Static Response Analysis of a FGM Timoshenko s Beam Subjected to Uniformly Distributed Loading Condition

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1 Static Response Analysis of a FGM Timoseno s Beam Subjected to Uniformly Distributed Loading Condition 8 Aas Roy Department of Mecanical Engineering National Institute of Tecnology Durgapur Durgapur, Maatma Gandi Avenue West Bengal (INDIA) roymb.855@gmail.com Kallol Kan Department of Mecanical Engineering National Institute of Tecnology Durgapur Durgapur, Maatma Gandi Avenue West Bengal (INDIA) allol_rec@yaoo.co.in ABSTRACT Functionally graded material beams (FGMs) possess a smoot variation of material properties due to continuous cange in micro structural details. Te variation of material properties is along te beam ticness and assumed to follow te power-law. A cantilever beam subjected to uniformly distributed load as been cosen ere for te analysis. For numerical implementation Galerin s weigted residual metod (approximation) is used ere witin te framewor of Timoseno or first order sear deformation teory (FSDT). Timoseno beam teory is used to capture te sear deformation. Te governing equations and boundary conditions are derived from virtual wor principle. In tis study, te effect of various material distributions on te mid plane deflections and stresses distribution along te ticness of FGM beam are examined and compared wit isotropic beam. Tis study can be elpful for tose beam type structures were te need for minimum stresses and displacements are required. Keywords: FGM, Timoseno s beam, Power-law, Mecanical load, Galerin s weigted residual metod. 1. INTRODUCTION As tecnology progresses at an ever increasing rate, te need for advanced capability materials becomes a priority in te engineering of more complex and iger performance systems. Pure metals are of little use in engineering applications because of te demand of conflicting property requirement. For example, an application may require a material tat is ard as well as ductile, tere is no suc material existing in nature. To solve tis problem, te term alloying is discovered in material science. Bronze, alloy of copper and tin, was te first alloy tat appears in uman istory. Tis need for new materials different from te parent materials can be seen in many fields. Composite material is also a class of advanced material. It is made up of one or more materials combined in solid states wit distinct pysical and cemical properties. Composite material offers an excellent combination of properties wic are different from te individual parent materials and are also ligter in weigt. Wood is a composite material from nature wic consists of cellulose in a matrix of lignin. But Composite materials will fail under extreme woring conditions troug a process called delamination (separation of fibers from te matrix) used by M. Maamood et al. [1]. To solve tis problem, researcers in Japan in te mid 198s, invented Functionally Graded Materials (FGMs). In materials science functionally graded material (FGM) may be caracterized by te variation in composition and structure gradually over volume, resulting in corresponding canges in te properties of te material. Te materials can be designed for specific function and applications. In laminated composite materials provide te design flexibility to acieve desirable stiffness and strengt troug te coice of lamination sceme, te anisotropic constitution of laminated composite structures often result in stress concentrations near material and geometric discontinuities tat can lead to damage in te form of delamination, matrix cracing, and adesive bond separation. FGMs alleviate tese problems because tey consist of a continuous variation of material properties from one surface to te oter. Also te smoot transition troug te various material properties reduces bot termal and residual stresses. In most cases te material progresses from a metal on one surface to a ceramic or mostly ceramic on te opposite surface, wit a smoot transition trougout te centre of te material. Also te material properties can cange in any orientation across a material, but te majority of applications

2 81 to deal wit a material in wic te properties cange troug te ticness of te material. Static and dynamic analyses of FGM structures ave attracted increasing researc effort in te last decade because of te wide application areas of FGMs. In many oter static analyses te variation of poisson s ratio is ept constant but in tis study it is also varying along te beam ticness. For instance. Simse [], gave static analysis of a functionally graded simply-supported beam subjected to a uniformly distributed load wit te elp of Ritz metod witin te framewor of Timoseno and te iger order sear deformation beam teories. He examined te continuously variation of te material properties in te ticness direction according to te power-law form. In tis paper it is expressed tat te axial deflections and te rotation of te cross-section of te beam are expressed in trigonometric functions. B.V. Sanar [3], proposed an elasticity solution of a FG beam subjected to transverse loads. He assumed te Young s modulus of te beam is varying exponentially in te ticness direction, and te Poisson ratio is ept constant. By using simple Euler- Bernoulli beam teory e also developed tat stresses and displacements depend on a single non-dimensional parameter for a given variation of Young s modulus. He found tat te FG beam teory is valid for long, slender beams wit slowly varying transverse loading. H.J. Ding et al. [4], investigate te elasticity solutions for plane anisotropic FG beams. Tey gave te concept of Airy stress function to solve te partial differential equation by considering te effect of body force. Zeng Zong et al. [5], explained to get te analytical solution of a FG cantilever beam subjected to different loads. By assuming tat all te elastic modulus of te beam material ave te same variations along te beam ticness, tey presented a two dimensional solution for a beam by using Airy stress function. In tis paper te solution is obtained by means of te semi-inverse metod. De-jin Huang et al. [6], studied te bending problem of a functionally graded anisotropic cantilever beam subjected to a linearly distributed load is investigated. Te analysis is based on te exact elasticity equations for te plane stress problem. Te stress function is introduced and assumed in te form of a polynomial of te longitudinal coordinate. Syang-Ho Ci et al. (part-1) [7], investigate te mecanical beavior of an elastic, rectangular and simply supported FGM plate of medium ticness under transverse loading. Tey assumed Poisson s ratio of te plate is constant, but Young s modulus varies in te ticness direction according to te power law. According to teir investigation te formulations of te solutions of FGM plates and omogeneous plates are similar, except te bending stiffness of plates. S.S. Alieldin et al. [8], investigate a first order sear deformation finite element model for elasto static analysis. Tey carried out te mecanical beavior of laminated composite and FG plates. Tey used tree approaces to transform te laminated composite plates, wit stepped material properties, to an equivalent FG plate wit a continuous property function across te plate ticness. Trung-Kien Nguyen et al. [9], proposed te first order sear deformation plate models for modeling structures made of FGM. In tis paper te transverse sear stresses are derived from te expression of membrane stresses and equilibrium equations. Identification of transverse sear factors is investigated by energy equivalence. Caraborty et al. [1], gave a new beam finite element metod based on first order sear deformation teory to study te termo elastic beavior of FGM beam structures. Here tey used power law distribution of mecanical properties along te ticness and expressed mid plane deflections and rotation in terms of interpolating polynomials. C.M.Wang et al. [11], is used to find out differential governing equations for a beam by Timoseno beam teory. [1], (Sesu) is used to calculate te residuals and weigt functions by Galerin s metod. [13], (Reddy) is used to find out te governing equations by virtual wor principle. As it is nown, Timoseno beam teory (TBT) or te first order sear deformation teory in wic straigt lines are perpendicular to te mid-plane before bending remain straigt, but no longer remain perpendicular to te mid-plane after bending. In TBT, for isotropic material te distribution of te transverse sear stress wit respect to te ticness coordinate is assumed constant. Tus, a sear correction factor ( s ) is required to compensate for te error because of tis assumption in TBT. However, studies sow tat TBT gives satisfactory results and it is very effective to investigate beavior of beams and plates.. THEORY AND FORMULATION Te term functionally graded materials (FGMs) refers to solid objects or parts tat usually consist of multiple materials or embedded components, tat is, tey are materially eterogeneous. A FGM consists of a material wose properties cange from one surface to anoter according to a smoot continuous function based on te position trougout te ticness of te material. Most often, tis material consists of metallic and ceramic constituents. One surface is generally a pure metal wile te opposite surface is usually pure ceramic or a majority ceramic []. Te metal portion of te material acts in te role of a structural support wile te ceramic provides termal protection wen subjected to ars temperatures. Functionally graded materials (FGMs) are new advanced multifunctional composites were te volume fractions of te reinforcements pase(s) vary smootly. Tis is acieved by using reinforcements wit different properties, sizes, and sapes, as well as by intercanging te functions of te reinforcement and matrix pases in a continuous manner. Te result is a microstructure bearing continuous canges in termal and mecanical properties at te macroscopic or continuum scale. In oter words, FGM is usually a combination of two materials or pases tat sow a gradual transition of properties from one side of sample to te oter. Tis gradual transition allows te creation of superior and multiple properties witout any mecanically wea interface. Moreover, te gradual cange of properties can be tailored to different applications and service environments. Tis new concept of materials engineering inges on materials science and mecanics due

3 8 to te integration of te material and structural considerations into te final design of structural components. To study FGMs, a model must be created tat describes te function of composition trougout te material. In Fig. 1, te volume fraction, Vc(z), describes te volume of ceramic at any point z trougout te ticness according to a parameter (also called power law exponent) wic controls te sape of te function.vc(z) is given by V c ( z) Ê 1 z ˆ = Á + It follows tat te volume fraction of metal, Vm(z), in te FGM is 1-Vc (z). One of te most common metods to determine te effective properties of FGMs is te rule of mixtures, were te material properties troug te ticness vary as a function of te volume fraction and are given by (1) P( z) = ( P - P ) V ( z) + P () m c c c Ê z 1ˆ G( z) = ( G - E ) Á + + G m c c were is te non-negative variable parameter (power-law exponent) wic dictates te material variation profile troug te ticness of te beam, m and c stand for metal and ceramic constituents, respectively. It is clear from Eqs. (1-3) tat (3) E = E c, ν=ν c, G = G c at z = / (4) E = E m, ν=ν m, G =G m at z = / (5) Based on te first order sear deformation teory, Te axial and transverse displacement using FSDT for a beam is given by u(x,y,z) = u (x) + z ϕ(x) (6) w(x,y,z)= w (x) (7) were (x,y,z) are te coordinates of a point in te beam, (u,w) are te components of te displacement vector in te coordinate directions, ϕ(x) is te angle of rotation of te normal to te mid-surface of te beam, and u (x), w (x) are te displacement of te mid-surface in te axial and transverse directions. Figure 1: Material properties trougout te FGM layer A FGM cantilever beam of lengt L, widt b and ticness is sown in Figure. Te beam is subjected to uniformly distributed load. Figure 3: Deformation of Timoseno beam Figure : Cantilever beam subjected to U.D.L. In tis study, it is assumed tat te FG beam is made of ceramic and metal, and te effective material properties of te FG beam, i.e., Young s modulus E, Poisson s ratio ν and sear modulus G vary continuously in te ticness direction (z axis direction) according to power-law form introduced by [] Ê z 1ˆ E( z) = ( E - E ) Á + + E m c c z 1 n( z) ( n n ) Ê ˆ = - Á + + n m c c (1) () from equations (6) and (7), te linear strains displacements are: du ( x ) df ( x e ) x = + z (8) dx dx g xz dw ( x = f x + ) (9) dx According to te constitutive relation for FGM material: Ïs x ÈE( z) Ï e x Ì = t Í xz G( z) Ì (1) Ó Î Óg xz Te virtual strain energy or internal virtual wor [13] is given by: L ( x x xz xz ) dadx (11) du = s de + t dg A Te virtual potential energy of te beam is given by: L d dv = - q x w x dx (1)

4 83 Te principle of virtual displacements states tat if te beam is in equilibrium it must satisfy: du + dv = (13) Te boundary conditions of te cantilever beam are given by: u () =, ϕ() =, w () = (Essential b. c) and F x =, M x =, V x = at x = L (Natural b. c). were F x is net axial force, M x is bending moment and V x is sear force respectively and are given by: F = s da, M = z s da, V = t da X x X x X s xz A A A After solving equation (13): f 1 c d u x d x -c - = (14) dx dx f Ï 3 s 4 Ìf d u x d x dw x -c - c + c x + = (15) dx dx ÔÓ dx Ô df x d w x q x + + = dx dx b c were = = c b E z dz, c b ze z dz, 1 c = b z E z dz, c = b G( z) dz 3 4 s 4 (16) Equations (14) (16) are nown as are nown as governing differential equations of te beam. Tese tree governing equations are obtained in terms of assumed trial or guess solutions. Let te assumed solutions are [1]: u ( x) a1 ax a3x 3 = + + (17) w x = a + a x + a x + a x (18) ( x) a a x a x f = + + (19) Te order of interpolation of w (x) is one order iger tan slope ϕ(x). Tis is one of te requirements for te beam to be free of sear locing. Te exact solutions for te displacements ave a total ten constants and only six boundary conditions. But from essential boundary conditions a 1, a 4 and a 8 =, so remaining constants are seven and boundary conditions are tree. Hence, tere are only tree independent constants. Te additional four dependent constants can be expressed in terms of tree independent constants by applying Galerin s weigted residual metod [1]. In general te guess solutions in equations (17-19) will satisfy neiter te differential equation witin te domain nor te boundary conditions. By substituting te assumed function in te differential equation and te boundary conditions of te problem, find te error in satisfying tese (also nown as domain residual and boundary residual ). Te constants (in equations (17-19) are calculated by maing tese residuals as low as possible. We see tat te approximation functions are dependent not only on te beam lengt but also on its material properties and cross-sectional properties. 3. RESULT AND DISCUSSION In tis section, te results are discussed for te FGM cantilever beam (L=1 mm, = mm, b=1mm) subjected to a uniformly distributed load (q (x) =5 N/mm). Te beam is composed of steel (E=1 1 3 N/mm, G=8 1 3 N/mm ) and alumina (Al O 3 ) (E= N/mm, G= N/mm ) and its properties canges troug te ticness of te beam according to te power-law []. Te sear correction factor is taen as s =5/6 for TBT. For te calculation of parameters c 1 to c 4 coding is done in FORTRAN 9. In tis study, te compressive and tensile stresses are sown by positive and negative signs respectively. Figure 4 sow te variation of modulus of elasticity along te beam ticness for different values of, as we now tat modulus of elasticity for full metal (z/ =.5) is 1x1 3 and for full ceramic (z/ = -.5) is 1x1 3. Similarly te variation for sear modulus and poisson s ratio along te ticness of te beam can be plotted. Figure 4: Variation of modulus of elasticity along te ticness of te beam Figures 5 and 6 sow te mid plane transverse and mid plane axial (stretcing) deflections of te beam along te beam lengt. In transverse deflection we see tat for full metal (=) te deflection is more tan te FGM composition and as te value of increases, te deflections of te beam decreases. Tis is due to te fact tat an increase in power-law exponent yields a decrease in te bending rigidity of te beam. In axial deflection, for full metal (=) te deflection is zero. Because in isotropic beams, tere is no coupling between te bending and te stretcing. It is also to be noted tat as te value of power-law exponent increases, te composition of te FG beam approaces to te composition of te full ceramic beam.

5 84 Figure 5: Mid plane transverse deflection along te beam lengt Figure 7: (a) Axial stress distribution along te beam ticness for FGM (b) Axial stress distribution along te beam ticness for full metal Figure 6: Mid-plane axial deflection (stretcing) along te beam lengt Figures 7 and 8 sows te distribution of axial and sear stresses at te fixed end (origin) for different values of. As seen from Fig. 7 (b), te axial stress distribution is linear only for full metal (=) and also te values of tensile and compressive stresses are equal for full metal (=). But for oter values of as sown in Fig.7 (a) te axial stress distribution is not linear and also te values of compressive stresses are greater tan tensile stresses. Also from Fig.7 (b), te value of axial stress is zero at te mid-plane but it is clearly visible from Fig.7 (a) tat te values of axial stresses are not zero at te mid-plane of te FG beam for te oter values of, it indicates tat te neutral plane of te beam moves towards te lower side of te beam for FG beam. Tis is due to te variation of te modulus of elasticity troug te ticness of te FG beam. It is seen from Fig. 8 (a) (b) tat te sear stress is constant for = but for =1 it is linear and for larger values of sear stresses are neiter linear nor constant. Te sear stress distribution is greatly influenced by power-law exponent. 4. CONCLUSIONS Static analysis of FG cantilever beam subjected to mecanical load sows tat te material properties of te beam vary continuously in te ticness direction according to te power law form. Numerical results clearly sow tat te variation Figure 8: (a) Sear stress distribution along te beam ticness for FGM, (b) Sear stress distribution along te beam ticness for full metal

6 85 of te modulus of elasticity plays a major role on te axial stress distribution and te displacements of te FG beam. Also for FGM beams as te value of power-law exponent increases, te bending rigidity decreases and te sear stress distributions are greatly influenced by te power law exponent. So, by concluding all of tis it can be said tat in te design of structures, by coosing a suitable power law exponent, te material properties of te FG beam can be tailored to meet te desired goals of minimizing stresses and te displacements in a beam-type structure. REFERENCES [1] Maamood, M.R., Ainlabi, T.E., Sula, M and Pityana, S., Functionally Graded Material: An Overview. Proceedings of te World Congress on Engineering, 1 Vol III WCE 1, July 4-6, 1, London, U.K. [] Simse, M., Static Analysis of a Functionally Graded Beam under a Uniformly Distributed Load by Ritz Metod. International Journal of Engineering and Applied Sciences Vol. 1, 9. [3] Sanar, B.V., An elasticity solution for functionally graded beams. Composites Science and Tecnology, 61 (1), pp [4] Ding, H.J., Huang, D.J., Cen, W.Q., Elasticity Solutions for Plane Anisotropic Functionally Graded Beams. International Journal of Solids and Structures 44 (7), pp [5] Zong, Z., Yu T., Analytical solution of a cantilever functionally graded beam. Composites Science and Tecnology, 67 (7), pp [6] Huang, De-jin, Ding Hao-jiang, Cenwei-qiu., Analytical solution for functionally graded anisotropic cantileverbeam subjected to linearly distributed load. Applied Matematics and Mecanics (Englis Edition), 7, 8(7): pp [7] Ci.S-Ho., Cung.Y- Ling., Mecanical beavior of functionally graded material plates under transverse load- Part I: Analysis. International Journal of Solids and Structures 43 (6) [8] Alieldin, S.S., Alsorbagy A.E., Saat M., A first-order sear deformation finite element model for elastostatic analysis of laminated composite plates and te equivalent functionally graded plates. Ain Sams Engineering Journal (11), [9] Nguyen.T-Kien., Sab., K., Bonnet, G., First-order sear deformation plate models for functionally graded material. Composite Structures, 83 (8), pp [1] Caraborty, A., Gopalarisnan, S., and Reddy, J. N., 3., A new beam finite element for te analysis of functionally graded materials. International Journal of Mecanical Sciences, 45(3), pp [11] Wang, C.M., Reddy J.N., Lee K.H., Sear Deformable Beams And Plates Relationsips Wit Classical Solutions. Elsevier UK. [1] Sesu, P., Finite Element Analysis, PHI New Deli. [13] Reddy, J.N., Mecanics of Laminated Composite Plates and Sells, Teory and Analysis. CRC PRESS Wasington.