frequency is one of the most common modal features used in crack detection because it can be measured most conveniently and accurately.However,natural frequency changes alone may not be sufficient for a unique identification of the structural damage.This is because cracks associated with different crack lengths but at two different conditions or with similar crack lengths but at two different locations may cause the same amount of frequency change.Therefore,for successful utilization of vibration data as an analytic tool for damage identification,it is also necessary to understand the effects of all possible damage events at various conditions on the structures.

This paper presents a study on the vibration behaviors of a cracked plate using the numerical method.The solutions compared with the finite element method are also analyzed.The effects of several parameters,including the crack size and the aspect ratio,the compression and tension load are investigated.Some useful conclusions are drawn simultaneously.

2 Governing equation

The equations of motion for generally isotropic plates based on Von Karman’s plate theory are given by Dym and Shames[23],and can be reduced to the following set of equations:

A cracked isotropic rectangular plate is subjected to the in-plane forces,which are uniformly distributed along two opposite edges,as shown in Fig.1.This cracked plate is simply supported at all edges,and the boundary conditions can be written as follows:

The transverse deflection function w,satisfying the geometric boundary conditions,can be written as

where k=［ρh/ D ］1/2,r1=［kω+（mπ/ L）2］1/2,r2=［-kω+（mπ/ L）2］1/2andωis the fundamental natural frequency of the cracked plate without in-plane forces.ρis the mass density per unit volume,h is the thickness of the plate.The boundary conditions in equation(2)are satisfied by the deflection function automatically.Applying the boundary conditions(3)and(4),the three constants Am,Bmand Dm are determined in terms of Cm,and can be written as:

（1）要求學生自學教材后，仿照教師歸納“驗證綠葉在光下合成淀粉”的6個步驟（暗處理→設(shè)置對照→接受光照→酒精脫色→清水漂洗→滴碘檢驗）的方法，對其他3個實驗步驟進行歸納總結(jié)。

(1)for-kω+（mπ/L）2<0

Taking Cm=1 and substituting the fundamental natural frequency into equations(11)and(12),then the admissible shape function in the y-direction,Ym（y）,Am,Bmand Dmare obtained.Substituting theωvalues into the deflection function,the boundary conditions(5)and(6)are satisfied automatically.

Using the results of the ANSYS 5.5[24],the polynomial of the relative fundamental natural frequency is defined as.The curve fitting functions of the frequency parameters for the first symmetric mode have been obtained with aspect ratio(L/W=0.5-2.0)and crack ratio(c/W=0.0-0.5)as follows.The fundamental natural frequencies of the problem varying with aspect ratios,different boundaries,aspect ratios and in-plane loads obtained from this study agreed well with the solution of the finite element method.The functions of frequency parameter for the first mode(m=1)are obtained as

whereξ=c/W,λ=L/W.

In Tab.1,the dimensionless fundamental natural frequencies for a rectangular crack plate are compared with the analytical results from the finite element method.

Tab.1 Frequency parameterω*at aspect ratios 0.5 and 2.0 versus crack ratios for a simply supported plate

3 Finite element modeling of cracked plates

ANSYS is used in this study to analyze a thin plate with a central crack.The finite element model is shown in Fig.2.The eight-node isoparametric quadratic shell element in ANSYS is used to model the structure.The element is suitable for modeling thin shell structures.Two boundary conditions are considered:S-S type(all edges supported)and C-C type(all edges clamped).The meshing near the crack tip is controlled by two parameters:the zoom size and the zoom factor.The zoom size is the number of rings of elements around the crack tip before all elements become regular and near square.Based on the experience gained from linear elastic analysis of cracked plates and shells[25],the zoom size is set equal to 8 in this study.The zoom factor is the size ratio of elements in a ring around the crack tip to those in the next(outer)ring.Numerical results show that a zoom factor of 0.5 generally results in well-conditioned FE meshes,so this value is used in this study.

4 Results and discussion

A fully supported and clamped plate with a central crack subjected to a pair of opposite uniform axial loading on the two opposite edges was studied.Different crack lengths,magnitudes of the axial forces were considered.The results are discussed as follows.

Figs.4-9 show the effect of a central crack on the first three natural frequencies of free vibration.The natural frequencies have been normalized by the fundamental frequency of the corresponding non-cracked plateωμ.The presence of the crack reduces the first three frequencies.This reduction is initially gradual as the crack length increases,but becomes very rapid when the crack length exceeds about 30% of the width of the plate.This effect is more remarkable for lower vibration modals and longer cracks.

Figs.10-13 show the effect of the applied compressive load N on the fundamental fre quencyωof an all-supported or clamped plate with a central crack.For ease of description,values of N andωare scaled by the critical buckling load Nμand the fundamental frequency of vibration of the plate without a crack,ωμ,respectively.For a non-cracked plate （c=0）,as the compressive axial load increases up to 95% of the static buckling load （N /Nμ=0.95）,the frequency decreases almost linearly to about 20% of the frequency when there is no axial compressive load（ω /ωμ?0.2）.When the compressive load further increases beyond N/Nμ=0.95,the frequency dramatically reduces to zero at N/Nμ=1.This rapid reduction of frequency occurs at lower compressive loading levels in cracked plates.The longer the crack,the lower the compressive loading.

The effect of tensile load on the fundamental natural frequency for a central cracked plate is shown in Figs.12-13.The frequency increases with the load in a non-cracked shell because the tensile load makes the structure stiffer.The loading level at which the maximum fundamental frequency occurs naturally depends on the length of the crack.The longer the crack the more remarkable effect it has on the vibration behavior.

The effect of aspect ratio on the fundamental natural frequency for a central cracked plate is shown in Figs.14-15.The frequency decreases when the aspect ratioλ<1,but when 1<λ<2,the frequency increases.

4 Conclusions

In this paper,the vibration behavior of plates with different crack lengths subjected to axial compressive or tensile loading is examined.The results can be summarized as follows:

(1)The method employed in this study has the advantage of high accuracy accompanied by great savings in computation time and data storage as compared with the finite element method.

(2)The fundamental natural frequencies of the problem varying with aspect ratios,different boundaries,and in-plane loads obtained from this study agreed well with the solution of finite element method.

(3)Cracks can fundamentally decrease the natural frequency of vibration of plates or the certain value of crack size.

(4)Natural frequencies decrease as the compressive load increases and the frequency becomes zero at the buckling load of the plate.Natural frequencies under tensile loading rise with the load.

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