Thermomechanical Buckling of Temperaturedependent

Transcription

1 (3) 3 46 heroechanical Buckling of eperaturedependent FGM Beas Abstract Buckling of beas ade of functionally graded aterials (FGM) under theroechanical loading is analyzed herein. Properties of the constituents are considered to be functions of teperature and thickness coordinate. he derivation of the equations is based on the ioshenko bea theory, where the effect of shear is included. It is assued that the echanical and theral nonhoogeneous properties of bea vary soothly by distribution of the power law inde across the thickness of the bea. he equilibriu and stability equations for an FGM bea are derived and the eistence of bifurcation buckling is eained. he bea is assued under three types of theral loadings; naely, the unifor teperature rise, heat conduction across the thickness, and linear distribution across the thickness. Various types of boundary conditions are assued for the bea with cobination of roller, claped, and siply-supported edges. In each case of boundary conditions and loading, closed for solutions for the critical buckling teperature of the bea is presented. he results are copared with the isotropic hoogeneous beas, that are reported in the literature, by reducing the results of the functionally graded bea to the isotropic hoogeneous bea. Y. Kiani a and M.R. Eslai b, * a Ph.D. Student, Mechanical Engineering Departent, Airkabir University of echnology, 594 ehran, Iran. b Professor and Fellow of Acadey of Sciences, Mechanical Engineering Departent, Airkabir University of echnology, ehran 594, Iran. el: (+98-) , Fa: (+98-) Received Oct In revised for 3 Nov * Author eail: eslai@aut.ac.ir Keywords Buckling, ioshenko bea theory, Functionally graded aterial, eperature Dependency INRODUCION Functionally graded aterials, as a branch of new aterials, have attracted increasing attention in recent years. A survey in the literature reveals the eistence of wealth investigations on analysis of functionally graded aterial beas. Aong the, Kang and Lee [] presented eplicit epressions for deflection and rotation of an FGM cantilever bea subjected to an end oent. Considering the large deflection of the bea, they reported that an FGM bea can bear larger applied load than a hoogeneous bea. Free vibration analysis of siplysupported functionally graded aterial beas is reported by Aydogdu and askin []. hey

2 4 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas used both eponential and power law for of aterial properties distribution to derive the governing equations. heir study includes four types of displaceent fields naely, the classical bea theory, the first order theory, and the parabolic and eponential shear deforation bea theories. hey concluded that, in coparison with the classical bea theory, other three types of displaceent fields accurately predict the natural frequencies. Nirvana et al. [3] obtained analytical epressions for thero-elastic analysis of three layered beas, when the iddle layer is ade of FGMs. A unified ethod to study the dynaic and static analysis of FGM ioshenko beas is reported by Li [4]. He derived a fourth order differential equation and linked the other paraeters of the bea to the solution of fourth order differential equation. His study includes the siply supported and cantilever beas. he static and free vibration analysis of layered FGM beas based on a third order shear deforation bea theory is developed by Kapuria et al. [5]. A two nodes finite eleent ethod is adopted to solve the coupled ordinary differential equations. he echanical and theral buckling of beas, as a ajor solid structural coponent, have been the topic of any researches for a long period of tie. Developent of the new aterials, such as the functionally graded aterials, have necessitated ore research in this area. Huang and Li [6] obtained an eact solution for echanical buckling of FGM coluns subjected to unifor aial loading based on various bea theories. Zhao et al. [7] studied the post-buckling of siply supported rod ade of functionally graded aterials under unifor theral loading and nonlinear teperature distribution across the bea thickness using the nuerical shooting ethod. hey found that, under the sae teperature condition, the deforation of iovably siply supported FGM rod is saller than those of the two hoogenous aterial rods. Also, end constrained force of FGM rod is saller than the corresponding values of the two hoogenous aterial rods with the sall deforation. Accordingly, the stability of FGM rod is higher than those of the two hoogenous aterial rods when there is a teperature difference. Li et al. [8] presented the post-buckling behavior of fiedfied FGM beas based on the ioshenko bea theory under nonlinear teperature loading. hey found the effect of shear on the critical buckling teperature of beas and used the shooting ethod to predict the post-buckling behavior of beas. It was found that the non-diensional theral aial force increases along with increase of the power law inde, as the increent of etal constituent can produce ore theral epansion of bea under the sae value of theral load. Kiani and Eslai [9] discussed the buckling of functionally graded aterial beas under three types of theral loading through the thickness. hey eained the eistence of bifurcation type buckling for various edge supports and presented their results in closed-for epressions. A sei inverse ethod to study the instability and vibration of aially FGM beas is carried out by Aydogdu []. Ke et al. [] presented the postbuckling of a cracked bea for hinged-hinged and claped-hinged edge conditions based on the ioshenko bea theory. Also, Ke et al. [] presented the free vibration and echanical buckling of cracked beas using the first order shear deforation bea theory for three types of boundary conditions. hey found that FGM beas with a saller slenderness ratio and a lower Young's odulus ratio are uch ore sensitive to the edge crack. Ma and Lee [3] discussed the nonlinear behavior of FGM beas under in-plane theral loading by eans of Latin Aerican Journal of Solids and Structures (3) 3 46

3 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas 5 first order shear deforation theory of beas. he derivation of the equations is based on the concept of neutral surface, where the nuerical shooting ethod is used to solve the coupled nonlinear equations. heir study concluded that when a claped-claped FGM bea is subjected to unifor theral loading, it follows the bifurcation-type buckling while the siplysupported beas do not. his feature of FGM beas, however, is ignored in soe of the published works through the literature [9, 4, 5]. Kiani et al. [6,7] studied the effect of applied actuator voltage on the critical buckling teperature difference of FGM beas. It is reported that the effect of applied actuator sart layers is soehow negligible on theral buckling control. In an analytical study, Ma and Wang [8] analyzed the nonlinear response of FGM beas with shear deforation effects and obtained the eact closed-for solutions for equilibriu path of the bea. his analytical study also proves the iportance of the boundary conditions on the bea equilibriu path. Fu and his co-authors [9] obtained closed-for solutions for the free vibration of therally loaded and theral equilibriu path of a thin FGM bea with both edges claped. In this work the teperature dependency of the constituents is also taken into account. A single ter Ritz solution along with a finite eleent forulation is developed by Anadrao et al. []. In this work the cases of a bea with both edges claped and both edges siplysupported are analyzed. he present work deals with the buckling analysis of FGM beas subjected to theral or echanical loadings. Various types of boundary conditions are assued and the eistence of bifurcation type buckling in each case is eained. Based on the concept of virtual displaceents principle, three coupled differential equations are obtained as the equilibriu equations. In theral buckling analysis, the bea is under three types of theral loading distinctly, and closed-for solutions are obtained to evaluate the critical buckling teperatures/loads. FUNCIONALLY GRADED IMOSHENKO BEAMS Consider a bea of functionally graded aterial, where the graded properties are assued to be through the thickness direction. he volue fractions of the constituent aterials, which are assued to ceraic of volue V c and etal of volue V, ay be epressed using the power law distribution as [] z Vc V, V c h k () where h is the thickness of the bea and z is the thickness coordinate easured fro the iddle surface of the bea h / z h /, k is the power law inde which has the value equal or greater than zero. Variation of V c with k and z / h is shown in Figure. Latin Aerican Journal of Solids and Structures (3) 3 46

4 6 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas Figure Distribution of ceraic volue fraction through the thickness for various power law indices he value of k equal to zero represents a fully ceraic bea Vc and k equal to infinity represents a fully etallic beav. We assue that the echanical and theral c properties of the FGM bea are distributed based on Voigt's rule []. hus, the property variation of a functionally graded aterial using Eq. () is given by z Pz () P P c h k () where Pc Pc P, P c and P are the corresponding properties of the etal and ceraic, respectively. In this analysis the aterial properties, such as Young's odulus Ez (), coefficient of theral epansion () z and theral conductivity Kz () ay be epressed by Eq. (), whereas Poisson's ratio is considered to be constant across the thickness []. 3 GOVERNING EQUAIONS Consider a bea ade of FGMs with rectangular cross section. It is assued that the length of the bea is L, width is b, and the height is h. Rectangular Cartesian coordinates is used such that the ais is at the left side of the bea on its iddle surface and z is easured fro the iddle surface and is positive upward, as shown in Figure. Latin Aerican Journal of Solids and Structures (3) 3 46

5 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas 7 Figure geoetry and coordinate syste of an FGM bea he analysis of bea is based on the first order shear deforation bea theory using the ioshenko assuptions. According to this theory, the displaceent field of the bea is assued to be [4] uz (, ) u z wz (, ) w (3) where uz (, ) and wz (, ) are displaceents of an arbitrary point of the bea along the and z -directions, respectively. Here, u and w are the displaceent coponents of iddle surface and is the rotation of the bea cross-section, which are functions of only. he strain-displaceent relations for the bea are given in the for [] u w z u w z (4) where and z are the aial and shear strains. Substituting Eq. (3) into Eq. (4) gives du dw d z dw z he constitutive law for the aterial, using the linear thero-elasticity, is given by [8] (5) Latin Aerican Journal of Solids and Structures (3) 3 46

6 8 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas E z E In Eqs. (6), and z are the aial and shear stresses, is the reference teperature, and is the teperature distribution through the bea. Eqs. (5) and (6) are cobined to give the aial and shear stresses in the bea in ters of the iddle surface displaceents as z (6) du dw d E z z E dw (7) he stress resultants of the bea epressed in ters of the stresses through the thickness, according to the ioshenko bea theory, are [8] N h/ h / dz h/ M z dz (8) h / h/ Q K dz z s h / z where K s is the shear correction factor. he values of 5/6 or / are used as its approiate value for the coposite and FGM beas with rectangular cross section []. he shear correction factor is taken as K / for the FGM bea in this study. s Using Eqs. (), (7), and (8) and noting that uw,, and are functions of only, the epressions forn, M, and Q z are obtained as du dw d N E E N du dw d M E E3 M (9) Latin Aerican Journal of Solids and Structures (3) 3 46

7 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas 9 Q z EK s dw dz wheree, E, and E 3 are stretching, coupling stretching-bending, and bending stiffnesses, respectively, and N and M are theral force and theral oent resultants, which are calculated using the following relations h / E c E Ezdz () h E h / k h/ E ze() zdz he h / c k k E3 zezdz h E E k 3 k 4k 4 h/ 3 () h / c () h/ N Ez () () z dz h/ h/ M ze () z () z dz h/ Note that to find the theral force and oent resultants, the teperature distribution through the bea should be known. he equilibriu equations of an FGM bea ay be obtained through the static version of virtual displaceent principle. According to this principle, since the eternal load is absent, an equilibriu position occurs when the first variation of strain energy function vanishes. hus, one ay write L b/ h/ s z z () U K dzdy b/ h/ With the aid of the stress resultant definition (9), and perforing the integration by part technique to relieve the displaceent gradients, the following syste of equilibriu equations is obtained Latin Aerican Journal of Solids and Structures (3) 3 46

8 3 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas dn dm Qz and the boundary conditions for each side of the bea are () dq z dw N N or u M or (4) dw Qz N or w 4 EXISENCE OF BIFURCAION YPE BUCKLING 4. heral Loading Consider a bea ade of FGMs subjected to a transversely teperature distribution. When the aial deforation is prevented in the bea, an applied theral loading ay produce an aial load. Only perfectly flat pre-buckling configurations are considered in the present work, which lead to bifurcation type buckling, otherwise bea undergoes a unique and stable equilibriu path. Now, based on Eq. (9), in the pre-buckling state, when bea is copletely undefored, and both edges are iovable, the generated pre-buckling force through the bea is equal to N N (4) Here a subscript is adopted to indicate the pre-buckling state deforation. Also, according to Eq. (9), an etra oent is produced through the bea which is equal to M M (5) In general, this etra oent ay result in deforation through the bea, ecept when it is vanished for soe especial types of theral loading or when boundary conditions are capable of handling the etra oents. he claped and roller (sliding support) boundary conditions are capable of supplying the etra oents on the boundaries, while the siplysupported edge does not. herefore, the C C and C R FGM ioshenko beas reain un-defored prior to buckling, while for the other types of beas with at least one siply supported edge bea coence to deflect. Also, an isotropic hoogeneous bea reains un- Latin Aerican Journal of Solids and Structures (3) 3 46

9 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas 3 defored when it is subjected to unifor teperature rise, because theral oent vanishes through the bea. herefore, bifurcation type buckling eists for C C and C R FGM beas subjected to arbitrary transverse theral loading. he sae is true for the isotropic hoogeneous beas subjected to unifor teperature rise with arbitrary case of boundary conditions. 4. Mechanical Loading in heral Field Consider an FGM bea in theral field which is subjected to an in-plane aial loadp, and operates in theral field. he left side of the bea is iovable, while the right hand side is ovable and undergoes an in-plane force P When the bea ehibits the bifurcation-type of buckling, it reains un-defored in priary equilibriu path. Based to the first equilibriu equation, the pre-buckling force resultant is equal to N = P (6) b Based on the definition of force resultants, when the lateral deflection is ignored, the induced echanical oent due to the applied in-plane force is equal to P E E M N M (7) = be E he eistence of bifurcation type buckling depends on the vanishing of the etra bending oent in Eq. (7). In the following general cases are studied Case : For the case when an FGM bea is subjected to aial load only, C C and C R cases follow the branching type of buckling. Otherwise the induced oent in Eq. (7) results in the initial deflection. Case : For the case of reduction of an FGM bea to an isotropic hoogeneous one that is subjected to unifor teperature rise loading, M and bifurcation occurs for any arbitrary case of out-of-plane boundary conditions. Case 3: For the case of reduction of an FGM bea to an isotropic hoogeneous one that is subjected to heat conduction across the thickness, M M and bifurcation occurs only for the especial cases of C C and C R end supports. his is due to the ability of claped and roller edges to supply the etra oent in pre-buckling state. Latin Aerican Journal of Solids and Structures (3) 3 46

10 3 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas Case 4: For the case of an FGM bea that is subjected to arbitrary case of theral loading in the presence of aial in-plane load, M is given by Eq. (7). Generally this oent does not vanish and siilar to the previous case only C C and C R end supports ehibit the bifurcation-type of buckling. In other cobinations of edge supports, bea initially start to lateral deflection at the onset of theral loading. 5 SABILIY EQUAIONS o derive the stability equations, the adjacent-equilibriu criterion is used. Assue that the equilibriu state of a functionally graded bea in pre-buckling state is defined in ters of the displaceent coponentsu, w and. he displaceent coponents of a neighboring stable state differ byu, w and with respect to the equilibriu position. hus, the total displaceents of a neighboring state are [3] u u u w w w (8) Siilar to the displaceents, the force and oent resultants of a neighboring state ay be related to the state of equilibriu as N N N M M M (9) Q Q Q z z z Here, stress resultants with subscript represent the linear parts of the force and oent resultant increents corresponding tou, w and. he stability equations ay be obtained by substituting Eqs. (8) and (9) in Eq. (). Upon substitution, the ters in the resulting equations with subscript satisfy the equilibriu conditions and therefore drop out of the equations. Also, the non-linear ters with subscript are ignored because they are sall copared to the linear ters. he reaining ters for the stability equations as E du d E Latin Aerican Journal of Solids and Structures (3) 3 46

11 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas 33 E du d EK s dw E 3 EK s d dw N dw () Cobining Eqs. () by eliinating u and provides an ordinary differential equation in ters of w which is the stability equation of an FGM bea under transverse theral loadings with 4 dw 4 dw () EN EE 3 E N EK s () he stress resultants with subscript are linear parts of resultants that correspond to the neighboring state. Using Eqs. (9) and (8) the epressions for, N, M and Q z becoe du d N E E Q du d M E E3 (3) EK dw s z When teperature distribution through the bea is along the thickness direction only, the paraeter is constant. In this case the eact solution of Eq. () is w ( ) C sin C cos C C (4) 3 4 Using Eqs. (), (3), and (4), the epressions foru,, N, M, and Q z becoe Latin Aerican Journal of Solids and Structures (3) 3 46

12 34 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas ( ) S C cos C sin C 3 E u S C C C C () cos sin 5 6 E EE 3 E () ( ) sin cos 5 E M S C C EC (5) EK s Qz() S Ccos C sin N () EC 5 With S( ) EE 3 E EK s (6) he constants of integration C to C 6 are obtained using the boundary conditions of the bea. Also, the paraeter ust be iniized to find the iniu value of N associated with the theral or echanical buckling load. Five types of boundary conditions are assued for the FGM or hoogeneous bea with cobination of the roller, siply supported, and claped edges. Boundary conditions in each case are listed in able. able Boundary conditions for FGM ioshenko beas under theral loading. C indicates claped, u should be replaced by N S shows siply-supported and R is used for roller edge. For echanical buckling case, Edge supports B.Cs at B.Cs at L C C u w u w S S u w M u w M C S u w u w M C R u w u Q dw z N S R u w M u Q dw z N Latin Aerican Journal of Solids and Structures (3) 3 46

13 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas 35 Let us consider a bea with both edges claped under theral loading. Using Eqs. (4) and (5), the constants C to C 6 ust satisfy the syste of equations sinl cosl L C S C ScosL SsinL C 3 E C 4 S E C 5 E E C 6 ScosL SsinL L E E (7) o have a nontrivial solution, the deterinant of coefficient atri ust be set equal to zero, which yields S L cos L LS sin L (8) he sallest positive value of which satisfies Eq. (8) is in. It can be seen L easily that for the other types of boundary conditions, ecept C S case, the nontrivial solution leads to an eact paraeter for. Using an approiate solution given in [4] for the critical aial force of C S beas, the critical force for an FGM ioshenko bea with arbitrary boundary conditions can be epressed as below N p E E 3 L E, cr E 3 E q KL E s E (9) where p and q are constants depending upon the boundary conditions and are listed in able able Constants of forula (9) which are related to boundary conditions. Paraeter C C S S C S S R C R p q Latin Aerican Journal of Solids and Structures (3) 3 46

14 36 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas When the critical buckling force resultant is obtained, in the case of echanical loading, the total copressive load ay be evaluated by Eq. (6). For the case of theral buckling analysis, however, teperature profile should be known. 6 YPES OF HERMAL LOADING 6. Unifor teperature rise (UR) Consider a bea which is at reference teperature. When the aial displaceent is prevented, the unifor teperature ay be raised to Substituting into Eq. () gives such that the bea buckles. E = c E c E c N c h E k k (3) Considering Eq. (3), the critical buckling teperature difference the for UR cr is epressed in UR cr p h Fk (, ) L = h Gk (,, ) q Ek (, ) Ks L (3) E where = c and = c. Also, the functions Ek (, ), Fk (, ), and Gk (,, ) are defined E as ( k k ) k (, )= 4( k )( k )( k 3) 4( k )( k ) ( k ) Fk k ( k k ) k Ek (, )= ( k ) 4( k )( k 3)( k ) 4( k ) ( k ) (3) Gk (,, )= k k Latin Aerican Journal of Solids and Structures (3) 3 46

15 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas Linear teperature distribution (LD) Consider a thin FGM bea which the teperature in ceraic-rich and etal-rich surfaces are c and, respectively. he teperature distribution for the given boundary conditions is obtained by solving the heat conduction equation along the bea thickness. If the bea thickness is thin enough, the teperature distribution is approiated linear through the thickness. So the teperature as a function of thickness coordinate z can be written in the for z = ( c )( ) (33) h Substituting Eq. (33) into Eq. () gives the theral force as E = ( ) c E c E c N c h E k k E Ec Ec Ec c h k k (34) where c. Cobining Eqs. (9) and (34) gives the final for of the critical buckling teperature difference through the thickness as p h Fk (, ) L h Hk (,, ) q Ek (, ) K s L LD cr = ( ) Hk Gk (,, ) (,, ) (35) Here, the functions Ek (, ), Fk (, ), and Gk (,, ) are defined in Eq. (3) and function Hk (,, ) is defined as given below (,, )= Hk k k (36) Latin Aerican Journal of Solids and Structures (3) 3 46

16 38 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas 6.3 Nonlinear teperature distribution (NLD) Assue an FGM bea where the teperature in ceraic-rich and etal-rich surfaces are c and, respectively. he governing equation for the steady-state one-diensional heat conduction equation, in the absence of heat generation, becoes d d Kz () = dz dz h h ( )= c, ( )= (37) where Kz () is given by Eq. (). Solving this equation via polynoial series yields the teperature distribution across the bea thickness as = N ( ) i i ik c ( ) K c z D i= ik K h (38) with D N i ( ) K c = ik (39) K i= i Here N is the nuber of ters which should be taken into account to assure the convergence of the series. Evaluating the teperature difference as N and solving for gives the critical bucking value of p h Fk (, ) NLD L Gk (,, ) cr h Ik Ik (,,, ) q Ek (, ) K s L = ( ) (,,, ) (4) Latin Aerican Journal of Solids and Structures (3) 3 46

17 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas 39 K In this relation, = c and the function Ik (,,, ) is defined as K N i Ik (,,, )= D i ik ik ik k ik k (4) It should be pointed out that in each case of theral loading, an iterative process should be ipleented to calculate the critical buckling teperature difference. o this end, properties are evaluated at reference teperature and the critical buckling teperature difference is calculated. Properties of the constituents are then evaluated at the current teperature and again critical buckling teperature difference is obtained. his process should be continued to obtain the convergent critical buckling teperature difference. 7 RESULS AND DISCUSSION Consider a ceraic-etal functionally graded bea. he cobination of aterials consist of Silicon-Nitride as ceraic and stainless steel as etal. he elasticity odulus, the theral epansion coefficient, and the theral conductivity coefficient for these constituents are highly dependent to the teperature and their properties ay be evaluated in any teperature based on oloukian odel. Each property of the constituents follow the net dependency to the teperature 3 3 P ( ) P P P P P (4) In this equation is easured in Kelvin. he constants P i are unique for the constituents and for the constituents of this study are given in able 3. Poisson s ratio for siplicity is chosen as.8. able 3 Introduced coefficients of Eq. (4) E c E c K c P e 9.4e e 6.33e P 3.7e e e e 4.3e 3.64e 3 P.6e e e 7.9e 6 P 8.946e 7.876e 7.3e 3 K Latin Aerican Journal of Solids and Structures (3) 3 46

18 4 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas o validate the results, the effect of shear is plotted in Figure 3, for an isotropic hoogeneous bea with teperature independent aterial properties. For this purpose, the results are copared between the Euler and ioshenko bea theories. he bea is under the unifor teperature rise loading. Non-diensional critical buckling teperature is defined by = UR ( L / h). It is apparent that the critical buckling teperature for beas cr cr with L / h ratio ore than 5 is identical between the two theories. But, for L / h ratio less than 5, the difference between the two theories becoe larger, and it will becoe ore different for L / h values less than. he sae graph is reported in [8] based on the nuerical shooting ethod. Figure 3 Effect of transverse shear on critical buckling teperature difference In Figure 4, the critical buckling teperature difference of an FGM bea under the unifor teperature rise loading is depicted. Both edges are claped. D case indicates that properties are teperature dependent, whereas ID indicates that properties are evaluated at reference teperature. As seen, as the power law inde increases, the critical buckling teperature decreases peranently. When it is copared to the D case, ID case overestiates the buckling teperatures. Difference between ID and D cases is ore pronounced at higher teperatures Latin Aerican Journal of Solids and Structures (3) 3 46

19 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas 4 Figure 4 Effects of power law inde and teperature dependency on cr In Figure 5, two other cases of theral loadings are copared with respect to each other. As seen in both of these cases, also, an increase in power law inde results in lower buckling teperature. LD case as an approiate solution of the NLD case underestiates the critical buckling teperatures ecept for the case of reduction of an FGM bea to the associated hoogeneous cases. his is epected since in these cases, the eact solution of the heat conduction equation is also linear. Figure 5 Effects of power law inde on cr of FGM beas under LD and NLD cases Latin Aerican Journal of Solids and Structures (3) 3 46

20 4 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas he influence of boundary conditions on buckling teperature difference is plotted in Figure 6. he unifor teperature rise case of loading is assued and properties are assued to be D. he case of a hoogeneous bea is chosen. As epected, the higher buckling teperature belongs to a bea with both edges claped and the lower one is associated to a bea with one side siply supported and the other one roller. he critical buckling teperature of S S and C Rcases are the sae. Figure 6 Boundary conditions effect on cr he effect of unifor teperature rise field on the aial buckling load of C R and C C beas is deonstrated in Figure 7. he obtained buckling loads are noralized by the equationn cr c PcrL ref, where E ref 3 c is the ceraic elasticity odule at reference tepera- E bh ture. As epected, an increase in the power law inde results in the lower buckling load. his is epected since as power law inde decreases, FGM bea tends to a ceraic bea which is stiffer than etal. With the increase of teperature rise paraeter, buckling load decreases since the constituents lose their stiffnesses. Latin Aerican Journal of Solids and Structures (3) 3 46

21 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas 43 Figure 7 Effect of theral environent on echanical buckling of FGM beas 8 CONCLUSION In the present article, the equilibriu and stability equations for the FGM beas with various types of boundary conditions are obtained. he derivation is based on the ioshenko bea theory, with the assuption of power law coposition for the constituent aterial. he buckling analysis under three types of theral loadings is presented. Also, the echanical buckling analysis under theral loads is studied. Closed for solutions are derived for the critical teperature/load. It is concluded that: ) he C C and C R functionally graded beas ehibit the bifurcation type buckling while thes S, C S and S R FGM beas coence to deflect with the initiation of theral loading. ) In each case of theral loading, the critical buckling teperature for FGM beas is lower than fully ceraic bea but greater than fully etallic bea. 3) According to the Euler and ioshenko bea theories, the critical buckling teperature of isotropic hoogeneous bea is independent of elasticity odulus; but for an FGM bea the elasticity odulus of the constituent aterials have significant effect on critical buckling teperature. Latin Aerican Journal of Solids and Structures (3) 3 46

22 44 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas 4) he critical buckling teperature of C R and S S hoogeneous beas are identical for studied cases of theral loading, while S S andc R FGM beas reveal different behaviors when are subjected to in-plane theral loading. 5) he Euler bea theory over-predicts the critical teperature of thick beas, especially for h / L greater than.5. 6) eperature dependency of the constituents has significant effect on critical buckling teperature difference. he value of cr is overestiated when the properties are assued to be independent of teperature. 7) In each case of theral loading, the ioshenko bea theory predicts lower values for critical buckling teperature in coparison with the Euler bea theory. Acknowledgent he financial support of the National Elite Foundation is gratefully acknowledged. References [] Kang Y.A., Li X.F., () Large Deflections of a Non-linear Cantilever Functionally Graded Bea, Journal of Reinforced Plastics and Coposites, 9: [] Aydogdu M., askin V., (7) Free Vibration Analysis of Functionally Graded Beas with Siply Supported Edges. Material and Design, 8: [3] Nirala K., Upadhyay P.C., Prucz, J., Lyons D., (6), hero-elastic Stresses in Coposite Beas with Functionally Graded Layer, Journal of Reinforced Plastics and Coposites, 5: [4] Li X.F., (8) A Unified Approach for Analyzing Static and Dynaic Behaviors of Functionally Graded ioshenko and Euler Bernoulli Beas, Journal of Sound and Vibration, 38: -9. [5] Kapuria S., Bhattacharyya M., Kuar A.N., (8) Bending and Free Vibration Response of Layered Functionally Graded Beas: A heoretical Model and Its Eperiental Validation. Coposite Structures, 8: [6] Huang Y., Li X.F., () Buckling of Functionally Graded Circular Coluns Including Shear Deforation. Materials and Design, 3: [7] Zhao F.Q., Wang Z., Liu, H., (7) heral Post-buckling Analyses of Functionally Graded Material Rod. Applied Matheatics and Mechanics, 8: [8] Li S., Zhang J., Zhao Y., (6) heral Post-buckling of Functionally Graded Material ioshenko Beas. Applied Matheatics and Mechanics, 7: [9] Kiani Y., Eslai M.R., () heral buckling analysis of functionally graded aterial beas. International Journal of Mechanics and Materials in Design, 6: 9-38 Latin Aerican Journal of Solids and Structures (3) 3 46

23 Y. Kiani et al / heroechanical Buckling of eperature-dependent FGM Beas 45 [] Aydogdu M., (8) Sei-inverse Method for Vibration and Buckling of Aially Functionally Graded Beas. Journal of Reinforced Plastics and Coposites, 7: [] Ke L.L., Yang J., Kitipornchai S., (9) Postbuckling Analysis of Edge Cracked Functionally Graded ioshenko Beas under End-shortening. Coposite Structures, 9: 5-6. [] Ke L.L., Yang J., Kitipornchai S., Xiang Y., (9) Fleural Vibration and Elastic Buckling of a Cracked ioshenko Bea Made of Functionally Graded Materials. Mechanics of Advanced Materials and Structures, 6: [3] Ma, L.S. and Lee, D.W. () A Further Discussion of Nonlinear Mechanical Behavior for FGM Beas under In-plane heral Loading. Coposite Structures, 93: [4] Fallah, A. and Aghda, M.M. () hero-echanical buckling and nonlinear free vibration analysis of functionally graded beas on nonlinear elastic foundation. Coposites Part B: Engineering, 43: [5] Wattanasakulpong N., Prusty B. G., Kelly D. W., () heral Buckling and Elastic Vibration of hird-order Shear Deforable Functionally Graded Beas, International Journal of Mechanical Sciences, 53: [6] Kiani Y., Rezaei M., aheri S., Eslai M. R., () hero-electrical buckling of piezoelectric functionally graded aterial ioshenko beas, International Journal of Mechanics and Materials in Design, 7 :85-97 [7] Kiani Y., aheri S., Eslai M. R., () heral Buckling of Piezoelectric Functionally Graded Material Beas, Journal of heral Stresses, 34 : [8] Ma L.S., Lee D. W., () Eact solutions for nonlinear static responses of a shear deforable FGM bea under an in-plane theral loading, European Journal of Mechanics A Solids, 3 :3- [9] Fu Y., Wang J., Mao Y., () Nonlinear analysis of buckling, free vibration and dynaic stability for the piezoelectric functionally graded beas in theral environent, Applied Matheatical Modelling, 36 : [] Anandrao K. S., Gupta R. K., Rachandran P., Rao G. V., (), heral post-buckling analysis of unifor slender functionally graded aterial beas, Structural Engineering and Mechanics, 36 : [] Praveen, G.N. and Reddy, J.N. (998) Nonlinear ransient heroelastic Analysis of Functionally Graded Ceraic-Metal Plates. International Journal of Solids and Structures, 35: [] Suresh, S. and Mortensen, A. (998) Fundaentals of Functionally Graded Materials, IOM Counications Ltd, London. [3] Brush, D.O. and Alorth, B.O. (975) Buckling of Bars, Plates, and Shells, McGraw-Hill, New York. [4] Wang, C.M., Wang, C.Y. and Reddy, J.N. (4) Eact Solutions for Buckling of Structural Mebers, CRC Press, Boca Raton. Latin Aerican Journal of Solids and Structures (3) 3 46

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