B-Spline Wavelet on Interval Finite Element Method for Static and Vibration Analysis of Stiffened Flexible Thin Plate

Xing Wei,Wen Chen,Bin Chen,3,Bin Chen,Bin Chen,Bin Chen

B-Spline Wavelet on Interval Finite Element Method for Static and Vibration Analysis of Stiffened Flexible Thin Plate

Xing Wei1,2,Wen Chen2,Bin Chen2,3,Bin Chen1,4,Bin Chen2,Bin Chen1

A new wavelet finite element method(WFEM)is constructed in this paper and two elements for bending and free vibration problems of a stiffened plate are analyzed.By means of generalized potential energy function and virtual work principle,the formulations of the bending and free vibration problems of the stiffened plate are derived separately.Then,the scaling functions of the B-spline wavelet on the interval(BSWI)are introduced to discrete the solving field variables instead of conventional polynomial interpolation.Finally,the corresponding two problems can be resolved following the traditional finite element frame.There are some advantages of the constructed elements in structural analysis.Due to the excellent features of the wavelet,such as multi-scale and localization characteristics,and the excellent numerical approximation property of the BSWI,the precise and efficient analysis can be achieved.Besides,transformation matrix is used to translate the meaningless wavelet coefficients into physical space,thus the resolving process is simplified.In order to verify the superiority of the constructed method in stiffened plate analysis,several numerical examples are given in the end.

B-spline wavelet on the interval;Wavelet finite element method;S-tiffened plate;Bending analysis;Vibration analysis.

1 Introduction

The finite element method(FEM)has been well accepted in numerous industrial areas for structural analysis and simulation[Zienkiewicz and Taylor(2005);Bathe(1996);Song,Noh,and Choi(2003);Atluri,Gallagher,and Zienkiewicz(1983);Dong,EI-Gizawy,Juhany,and Atluri(2014)].It has helped many engineers and scholars to promote their work both in design and application.For example,the structural analysis results by FEM helped the engineers to improve their design.It can test the characteristics of the new theoretical achievements or simulate the experimental environment before it is used in experiment or application.

Stiffened plate is a typical structure used in various mechanical equipment and architecture structures.It can enhance the component and can improve the characteristics of the equipment.Therefore,many scholars and engineers spent much energy and time on precise analysis and test of stiffened structures.For example,based on the work of Han and Atluri(2003),Dong and Atluri developed the twodimensional weakly-singular symmetric galerkin boundary elements for fatigue crack growth analyses in stiffened panels[Dong and Atluri(2012)].Patel,Bisagni,and Datta(2011)studied the dynamic buckling behavior of laminated composite stiffened cylindrical shell by using ABAQUS.Fenner and Watson(2012)tackled the buckling problem of stiffened plate with filleted junction by three dimensional fi nite element in order to optimize the fi llet radius along the line junction.[Duran,Rodriguez,and Sanhueza(2012)]applied a low order finite element method to analyzing a stiffened plate based on Reissner-Mindlin equations and Timoshenko beams equations.Fernandes and Neto(2014)analyzed stiffened plates composed by beams and slabs with different materials based on boundary element method.[Golmakani and Mehrabian(2014)]studied the elastic large deflection problem of axisymmetric ring-stiffened circular and annular general angle-ply laminated plates by first order shear deformation theory and the dynamic relaxation method.Askari,Saadatnia,Esmailzadeh,and Younesian(2014)investigated the free and forced vibrations of stiffened triangular plate based on Galerkin approach,the energy balance method and the variational approach.Zhu,Chen,Kong,and Zhang(2014)developed a hybrid method combining finite element analysis and energy finite element analysis together to predict the vibrations of stiffened built-up structure.Shi,Kapania,and Dong(2015)developed a finite element method for the static,vibration and buckling behaviors of curvilinearly stiffened plates.Although many achievements have been obtained for stiffened structure analysis,the precision or efficiency is limited.This is because the traditional finite element or hybrid method with the traditional polynomial function or other traditional interpolation function being used to interpolate the solving field cannot achieve multi-scale analysis for high efficiency and precision.

Wavelet finite element method is a new numerical method developed in the last two decades,which takes wavelet functions to replace traditional interpolation in field variables discretion.Due to excellent features of the wavelet function,such as mul-tiresolution,multi-scale and localization etc.,the Wavelet Finite Element Method(WFEM)possesses excellent numerical analysis ability.As an example,[Ko,Kurdila,and Pilant(1995)]constructed Daubechies wavelet finite element method and applied it in 1D and 2D elliptic partial differentiation equations with Neumann boundary conditions in 1995.This paper uncovers the prelude on WFEM.Li and Chen(2014)reviewed the recent development of wavelet numerical method,including wavelet weighted residual method,wavelet finite element method,wavelet boundary method and wavelet meshless method.Mitra constructed the 2-D wavelet spectral finite element method for analysis of wave propagation in an isotropic plate[Mitra and Gopalakrishnan(2006)].Xiang implemented damage detection through a hybrid method by combining the interval wavelets and wavelet finite element model together[Xiang,Matsumoto,Wang,and Jing(2011)].Li et al.achieved multiple cracks quantitative identification in a rotor by using wavelet finite element method in forward problem to construct the model of the rotor[Li and Dong(2012)].Yang analyzed the vibration problem of shell structures by using B-spline wavelet on the interval finite element and general shell theory[Yang,Chen,Li,He,and Miao(2012)].Zhang,Chen,Yang,Li,and He(2014a)developed a stochastic finite element method based on B-spline wavelet on the interval for static analysis of 1D and 2D structures.Wang and Wu(2013)investigated a new kind of operator-orthogonal wavelet-based element for adaptive analysis of thin plate bending problems.Liu,Xiang,Gao,Jiang,Zhou,and Li(2014)implemented WFEM to analyze the dispersion relation for one dimensional phononic crystals.Yang,Chen,Li,Miao,and He(2014)applied BSWI finite element method for wave motion analysis in arch structures.Xue,Zhang,Li,Qiao,and Chen(2014)proposed a multi-scale wavelet-based numerical method for wave propagation and load identification by introducing modified Hermitian cubic spline wavelets on the interval.Alm,Harbrecht,andKrämer(2014)investigated the fast solution for nonlocal operator equations based on H2-wavelet method.Zuo et al.studied the static,vibration and buckling problems of functionally graded beams and plates[Zuo,Yang,Chen,Xie,and Zhang(2014a,2014b)].Zhao(2015)studied the temperature-pressure coupled field analysis of liquefield petroleum gas tank under jet fire by WFEM.Samaratunga,Jha,and Gopalakrishnan(2015)]developed a wavelet spectral model for studing transient dynamics and wave propagation in adhesively bonded composite joints.In order to improve the calculation precision of generalized stress and strain,[Zhang,Chen,and He(2011a,2012);Zhang,Chen,He,and Yang(2011b);Zhang,Chen,Yang,and Shen(2014b);Zhang,Zuo,Liu,Chen,and Yang(2016)]constructed a series of multivariable wavelet finite elements for the structures of beam,plate and shell,and static and dynamic characteristics are analyzed and investigated.[Wang,Wu,and Wang(2015)]proposed a design method of second generation wavelet-based multivariable finite element for static and vibration anal-ysis of beam.

In this paper,the wavelet finite element for stiffened plate is investigated to improve the solving precision and efficiency in bending and free vibration analysis.First,the FEM formulation of the stiffened plate is derived from generalized potential energy function.Then,taking BSWI scaling functions as interpolation function,the solving field variables are discretized.In the meantime,the transformation matrix is constructed to translate the meaningless wavelet coefficients into physical space.Finally,the static and dynamic problems can be resolved by solving the FEM matrix equations.

2 Two-dimensional BSWI

B-spline wavelet on the interval was constructed by Chui and Quak(1992),and the decomposition and reconstruction algorithm was derived in 1994 Quak and Weyrich(1994).The even order BSWI is frequently chosen in practical numerical calculation.

At any scalej,themth order BSWI scaling functions must satisfy the following condition[Goswami,Chan,and Chui(1995)]in order to have at least one inner wavelet on the interval[0,1].

Since the 0 scalemth order BSWI scaling functions and wavelet functions had been derived by Goswami,Chan,and Chui(1995),thejscalemth order BSWI scaling functionsand wavelet functionscan be evaluated by the following formulas.

The 11 BSWI43scaling functions are given here for the use in the following sections.

The vector form of the BSWI scaling functions is:

The vector form of BSWI wavelet functions is:

Two dimensional BSWI can be generated from the BSWI vectors in Eqs.(4-5)by tensor product.The two dimensional BSWI scaling functions are:

where,Φ1and Φ2are BSWI scaling function vector in Eq.(4).

The two dimensional BSWI wavelet functions are:

where,ψ1and ψ2are BSWI wavelet function vector in Eq.(5).

The corresponding two dimensional BSWI scaling functions and wavelet functions are shown in Fig.1.

Figure 1:Two dimensional BSWI scaling and wavelet functions.(a)scaling functions Φ =Φ 1⊗Φ 2and(b)wavelet functions ψ 1=Φ 1⊗ψ 2;(c)wavelet functions ψ2=ψ 1⊗Φ 2and(d)wavelet functions ψ 3=ψ 1⊗ψ 2.

Figure 2:Stiffened plate model and node displacement.

3 BSWI element for bending analysis of stiffened plate

3.1 BSWI element construction

As shown in Fig.2,there are three Degrees of Freedom(DOF)in each node.The total potential energy of the stiffened flexible thin plate is consisted with strain energy of the flexible thin plate,strain energy of the stiffeners and the potential energy of the load.Therefore,the total potential energy function of the stiffened fl exible thin plate can be obtained as Shen(1991):

where,

Ubis the bending strain energy of the flexible thin plate:

Usis the tensile strain energy of the flexible thin plate.

Uiis the strain energy of stiffeners along x direction.

Ujis the strain energy of stiffeners along y direction.

Vpis the potential energy of load.

Where,u,v,ware the displacement of flexible thin plate in middle plane along the direction ofx,yandzrespectively shown in Fig.2.Dis the bending stiffness.µis poisson ratio.E,EiandEjare the elastic modulus of flexible thin plate,stiffeners alongxdirection and stiffeners alongydirection respectively.AiandAjare the cross area of the stiffeners alongxandydirection.IiandIjare the inertia moment of stiffeners alongxandydirection.GiandGjare the shear modulus of stiffeners alongxandydirection.qis the load.

The BSWI scaling function in Eq.(4)is used to discrete the displacement field variables.

where,Teis the transformation matrix,which can be expressed as follows.

Substituting Eqs.(11-20)into Eq.(10),and based on the minimum potential energy principle,the BSWI finite element for the stiffened flexible thin plate can be obtained as follows.

where,

where the integral terms are:

SubstitutingLxanddξ asLyanddη in(i,j=0,1,2),the expression of(i,j=0,1,2)can be obtained.

3.2 Numerical examples

3.2.1 Bending analysis example 1

As shown in Fig.3,the clamped flexible thin plate with two orthogonal stiffeners is considered in this numerical example for bending analysis.The corresponding parameters of this plate are:Elastic modulusE=2.116×104N/mm;Poisson ratioµ=0.3;Plate lengtha=b=201.8 mm;Plate thicknesst=2.817 mm;Distributed loadq=0.005 N/mm2.The parameters of the two stiffeners are:Moment of inertiaI=59.6 mm4;Offsete=0.

Figure 3:Clamped flexible thin plate with two orthogonal stiffeners.

Table 1:Displacement and moment of the stiffened flexible plate with two orthogonal stiffeners.

The static bending problem of the stiffened flexible thin plate with two orthogonal stiffeners clamped with distributed load is analyzed in this example.The displace-ment of middle plane and moment results are compared with spline FEM[Shen,Huang,and Wang(1987)],BSWI FEM and experimental results[Shen(1991)]in Table 1.BSWI43is used to discrete the displacement field variables,so there are 363 DOFs in each BSWI element.The results of these three elements are in consistent with each other,so the correctness of the proposed BSWI FEM is verified.With a relatively small computational cost,it can achieve similar precision as spline FEM and experiment do.Besides,the isolines and global deformation in Fig.4 further demonstrated that the analyzed results are reasonable.Therefore,the constructed BSWI FEM is an efficient way in bending analysis for stiffened flexible thin plate.

Figure 4:Deformation of the stiffened flexible thin plate with two orthogonal stiffeners(Isoline along y=a/2).

Figure 5:Clamped flexible thin plate with four orthogonal stiffeners.

3.2.2 Bending analysis example 2

As shown in Fig.5,the square stiffened flexible thin plate with four orthogonal stiffeners is considered in this example.The corresponding parameters are:Elastic modulusE=2.116×104N/mm;Poisson ratioµ=0.3;Plate lengtha=b=201.8 mm;Plate thicknesst=2.817 mm;Distributed loadq=0.005 N/mm2.The parameters of the four stiffeners are:Moment of inertiaI=45.0702 mm4;Offsete=0.

The bending problem of the stiffened flexible thin plate with four orthogonal stiffeners is analyzed in this numerical example.The displacement of middle plane and moment results are shown in Table 2 and the deformation of the stiffened thin plateare drawn in Fig.6.It can be seen from the comparison among spline FEM[Shen(1991)],BSWI FEM and experiment[Shen(1991)]in Table 2 that the results of the three methods are in consistent with other.For BSWI FEM,only one element is used in this example,so the proposed BSWI FEM is an efficient way in structural analysis.Besides,the global deformation and isolines in Fig.6 further demonstrate the correctness of the BSWI FEM.

Table 2:Displacement and moment of the stiffened flexible plate with four orthogonal stiffeners.

Figure 6:Deformation of the stiffened flexible thin plate with four orthogonal stiffeners(Isoline along y=a/2).

4 Bswi element for free vibration analysis of stiffened plate

4.1 BSWI element construction

The total potential energy of the stiffened flexible thin plate in Fig.2 is[Shen(1991)]:

Where,Δ=[u v w]is the displacement in middle plane.

According to minimum potential energy principle

the frequency equation to free vibration can also be obtained.

By solving the frequency equation in Eq.(24),the natural frequencies ω and corresponding mode shapes can be obtained.

Kcan be solved and obtained as shown in section 3.1,so the next problem is how to getM.

The maximum kinetic energy of the stiffened flexible thin plate is:

where,

where,,andare the density of thin plate and ribs alongxandydirections.ω is circular frequency.Other symbols are the same as those in Eq.(10).

Then,according to Eq.(25),the mass matrix for vibration problem of the stiffened fl exible thin plate can be obtained.

where,

The integral terms above are the same as those in section 3.1.

Figure 7:Simply supported flexible thin plate with six orthogonal stiffeners.

4.2 Numerical example

As shown in Fig.7,the square stiffened flexible thin plate with six orthogonal stiffeners is considered in this numerical example for free vibration analysis.The corresponding material parameters of the plate are:Elastic modulusE=10920;Poisson ratioµ=0.3;Plate lengtha=b=1;Plate thicknesst=0.1;Bending stiffnessD=1;Density¯m=1.The parameters of the eight stiffeners are:Cross areaA=0.03;Moment of inertiaI=0.00025;Offsete=0.

The free vibration analysis results of simply supported stiffened flexible thin plate are shown in Table 3.The first four circular frequencies are compared with spline FEM[Shen,Huang,and Wang(1987)],BSWI FEM,orthotropic plate[Shen(1991)]and Reference[Cao(1983)].It can be seen from the comparison that the precision of the constructed BSWI FEM is in consistent with other three methods,while only one BSWI element is used in this free vibration analysis.Therefore,the proposed BSWI FEM is an efficient way in structural analysis,including bending analysis and vibration analysis,which can achieve the results with good precision and small computational cost.

Table 3:Free vibration analysis results of simply supported stiffened flexible thin plate.

5 Conclusion

Based on generalized potential energy function and B-spline wavelet on the interval,a novel BSWI finite element method is constructed for static and free vibration analysis of the stiffened flexible thin plate in this paper.The FEM formulations are derived from the generalized potential energy function according to generalized potential energy function and virtual work principle.BSWI is used to replace polynomial interpolation function in traditional FEM to discrete the solving field variables,and the transformation matrix is constructed to transfer the meaningless wavelet coefficients into physical space.Due to the excellent features of BSWI,the efficient and precise analysis results can be obtained.Through several numerical analysis examples,it proved that the proposed BSWI FEM can achieve static and vibration analysis with high efficiency and precision.

Acknowledgement:This work was supported by the National Natural Science Foundation of China(Nos.51405370&51421004),the National Key Basic Research Program of China(No.2015CB057400),the project supported by Natural Science Basic Plan in Shaanxi Province of China(No.2015JQ5184),the Fundamental Research Funds for the Central Universities(xjj2014014),and Shaanxi Province Postdoctoral Research Project.

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1State Key Laboratory for Manufacturing System Engineering,School of Mechanical Engineering,Xi’an Jiaotong University,Xi’an 710049,PR China

2Department of Mechanical and Aerospace Engineering,Case Western Reserve University,Cleveland Ohio 44106,USA

3School of Instrument Science and Engineering,Southeast University,Nanjing 210096,PR China

4Corresponding author.Tel:+86 29 82667963;fax:+86 29 82663689.E-mail address:chenxf@mail.xjtu.edu.cn(X.F.Chen)