Australian Journal of Basic and Applied Sciences, 5(7): 743-747, ISSN 99-878 Stability of Smart Beams wit Varying Properties Based on te First Order Sear Deformation Teory ocated on a Continuous Elastic Foundation M. KaramiKorramabadi, A.R. Nezamabadi Department of Mecanical Engineering, Korramabad Branc, Islamic Azad University, Korramabad, Iran. Department of Mecanical Engineering, Arak Branc, Islamic Azad University, Arak, Iran. Abstract: Tis paper studies stability of functionally graded beams wit piezoelectric layers subjected to axial compressive load tat is simply supported at bot ends lies on a continuous elastic foundation. Te displacement field of beam is assumed based on first order sear deformation beam teory. Applying te Hamilton's principle, te governing equation is establised. Te influences of applied voltage, dimensionless geometrical parameter, functionally graded index, foundation coefficient and piezoelectric tickness on te critical buckling load of beam are presented. To investigate te accuracy of te present analysis, a compression study is carried out wit a known data. Key words: Mecanical Buckling, Functionally graded beam- Piezoelectric layer. INTRODUCTION Structural stability is considered to be one of te most important engineering issues in te design and application of slender structures. Buckling and postbuckling are te two main types of structural instability, tey often govern te failure of structures under static or dynamic compressive loading conditions, tus, ave been investigated by several researcers in te past decades (Ari-Gur, J., 98). Piezoelectric materials ave been used in te past few years in a variety of applications ranging from active control to noise suppression. In all tese applications, piezoelectric actuators are used to enance te performance of a structural system by inducing a favorable structural deformation. Detailed models on te iteration between piezoelectric sensors or actuators wit te structure to wic tey are bonded or embedded ave been developed (Tzou, H.S., G.C. Wan, 99). To te autor's knowledge, tere is no analytical solution available in te open literatures for stability of functionally graded beams wit piezoelectric layers located on a continuous elastic foundation based on first order sear deformation beam teory. In te present work, te stability of a functionally graded beam wit piezoelectric actuators subjected to axial compressive loads located on a continuous elastic foundation based on first order sear deformation beam teory is studied. Appling te Hamilton's principle, te equilibrium equations of beam are derived and solved. Te effects of dimensionless geometrical parameter, foundation coefficient and functionally graded index on te critical buckling load of beam are presented. To investigate te accuracy of te present analysis, a compression study is carried out wit a known data. Formulation Te formulation tat is presented ere is based on first order sear deformation beam teory. Based on tis teory, te displacement field can be written as (Wang C.M., J.N. Reddy, ): uxz (, ) z ( x), wxz (, ) w( xz, ) () In view of te displacement field, te strain displacement relations are given by (Wang C.M., J.N. Reddy, ): u d xx z, u w dw xz x dx z x dx () Corresponding Autor: M. Karami Korramabadi, Department of Mecanical Engineering, Korramabad Branc, Islamic Azad University, Korramabad, Iran. 743
Aust. J. Basic & Appl. Sci., 5(7): 743-747, Consider a functionally graded beam wit piezoelectric actuators and rectangular cross-section as sown in Fig.. Te tickness, lengt, and widt of te beam are denoted, respectively, by, and b. Also, T and B are te tickness of top and bottom of piezoelectric actuators, respectively. Te x-y plane coincides wit te midplane of te beam and te z-axis located along te tickness direction. Fig. : Scematic of te problem studied. Te Young's modulus E is assumed to vary as a power form of te tickness coordinate variable z (-/< z < /) as follow (Karami, 8): z Ez ( ) ( Ec Em) VEm, V k (3) were k is te power law index and te subscripts m and c refer to te metal and ceramic constituents, respectively. Te constitutive relations for functionally graded beam wit piezoelectric layers based on first order sear deformation beam teory are given by (Wang C.M., J.N. Reddy, ): Q ( z) e E, Q ( z) e E xx xx 3 z xz 55 xz 5 x (4) V were Ei and xx, xz,q ( z), Q 55( z) are te normal, sear stresses and plane stress-reduced stiffnesses and i e, e 3 5 are piezoelectric elastic stiffnesses respectively. Also, u and w are te displacement components in te x- and z- directions, respectively. Te potential energy can be expressed as (Wang C.M., J.N. Reddy, ): U ( xxxx xz xz ) d v v Substituting Eq. () and Eq. (4) into Eq. (5) and neglecting te iger-order terms, we obtain [ ( ) ( ( ) 3( T T B B) 5( T B)( ) dx dx dx dx dx b d dw dw d dw U D A e V V e V V (5) (6) T were A Q ( z)dz, D z Q( z)dz and VT, VB are te applied voltages on te top and B 55 T B bottom actuators respectively. Te beam is subjected to te axial compressive loads. Te work done by te axial compressive load can be expressed as (Wang C.M., J.N. Reddy, ): w W P dx x We apply te Hamilton's principle to derive te equilibrium equations of beam, tat is (Wang C.M., J.N. Reddy, ): (7) 744
Aust. J. Basic & Appl. Sci., 5(7): 743-747, t ( T U W) dt (8) Substitution from Eqs. (6) and (7) into Eq. (8) leads to te following equilibrium equations of te functionally graded Engesser-Timosenko beam wit piezoelectric layers. Assume tat a FG beam wit piezoelectric actuators tat is simply supported at bot ends lies on a continuous elastic foundation, wose reaction at every point is proportional to te deflection (Winkler foundation). Te equilibrium equation of te FG beams wit piezoelectric layers located on a continuous elastic foundation subjected to axial compressive load is obtained from equilibrium equations by te addition of ηw for te foundation reaction as dw d dw d ( PbA) ba( ) w, A( ) e 5VT D( ) dx dx dx dx (9) were η is te foundation coefficient. Stability Analysis: Te boundary conditions for te pin-ended Timosenko column are given by: dw d w, at x and x dx dx () Substituting Eq. () into (9) and by equating power-law index to zero and neglecting te piezoelectric effect and foundation coefficient, te critical Engesser-Timosenko buckling load of a omogeneous beam will be derived, tat is: p cr b Q Q (( ) ) /( ( ) ) 3 55 b Q () Te above equation as been reported by Wang and Reddy (Wang C.M., J.N. Reddy, ). Numerical Results: Te mecanical buckling beaviors of simply supported functionally graded Engesser-Timosenko beams wit piezoelectric actuators located on a continuous elastic foundation are studied in tis paper. Te material properties of te beam are listed in Table. Table : Material properties. Property Piezoelectric layer FGM layer ----------------------------------------------------------- Stainless steel Nickel Young's modulus 63.4 3.95 Poisson's ratio.3.3.3 engt.3.3.3 Tickness.5.. Density 76 866 89 Piezo constant 7.6 - - Te Poisson s ratio is cosen to be.3 for bot materials. Te critical buckling loads for Bernoulli-Euler beam (BEB) and Engesser-Timosenko beam (ETB) evaluated considering of a / =., b/ =, =, V=v and several values of dimensionless geometrical parameter / are sown in Fig.. It is seen tat te critical buckling loads for Engesser-Timosenko beam are generally lower tan corresponding values of Bernoulli- Euler(Karami, 8) beam. Fig. 3 demonstrates te buckling loads for functionally graded Engesser-Timosenko beam. It is seen tat te critical buckling loads for Engesser-Timosenko beam increased wit an increase of te ratio / and decreased wit an increase of power-law index of constituent volume fraction. 745
Aust. J. Basic & Appl. Sci., 5(7): 743-747, Fig. : Critical Buckling oad of FG Smart Beam Versus /. Fig. 3: Critical Buckling oad of FG Smart Beam Versus / for V= v. Conclusion: Te stability of a functionally graded Engesser-Timosenko beam wit piezoelectric actuators located on a continuous elastic foundation subjected to axial compressive loads is studied. It is conclude tat:. Te piezoelectric actuators induce tensile piezoelectric force produced by applying negative voltages tat significantly affect te stability of te functionally graded Engesser-Timosenko beam wit piezoelectric actuators.. Te critical buckling loads of FG Engesser-Timosenko beam are generally lower tan corresponding values for te omogeneous Engesser-Timosenko beam. 3. Te critical buckling loads of FG Engesser-Timosenko beam under axial compressive load generally increases wit te increase of relative tickness /. 4. Te accuracy of Engesser-Timosenko beam teory is more tan Bernoulli-Euler beam teory. REFERENCES Ari-Gur, J., T. Weller, J. Singer, 98. Experimental and teoretical studies of columns under axial impact, Int. J. Solid Struct., 8(7): 69-64. 746
Aust. J. Basic & Appl. Sci., 5(7): 743-747, Karami Korram Abadi M., P. Kazaeinejad and J. Jenabi, 8. "Mecanical Buckling of Functionally graded Beam wit Piezoelectric Actuators", NCME. Tzou, H.S., G.C. Wan, 99. Distributed structural dynamics control of flexible manipulators-i. Structural dynamics and distributed viscoelastic actuator. Computers and Structures, 35(6): 669-677. Wang C.M., J.N. Reddy,. "Sear Deformable Beams and Plates", Oxford, Elsevier. 747