Effects of Boundary Conditions on Cross-Ply Laminated Composite Beams

Transcription

1 Internatonal Journal of Engneerng Research And Advanced Technology (IJERAT) DOI: E-ISSN : Vol.3 (0) Oct -07 Effects of Boundary Condtons on Cross-Ply Lamnated Composte Beams Imad-Eldn Mahmoud Mahd, Osama Mohammed Elmard Suleman and Ahmed F. Algarray 3 Assstant Professors, Department of Mechancal Engneerng, Faculty of Engneerng and Technology, Nle Valley Unversty, Atbara, Rver Nle State, Sudan Lecturer 3 Department of Mechancal Engneerng, Faculty of Engneerng, Red Sea Unversty, Port Sudan, Red Sea State, Sudan ABSTRACT In ths study, the effect of the end condtons of cross-ply lamnated composte beams (CLCB) on ther nondmensonal natural frequences of free vbraton was nvestgated. The problem s analyzed and solved usng the energy approach whch s formulated by a fnte element model. In that model, a three-noded element wth three degrees of freedom at each node s assumed. Numercal results were verfed by comparsons wth other relevant works. The end condtons of beams are: clamped -free (CF), hnged -hnged (HH), clamped -clamped (CC), hnged - clamped (HC), hnged -free (HF), free -free (FF). Each beam has ether movable ends or mmovable ends. It s found that the more constraned beams have the hgher values of natural frequences of transverse vbraton. However, the free-free and hnged-free beams are found to have the hghest frequences of transverse vbraton amongst all beams although they look less constraned. Ths behavor s due to the fact that the frst mode of the two beams s equal zero (rgd body moton), and replaced by the second mode to be the fundamental mode. The values of the natural frequences of longtudnal modes are found to be the same for all beams wth movable ends snce they are generated by longtudnal movements only. But for mmovable ends, the clamped-free and hnged-free beams have equal frequences n longtudnal vbraton, and those of the other beams are also the same. Keywords: Fnte Element Method, End Condtons, Cross Ply Lamnates, Frst Order Shear Deformaton, Natural Frequences.. INTRODUCTION Cross-ply lamnated composte beams (CLCB) are those havng alternatng layers of materal bonded together n some manner, and found n many of the products used n our day-to-day lves, lke cars, boats, machnes. Addtonally, they are used n many crtcal ndustral, aerospace and mltary applcatons. The benefts of compostes have fueled growth of new applcatons n Page 5

2 Osama Mohammed Elmard Suleman et al., Effects of Boundary Condtons on Cross-Ply Lamnated. markets such as transportaton, constructon, corroson-resstance, marne, nfrastructure, consumer products, electrcal, arcraft and busness equpment's[], [], [3], [4], [5], [6] and [7]. It s very mportant to estmate the values of natural frequences of CLCB n order to avod falure by avodng resonance when these beams subjected to dynamc loads. Usually the non-dmensonalzed frequences are computed and can then be appled for any beam sze. Many parameters are affectng the values of natural frequency; one of them s the end condtons. The frst-order shear deformaton theory was used by Teboub and Hajela [8] to analyze the free vbraton of generally layered composte beams. Banerjee J.R. [9] appled an exact soluton to predct the frequency equatons and mode shapes of free vbraton of lamnated composte beams. Also, Abramovch and Lvshts [0] presented exact solutons for the free vbraton of non-symmetrcally lamnated cross-ply composte beams. A beam fnte element based on layer wse trgonometrc shear deformaton theory s presented by Shmp R.P., and Anapure A.V. []. A free vbraton analyss of fber-renforced composte beams was carred out by Marur S.R. and Kant T. []. The authors appled the hgher-order theores and fnte element method to predct the values of natural frequences of lamnated composte beams.. MATERIAL AND GEOMETRY AS/350-6 graphte-epoxy materal was used for all numercal results because of ts wde applcatons n modern ndustres. The mechancal propertes of ths materal are tabulated n Table. Table Mechancal Propertes of AS/350-6 graphte-epoxy materal Property E E G G3 Magntude 45 GN/m 9.6 GN/m 4. GN/m 4. GN/m G3 Posson s rato 0.3 Densty 3.4 GN/m 50 kg/m3 Fgure below shows the geometry of a beam drawn n the three dmensons X, Y, and Z or,, and 3. Y, h X, b Z, 3 L Fgure The geometry of a lamnated composte beam. 3. MATHEMATICAL FORMULATIONS The tme-dependent axal and transverse dsplacements felds are: Journal Impact Factor:.45 Page 53

3 Internatonal Journal of Engneerng Research And Advanced Technology Vol.3 (0) Oct -07 U W x, z, t ux, t zx, t x, z, t wx, t () Where, u and w are the axal and transverse dsplacements at the md-plane, z s the perpendcular dstance from the md-plane to the layer plane, s the rotaton of a plane after deformaton, and t s the tme. The stran- dsplacement relatons are: U u z x x x W U w 5 x z x () Where the subscrpts have the same meanngs as those used n 3-D elastcty formulaton,.e. s the axal or longtudnal stran, and 5 s the through-thckness shear stran. The stress-stran relatonshp of a lamna can be shown as: j (3) C Where, T 5 C 0 C j (4) 0 C 55 5 t The elastc constants C and 55 C for orthotropc beams can be expressed as: C C 55 C C 44 4.cos C.sn C 55 4.sn ( C.cos. C 66 ).sn.cos (5) Where; C 66 G; C44 G3; C55 G3; C C C (6) 4. FINITE ELEMENT MODELING Applyng the energy approach for the beam element shown n Fgure, the stran energy stored s gven by: U s T e dv (7) Where, dv bdxdz and the subscrpt, e, means one element Page 54

4 Osama Mohammed Elmard Suleman et al., Effects of Boundary Condtons on Cross-Ply Lamnated. N u N u N 3 u 3 w w w 3 L e 3 X Fgure, A 3-noded fnte element wth nne degrees of freedom and shape functons. Also, the knetc energy s found as follows; K E. b e W U W U dx dz t t. (8) The degrees of freedom at each node are; the axal dsplacement u, deflecton w, and rotaton, and can be wrtten n terms of ther nodal values as follows: 3 u, w, N u, N w, N (9) Where, N s shape functon and assumed as a second-order polynomal as: N a b x c x,,3 (0) a The constants b, c and can be computed for each element from the followng data: N x at 0 x x x,,3 () Equatons (7), and (8) leads to the fnal form of the non-dmensonal element stffness and nerta matrces [K] e, and [M] e respectvely. These ndvdual element stffness and nerta matrces must be lnked together or assembled to form the global matrces and to characterze the unfed behavor of the entre beam. Therefore, the global stffness and nerta matrces are gven respectvely by, N K K n N M M n Where, N s the total number of beam elements. e e () The beam end condtons whch consdered are; () Clamped-free beam (CF), () Hnged-hnged beam (HH), (3) Clampedclamped beam (CC), (4) Hnged-clamped beam (HC), (5) Hnged-free beam (HF), and (6) Free-free beam (FF). Each beam has ether movable ends or mmovable ends. In the former group, the axal dsplacement at both beam-ends s consdered, whle neglected for the latter group. Journal Impact Factor:.45 Page 55

5 Internatonal Journal of Engneerng Research And Advanced Technology Vol.3 (0) Oct -07 The soluton can be obtaned after the ncorporaton of ends condtons whch wll modfy both stffness and nerta matrces. Thus, the non-dmensonal natural frequences can be determned from the relaton: K I 0 M (3) Where, I s an dentty matrx, and s the non-dmensonalzed natural frequences, whch can be computed by computng the square root of the egenvalues of the matrx M K usng a sutable computer program (Here MATLAB was used). 5. METHOD VALIDITY In order to check the valdty of the present method, some comparsons were performed. These comparsons were selected to cover the cases of symmetrcally and non-symmetrcally lamnated beams. Table presents a comparson wth Abramovch and Lvshts [0] of the non-dmensonal frequences of symmetrc [0/90/90/0] cross-ply graphte-epoxy beams for aspect rato of (L/h = 0). The end condtons consdered are hnged-hnged, fxed-free, and fxed-fxed wth mmovable ends. Here, the authors ntroduced the secondary effect of couplng between bendng and torson n ther analyss, whch s small, compared wth the other secondary effects. For the hnged-hnged beam, a percentage dfference of less than 0.6% was recorded for the fundamental frequency, and less than 0.54% for both fxed-free and fxed-fxed beams. Ths dfference was observed to ncrease as the mode order ncreases (less than.4%) for the seventh mode for all beams consdered. Table Non-dmensonal frequences [ Mode No.. L Eh ] of symmetrc [0/ 90/ 90/ 0] cross-ply beams (L/h = 0) Hnged-hnged Fxed-free Fxed-fxed Present Ref. [0] Present Ref. [0] Present Ref. [0] *.93* * 3.007* * 3.007* (*) Modes wth predomnance of longtudnal vbraton. Another comparson was carred out wth the results of Marur and Kant []. Table 3 compares the frst sx modes of the non- dmensonalzed frequences [. L E h ] of symmetrc [ 0 /90/90/ 0 ] cross-ply, graphte-epoxy, clamped-free beam wth aspect rato of (L/h = 5). Table 3Non-dmensonal natural frequences [. L E h ] of symmetrc [ 0 /90/90/ 0 ] cross-ply clamped- free beam. (L/h = 5). Mode No. Present Ref. [] * Page 56

6 Osama Mohammed Elmard Suleman et al., Effects of Boundary Condtons on Cross-Ply Lamnated (*) Modes wth predomnance of longtudnal vbraton. The authors appled the hgher-order shear deformaton theory (HOSDT) n the analyss, whereas, the present method uses frstorder shear deformaton theory. The comparson shows a percentage dfference of less than 0.03% for the fundamental mode of vbraton. Ths dfference ncreases wth the mode order to less than 3.6% for the sxth mode. 6. NUMERICAL RESULTS AND DISCUSSIONS Table 4 shows the frst ten modes of free vbraton of CLCB wth mmovable ends (.e. axal movement s restrcted), whle table 5 shows those for movable ends. Generally, t s found that more constraned beams have hgh values of natural frequences. However, the free-free and hnged-free beams are found to have the hghest frequences amongst all beams although they look less constraned. Ths behavor s due the fact that the frst mode of the two beams s equal zero and replaced by the second mode. The fundamental mode shapes of both beams are straght lnes, Fgure 3, and ths due to the rgd moton n ths mode where there s no vbratng moton. 4 Table 4Non-dmensonal natural frequences L Eh of a symmetrc cross-ply [90/-90/-90/90] lamnated beam wth mmovable ends, (L/h = 0). Mode No. Beam type CF HH CC HC HF FF * * * 8.077* 8.077* * * * 6.55* 6.55* 6.55* * * 4.334* * 4.334* 4.334* 4.334* (*) Modes wth predomnance of longtudnal vbraton. Also, t can be observed from the results n tables 4 and 5 that the values of the non-dmensonalzed natural frequences of the transverse modes are not affected by the longtudnal movements of the ends snce these modes are generated by lateral movements only. However, the values of the natural frequences of longtudnal modes are found to be the same for all beams wth movable ends snce they are generated by longtudnal movements only. It could be notced that the values of nondmensonalzed natural frequences of the longtudnal vbraton for the clamped-free (CF) and hnged-free (HF) beams are equal, and those of the other beams are also the same. Ths phenomenon occurs snce both clamped-free and hnged-free beams wth mmovable ends are the same when restrcted from executng longtudnal moton at the ends. Smlarly, the rest of beams wth mmovable ends have the same longtudnal end condtons. Journal Impact Factor:.45 Page 57

7 Internatonal Journal of Engneerng Research And Advanced Technology Vol.3 (0) Oct Table 5 Non-dmensonal natural frequences L Eh of a symmetrc cross-ply [90/-90/-90/90] lamnated beam wth movable ends, (L/h = 0). Mode No. CF HH CC HC HF FF * 8.077* 8.077* 8.077* 8.077* * * 6.55* 6.55* 6.55* 6.55* * * * 4.334* 4.334* 4.334* (*) Modes wth predomnance of longtudnal vbraton. 7. CONCLUSIONS In the present study, the effects of end condtons of CLCB on ther non-dmensonalzed natural frequences are studed. The problem s modeled and analyzed by the fnte element method. Twelve end condtons were studed whch are clamped-free, hnged-hnged, clamped-clamped, hnged-clamped, hnged-free, and free-free beams wth mmovable and movable ends. It s found that more constraned beams have hgh values of natural frequences. However, the free-free and hnged-free beams are found to have the hghest frequences amongst all beams although they look less constraned. The transverse modes are not affected by the longtudnal movements of the ends snce these modes are generated by lateral movements only. The values of the natural frequences of longtudnal modes are found to be the same for all beams wth movable ends snce they are generated by longtudnal movements only. But for mmovable ends, the clamped-free and hnged-free beams have equal frequences n longtudnal vbraton, and those of the other beams are also the same. Page 58

8 Osama Mohammed Elmard Suleman et al., Effects of Boundary Condtons on Cross-Ply Lamnated. ACKNOWLEDGEMENT The authors would lke to acknowledge wth deep thanks and profound grattude Mr. Osama Mahmoud of Danya Center for Publshng and Prntng Servces, Atbara, who spent many hours n edtng, re edtng of the manuscrpt n complance wth the standard format of Internatonal Journal of Engneerng Research and Advanced Technology (IJERAT). REFERENCES [] R.A. Jafar-Talookolae and M.T.Ahmadan ' Free Vbraton Analyss of a Cross-Ply Lamnated Composte Beam on Pasternak Foundaton', Journal of Computer Scence 3 (): pp. (5-56), (007). [] Mahmoud Yassn Osman and Osama Mohammed Elmard Suleman, 'Large deflecton of composte beams', Internatonal Journal of Engneerng Research and Advanced Technology(IJERAT), Vol.03, Issue 3, (07), pp. (6 39). [3] Osama Mohammed Elmard Suleman, Lterature revew on mperfecton of composte lamnated plates', Journal of Mcroscopy and Ultrastructure, Vol.03, 5 (07), pp. (9 ). [4] Mahmoud Yassn Osman and Osama Mohammed Elmard Suleman, Free vbraton of lamnated plates', Internatonal Journal of Engneerng Research and Advanced Technology (IJERAT), Vol.03, Issue 4, (07), pp. (3 47). [5] Mahmoud Yassn Osman and Osama Mohammed Elmard Suleman, Free vbraton analyss of lamnated composte beams usng fnte element method', Internatonal Journal of Engneerng Research and Advanced Technology (IJERAT), Vol.03, Issue, (07), pp. (5 ). [6] Mahmoud Yassn Osman and Osama Mohammed Elmard Suleman, Bucklng analyss of thn lamnated composte plates usng fnte element method', Internatonal Journal of Engneerng Research and Advanced Technology (IJERAT), Vol.03, Issue 3, (07), pp. ( 8). [7] Osama Mohammed Elmard Suleman, Verfcaton of dynamc relaxaton (DR) method n sotropc, orthotropc and lamnated plates usng small deflecton theory', Internatonal Journal of Advanced Scence and Technology (IJAST), Vol. 7 (07), pp. (37 48). [8] Teboub Y. and Hajela P. ' Free vbraton of generally layered composte beams usng symbolc computatons', Composte Structure 33, pp. (3-34), (995). [9] Banerjee J.R. 'Explct analytcal expresson for frequency equaton and mode shapes of composte beams', Internatonal Journal of Sold and Structures 38, pp. (45-46), (00). [0] Abramovch H. and Lvshts A. ' Free vbratons of non-symmetrc cross-ply lamnated composte beams', Journal of sound and vbraton 76(5), pp. (597-6), (994). [] Shmp R.P. and Anapure A.V. ' A beam fnte element based on layer wse trgonometrc shear deformaton theory', Composte Structures 53, pp. (53-6), (00). [] Marur S.R. and Kant T. 'Free vbraton analyss of fber-renforced composte beams usng hgher-order theores and fnte element modelng', Journal of sound and vbraton 94, pp. (337-35), (996). Journal Impact Factor:.45 Page 59