多變量分層靜態(tài)和振動梁有限元分析

1、 A multivariable hierarchical nite element for static and vibration analysis of beamsZhigang Yu a,b , Xiaoli Guo b , Fulei Chu a, Ãa Department of Precision Instruments, Tsinghua University, Beijing 100084, China bQinghe Dalou Zi, Beijing 100085, Chinaa r t i c l e i n f oArticle history:Receiv

2、ed 25April 2008Received in revised form 4March 2010Accepted 14March 2010Available online 2April 2010Keywords:Generalized variational principle MultivariableHierarchical nite element method BeamBending problem Vibration analysisa b s t r a c tFormulations of a multivariable hierarchical beam element

3、for static and vibration analysis are presented based on the generalized variational principle with two kinds of variables. Two forms of shifted Legendre hierarchical polynomials are used as interpolating basis functions of displacement and generalized force eld functions for the beam element respec

4、tively, which will simplify the computations of the relevant matrices. The multivariable hierarchical beam element formulations, in which the displacement and generalized force eld functions are independently constructed, are derived by applying the generalized variational principle with two kinds o

5、f variables. Since differential operations to obtain stress elds in conventional displacement based nite element methods are not required, the present method has very high accuracy for the two kinds of independent variables simultaneously, especially for the generalized forces. Static and vibration

6、numerical examples demonstrate the applicability of the proposed method. The proposed method can be easily extended to deal with structural analysis of shells or plates.&2010Elsevier B.V. All rights reserved.1. IntroductionThe nite element method (FEMhas been a powerful numerical tool for struct

7、ural analysis up to date. It successfully integrated merits of the traditional energy method and the nite difference method. During the development of FEM, energy variational principles have played a substantially important role and provided a theoretical basis to establish various kinds of nite ele

8、ment formulations 1,2. Melosh 3constructed a displace-ment nite element model by using the potential energy principle. Elias 4and Pian 5presented a stress and hybrid stress nite element model by employing complementary energy principle, respectively. In general, the nodal values of the independent v

9、ariable in the single variable based nite element formulations are quite accurate whereas the rest eld functions, for example, stress and strain in the displacement-based FEM will loss accuracy due to differential or integral operations. In addition, the solutions of these eld functions would not be

10、 continuous at nodal positions if low order approximation functions are used as did in the conventional FEM. However, in some practical situations accurate predictions of displacement as well as higher derivative quantities are of considerable value, such as in structural topology optimization for t

11、he imposition oflocal stress constraints 6, damage detection by using static or dynamic strain 7,8, and stress monitoring in rotating turbine blades. As a consequence, multivariable nite element models have been developed. In these formulations, more than one eld variables are constructed independen

12、tly and can achieve very high accuracy simultaneously. The theoretical foundations of the multivariable FEM are the generalized variational principles with multiple eld variables. The energy variational principle with two kinds of independent variables was rst proposed by Hellinger in 1914. Reissner

13、 9further claried the boundary condition of the principle in 1953. This two-eld variational principle can be derived from the variational principle with one kind of variable and Lagrange multiplier method.The FEM with two kinds of variables have been investigated by many researchers and there are nu

14、merous signicant work reported. An important category of the efforts were concerned with employing more approximate basis functions to interpolate these eld variables and establishing the multivariable FEM with higher accuracy and fewer degrees of freedom. Shen 1014and his collaborators proposed a m

15、ultivariable FEM by using the cubic B spline functions as interpolation functions as well as the HellingerReissner variational principle and applied the formula-tion to tackle with bending, vibration and stability problems of thin plates, thick plates and beams. Han, Reng and Huang 15,16studied thic

16、k plate problems by adopting interpolating wavelet basis functions and the HellingerReissner variational principle. This work advanced the application scope of FEM, but someContents lists available at ScienceDirectjournal homepage:Finite Elements in Analysis and Design0168-874X/$-see front matter &a

17、mp;2010Elsevier B.V. All rights reserved. doi:10.1016/j.nel.2010.03.002ÃCorresponding author.E-mail address:chu(F.Chu.Finite Elements in Analysis and Design 46(2010625631improvements should be carried out. For instance, when for-mulating the multivariable nite element equations, they ex-pressed

18、 the two independent variables using interpolating coefcients instead of nodal quantities. This would lead to intrinsic cumbersomeness in imposing boundary conditions and external loads on the analyzed structure. Furthermore, the results from the nite element equations did not correspond to the noda

19、l displacement and stress and some post-processing work has to be done.In this paper, a multivariable hierarchical element formulation is presented to solve the static bending and free vibration problems of beam structures. The main advantages of this hierarchical strategy are that it keeps a xed co

20、arse mesh and any desired accuracy can be obtained by simply increasing the number of hierarchical shape functions, and the time cost to generate a rened mesh can be saved 1722. The performance of the hierarchical method is usually better than that of the conventional FEM 23. Two different forms of

21、the shifted Legendre orthogonal polynomials are deliberately chosen to represent displacement and generalized force functions. Then the multivariable hierarchical nite element formulations for static bending and vibration are developed by using the HellingerReissner generalized variational principle

22、 with two kinds of independent variables. The explicit expressions of the resulting matrices in rational or integer forms are derived and hence the computations of the relevant matrices are simplied. The applicability of the proposed multivariable hierarchical FEM is demonstrated by determining the

23、static bending solutions for beams with different boundary conditions and distributed loads as well as dynamic characteristics of a cantilever beam. Accuracy and convergence of the proposed method can be observed. 2. Generalized variational principle with two kinds of variablesThe generalized potent

24、ial energy principle with two kinds of variables is the stationary condition of the following functional 1:dP p 2¼0, ð1ÞwhereP p 2ðu , r Þ¼Z Vr T L T u À12r T D À1er þf T u dvÀZ S u p T ðu ÀÞds ÀZS sds :ð2Þin which the d

25、isplacement vector u and the stress vector r are the two categories of unknown independent variable functions, L is the differentiation operator matrix, D e is the elasticity matrix, f is body force vector, P is the traction vector, V , S u and S s are boundaries of the volume, displacement eld and

26、stress eld, and the prescribed boundary traction and displacement vector, respectively. The prescribed surface tractions are applied over the surface S s and the prescribed boundary displacements over S u . S s along with S u make up the total surface of the solid.For bending and vibration problems

27、of the beam shown in Fig. 1, the following relations hold,s x ¼M I y , L T u ¼Àyd 2wdx 2,12r T D À1er ¼s 2x2E, ð3Þwhere M denotes external bending moment, I is the second moment of the cross-section, E is the elastic modulus, w is transverse displacement of the neu

28、tral axis of the beam. Then the extended generalized potential energy function with two kinds of variables for the beam considering inertia effect under homo-geneous boundary conditions is dened as follows:P p 2¼Z LZ h =2À2=hÀM d 2wdxÀM 22EIy 2dy dx ÀZ Lw T q dxÀ1Z Lo 2

29、r Aw 2dx , ð4Þwhere q is the transverse distributed load over the beam, o is the vibration eigenvalue, r is the mass density of the beam material, A is the area of the cross-section. It is worth noting that in Eq. (4, bending moment M and transverse displacement w are inde-pendent variable

30、s and now between them there are no additional constraints.3. Hierarchical interpolating functionsFor simplicity and convenience in mathematical formulation, the local non-dimensional coordinate x , which is related to the element length L through x i ¼2ðx Àx i Þ=L À1, is in

31、troduced.The transverse displacement w within a beam element is suggested to take the form ofw ¼w 1j 1þy 1j 2þw 2j 3þy 2j 4þX ni ¼1w i þ2j i þ4, ð5Þwhere w 1, y 1, w 2and y 2are the transverse displacement and rotation at the two ends of the beam ele

32、ment, w i are generalized internal degrees of freedom, j 1 j 4are the four standard Hermite cubics having the following forms:j1¼14ð2À3x þx 3Þ, j 2¼L8ð1Àx Àx 2þx 3Þ,j3¼1ð2þ3x Àx 3Þ, j 4¼LðÀ1Àx þx

33、 2þx 3Þ,j i are hierarchical interpolating functions which should possess C 1continuity and form a complete set. Here the shifted Legendre polynomials are adopted since they are second derivative orthogonal with respect not only to themselves but also to the rst four Hermite cubics. Furthe

34、rmore, j i and its rst derivatives contribute zero at each end of the element, hence not affecting the imposition of boundary conditions through nodal constraints alone. These functions can be explicitly written as follows:ji þ4ðx Þ¼X½ði À1Þ=2k ¼0ð&#

35、192;1Þk ð2i À2k À7Þ!2k k ! ði À2k À1Þ!x i À2k À1, i Z 1, ð6Þin which k ! ¼k ðk À2Þ. 2or 1, 0! ¼ðÀ1Þ! ¼1, and Ádenotes taking integer part. The curves of the rst six shifted Legendre pol

36、ynomials are plotted in Fig. 2(a.The bending moment distribution within the element can be independently assumed as follows:M ¼M 1c 1þM 2c 2þX mi ¼1M i þ2c i þ2ð7 ÞFig. 1. A beam element with rectangular cross-section.Z. Yu et al. /Finite Elements in Analysis

37、and Design 46(2010625631 626in which M 1and M 2are the bending moments at each end of theelement, M i are generalized internal bending moments, c 1and c 2are linear functions given byc 1¼12ð1Àx Þ, c 2¼12ð1þx Þc i are the hierarchical terms and must have the pr

38、operty of C 0continuity. The following forms of the shifted Legendre poly-nomials, as shown in Fig. 2(b,are employed,c i þ2ðx Þ¼X ½ði À1Þ=2 k ¼0ðÀ1Þk ð2i À2k À5Þ! 2kk ! ði À2k À1Þ!x i À2k À1

39、, i Z 1:ð8ÞThey have similar properties of Eq. (4except for the secondderivative orthogonality.The shape functions in Eqs. (5and (7can be written in row matrix form asN w ¼½j 1j 2j 3j 4j 5. j n , ð9ÞN M ¼½c 1c 2c 3c 4c 5. c m :ð10ÞIt should be noted

40、that the necessary condition of m Z n Àl must besatised to guarantee the rank sufciency of the resulting matrix equations, where l is the number of rigid body modes 23,24.4. Multivariable hierarchical nite element formulations Substituting Eqs. (9and (10into Eq. (4yields the generalized potenti

41、al energy functional in the discrete form asP p 2¼wK wM M T À12MK M M T ÀwP T À12o 2r A wK w w T , ð11Þwherew ¼½w 1, y 1, w 2, y 2, w 5, . , w n , M ¼½M 1, M 2, M 3, . , M m , K wM ¼2LZ1À1N T Md 2N w d x2d x ,K M ¼L2EIZ1À1N T M N

42、M d x ,K w ¼L 2Z1À1N T w N wd x ,P ¼L2Z1À1N T w q d z :The explicit expressions of the preceding matrices are obtained byusing the symbolic computation software Maple 25and given in the Appendix A. It can be seen that the coefcients of the elements in these matrices are in either

43、 fraction or integer form, which can simplify the computation of the relevant matrices.Now by taking Eq. (11to be stationary with respect to generalized variables, namely, displacement and bending moments, P p 2w¼0, P p 2M¼0, ð12Þthe multivariable hierarchical beam element can be

44、 formulated as follows:0K wM K T wM ÀK M " #w M ¼P 0 þo 2r A K w 000" #wM :ð13ÞEliminating the inertial part from Eq. (13will lead to the formulations for static analysis in which the displacement and bending moment can be calculated directly.Based on Eq. (13,the f

45、ollowing equation can be obtained:M ¼K À1M K TwM w :ð14ÞThus the static linear equation in terms of the displacement variable can be written asK wM K À1M K TwM w ¼P ,ð15Þfrom which the elastic stiffness matrix can be extractedK ¼K wM K À1M K TwM :

46、40;16ÞSubstitution of expressions of K wM , K M into Eq. (16yields the following form,K ¼8EI L Z1À1d 2N w d x d 2N wd x " #T d x :ð17ÞThe expression of the above stiffness matrix is same as the oneobtained by the high-order nite element with a single variable when the s

47、ame shape functions are used to approximate the displacement eld within elements.The modal equation can be obtained by substituting Eq. (14into Eq. (13as follows:ðK wM K À1M K T wM Ào 2r A K w Þw ¼0:ð18ÞThen the frequency equation is obtained from the necessary con

48、dition of non-trivial solution for w asj K wM K À1M K T wM Ào 2r A K w j ¼0:ð19ÞSubstituting of eigenfrequencies solved from the above equationinto Eqs. (15and (18,one can calculate the corresponding mode shape of displacement, rotation, bending moment and shear force withou

49、t differentiation operations.In conventional single variable based FEM, the displacement eld variable of the beam element is approximated by cubic polynomials, and after differentiation two and three times the moment and shear force variables are obtained, which are linear function and constant, res

50、pectively. As the order of differentiation increases, precision of the corresponding results would become worse. To improve the computation precision, adequate elements have to be used. However, it is still difcult to make the shear forces converge to exact solutions due to they are expressed by con

51、stants. In the multivariable hierarchical FEM, the generalized displacement and force variables are approximated with higher-order polynomials and solved simultaneously without differentia-tion. Therefore, accuracy of the generalized force results can be guaranteed.Fig. 2. Hierarchical shape functio

52、ns for (adisplacement, and (bbending moment.Z. Yu et al. /Finite Elements in Analysis and Design 46(20106256316275. Numerical resultsIn this section, the applicability of the multivariable hierarch-ical nite element formulations is veried by analysis of static and vibration problems, respectively. T

53、he corresponding compu-tations are implemented in mathematics software Maple in terms of the above multivariable hierarchical nite element formulations.5.1. Example 1:static analysisThe multivariable hierarchical FEM is rstly applied to static analysis of a uniform beam of Length L and exural rigidi

54、ty EI . Different boundary conditions, i.e., clampedclamped, cantilev-ered and simply supported, as well as exponentially, uniformly, linearly distributed external loads have been investigated, as shown in Fig. 3 .Fig. 3. Beam models of (adifferent boundary conditions, subjected to (bdifferent loads

55、.Table 1Results of cantilevered beams subjected to various external loading. x /LThe present method The conventional FEM Exact solutionsn ¼m ¼3n ¼m ¼5NE ¼5NE ¼7Exponentially distributed load 0.0w 0000y 0000M 1.01291.00010.98940.99821q À1.8671À1.7197À1.533

56、6À1.6436À1.71830.5w 0.09210.09210.09200.09210.0921y 0.30810.30890.30560.30890.3089M 0.28920.28950.29720.29090.2895q À1.0249À1.0697À1.0679À1.0687À1.06951.0w 0.26520.26520.26520.26520.2652y0.35910.35910.35910.35910.3591M À0.01490À0.0220À0.00430q À

57、0.1887À0.0018À0.4109À0.18600Uniformly distributed load 0.0w 00000y 0000M 0.50.50.49070.49820.5q À1À1À0.8333À0.9285À10.5w 0.04420.04420.04420.04420.0442y 0.14580.14580.14580.14580.1458M 0.12500.12500.12960.12580.1250q À0.5À0.5À0.5À0.5À0

58、.51.0w 0.12500.12500.12500.12500.1250y 0.16660.16660.16660.16660.1666M 00À0.0092À0.00170q 00À0.1666À0.07140Linearly distributed load 0.0w 00000y 000M 0.34160.33330.33200.33320.3333Q À0.6000À0.5000À0.4833À0.4969À0.50000.5w 0.03150.03150.03140.03150.0315y 0

59、.10620.10670.10670.10670.1067M 0.10410.10410.10640.10450.1041q À0.3500À0.3750À0.3722À0.374À0.37501.0w 0.0916670.09160.09160.09160.0916y0.12500.12500.12500.12500.1250M À0.00830À0.0080À0.00160q À0.10000À0.1500À0.0683Table 2Results of xed-xed beams

60、 subjected to various external loading. x /LThe present method The conventional FEM Exact solutionsn ¼m ¼3n ¼m ¼5NE ¼5NE ¼7Exponentially distributed load 0.0w 00000y 000M 0.13980.12690.11620.12500.1268q À0.8391À0.6918À0.5056À0.6157À0.69030.5w 0.

61、00430.00430.00430.00430.0043y0.00080.00080.00080.00080.0008M À0.0699À0.0695À0.0619À0.0681À0.0695q À0.0417À0.0417À0.0370À0.0407À0.04151.0w 00000y 000M 0.13980.15470.13270.15040.1548q 0.83911.02610.61700.84181.0279Uniformly distributed load 0.0w 00000y

62、 000M 0.08330.08330.07400.08160.0833q À0.5À0.5À0.3333À0.4285À0.50.5w 0.00260.00260.00250.00260.0026y000M À0.0416À0.0416À0.0370À0.0408À0.0416q 000001.0w 00000y 000M 0.08330.08330.07400.08160.0833q 0.50.50.33330.42850.5Linearly distributed load0.0w 00000y 00M 0.04160.03330.03200.03320.0333q À0.2500À0.1500À0.1333À0.1469À0.15000.5w