The Buckling Analysis of the Composite Plates with Different Orientations of Layers

Transcription

1 Amercan Journal of Mechancal Engneerng, 2016, Vol. 4, No. 7, Avalable onlne at Scence and Educaton Publshng DOI: /ajme The Bucklng Analyss of the Composte Plates wth Dfferent Orentatons of Layers Pavol Lengvarský *, Jozef Bocko, Martn Hagara Department of Appled Mechancs and Mechancal Engneerng, Faculty of Mechancal Engneerng, Techncal Unversty of Košce, Košce, Slovaka *Correspondng author: Abstract Ths paper deals about a bucklng analyss of composte plates wth four dfferent layers orentatons and wth three, sx and twelve layers. Composte plate was modeled by the fnte element method. Compresson load, shear load and combnaton of both loads were appled to the composte plate. Then the computatonal bucklng analyss for the all combnatons of load was performed for four dfferent orentatons of layers and for plates wth dfferent numbers of layers. The computed crtcal bucklng loads for the appled loads were compared for all orentatons of layers. From the results the napproprate orentatons of layers for dfferent loads were obtaned. The crtcal bucklng load ncreased almost ten tmes for the composte plates wth two tmes hgher thckness. Keywords: bucklng, composte plate, orentaton of layers, fnte element method, compresson load, shear load Cte Ths Artcle: Pavol Lengvarský, Jozef Bocko, and Martn Hagara, The Bucklng Analyss of the Composte Plates wth Dfferent Orentatons of Layers. Amercan Journal of Mechancal Engneerng, vol. 4, no. 7 (2016): do: /ajme Introducton The ncreasng labor productvty n machnery puts hgher demands on the strength and stffness propertes of the structural members. Snce the strength and rgdty depends on the mechancal propertes of the used materal, the development of new materals s one way of mprovng the requred parameters. The example of these materals are compostes. These exhbt extraordnary materal propertes dependng on the used components. The compostes are often used n modern cars, shps and n others modern equpments. The bucklng of the composte plates s a very complcated subject and more detals can be seen n references [1-7]. In these publcatons, the authors nvestgated pre- and post- bucklng of the composte plates usng varous methods (expermental, numercal and computatonal). For analyzng the bucklng behavor of lamnated composte plates, a recently proposed smoothed fnte element method named the cell-based smoothed dscrete shear gap method (CS-DSG3) s employed n [8]. Researchers n publcaton [9] ntroduced an explct procedure n desgnng lamnated composte plates for maxmum bucklng load subjected to strength and stffness constrants usng the polar formalsm, whch relates stffness to the crtcal bucklng load n order to smplfy the calculaton and optmzaton of the bucklng load. Desgn for maxmum flexural stffness as well as lamnate effectve elastc modulus was developed n order to acheve the optmal orentaton of the layers gvng the maxmum bucklng strength. In the paper [10], the thckness of lamnated composte plates s mnmzed by optmzng the fber orentaton angles for dfferent load cases. The nfluence of materal propertes was nvestgated n papers [11,12]. The strength and stffness of compostes can be evaluated usng expermental, analytcal or computatonal methods. In ths paper the lnear bucklng analyss of the composte plates wth dfferent three, sx and twelve layers and four orentatons of layers s nvestgated by computatonal method. The composte plate s modeled as shell model and then t s loaded by compresson load, shear load and combnaton of these two loads. Then the obtaned crtcal loads are compared for all orentatons of layers and for all numbers of layers. 2. Bucklng Analyss The bucklng analyss of the square composte plate (Fgure 1) wth 3, 6 and 12 ples, and four dfferent layouts of ples s performed. The materal propertes of composte plate are: E x = 50 GPa, E y = 15 GPa, E z = 15 GPa, G xy = 5 GPa, G xz = 5 GPa, G yz = 4.5 GPa, µ xy = 0.3, µ yz = 0.4 and µ xz 0.3. The thckness of one ply s t = 0.5 mm. The orentatons of ples α n composte wth three ples are defned as [0 45 0], [ ], [0 90 0] and [ ], for more ples than three are used symmetrc orentatons. The composte layup for orentaton of ples defned as [0 45 0] s shown n Fgure Boundary Condtons and Appled Load The boundary condtons appled to composte plate are consdered to be zero and they are defned as follows: - on the edges parallel wth x axs: u z, r x, r y ;

2 Amercan Journal of Mechancal Engneerng on the edges parallel wth y axs: u y, u z r x, r y, r z. Then the bucklng analyss for three dfferent load states s performed for: - plate loaded by compresson load p (Fgure 3), - plate loaded by shear load q (Fgure 4), - plate loaded by combnaton of compresson p and shear load q (Fgure 5). The magntude of all loads s p = q = 1 N/mm. Fgure 4. Composte plate loaded by shear load Fgure 1. Model of composte plate Fgure 5. Composte plate loaded by combnaton of compresson and shear load 2.2. Fnte Element Model For a lnear bucklng analyss, the egenvalue problem s solved (below) to get the bucklng load multpler λ and Fgure 2. Composte layup for [0 45 0] Fgure 3. Composte plate loaded by compresson load bucklng modes ψ [ ] λ [ ] { ψ } ( K + S ) = 0, (1) where [K] s the stffness matrx and [S] s the stress stffness matrx. Ths matrx ncludes the effects of the membrane loads on the stffness of the structure. The stress stffenng matrx s assembled based on the results of a prevous lnear statc analyss. The bucklng soluton uses an algorthm that extracts frstly the load multpler λ and secondly the bucklng modes ψ that defne the correspondng mode shape. The bucklng load multpler λ s defned as λ = Bucklng Load / Appled Load. (2) The fnte element model s created usng shell elements wth defned composte layout (Fgure 6). The fnte element mesh s created wth element of sze 10 mm. The fnte element mesh conssts of 4961 nodes and 1600 elements. For the study effect of element sze on the magntude of the crtcal bucklng load, the fnte element model s meshed usng shell elements wth sze 20 mm and 5 mm, respectvely. The results show that the dfferences between crtcal bucklng loads are less than 2 %.

3 415 Amercan Journal of Mechancal Engneerng 2.2. Results and Dscusson We nvestgated bucklng behavor of rectangular composte plate under dfferent load usng the fnte element method. From computaton we get the load multpler λ. Ths parameter has to be premultpled by appled load n order to get bucklng load. Accordngly, the crtcal bucklng load for four orentatons of the layers and for three numbers of the layers are obtaned and are shown n Table 1. Fgure 6. Fnte element mesh The appled boundary condtons on the fnte element model are shown n Fgure 7. Fgure 8. The 1 st bucklng mode of the composte plate wth twelve layers and orentaton of layers [0 45 0] and loaded by a) compresson load, b) shear load, combnaton of compresson and shear load Fgure 7. Appled boundary condtons and load for a) compresson, b) shear, combnaton of compresson and shear Table 1. Values of the crtcal bucklng load Nr. α [ ] Crtcal bucklng load [N/mm] C S C & S 3 [0 45 0] [ ] [0 90 0] [ ] [0 45 0] S [ ] S [0 90 0] S [ ] S [0 45 0] S [ ] S [0 90 0] S [ ] S Nr. number of the composte layers, α orentatons of the composte layers, C compresson load, S shear load. Fgure 9. The 1 st bucklng mode of the composte plate wth twelve layers and orentaton of layers [ ] and loaded by a) compresson load, b) shear load, combnaton of compresson and shear load The resstance to the loss of stablty s obtaned from the computatons for the three dferent number of layers and four orentatons of layers of the rectangular composte plate. The napproprate orentatons of layers for composte plate loaded by comppreson load are: a) [ ] for the composte plate wth three layers (t = 1.5 mm), b) [ ] S for the composte plate wth sx (t = 3 mm) and twelve (t = 4.5 mm) layers. For composte plate loaded by shear load

4 Amercan Journal of Mechancal Engneerng 416 the napproprate orentatons are: a) [ ] for three layers, b) [ ] S for sx and twelve layers. The composte plate loaded by compresson and shear loads have the smallest crtcal bucklng load for the orentaton of layers: a) [0 90 0] for the composte plate wth three of layers, b) [ ] S for the composte plate wth sx and twelve layers. The crtcal bucklng load of the composte plate wth sx layers s almost ten tmes hgher than the crtcal bucklng load of the composte plate wth three layers. For the composte plate wth twelve layers s the crtcal bucklng load aproxmately sxty tmes hgher than the crtcal bucklng load of the composte plate wth three layers. The thckness of composte plate wth three, sx and twelve layers s 1.5mm, 3 mm and 4.5 mm, respetvely. The frst bucklng modes for composte plates wth twelve layers and for the dfferent orentatons of layers are shown n Fgure 8-Fgure 11. The bucklng shapes are slghtly effected by orenatons of layers and number of layers of the composte plate. 3. Concluson The paper studed the bucklng behavor of the rectangular composte plates wth four dfferent layer orentatons and three, sx and twelve layers. The all composte plates were modeled usng fnte element method. The fnte element model of composte plate conssted from shell elements, whch had defned the materal of composte plate, the thckness of composte plate and layout of layers. The composte plates were loaded by three types of loads (compresson load, shear load and combnaton of both). The boundary condtons on the parallel edges wth x and y were appled. The computed crtcal bucklng loads for all confguratons showed that: - crtcal bucklng force ncrease wth ncreasng numbers of layers as well as thckness of the composte plate; two tmes hgher thckness of layers leads to almost ten tmes hgher crtcal bucklng load and three tmes hgher thckness leads to sxty tmes hgher crtcal bucklng force, - the orentatons of composte layers effect the value of crtcal bucklng load, for example the composte plate wth three layers and orentaton of layers [ ] s more senstve to compresson load than shear load, - the bucklng modes are slghtly effected by all orentatons of layers and number of layers, the vsble change s obtaned at frst mode shape from the crtcal bucklng compresson load. Acknowledgements Fgure 10. The 1 st bucklng mode of the composte plate wth twelve layers and orentaton of layers [0 90 0] and loaded by a) compresson load, b) shear load, combnaton of compresson and shear load Ths research was supported by grant from the Slovak Grant Agency VEGA no. 1/0731/16 - Development of Modern Numercal and Expermental Methods of Mechancal System Analyss, VEGA no. 1/0751/16 - Development of a methodology for determnng resdual stresses usng modern expermental methods n mechancs and KEGA no. 054TUKE-4/2014 Usng of modern numercal methods of mechancs as a base of scentfc desgn to the development of knowledge base of students at the second and thrd level of unversty study. References Fgure 11. The 1 st bucklng mode of the composte plate wth twelve layers and orentaton of layers [ ] and loaded by a) compresson load, b) shear load, combnaton of compresson and shear load [1] Laredj, A., Bouazza, M., Amara, K. and Chaban, A., Behavors of smply supported cross-ply rectangular symmetrc lamnates plates wth consderaton of prebucklng n-plane deformaton, Acta Mechanca Slovaca, 19 (1) Mar [2] Lessa, A.W., Bucklng of composte plates, Composte Structures, 1 (1) [3] Dash, P. and Sngh, B.N., Bucklng and post-bucklng of lamnated composte plates, Mechancs Research Communcatons, Dec [4] Jančo, R., Numercal and exact soluton of bucklng load for beam on elastc foundaton, n Sborník vědeckých prací Vysoké školy báňské-techncké unverzty Ostrava, Techncal Unversty West Bohema, [5] Shufrn, I., Rabnovtch, O. and Esenberger, M., Bucklng of lamnated plates wth general boundary condtons under combned compresson, tenson, and shear A sem-analytcal soluton, Thn-Walled Structures, 46 (7-9) Jul-Sep

5 417 Amercan Journal of Mechancal Engneerng [6] Park, C.H., Lee., W., Han, W.S. and Vautrn, A., Improved genetc algorthm for multdscplnary optmzaton of composte lamnates, Computers & Structures, 86 (19-20) Oct [7] Mohammadmehr, M., Salem, M. and Nav, B.R., Bendng, bucklng, and free vbraton analyss of MSGT mcrocomposte Reddy plate renforced by FG-SWCNTs wth temperaturedependent materal propertes under hydro-thermo-mechancal loadngs usng DQM, Composte Structures, Mar [8] Ho-Huu, V., Do-Th, T.D., Dang-Trung, H., Vo-Duy, T. and Nguyen-Tho, T., Optmzaton of lamnated composte plates for maxmzng bucklng load usng mproved dfferental evoluton and smoothed fnte element method, Composte Structures, Jun [9] Kazem, M. and Verchery, G., Desgn of composte lamnated plates for maxmum bucklng load wth stffness and elastc modulus constrants, Composte Structures, Jul [10] Khandan, R., Norooz, S., Sewell, P., Vnney, J. and Koohglan, M., Optmum desgn of fbre orentaton n composte lamnate plates for out-plane stresses, Advances n Materals Scence and Engneerng, [11] Delyová, I., Svák, P., Trebuňa, F. and Hrcová, B., Influence of materal structure and ts propertes on predctng lfe of pressure ppelnes, Appled Mechancs and Materals, [12] Delyová, I., Svák, P., Hroncová, D. and Kováč, J., Stress analyss of the thn-walled vessels, Appled Mechancs and Materals,