HIGH ORDER SHEAR DEFORMATION THEORY FOR ISOTROPIC BEAMS WITH USING MIXED FINITE ELEMENT METHOD

Transcription

1 Held on 1-, Nov, 15, in Dubai, ISBN: HIGH ORDER SHEAR DEFORATION THEORY FOR ISOTROPIC BEAS WITH USING IXED FINITE ELEENT ETHOD Emra adenci, Department of Civil Engineering, Necmettin Erbakan University, Köyceğiz Campus 49 eram, Konya, TURKEY Atilla Özütok Department of Civil Engineering, KTO Karatay University, Akabe Str. 4 Karatay, Konya, TURKEY Abstract Tis paper a mied finite element metod for te static analysis of isotropic beams are investigated wit considering ig order sear deformation beam teory. By using virtual displacement principle, field equation of beam wic is including parabolic sear deformation function is derived. In te transformed differential equation of ig order beam element to functional form, te Gâteau differential metod is used tat functional as ten independent variables. Different tree beam model about parabolic sear function presented for tree beam models and HOBT1 beam element is developed. Te element matri is formulated by using mied finite element metod. A computer program using FORTRAN language is developed to eecute te analyses for te present study. Te performance of te beam elements for static analysis is verified wit a good accuracy by te solution of numerical eamples present in te literature. Keywords Hig order sear deformation teory, finite element metod, Gâteau differential metod, Energy principle. INTRODUCTION In te beam bending, te investigations of problems ave been limited to te elementary Classical beam teory and First order beam teory [1]. Te classical beam teory (EBT) is inaccurate for reasonably tick beams []. It is widely used for tin beams and inapplicable to neglect te transverse sear deformation. By Timosenko, due to drawbacks of te classical beam teory developed a beam teory wic is as known FSDT to include te effect of te transverse sear deformation. In te FSDT, assumes a constant sear strain across te tickness of te beam and requires a problem dependent 6 sear correction factor [3]. To overcome te limitations in te EBT and FSDT, many te ig order beam teories ave been developed. Hig order teories account for sear deformation effects by iger order variations of inplane and transverse displacements troug te tickness and satisfy te equilibrium conditions on te top and bottom surfaces of te beam [4]. By using ig order teory, te sear deformation factors or effects ave been represented using parabolic, yperbolic, trigonometric and eponential functions troug te tickness of te beam [5-9]. On te contrary, Reddy [1] epresses tat: Higerorder teories can represent te kinematics better, may not require sear correction factors, and can yield more accurate interlaminar stress distributions. However, tey involve iger-order stress resultants tat are difficult to interpret pysically and require considerably more computational effort. Terefore, suc teories sould be used only wen necessary. Solution to differential equations is not always possible due to compleity of boundary conditions. Te finite element metod (FE) is te most suitable coice for structural analysis, because of its versatility in andling comple geometries and boundary conditions. Te mied type FE metod proved itself for static and dynamic analyses of various structural members and etensively used by researcers because mied-type FE model as greater importance since te forces and moments can be calculated wit less number of elements but more sensitivity [11]. In te FE, starting point is obtaining te function wic is corresponding to differential equation. Recently te Gâteau differential (GD) metod as been employed by Aköz and oter autors [1-17] to construct te functional for various problems. Te GD as been successfully applied to obtaining te functionals for te static and dynamic analysis beams, plates and sells. In tis study, te static responses of isotropic

2 Held on 1-, Nov, 15, in Dubai, ISBN: beam basis on ig order sear deformation beam models are presented using mied type FE formulation. Te virtual displacement principle was applied to obtaining te field equations written in operator form by including te dynamic and geometric boundary conditions. Te GD approac is employed to construct te functional using te variational metod. So using te mied type FE, te internal forces, moments, rotations, displacements of beams can be calculated easily. In order to model te beam wit two nodes, namely HOBT1 was derived wit ten degrees of freedom. A computer program was developed by using FORTRAN for analysis. Formulation In tis study, te displacement field of te eisting HOBT is given by function form; U (, y, z) zw f ( z) V (, y, z), W (,, y, z) w( ) were w and are te unknown displacement along te beam plane and rotation of te normal to te cross section about y-ais at te plane, respectively as illustrate in Fig.1. And f( z) represent parabolic sape functions determining te distribution of te transverse sear strains and stresses along te tickness. z z f( z) ( ) (a) 4 3 5z 4z f( z) (1 ) (b) 4 3 4z f ( z) z(1 ) (c) 3 In Eq.. Te contributions to constructing te functions are respectively ascribed to Ambartsumain [18] (Eq. a); Kaczkowski [19], Panc [] and Reissner [1] (Eq. b); Reddy [], urty [3] and Levinson [4]. (1) Fig. 1 Coordinate system of beam element Te strain-displacement relationsip under te assumption of small deformation and small rotations for te teories is given as U zw f ( z) U W z f ( z ), z z,, Te state of stress is given by te Hooke s Law as follows [1] Q z Were y 11 Q55 z Qij are te well-known reduced stiffnesses. Te principle of virtual displacements is used to derive te governing equations wit te displacement formulations as following [16] L z z dad q wd Ω ( ) ( ) (5) Were q ( ) is te distributed transverse load per unit lengt and is variational symbol; substituted from Eq. 4, and z and and (3) (4) z are z are substituted from Eq. 3, Euler-Lagrange equations are obtaining as follow L b 63

3 Held on 1-, Nov, 15, in Dubai, ISBN: w: q, : Q Here, is bending moment, moment and z (6) is refined bending Qz is refined sear force. Force and moment resultants of present teory are defined in te following form [7] zdz f ( z) dz Q f ( z) dz z z, z Using te parabolic sear deformation teory, by substituting te stress-strain relations into te definitions of force and moment resultants given in Eq. (7) te following te constitutive equations of te beam are obtained D11 F11 w, F H 11 11, Q z A 55 Te stiffness coefficients D 11, F11and H11 sear stiffness A55 are given by ( ) 11( ( ) ( ) ) D F H Q z f z z f z dz A Q f ( z) dz 55 55, z (7) (8) and transverse Dynamic and geometric boundary conditions are written in symbolic form as Eq. (1). Eplicit forms of te boundary condition will be obtained after some variational manipulations. Tere representing te moment, force, deflection and rotation vectors [1]. Dynamic boundary conditions are (9) R R Geometric boundary conditions are uu ΩΩ (1a) (1b) Te Eq. (6) and Eq. (8) can be written togeter wic are field equations q, Q, D w F 11, 11, F w H 11, 11, Q A z 55 z (11) Tese equations can be written in operator form similar to elastic beam [11] Q Py f (1) Te eplicit form of te operator is Pij yi fi 1 u R 1 Ω 1 Ω 1 R u If Q is a potential operator, te equality (13) * * dq(y,y),y dq(y,y ),y (14) ust be satisfied [5]. Tere are te Gâteau derivatives of te operator wic are defined Q( y τ y) dq( y, y ) (15) τ τ Were is τ is a scaler. Since te operator is potential ten te functional corresponding to te field equations is obtained as [5] 64

4 Held on 1-, Nov, 15, in Dubai, ISBN: ( y) 1 I Q(sy),y ds (16) Were is s is scaler quantity. Te functional of ig order sear deformation beam is in similar closed form I( y),, w,,,,,, Qz, Qz,, boundary conditions Q q w z (17) Initially te interpolation function must be cosen to derive te mied finite element formulation [8]. If a one dimensional element wit a parent sape function 1 1 i 1, j 1, i 1, (18) is used, te element matri can be obtained eplicitly as 1 1 k d k d 1 i j i, j, 1 1 k 1 3 i j, 1 d Analysis of Results (19) In tis study, te numerical results computed using te finite elements developed based on te above ig order sear deformation teories are compared wit various parabolic sear deformation functions wic are model1 is about Eq. (a), model is about Eq. (b) and model3 is about Eq. (c). Teoretical results calculated for various lengts to tickness ratio. Te boundary conditions of te beams were denoted by S for simply supported edge. Wit te application of te mied-type FE metod, HOBT1 beam elements were derived for HOBT were te HOBT1 elements ave a total of ten unknowns of displacements, bending moments, refined bending moments, refined sear forces and rotations. Troug te numerical eamples, te static analyses of beams were investigated using HOBT1 finite elements. TABLE I. THE AXIU DISPLACEENT OF AN ISOTROPIC SS THICK BEAS Teory L/ 1/1 4/1 8/1 16/1 HOBT1 model HOBT1 model HOBT1 model [1] [6] [7] In te Table 1., te maimum displacements of isotropic simply supported tick beams are calculated, under uniform distributed load q 1 and various L/ are considered for different beam teories and solutions of oter studies in literature. Te material and geometric properties are E=9, v=.3, b=1. Te values for te ig order elements are taken from te paper by [1], [6] and [7]. And also given in Table., same analysis presented for tin beams and te results are compared oter references [1], [6]. Eac model is a good comparison for various cross section states. Te ig order beam element by using mied finite element metod gives ecellent results wen te Gâteau differential metod is used for tick and tin beams. TABLE II. THE AXIU DISPLACEENT OF AN ISOTROPIC SS THIN BEAS Teory L/ 1/1 4/1 8/1 16/1 HOBT1 model HOBT1 model HOBT1 model [1] [6]

5 Held on 1-, Nov, 15, in Dubai, ISBN: Conclusions In tis study, te GD approac is employed to construct te functional using te variational metod for static analysis of isotropic beams for realized to solving mied finite element formulation. In order to develop a mied finite element model for applied to beams, it is necessary to establis a functional. Using te mied type FE, te internal forces, moments, rotations, displacements of isotropic beams can be calculated easily. Te Gâteau differential approaces ave some important advantages [15]. Results of te isotropic beam elements are verified by te oter epression given by [1], [6] and [7]. Te comparison study sows tat results of free vibration of ig order isotropic beams better tan te oter teories and te finite elements models. REFERENCES [1] P.R. Heyliger, and J.N.Reddy, A Higer Order Beam Finite Element for Bending and Vibration Problems, J.Sound. Vib., vol. 16, No., pp , [] L. Jun, and H. Honging Dynamic Stiffness Analysis of Laminated Composite Beams Using Trigonometric Sear Deformation Teory, Composite Strucutres, vol. 89, pp , 9. [3] P. Subramanian, Dynamic Analysis of Laminated Composite Beams Using Higer Order Teories and Finite Elements, Composite Structures, vol.73, pp , 6. [4] H.T. Tai, D.H. Coi, A Simple First-order Sear Deformation Teory for Laminated Composite Plates, Composite Structures, vol. 16, pp , 13. [5] F.G. Yuan, and R.E. iller, A Higer Order Finite Element for Laminated Beams, Composite Structures, vol. 14, no., pp , 199. [6] H. Yu, A Higer-order Finite Element for Analysis of Composite Laminated Structures, Composite Structures, vol. 8, no.4, pp , [7] G. Si, K. Lam, and T. Tay, On Efficient Finite Element odeling of Composite Beams and Plates Using Hig-order Teories and Accurate Composite Beam Element, Composite Structures, vol. 41, no., pp , [8] P. Subramanian, Fleural Analysis of Symmetric Laminated Composite Beams Using C1 Finite Element, Composite Structures, vol. 54, no.1, pp , [9].V.V.S. urty, D.R. aapatra, K. Badarinarayana and S.A. Gopalakrisnan, A Refined Higer Order Finite Element for Asymmetric Composite Beams Composite Structures, vol. 67, no.1, pp. 7 35, 5. [1] J.N. Reddy, ecanics of Laminated Composite Plates and Sells: Teory and Analysis, nd ed., Boca-Rotan: CRC Press, 4. [11] A.Y. Aköz and F. Kadioğlu, Te ied Finite Element etod for te Quasi-Static and Dynamic Analysis of Viscoelastic Timosenko Beam, Int. J. for Num. etods in Eng., vol. 44, no. 1, pp , [1] A.Y. Aköz,.H. Omurtag and A.N. Doğruoğlu, Te ied Finite Element Formulation for Tree Dimensional Bars, Int. J. Solids Structure, vol.8, pp. 5-34, [13].H. Omurtag and A.Y. Aköz, A Compatible Cylindrical Sell Element for Stiffened Cylindrical Sells in A ied Finite Element Formulation, Computers Structures, vol.49, no., pp , [14] A. Y. Akoz and F. Kadıoğlu, Te ied Finite Element Solution of Circular Beam on Elastic Foundation, Computers Structures, vol.6, no.4, pp , [15] A. Y. Akoz and A. Özutok, A Functional for Sells of Arbitrary Geometry and A ied Finite Element etod for Parabolic and Circular Cylindrical Sells, Int. J. for Num. etods in Eng., vol.47, no. 1, pp ,. [16] A. Ozutok and E. adenci, Free Vibration Analysis of Cross-ply Laminated Composite Beams by ied Finite Element Formulation, Int. J. of Struc. Stabil. and Dyn., Vol. 13, No., pp , 13. [17] A. Ozutok, E. adenci, and F. Kadıoğlu, Free Vibration Analysis of Angle-ply Laminate Composite Beams by ied Finite Element Formulation Using te Gâteau Differential, Science and Composite aterials, vol. 1, no., pp , 14. [18] S.A. Ambartsumian, On te Teory of Bending plates, Izv Otd Tec Nauk, ANN SSSR, vol. 5, pp , [19] Z. Kaczkowski, Plates-statistical calculations, Warsaw: Arkady, [] V. Panc, Teories of elastic plates, Prague, Academia; [1] E. Reissner, On Transverse Bending of Plates Including te Effects of Transverse Sear

6 Held on 1-, Nov, 15, in Dubai, ISBN: Deformation, Int. J. Solids Struc., vol. 5, pp , [] J.N. Reddy, A Simple Higer-order Teory for Laminated Composite Plates, J. Appl. ec., vol. 51, pp , [3].V.V. urty, An Improved Transverse Sear Deformation Teory for Laminated Anisotropic Plates, NASA Tecnical Paper, [4]. Levinson, An Accurate Simple Teory of Statics and Dynamics of Elastic Plates, ec.rese. Comm., vol. 7, pp , 198. [5] J.T. Oden and J.N. Reddy, Variational etods in Teoretical ecanics, Berlin: Sipringer-Verlag, [6] W.B. Bickford, A Consistent Higer Order Beam Teory, Development in Teo. App. ec., Vol. 11, pp , 198. [7] T.P. Vo, and H.T. Tai, Static Beavior of Composite Beams Using Various Refined Sear Deformation Teories, Composite Structures, Vol. 94, pp , 1. [8] J.N. Reddy, An Introduction to te Finite Element etod, New York: cgraw-hill,