A beam nite element based on layerwise trigonometric shear deformation theory

Transcription

1 A beam ite elemet based o layerwise trigoometric shear deformatio theory R.P. Shimpi *, A.V. Aiapure Departmet of Aerospace Egieerig, Idia Istitute of Techology, Bombay, Powai, Mumbai , Idia Abstract A simple oe-dimesioal beam ite elemet, based o layerwise trigoometric shear deformatio theory, is preseted. The elemet has two odes ad oly three degrees of freedom per ode. Yet, it icorporates through the thickess siusoidal variatio of i-plae displacemet such that shear-stress free boudary coditios o the top ad bottom surfaces of the beam elemet are satis ed ad the shear-stress distributio is realistic i ature. Costitutive relatios betwee shear-stresses ad shear-strais are satis ed i all the layers, ad, therefore, shear correctio factor is ot required. Compatibility at the layer iterface i respect of iplae displacemet is also satis ed. It is to be oted that the elemet developed is free from shear lockig. The results obtaied are accurate ad show good covergece. Ulike may other elemets, trasverse shear-stresses are evaluated directly usig costitutive relatios. The e cacy of the preset elemet is demostrated through the eamples of static eure ad free vibratio. Keywords: Beam ite elemet Lamiated beam Layerwise theory Shear deformatio 1. Itroductio Lamiated composite beams are dig icreasig use due to their high stregth to weight ad high sti ess to weight ratios. It is to be oted that, the topic of ite elemets pertaiig to lamiated beams does ot appear etesively i literature as has bee oted by Abramovich ad Livshits [1], Abramovich et al. [2], Maiti ad Siha [3], Eiseberger et al. [4]. Eve i case of lamiated plate, layerwise models have show the superiority of layerwise approach to predict accurately static ad dyamic respose of thick structures [5]. I layerwise theories, appropriate separate displacemet eld epressios are assumed for each material layer, providig a kiematically better represetatio of the strai eld i discrete layer lamiates [6]. Yua ad Miller [7] derived a ve oded beam ite elemet but, the umber of degrees of freedom is depedet o umber of layers havig 3N 7 total degrees of freedom per elemet for a N layered beam). Davalos et al. [8] preseted a oe-dimesioal three oded lamiated beam ite elemet havig 2 N degrees of freedom at each ode for a N-layered beam. The layerwise costat shear-stresses obtaied from costitutive relatios are trasformed ito parabolic shear-stress distributio through tedious post-processig operatio. I the preset paper, a simple oe-dimesioal beam ite elemet, based o variatioally cosistet layerwise trigoometric shear deformatio theory [9] is preseted. The elemet has two odes ad oly three degrees of freedom per ode. Yet, it icorporates through the thickess siusoidal variatio of i-plae displacemet such that shear-stress free boudary coditios o the top ad bottom surfaces of the beam elemet are satis ed ad the shear-stress distributio is realistic i ature. Compatibility at the layer iterface i respect of i-plae displacemet is satis ed. Preset elemet is free from shear lockig. Trasverse shear-stresses are evaluated directly usig costitutive relatios. The e ectiveess of the preset elemet is demostrated through illustrative eamples. 2. About layerwise trigoometric shear deformatio theory Layerwise trigoometric shear deformatio theory LTSDT) has bee discussed i [9]. The theory has bee

2 154 eteded to deal with free vibratio [10]. Here, essetial iformatio about LTSDT will be outlied i brief About the beam The beam uder cosideratio cosists of two layers, layer-1 ad layer-2. Layer-1 occupies the regio: L b=2 6 y 6 b=2 h=2 6 z 6 0: 1 Layer-2 occupies the regio: L b=2 6 y 6 b=2 0 6 z 6 h=2 2 where, y z are right-haded Cartesia co-ordiates, L is the legth, b is the width ad h is the total depth of beam. The layers of beam are perfectly boded to each other. Local pricipal material aes Cartesia) for each layer are deoted as 1, 2 ad 3. Directio 3 is such that it coicides with trasverse z-directio, ad directio 1 is parallel to the bers. The material desities of layer-1 ad layer-2 are deoted by q 1 ad q 2, respectively. Material used for layers of beam is specially orthotropic ad obeys Hooke's law [11]. The beam is subjected to lateral load q t o the top surface i.e. surface z ˆ h=2), where t deotes time Stress±strai relatios Based o the assumptios made i respect of LTSDT [9], the stress±strai relatios for the layers ca be writte as: r 1 s 1 z ˆ E 1 e 1 r 2 ˆ E 2 e 2 ˆ G 1 c 1 z s 2 z ˆ G 2 c 2 z where superscripts 1) ad 2) refer to layer-1 ad layer- 2, respectively. Elastic moduli E 1 E 2 ad shear moduli G 1 G 2 ca be epressed i the otatio used by Joes [11] as follows: E 1 ˆ E 1 ad G 1 ˆ G 31 if fibers i the layer-1 are alog -directio: E 1 ˆ E 2 ad G 1 ˆ G 23 if fibers i the layer-1 are alog y-directio: E 2 ˆ E 1 ad G 2 ˆ G 31 if fibers i the layer-2 are alog -directio: E 2 ˆ E 2 ad G 2 ˆ G 23 if fibers i the layer-2 are alog y-directio: 2.3. The displacemet eld The displacemet eld of the preset layerwise beam theory is give as follows: u 1 ˆ z ah ow o h C 1 C 2 si p 2 u 2 ˆ z ah ow o h C 3 si p 2 z=h a 0:5 a z=h a 0:5 a / t / t 3 4 w ˆ w t 5 where u 1 ad u 2 are the i-plae displacemets i the -directio, superscripts 1) ad 2) refer to layer-1 ad layer-2, w the trasverse displacemet i the z-directio, t deotes time, ad / is ukow fuctio. Trasverse displacemet w ad ukow fuctio / are same for both the layers. C 1 C 2 C 3 a are costats as give i Appedi A Goverig equatios ad boudary coditios Usig the priciple of virtual work, variatioally cosistet di eretial equatios ad associated boudary coditios for the beam uder cosideratio at ay istat of time are obtaied. Goverig di eretial equatios are: " o 2 M o I 2 1 ou 1 o om h s o I 3u 1 I 2 ou 2 o # w ˆ q t 6 I 4 u 2 i V s ˆ 0 7 where overdot deotes di eretiatio w.r.t. time t, ad M M s V s I 1 I 2 I 3 I 4 are as give i Appedi A. Ad, associated boudary coditios at ˆ 0 ad ˆ L are: om h i o I 1u 1 I 2 u 2 ˆ 0 OR w is prescribed 8 M ˆ 0 OR ow is prescribed 9 o M s ˆ 0 OR / is prescribed: Fiite elemet formulatio The ite elemet formulatio of two layered crossply lamiated beam based o LTSDT) usig Galerki weighted residual method is outlied below.

3 Domai of the ite elemet uder cosideratio The domai of the beam is divided ito umber of elemets say ˆ 1toN). The typical th beam ite elemet uder cosideratio cosists of two layers, i.e. layer-1 ad layer-2. Layer-1 occupies the regio: L b =2 6 y 6 b =2 h =2 6 z 6 0: Layer-2 occupies the regio: L b =2 6 y 6 b =2 0 6 z 6 h =2 where y z are elemet local co-ordiates Cartesia), L is the legth, b is the width ad h is the total height of the th beam ite elemet. Elemet local coordiates are related to global co-ordiates by ˆ for ˆ 1 ˆ X 1 y ˆ y z ˆ z L m mˆ1 for ˆ N for ˆ N for ˆ N: The elemet is subjected to trasverse ormal load q t o top surface i.e. z ˆ h =2). The elemet has two odes, i.e. ode 1 ad ode 2. Local co-ordiates of ode 1 ad ode 2 are ad L 0 0 respectively Weak forms of the di eretial equatios Usig goverig di eretial equatios 6) ad 7) ad Galerki weighted residual method, we get the weak form of di eretial equatios for the th elemet as Z ˆL o 2 u M 1 I o 2 1 u 1 ou 1 o wu 1 q t u 1 d ˆ0 I 1 u 1 Z ˆL ˆ0 I 2 u 2 u 1 ˆL ou M 2 s V s u o 2 I 3 u 1 u 2 ˆ0 I 2 u 2 ou 1 o om u o 1 ou M 1 o ˆL ˆ0 ˆ 0 11 I 4 u 2 u 2 d M s u 2 Š ˆL ˆ0 ˆ 0 12 where u 1 u 2 are weighig fuctios ad are fuctios of Shape fuctios Eamiatio of Eqs. 11) ad 12) suggests that, the shape fuctios for the trasverse displacemet w should be twice di eretiable w.r.t., whereas the shape fuctios for / should be di eretiable oly oce. Sice, i reality, slope is cotiuous across the elemets, it is better to approimate ot oly w but also slope to be cotiuous across the elemets. Therefore, Hermite cubic shape fuctios are used for approimatig trasverse de ectio w, ad Lagrage liear shape fuctios are used for approimatig fuctio /. Now w ad / ca be approimated as: w ˆ N 1 w 1 N 2 h 1 N 3 w 2 N 4 h 2 Š si t 13 / ˆ N 5 / 1 N 6 / 2 Š si t 14 where w 1 h 1 / 1 are odal degrees of freedom associated with ode 1 ad w 2 h 2 / 2 are odal degrees of freedom associated with ode 2. I case of statics, the term si t will be abset i epressios 13) ad 14). Hermite shape fuctios N 1 N 2 N 3 N 4 ad Lagrage shape fuctios N 5 N 6 are give as: N 1 ˆ 1 3 =L 2 2 =L 3 15 N 2 ˆ 2 2 =L 3 =L2 16 N 3 ˆ 3 =L 2 2 =L 3 17 N 4 ˆ 2 =L 3 =L2 18 N 5 ˆ 1 =L 19 N 6 ˆ =L : Fiite elemet equatios I Eq. 11), usig epressios 13) ad 14) ad replacig weighig fuctio u 1 successively by shape fuctios N 1 N 2 N 3 N 4 give by epressios 15)± 18)), oe obtais four equatios. Similarly, i Eq. 12), usig epressios 13) ad 14) ad replacig weighig fuctio u 2 successively by shape fuctios N 5 N 6 give by epressios 19) ad 20)), two additioal equatios are obtaied. These si equatios ca be writte i matri form as ite elemet equatio for the th elemet: kš fdg mš fdg ˆffg 21 where kš is elemet sti ess matri, mš the elemet mass matri, fdg the elemet displacemet vector, fdg the elemet acceleratio vector ad ff g is elemet force vector. It should be oted that elemet sti ess ad mass matrices are symmetric i ature. The elemets of the elemet sti ess matri kš are give by: Z ˆL d 2 N i d 2 N j k ij ˆ I c1 d ˆ0 d 2 d 2 for i ˆ ad j ˆ Z ˆL d 2 N i dn j ˆ I c2 d ˆ0 d 2 d for i ˆ ad j ˆ 5 6

4 156 Z ˆL Z dn i dn ˆL j ˆ I c3 d I c4 N i N j d ˆ0 d d ˆ0 for i ˆ 5 6 ad j ˆ where costats I c1 I c2 I c3 I c4 are give i Appedi A. The elemets of the elemet mass matri mš are give by: Z ˆL Z dn i dn ˆL j m ij ˆ I m1 d I m4 N i N j d ˆ0 d d ˆ0 for i ˆ ad j ˆ Z ˆL dn i ˆ I m2 N j d ˆ0 d for i ˆ ad j ˆ 5 6 Z ˆL ˆ I m3 N i N j d ˆ0 for i ˆ 5 6 ad j ˆ where costats I m1 I m2 I m3 I m4 are give i Appedi A. The elemet displacemet vector fdg is such that: fdg T ˆ w 1 h 1 / 1 w 2 h 2 / 2 Š: 24 The elemets of the elemet force vector ff g are give by: f i ˆ Z ˆL ˆ0 q t N i d Q i for i ˆ ˆ Q i for i ˆ where epressios for Q i i ˆ ) are give as: Q 1 ˆ om u o 1 I 1 u 1 I 2 u 2 u 1 ˆ0 ou Q 2 ˆ M 1 o ˆ0 Q 3 ˆ om u o 1 I 1 u 1 I 2 u 2 u 1 ˆL ou Q 4 ˆ M 1 o Q 5 ˆ M s u 2 Š ˆ0 Q 6 ˆ M s u 2 Š ˆL : ˆL 4. Illustrative eamples The e ectiveess of the preset elemet is demostrated through followig eamples of static eure ad free vibratio. It may be oted that for obtaiig umerical results of the eamples, the beam was divided ito 2, 4, 10, 14, 20, 30 ad 40 elemets. For the sake of brevity, results obtaied usig 20 elemets are give here, ad these results hardly di er from those correspodig results obtaied usig 30 ad 40 elemets Eample 1: Fleure of two layered simply supported beam A two layered cross-ply beam with layer-1 90 layer) ad layer-2 0 layer), occupies the regios give by epressios 1) ad 2), respectively. Both the layers are of uidirectioal graphite-epoy. The beam is subjected to trasverse ormal siusoidal load q ˆq si p =L actig alog z-directio, where q is magitude of the siusoidal loadig per uit legth at midspa. This is the same eample, which was also cosidered by Pagao [15] ad may others e.g. [3,12,14]). The material properties of the beam material are such that: E 2 G ˆ 25 E 1 1 G ˆ 0:20 E 1 2 ˆ 0:02: E 2 For the discretised lamiated beam, all the elemet sti ess matrices ad force vectors are assembled usig routie procedure. This assembly gives global ite elemet equatio for bedig: KŠfdg ˆfFg 26 where KŠ fdg ff g are global sti ess matri, global displacemet vector ad global force vector, respectively. Solutio of the global ite elemet equatio 26) is achieved after applyig appropriate boudary coditios Eqs. 8)± 10)). Results obtaied for displacemets ad stresses at saliet poits are preseted usig odimesioalised parameters i Tables 1±3, ad Figs. 1± No-dimesioalised parameters The results i respect of trasverse displacemet, iplae ormal bedig stress, trasverse shear-stress are preseted i the followig o-dimesioalised form i this paper. w ˆ 100E 1 bh 3 w q L 4 r ˆ br q s z ˆ bs z q : The percetage di erece % Di ) i results obtaied by models of various researchers with respect to the correspodig results obtaied by the LTSDT±CFS are calculated as follows: % Diff ˆ f value obtaied by a model value by LTSDT±CFS Š = value by LTSDT±CFS g 100: 4.2. Eample 2: Free vibratio of two layered simply supported beam A simply supported two layered cross-ply [90/0] composite beam is cosidered. The two layered beam

5 157 Table 1 Compariso of o-dimesioalised maimum trasverse displacemet w of Eample 1 Source ad model At ˆ 0:5L S ˆ 4 S ˆ 10 w % Di a w % Di a Beam bedig models Preset LTSDT±FEM Shimpi ad Ghugal [9] LTSDT±CFS Majuatha ad Kat [12] ) )2.55 HOSTB5±FEM Maiti ad Siha [3] )25.49 ) ) HST±FEM Viayak et al. [14] )3.83 ) ) HSDT±FEM Euler-Beroulli ) )11.64 ETB Cylidrical plate bedig models Lu ad Liu [13] HSDT±CFS Pagao [15] ) )0.59 Elasticity a % Di i.e. Percetage di erece quoted is with respect to the correspodig value obtaied usig LTSDT±CFS. Table 2 Compariso of o-dimesioalised trasverse shear stress s z of Eample 1 Source ad model For S ˆ 4atˆ0:0 ad z ˆ ah s z CR % Di a s z EQM % Di a Beam bedig models Preset ) ) LTSDT±FEM Shimpi ad Ghugal [9] LTSDT±CFS Majuatha ad Kat [12] ) HOSTB5±FEM Maiti ad Siha [3] ) ) )9.45 HST±FEM Viayak et al. [14] ) ) HSDT±FEM Euler±Beroulli ) ) ETB Cylidrical plate bedig models Lu ad Liu [13] )6.59 ) ) HSDT±CFS Pagao [15] ) ) Elasticity a % Di i.e. Percetage di erece quoted is with respect to the correspodig value obtaied usig LTSDT±CFS. with layer-1 ad layer-2, occupies the regios give by epressios 1) ad 2), respectively. The layer properties of the beam material uder cosideratio are: E 1 ˆ 9:65 GPa G 1 ˆ 3:45 GPa q 1 ˆ q 2 ˆ 1389:23 kg=m 3 : E 2 ˆ 144:80 GPa G 2 ˆ 4:14 GPa For the discretised lamiated beam, all the elemet sti ess matrices ad force vectors are assembled usig routie procedure. The assembly gives global ite elemet equatio for free vibratio KŠfdg MŠf dgˆf0g 27 where KŠ MŠ fdg f dg are global sti ess matri, global mass matri, global displacemet vector ad global acceleratio vector, respectively. The global ite elemet equatio 27), after applyig appropriate boudary coditios Eqs. 8)± 10)), is solved for eigevalues

6 158 Table 3 Compariso of o-dimesioalised i-plae stress r of Eample 1 Source ad model For S ˆ 4, at ˆ 0:5L ad z ˆ h=2 z ˆ h=2 r % Di a r % Di a Beam bedig models Preset ) LTSDT±FEM Shimpi ad Ghugal [9] ) LTSDT±CFS Majuatha ad Kat [12] ) ) )11.50 HOSTB5±FEM Maiti ad Siha [3] ) ) )15.64 HST±FEM Viayak et al. [14] ) )11.66 HSDT±FEM Euler-Beroulli ) ) )8.54 ETB Cylidrical plate bedig models Lu ad Liu [13] ) ) )1.85 HSDT±CFS Pagao [15] ) ) )1.75 Elasticity a % Di i.e. Percetage di erece quoted is with respect to the correspodig value obtaied usig LTSDT±CFS. Fig. 1. Plot of maimum o-dimesioal trasverse de ectio w at ˆ 0:5L versus aspect ratio S. Fig. 2. Variatio of o-dimesioalised trasverse shear stress s z through the thickess at ˆ 0 for aspect ratio S ˆ 4). usig geeral solutio procedure. Results obtaied for atural frequecies are preseted i Table Eample 3: Free vibratio of sigle layered simply supported beam A simply supported sigle layered orthotropic beam, which is a special case of two layered orthotropic [0/0] composite beam, is cosidered here. This is the same eample, which was also cosidered i [16,17]. This eample ca be posed as a two layered beam eample. Layer-1 ad layer-2 of the beam occupy the regios give by epressios 1) ad 2), respectively. The layer properties are: Fig. 3. Variatio of o-dimesioalised i-plae stress r through the thickess at ˆ 0:5L for aspect ratio S ˆ 4).

7 159 Table 4 Compariso of atural frequecies of 90/0) beam Eample 2 S Mode o. Natural frequecy i khz Preset LTSDT FSDT ETB Table 5 Compariso of atural frequecies of orthotropic beam, Eample 3 S Mode o. Natural frequecy i khz Preset LTSDT [10] FSDT [17] HSDT [16] CLT [16] E 1 ˆ E 2 ˆ 144:80 GPa G 1 ˆ G 2 ˆ 4:14 GPa q 1 ˆ q 2 ˆ 1389:23 kg=m 3 : Followig similar procedure as that of Eample 2, atural frequecies of the simply supported beam ca be obtaied. Results obtaied for atural frequecies are preseted i Table Discussio The eact results i respect of static eure ad free vibratio eamples studied here, are ot available i the literature to the best of the kowledge of authors. I the absece of eact results, i respect of static eure, may researchers e.g. [3,12,14]) have compared their results with the eact solutio of cylidrical bedig of composite plate by Pagao [15]. However, it is to be oted that the di erece betwee cylidrical bedig ad beam bedig is aalogous to the di erece betwee plae strai ad plae stress i classical theory of elasticity [18]. Therefore, cylidrical plate bedig results are quoted here oly for the sake of iformatio. I case of free vibratio, researchers e.g. [16,19]) have compared their results with the rst order shear deformatio theory FSDT) results, probably i the absece of eact results. I this paper, results obtaied i various refereces are compared with respect to the close form solutio CFS) results obtaied by layerwise trigoometric shear deformatio theory i [9,10] because of the followig: 1. I both the layers of the beam, costitutive relatios are satis ed betwee: 1.1. i-plae stress r ad i-plae strai e, 1.2. trasverse shear-stress s z ad trasverse shearstrai c z. 2. Cotiuity coditios betwee the layers are satis ed a priori for: 2.1. i-plae displacemet u, 2.2. trasverse shear-stress s z. 3. I-plae displacemet u is such that the resultat of i-plae stress r actig over the cross-sectio is zero. Whereas, higher order shear deformatio theories HSDT) by Maiti ad Siha [3], Majuatha ad Kat [12], Viayak et al. [14], Chadrashekhara ad Bagera [16] do ot satisfy all the characteristics. From the illustrative eamples studied i.e. Eamples 1±3), for bedig ad free vibratio, followig observatios are made for displacemet, stresses ad atural frequecies: 1. Trasverse Displacemet w: a) For aspect ratio S equal to 4, it is observed from Table 1, that the maimum values of o-dimesioalised trasverse displacemet obtaied by the preset approach LTSDT±FEM) are matchig upto three

8 160 Table 6 Compariso of di eret beam ite elemets i respect of degrees of freedom dof) ad method of obtaiig shear stresses Source Type of elemet dof per Shear stress obtaied by usig elemet Preset 1D ± 2 oded, C 1 type 6 Costitutive relatios Shi ad Lam [19] 1D ± 2 oded, C 1 type 8 ) Majuatha ad Kat [12] 2D ± 4 oded, C 0 type 20 Costitutive relatios or equilibrium equatios Viayak et al. [14] 1D ± 3 oded, C 0 type 21 Equilibrium equatios Maiti ad Siha [3] 2D ± 9 oded, C 0 type 108 Equilibrium equatios decimal places) with LTSDT±CFS. Whereas, results obtaied by other researchers usig higher order theories are di erig by 3% to 9% w.r.t. LTSDT±CFS. b) From Fig. 1, it is see that LTSDT±FEM results coverge to elemetary theory of bedig ETB) results as the aspect ratio icreases, which demostrates, the preset elemet is free from shear lockig. 2. Trasverse shear-stress s z : To the best of the kowledge of authors, very few ite elemets are available for beams which evaluate shear-stresses usig costitutive relatios. Evaluatio of trasverse stresses from itegratio of equilibrium equatios is a veig problem as per [12]. But Viayak et al. [14] epressed di erece of opiio regardig this evaluatio method. I this paper trasverse shear-stresses are evaluated directly usig costitutive relatios oly. From Table 2 ad Fig. 2, the followig eeds to be oted about trasverse shear-stress s z : a) Results of maimum o-dimesioalised trasverse shear-stress, obtaied directly usig costitutive relatio, for the preset elemet are typically di erig by about 0.19% w.r.t. LTSDT±CFS for aspect ratio 4, whereas, results of Majuatha ad Kat [12] are di erig by 35.54%. b) Results of o-dimesioalised trasverse shearstress, obtaied idirectly by usig equilibrium equatios, by Maiti ad Siha [3], Majuatha ad Kat [12], Viayak et al. [14] are di erig by about 6.0% w.r.t. LTSDT±CFS for aspect ratio 4. Thus, it is see that the preset elemet gives accurate results for trasverse shear-stress directly usig costitutive relatios ad thus avoidig the tedious way of obtaiig the shear-stress idirectly by the use of equilibrium equatio for evaluatig trasverse shear-stress). Thus, the elemet has the added advatage of simplicity. 3. I-plae stress r : It is observed, from Table 3 ad Fig. 3, that the maimum values of i-plae stress r obtaied by LTSDT±FEM are typically di erig by 0.10% with LTSDT±CFS. Whereas, results obtaied by other researchers [12,14,15] are di erig typically by about 10% with LTSDT±CFS. Thus, it is see that the preset elemet gives accurate results for i-plae stress. 4. Natural frequecy : a) It ca be see from Table 4, results of atural frequecies usig the preset elemet LTSDT±FEM) are i close agreemet matchig upto three decimal places) with LTSDT±CFS results for aspect ratios 10, 15, 120. Whereas, results of rst order shear deformatio theory are matchig upto secod decimal poit of LTSDT±CFS ad as epected this di erece is icreasig i case of higher modes, which idicates the e ect of shear deformatio. b) LTSDT±FEM results coverge to ETB results as the aspect ratio icreases, which demostrates, the preset elemet is free from shear lockig. c) It ca be see from Table 5, amog the available atural frequecy results i respect of sigle layer orthotropic simply supported beam, preset elemet gives most accurate results matchig upto three decimal places) w.r.t. LTSDT±CFS. It is to be oted that i Table 5 results of atural frequecies for CLT are take from [16]. d) From Tables 4 ad 5, it ca be see that, as epected, the elemetary theory of bedig overestimates the atural frequecies. I additio to above discussio, it is iterestig to ote the compariso of preset beam ite elemet ad various beam ite elemets i respect of degrees of freedom ad method of obtaiig shear-stresses. It ca be clearly see from Table 6 wherei preset elemet ad various elemets are compared) that: 1. The preset elemet has oly si degrees of freedom. All other elemets have substatially more degrees of freedom per elemet. Therefore, the preset elemet is computatioally more e ciet. 2. Oly two refereces the preset oe ad the referece [12]) have reported obtaiig shear-stresses directly by usig costitutive relatios. Therefore, cumbersome post-processig for obtaiig shearstresses i a idirect way by usig equilibrium equatios) is avoided. 6. Cocludig remarks I this paper, a ew displacemet type beam ite elemet, which takes ito accout shear deformatio e ects, is developed for two layered cross-ply composite beams. The preset ite elemet has followig features: 1. It is oe-dimesioal, two oded ite elemet with oly three degrees of freedom at each ode. Yet,

9 the costitutive relatios betwee shear-stress ad shear-strai are satis ed i both the layers, 1.2. shear-stress free boudary coditios o the top ad bottom surfaces of the beam elemet are satis ed, 1.3. compatibility at the layer iterface i respect of i-plae displacemet, as well as trasverse shear-stress is satis ed, ad 1.4. the shear-stress distributio is realistic i ature. 2. The goverig di eretial equatios ad boudary coditios, o which the ite elemet formulatio is based, are variatioally cosistet. 3. From the illustrative eamples dealig with static eure ad free vibratio of two layered simply supported beams), it ca be see that: 3.1. Preset elemet gives accurate results Ulike other elemets, trasverse shear-stresses are obtaied directly usig costitutive relatios istead of obtaiig them i a tedious ad idirect maer usig equilibrium equatios) The use of elemet results i fast covergece It does ot su er from shear lockig. I geeral, preset elemet is simple, accurate ad easy to use. Appedi A Costats associated with displacemet eld as referred i Eqs. 3)± 5)): a ˆ 1 E 2 E 1 4 E 2 E 1 E 2 C 1 ˆ si C E 1 E 2 2 si 1 2a 1 2a E 1 1 2a pa 2C 2 cos E 2 p 1 2a 1 2a pa 2 cos p 1 2a C 2 ˆ G2 0:5 a cos 1 2a G 1 0:5 a cos 1 2a E 1 C 3 ˆ si E 1 E 2 1 2a 2C 2 si 1 2a 2 E 2 1 2a pa cos E 1 p 1 2a 1 2a pa 2C 2 cos : p 1 2a M M s V s I 1 I 2 I 3 I 4 as referred i Eqs. 6) ad 7) are give as: M ˆ Z zˆ0 Z yˆb=2 zˆ h=2 yˆ b=2 Z zˆh=2 Z yˆb=2 zˆ0 yˆ b=2 r 1 z ah dy dz r 2 z ah dy dz Z zˆ0 Z yˆb=2 M s ˆ r 1 h C 1 zˆ h=2 yˆ b=2 C 2 si p Z z=h a zˆh=2 dy dz 2 0:5 a zˆ0 r 2 h C 3 si p z=h a dy dz 2 0:5 a Z zˆ0 Z yˆb=2 V s ˆ s 1 z cos p z=h a zˆ h=2 yˆ b=2 2 0:5 a Z pc zˆh=2 Z yˆb=2 2 dy dz 1 2a zˆ0 yˆ b=2 s 2 z cos p z=h a p dy dz 2 0:5 a 1 2a I 1 ˆ I 2 ˆ Z zˆ0 Z yˆb=2 zˆ h=2 yˆ b=2 Z zˆh=2 Z yˆb=2 zˆ0 yˆ b=2 Z zˆ0 Z yˆb=2 q 1 zˆ h=2 yˆ b=2 q 1 z ah dy dz q 2 z ah dy dz I 3 ˆ h C 1 C 2 si p z=h a 2 0:5 a Z zˆh=2 Z yˆb=2 I 4 ˆ q 2 h C 3 si p 2 zˆ0 yˆ b=2 dy dz z=h a 0:5 a Z yˆb=2 yˆ b=2 dy dz: Itegratio costats associated with elemet sti ess matri refer Eq. 22)) are 8 9 I c1 >< I >= c2 ˆ I >: c3 > I c4 Z zˆ0 z ˆ h =2 Z zˆh =2 z ˆ0 8 E 1 z ah 2 h >< E 1 h z ah C 1 C 2 si p 2 h i 2 E 1 h 2 C 1 C 2 si p z =h a 2 0:5 a >: G 1 p 2 C 2 2 cos 2 p z =h a 1 2a 2 2 0:5 a 8 E 2 z ah 2 h >< E 2 h z ah C 3 si p 2 h i 2 E 2 h 2 C 3 si p z =h a 2 0:5 a >: G 2 p 2 cos 2 p z =h a 1 2a 2 2 0:5 a z =h a 0:5 a z =h a 0:5 a 9 i >= dz > 9 i >= dz : >

10 162 Itegratio costats associated with elemet mass matri refer Eq. 23)) are 8 9 I m1 >< >= I m2 ˆ I >: m3 > I m4 8 9 q 1 z ah 2 h i Z zˆ0 q 1 h z ah C 1 C 2 si p z=h >< a >= 2 0:5 a h i 2 dz z ˆ h =2 q 1 h 2 C 1 C 2 si p z =h a 2 0:5 a >: q 1 h b > q 2 z ah 2 h i Z zˆh =2 q 2 h z ah C 3 si p z >< =h a >= 2 0:5 a h i 2 dz : z ˆ0 q 2 h 2 C 3 si p z =h a 2 0:5 a >: q 2 h b > 2 Refereces [1] Abramovich H, Livshits A. Free vibratios of o-symmetric cross-ply lamiated composite beams. J Soud Vib ):597±612. [2] Abramovich H, Eiseberger M, Shulepov O. Vibratios ad bucklig of cross-ply osymmetric lamiated composite beams. AIAA J ):1064±9. [3] Maiti DK, Siha PK. Bedig ad free vibratio aalysis of shear deformable lamiated composite beams by ite elemet method. Compos Struct :421±31. [4] Eiseberger M, Abramovich H, Shulepov O. Dyamic sti ess aalysis of lamiated beams usig a rst order shear deformatio theory. Compos Struct :265±71. [5] Carrera E. A study of trasverse ormal stress e ect o vibratio of multilayered plates ad shells. J Soud Vib ):803±29. [6] Reddy JN. A evaluatio of equivalet-sigle-layer ad layerwise theories of composite lamiates. Compos Struct :21±35. [7] Yua F, Miller RE. A ew ite elemet for lamiated composite beams. Comput Struct ):737±45. [8] Davalos JF, Kim Y, Barbero EJ. Aalysis of lamiated beams with a layerwise costat shear theory. Compos Struct :241±53. [9] Shimpi RP, Ghugal YM. A layerwise trigoometric shear deformatio theory for two-layered cross-ply lamiated beams. J Reiforced Plastics Compos ):1516±42. [10] Shimpi RP, Aiapure AV. Free vibratio of two layered cross-ply lamiated beam usig layerwise trigoometric shear deformatio theory. J Reiforced Plastics Compos [commuicated]. [11] Joes RM. Mechaics of composite materials. Tokyo: McGraw- Hill Kogakusha p. 31±56. [12] Majuatha BS, Kat T. Di eret umerical techiques for the estimatio of multiaial stresses i symmetric/usymmetric composite ad sadwich beams with re ed theories. J Reiforced Plastics Compos :2±37. [13] Lu X, Liu D. A iterlamiar shear stress cotiuity theory for both thi ad thick composite lamiates. J Appl Mech :502±9. [14] Viayak RU, Prathap G, Nagaarayaa BP. Beam elemets based o a higher order theory -I. Formulatio ad aalysis of performace. Comput Struct ):775±89. [15] Pagao NJ. Eact solutio for composite lamiates i cylidrical bedig. J Compos Mater 19693:398±410. [16] Chadrashekhara K, Bagera KM. Free vibratio of composite beams usig a re ed shear eible beam elemet. Comput Struct ):719±27. [17] Chadrashekhara K, Krishamurthy K, Roy S. Free vibratio of composite beams icludig rotary iertia ad shear deformatio. Compos Struct :269±79. [18] Whitey JM. Structural aalysis of lamiated aisotropic plates. Lacaster: Techomic p. 69. [19] Shi G, Lam KY. Fiite elemet vibratio aalysis of composite beams based o higher-order beam theory. J Soud Vib ):707±21.