Exact Shape Functions for Timoshenko Beam Element

Transcription

1 IOSR Journal of Computer Engineering (IOSR-JCE) e-iss: 78-66,p-ISS: , Volume 9, Issue, Ver. IV (May - June 7), PP - Exact Shape Functions for Timoshenko Beam Element Sri Tudjono, Aylie Han, Dinh-Kien guyen, Shota Kiryu, Buntara S. Gan * Department of Civil Engineering, Diponegoro University, Indonesia Institute of Mechanics, Vietnam Academy of Science and Technology, Vietnam Department of Architecture, College of Engineering, ihon University, Japan Astract: This paper derives exact shape functions for oth non-uniform (non-prismatic section) and inhomogeneous (functionally graded material) Timoshenko eam element formulation explicitly. In this paper, the shape functions formula emedded the explicit functions and its derivatives descriing the non-uniformity and inhomogeneity of a eam element. The shape functions are made interdependent y requiring them to satisfy three homogeneous differential equations associated with the Timoshenko s eam theory. With the formulated axial, transverse and rotational displacement shape functions, the stiffness and mass matrices and consistent force vector for a two-node Timoshenko eam element are developed ased on Hamilton s principle. Comparison studies with reference work on the accuracy and computational efficiency for non-uniform and inhomogeneous Timoshenko eam structures are highlighted. Static and virational analyses of the eams element y using the exact shape functions can predict the displacement, and natural frequencies of nonuniform and inhomogeneous Timoshenko eams y using only one/the least element accurately. Keywords : Exact shape functions, Timoshenko eam, FEM, Functionally graded material I. Introduction In the state-of-the-art of computational eam element, the polynomial function assumptions are most usually used. However, these functions were developed ase on the uniform cross-section and homogenous material. By using these shape functions, the analyses of eam elements require a significant numer of element divisions to study the ehavior of eam with non-uniformity and inhomogeneity prolems accurately [-]. The latest accomplishment for developing the exact solutions of the shape functions makes use of the power series method [-], asic displacement function [5] and Taylor series expansion [6-7] approximations are reported. Given the fact that the finite element method has een a widely used technique for structural analysis, it is necessary to develop reliale and accurate displacement functions [8-6] for the eam element. With the proper shape functions in the finite element formulations, the mass, stiffness matrices and loading vectors can e constructed consistently. In this paper, the formulation for exact shape functions for oth non-uniform and inhomogeneous Timoshenko eam element are derived ased on Hamilton s principle. II. Formulation of Exact Displacement Functions The Euler-Lagrange equations [7] for the Timoshenko eams can e otained from u E( x) A( x) x x w G( x) A( x) x x (a) () w E( x) I( x) G( x) A( x) x x x where, uw, and are the axial displacement, transversal displacement and rotation. E( x), A( x), G( x) and are the elastic modulus, section area and shear modulus as the function of x and shear correction factor, respectively. By integration, (a) ecomes, u E( x) A( x) k x where k is the indefinite integration integral constant. ext, solving for u(x), (c) () DOI:.979/ Page

2 u( x) k dx k k k () E( x) A( x) defining, E( x) A( x) dx. () Imposing the oundary conditions of horizontal displacements at oth end nodes of the eam results in, u( x ) u, u( x L) u. Solving for k, k from () and (), then sustituting ack the results into (), the exact displacement functions and its first order derivative of the axial displacement of the eam can e otained as u u u( x) u u d dx d dx u L L L u L u L u (5),. where x L xl The transverse displacement is assumed in polynomial equation as follows a a 6 w a a a x x x w a a x x w( x) a ax x x The solution of transverse homogeneous equilirium equation of () can e otained from w G( x) A( x) c x w( x) c ( x) dx c (7) w ( x) c G( x) A( x) x Replacing (6) into (7), results in (6) ( ) x c a ax x c a ax x c a x a (8) DOI:.979/ Page

3 Replacing (8) into (7), results in Hence, (c) can e written as w( x) c ( x) dx c Exact Shape Functions for Timoshenko Beam Element a c c a ax x dx c (9) a a ax x x c 6 E( x) I( x) c x x x c c c c x E( x) I( x) E( x) I( x) ( x) c c c the second derivative can e given y, c c x where c i (i=,,,) and a i (i=,,,) are the polynomial constants. Defining, dx, E( x) I( x) x dx, E( x) I( x). dx G( x) A( x) Collecting rotation and its derivatives from (8) and (), as well as transverse displacement etween (6) and (9), we have relationship etween a i and c i c M a () ac Imposing the oundary conditions of vertical displacements and rotations at oth end nodes of the eam results in, w( x ) w, w( x L) w and( x ), ( x L). c w c M wθc () c w c By sustituting c i, the expressions for transverse and rotational shape functions are as follow, w( x) w w w w w ( x) w w w () () where the rotational shape functions and their derivatives are given y DOI:.979/ Page

4 L L L L L L L LL L)) L (6 L( L LL L)) 6( ( ) ( ) ( )) L ( ) L( L) ( 6 L( 6 L ( ) 6 6 L 6 L L L L LL L L L L L LL L ( 6 6 L 6 LL L LL L ) x 6 6 x( L ) 6 x( L L ) w x x x L ( L )(6 x x x ) LL LL L 6 LL L L L ( x x ) L LL 6x L LL L w L 6 (6 L( L L L L) 6 L( L L 6 L x( L L ) LL L) L x x L L w L x x x ) 6( L L ) w x(( )(6 x x ) ( )(6 6 x 6 6 L 6 6 L 6L (6 x ) LL L ( L L ) 6 L x x LL L x L x x 6 6 LL 6 6 L 6 ( x 6 ) L 6( L ) x x LL L x x () (5) with their derivatives, 6(( L ) ( L )) x L 6 L( L L ( L LL) x LL L) x x 6 6 L 6 6 L 6 LL x LL LL L LL L (6) DOI:.979/ Page

5 where, w 6( ( L ) ( L ) L ( )) x L 6 L( L ( L ) L( L LL) w LL L) x L6 L( L LL L) w w x x (6 L 6 L)( ) w ( ) 6 LL L LL L x 6 ( L ) L L L ( ) L( L) L L L ( 6 L( L L)) L L L (6 L( L L)) L L,,,,, x L xl x L xl x L xl,,,,, x L xl x L xl x L xl x, x L, x, xl, x,, x xl III. Governing Equations The governing equation can e derived via Hamilton's principle as follow: xl (7) t t H SE K E WE dt (8) where, H, SE, KE and W are the variation of total energy, strain energy, kinetic energy and external E work of the eam, respectively. The undamped equilirium equation can e otained y sustituting (5) and () into the variational equations of (8), which results in Kd Md f (9) where, K, M, f, d and d are the stiffness matrix, mass matrix, loading vector, general displacement vector and general acceleration vector, respectively. The stiffness matrix, mass matrix and loading vector are given y T u E( x) A( x) u L K ( ) ( ) v G x A x v dx E( x) I( x) T u ( x) A( x) u L M ( ) ( ) v x A x vdx ( x) I( x) u P u p( x) Q q( x) v v M L m( x) f dx u P u p( x) Q q( x) v v M m( x) () () () DOI:.979/ Page

6 where, P, Q and M are the concentrated axial load, transversal load and moment at node i, respectively. The i i i terms p( x), q( x) and m( x ) are the distriuted axial load, transversal load and moment, respectively. In constructing the matrices and vector of (-), the Gaussian quadrature integration scheme is recommended. Also, the shape functions,,, and their derivatives inside the equations can e evaluated numerically y using the Gaussian quadrature integration scheme. These shape functions will carry on the non-uniformity and inhomogeneity characteristics of the eam. IV. umerical Examples In this section, illustrative examples are solved y using the exact displacement functions derived in previous sections to show the validity and novelty of the present study. Comparisons with the numerical results reported y other studies are highlighted.. Taper Cantilever Beam A homogeneous non-dimensional unit tapered cantilever eam loaded at the free end tip as shown in Fig. l is considered. P=.75 E= G= κ=5/6... L= Figure. Taper cantilever eam example The eam is analyzed y using one element y using the exact displacement functions to construct the stiffness matrices of the static equilirium equation. Tale. Results of static analysis of taper cantilever eam example Reaction at fixed end Vertical displacement at free end Shear Deformation Shearing Force Bending Moment Present Study [5] Consideration Q M w Considered.... ot considered As shown in Tale., the reaction forces and displacement resulting from the shear deformation consideration are in good agreement with the theoretical results of [5].. Various Taper Clamped Beam A homogeneous non-dimensional unit various tapered clamped eam loaded at the mid-span as shown in Fig. is considered. Two different heights h and various values of the lengths L are analyzed. P=, w E= G= κ=5/6 L. Figure. Clamped taper eam example L. h Deflections which include shear deformation (w s ) and neglect shear deformation (w ns ) at the mid-span of the eams are listed in Tales, respectively. The present computed deflections are compared with the results of [8]. Only using two element divisions, the present results gave an excellent agreement with the reference (twelve elements) which can e oserved from the tale. DOI:.979/ Page

7 Tale. Deflection of clamped taper eam example h=.6 h=.9 L w s w ns w s w ns Present [8] Present [8] Present [8] Present [8] Free Viration of Various Boundary Conditions of Taper Beams Various oundary conditions of taper eams as shown in Fig.. The geometry and material data of the eam are given in the figure. h h a) Clamped-Free (C-F) E= Gpa E/(κG)=. Κ=/ h ) Clamped-Clamped (C-C) h L=.8665 m ρ=785 kg/m r =I /(A L )=.8 h h c) Clamped-Pinned (C-P) L Figure. Various oundary conditions of taper eam free viration example The cross-session is rectangular with constant width and linear-varying height h defined as h h () h The non-dimensional frequencies of the eam that are defined as AL i i () EI From Tale, the results of the non-dimensional frequencies of the present formulation show lower convergence results under various oundary conditions compared to the results in [5]. In the free viration prolem, the eam element should e divided into su-elements to accommodate the free degree of freedoms. In this study, only four element divisions for the eam were employed to otain the good results. DOI:.979/ Page

8 Tale. on-dimensional natural frequency of tapered eam with various oundary conditions β Boundary Condition µ µ µ µ µ 5. C-F Present [5] C-P Present [5] C-C Present [5] C-F Present [5] C-P Present [5] C-C Present [5] Functionally graded eam The viration of a uniform cross-section of an inhomogeneous Functionally Graded Material (FGM) eam shown in Fig. is investigated. The eam is made of two materials from the left to the right ends with a constant value of mass density, the effective elastic modulus, E and shear modulus, G of the eam are assumed to vary in the eam axis direction according to the following E( x) ( E E x ) E L G( x) ( G G x ) G L left right right left right right The Simply-Supported (S-S) rectangular - cross-section eam with geometric data (width =. m; height h=.9 m and the total length L= m) is employed in the computation of fundamental frequencies. The results (with only one element division) are compared with [] which is computed with ten element divisions. Very good agreement etween the frequencies otained in the present work with that of the reference is given in Tale where E r =E left /E right ; μ, μ are respectively the first and the second nondimensionalized fundamental frequencies of the eam. The non-dimensionalized fundamental frequency is given y AL i i (6) E I left (5) A,I A,I left right L Figure. Simply supported rectangular FGM eam example Tale. on-dimensional fundamental frequencies of FGM eam example E R μ μ Present [] Present [] DOI:.979/ Page

9 V. Conclusions Exact shape functions of a non-uniform and inhomogeneous Timoshenko eam element has een formulated. These shape functions can e implemented in finite-element codes to create the mass and stiffness matrices within the context of Timoshenko eam element where the shear deformation can e taken into account in the analyses. The correctness of the formulated shape functions was verified through the numerical examples of the static and free viration analyses of non-uniform and inhomogeneous (FGM) eam elements. With the formulated shape functions, highly accurate results can e otained y using the least element numer of element division. References [] D.K. guyen, B.S. Gan and T.H. Le, Dynamic Response of on-uniform Functionally Graded Beams Sujected to a Variale Speed Moving Load, Journal of Computational Science and Technology, 7(),, -7. [] D.K. guyen and B.S. 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