Buckling analysis of plate structures in composite materials with a new theory of trigonometric deformation

Transcription

1 Civil and nvironmental Reearch Vol.5 3 Special Iue for International Congre on aterial & Structural Stailit Raat orocco 7-3 ovemer 3 Bucling anali of plate tructure in compoite material with a new theor of trigonometric deformation L ICH oureddine BSL Khadidja delouahed Touni edia el aa Univerit de acara epartment of cience and technologie acara lgeria Univerit de jillali Liae epartment of Civil ngineering Sidi Bel é lgeria tract. In thi invetigation a new trigonometric hear deformation plate theor involving onl four unnown function a againt five function in cae of other hear deformation theorie The theor preented i variationall conitent doe not require hear correction factor and give rie to tranvere hear tre variation uch that the tranvere hear tree var paraolicall acro the thicne atifing hear tre free urface condition. Governing equation are derived from the virtual wor principle. The cloed-form olution of a impl upported rectangular plate ujected to in-plane loading ha een otained uing the avier method. The effectivene of the theorie i rought out through illutrative eample. Introduction Compoite material have man advantage over conventional material uch a teel and concrete in tructural performance with their uperior trength-toweight ratio a well a tiffne-to-weight ratio. For thi reaon compoite material epeciall in the form of laminated compoite plate are ecoming increaingl ued in man tructural application for civil infratructure tem. Therefore it require etter undertanding of their tructural ehaviour and failure condition for afer and more economical deign. One of the main failure mechanim in laminated compoite plate i ucling. in the cae of an iotropic homogeneou plate the preence of in plane loading ma caue ucling in orthotropic laminated plate. The laminate unlie plate made of conventional material are inherentl aniotropic and inhomogeneou there maing their ucling deformation more complicated. Thu the accurate nowledge of critical ucling load i eential for reliale and lightweight tructural deign. an eact olution for iotropic and orthotropic plate have een developed mot of them can e found in Timoheno and Woinow-Krieger [] Timoheno and Gere [] Ban and Jin [3] and Kang and Leia [4]. In compan with tudie of ucling ehaviour of plate man plate theorie have een developed. The implet one i the claical plate theor () which neglect the tranvere normal and hear tree. Thi theor i not appropriate for the thic and orthotropic plate with high degree of modulu ratio. In order to overcome thi limitation the hear deformale theor which tae account of tranvere hear effect i recommended. The Reiner [5] and indlin [6] theorie are nown a the firt-order hear deform- ale theor (FST) and account for the tranvere hear effect the wa of linear variation of in-plane diplacement through the thicne. However thee model do not atif the zero traction oundar condition on the top and ottom face of the plate and need to ue the hear correction factor to atif the contitutive relation for tranvere hear tree and hear train. For thee reaon man higher-order theorie have een developed to improve in FST uch a Levinon [7] and Redd [8]. Indeed Redd [8] put forward a paraolic hear deformation theor () which conider not onl the tranvere hear train ut alo their paraolic variation acro the plate thicne. Recentl Touni and hi coworer [9 ] developed a new imple higher order theorie involving onl four unnown function a againt five function in cae of Reiner and indlin theorie. The accurac of thi theor ha een demontrated for tatic ending of plate erdaci et al. [9] and meur et al. [] therefore it eem to e important to etend thi theor to the tatic ucling ehaviour of plate. In thi paper the new trigonometric hear deformation plate theor developed l meiche et al. [9] and meur et al. [] ha een etended to the ucling ehaviour of orthotropic plate ujected to the in-plane loading. Uing the avier method the cloed-form olution have een otained.umerical eample involving ide-to-thicne ratio and modulu ratio are preented to illutrate the accurac of the preent theor in predicting the critical ucling load of iotropic and orthotropic plate. The reult aed on the preent theor are compared with thoe otained the firt-order hear deformation plate theorie and claical plate theor. The influence of everal parameter are dicued. Preent refined hear deformation theor Unlie the other theorie the numer of unnown function involved in the new trigonometric hear deformation plate theor i onl four a againt five in 86

2 Civil and nvironmental Reearch Vol.5 3 Special Iue for International Congre on aterial & Structural Stailit Raat orocco 7-3 ovemer 3 cae of other hear deformation theorie. The theor preented i variationall conitent doe not require hear correction factor and give rie to tranvere hear tre variation uch that the tranvere hear tree var paraolicall acro the thicne atifing hear tre free urface condition.. umption of the preent plate theor umption of the preent plate theor are a follow on the top and ottom urface of the plate are conidered. Baed on the aumption made in preceding ection the diplacement field can e otained uing q. () (4) a u( u ( ) z f ( v( v ( ) z f ( w( w ( ) + w ( ) (4) (i) (ii) The diplacement are mall in comparion with the plate thicne and therefore train involved are infiniteimal. The tranvere diplacement w include two component of ending w and hear w. Thee component are function of coordinate onl. Where the function f ( i choen in the form h π z f ( z in π h The train aociated with the diplacement in q. (4) are (5) w( w ( ) w ( ) + (iii) The tranvere normal tre σ z i negligile in comparion with in-plane tree σ and σ. (iv) The diplacement u in -direction and v in -direction conit of etenion ending and hear component. U u + u + u V v + v + v The ending component u and v are aumed to e imilar to the diplacement given the claical plate theor. Therefore the epreion for u and v can e given a u z v z The hear component u and v give rie in conjunction with w to the paraolic variation of hear train z z and hence to hear tree τ z τ z through the thicne of the plate in uch a wa that hear tree τ τ z are zero at the top and ottom face of z the plate. Conequentl the epreion for can e given a u f ( v f ( u and () () (3) v Where z z ε z u ε v ε u v + z ε ε + ε ε + + g( g( z z z z z + f ( + f ( w w + f ( w w w w z g( f '( and df ( f '( dz For an orthotropic plate the contitutive relation can e written a σ σ τ τ z 44 τ z ε ε and z 55 z (8) (4) (6) (7). iplacement Field and Contitutive quation In the preent anali diplacement field model atifing the condition of zero tranvere hear tree 87 where ( σ σ τ τ z τ ) and ( ε ε z ) are the tre and train component repectivel. are the plane tre reduced elatic contant in the material ae of the plate and are defined a

3 Civil and nvironmental Reearch Vol.5 3 Special Iue for International Congre on aterial & Structural Stailit Raat orocco 7-3 ovemer 3 ν ν ν 66 G ν 44 G3 55 G3 in which are Young modulu G G 3 G 3 are hear modulu and ν ν are Poion ratio. For the iotropic plate thee aove material propertie reduce to G G3 G3 G ν ν ν. The ucript 3 correpond to z direction of Carteian coordinate tem repectivel. 3 Governing equation When a plate i ujected to in-plane compreive force (Fig. ) and if the force are mall enough the equilirium of the plate i tale and the plate remain flat until a certain load i reached. t that load called the ucling load the tale tate of the plate i ditured and plate ee an alternative equilirium configuration accompanied a change in the load-deflection ehaviour. Fig.. Rectangular plate under in-plane force. (9) h / / h / ( S z S z ) ( τ z τ z ) / ( σ σ τ ) z dz g( dz. ( ) f z Sutituting quation (8) into quation (3) and integrating through the thicne of the plate the tre reultant are given a ε ε H H H H H S z 44 z S z 55 z () () and tiffne component are given a h / ( H ) ( z f ( ) dz ( i j 6 ) / h / g ( dz ( i j 45) / (3) Collecting the coefficient of δ u δ v δ w and δ w in quation () the governing equation are otained a The governing equation are derived uing the virtual wor principle which can e written for the plate a [ σ δε + σδε + τδ + τzδ z+ τzδ z] dv+ [ δ ( w + w )] d V ( w + w ) ( w + w ) ( w + w ) + + () δ u δ v δ w δ w S z + S z (4) + The tre reultant and S are defined The diplacement function that atif the equation of oundar condition (8) are elected a the following Fourier erie 88

4 Civil and nvironmental Reearch Vol.5 3 Special Iue for International Congre on aterial & Structural Stailit Raat orocco 7-3 ovemer 3 w w W in( λ ) in( µ ) mn W in( ) in( ) m n mn λ µ 4 umerical reult and dicuion (5) Fig.. The effect of modulu ratio on the critical ucling load of quare plate with a / h (a) plate ujected to uniaial compreion; () plate ujected to iaial compreion; (c) ujected to tenion in the direction and compreion in the direction. 4 In thi tud ucling anali of impl upported rectangular plate the new trigonometric hear deformation plate theor (with four unnown function) i uggeted for invetigation. Plate ujected to the loading condition a hown in Fig. i conidered to illutrate the accurac of the preent model in predicting the ucling ehavior of the plate. Comparion are made with olution otained uing the firt and paraolic hear deformation theor (with five unnown function). In order to invetigate the effect of ide-to-thicne ratio and modulu ratio the preent eample i applied for iotropic and orthotropic quare plate. The following engineering contant are ued [] / Varied G / G3 /. 5 G 3 /. ν. 5 For implicit the non-dimenional critical ucling parameter i defined a cra (7) 3 h (6) Where a the length of the quare i plate and h i the thicne of the plate. Fig how that when the orthotropic plate i ued the difference etween the preent model and will increae with the increae of modulu ratio /. Furthermore the preent model (with onl four unnown function) give identical reult to thoe otained with (with five unnown function) for all value of / Preent (a) / Preent () / 8 6 Preent (c) / Provide critical ucling load for impl upported quare plate uing variou plate theorie. the reult of the preent theor are cloe to the reult of the firt hear deformation theor (FST). Hence the preent new trigonometric hear deformation plate theor (with four 89

5 Civil and nvironmental Reearch Vol.5 3 Special Iue for International Congre on aterial & Structural Stailit Raat orocco 7-3 ovemer 3 unnown function) give comparale reult to thoe otained with FST (with five unnown function). preented in Fig and Fig 3 the difference etween the preent model and FST with the hear correction factor 5/6 and the preent model and FST with the hear correction factor are 6.8 % and.8 % repectivel for the ame cae of quare plate (a 5h and / 4 ). It can e een from figure 3 that the difference of critical ucling load etween the preent model and FST depend on not onl the ide-tothicne and modulu ratio ut alo the in-plane loading condition. meur. Touni. echa I. dda Bedia.. new trigonometric hear deformation theor for ending anali of functionall graded plate reting on elatic foundation. KSC Journal of Civil ngineering 5(8) Concluion The ucling anali of iotropic and orthotropic plate uing a new trigonometric hear deformation plate theor i preented. The numer of primar variale in thi theor i even le than that of firt- and higher-order hear deformation plate theorie. The theor tae account of tranvere hear effect and paraolic ditriution of the tranvere hear train through the thicne of the plate hence it i unnecear to ue hear correction factor. It can e concluded that the preent new plate theor can accuratel predict the critical ucling load of the iotropic and orthotropic plate. Reference. Timoheno SP Woinow-Krieger S. Theor of plate and hell. ew Yor cgraw-hill (959).. Timoheno SP Gere J. Theor of elatic tailit. ew Yor cgraw-hill(96). 3. Ban L Yin J. Bucling of orthotropic plate with free and rotationall retrained unloaded edge. Thin Wall Struct 4() (996) 4. Kang JH Leia W. act olution for the ucling of rectangular plate having linearl varing in-plane loading on two oppoite impl upported edge. Int J Solid Struct 4(4) 4 38(5) 5. Reiner. The effect of tranvere hear deformation on the ending of elatic plate. J ppl ech-t S () 69 77(945) 6. indlin R. Influence of rotar inertia and hear on fleural motion of iotropic elatic plate. J. ppl. ech 8() 3 8(95) 7. Levinon. n accurate imple theor of the tatic and dnamic of elatic plate. ech Re Commun 7(6) 343 5(98) 8. Redd J.. imple higher order theor for laminated compoite plate. J. ppl. ech (984) 9. l eiche Touni Ziane echa I dda B. new hperolic hear deformation theor for ucling and viration of functionall graded andwich plate. Int J of ech 53(4) () 9