Buckling Capacity Optimization of Stiffened Rectangular Plate under Uniform Normal Compression

Transcription

1 JOURNAL OF COMPUTERS, VOL. 9, NO., MARCH 4 8 Buckling Capacity Optimization of Stiffened Rectangular Plate under Uniform Normal Compression Haifeng Fang a,b, Liua Cai a,b a Mecatronics & Automotive Engineering Scool, Jiangsu University of Science & Tecnology, Zangjiagang 6, Jiangsu, Cina b Suzou Institute of Tecnology, Zangjiagang 6, Jiangsu, Cina fangale@6.com Abstract Te stiffened rectangular plate was usually adopted in te blast airtigt doors. In order to improve te buckling capacity of stiffened rectangular plate under uniform normal compression, te optimization model of stiffened rectangular plate was set up based on APDL and ANSYS commands, and te sequential linear programming metod was executed to optimize te tickness of plate and te sizes of stiffeners. Moreover, we compared te mecanical property of te optimized stiffened rectangular plate wit te teoretical value of no-stiffener plate wit equal volume, and obtained te reasonable stiffener distribution based on te optimization results of five different longitudinal and transverse stiffener patterns. Te results sowed tat te buckling capacity of stiffened rectangular plate under uniform normal compression could be improved by approximately % on te condition of reasonable stiffeners distribution. Index Terms buckling; normal compression; stiffened rectangular plate; optimization I. INTRODUCTION Te stiffened rectangular plate was usually adopted in te blast airtigt doors of coal mine refuge cambers. Usually, te stiffened plates ave bigger buckling capacity tan te no-stiffener plates wit equal quality, wic can reac te stress level at 4% to % of te material s yield stress. Meanwile, it as become a prominent problem tat ow to distribute te stiffeners to improve te buckling capacity of plate structures under normal compression according to te plate structures and load caracteristics. Te teoretical and experimental researc on buckling of tin plates and sells was studied relatively early. And te rectangular plate limit condition only suffering te film stress as been applied to solve te ultimate load problems []. Te overall deformation process of elastic tin rectangular plate under symmetrical normal compression as been studied. And te buckling stress Tis paper is supported by Yout Science Fund of Jiangsu University of Science &Tecnology and Yout Science Fund of Zangjiagang Campus. Corresponding autor is Haifeng Fang(fangale@6.com) empirical formula for rectangular plate suffering normal compression at te axes as been suggested [-]. Recently, Some researcers ave studied te plastic buckling and post-buckling of tin plate and sell under normal compression. Moreover, te elastoplastic numerical solution of tin plate s buckling and postbuckling as been obtained [6]. Furtermore, based on te researc of te buckling deformation pattern of tis structure on combined load, te most important influence factor in stiffened plate and plate as been obtained [7,8]. Recently, lots of researcers ave adopted different metods in te plate structure optimization, suc as energy principle metod, flexible tolerance polyedron metod, and full-stress standard metod [9-4]. In view of te above, we ave a lot of work to do to improve te buckling capacity of stiffened rectangular plate in te field of plate structure optimization [,6]. Te autor as studied te buckling capacity optimization of stiffened cylindrical sell under uniform axial compression [7]. In tis paper, we set up te optimization model of stiffened rectangular plate based on APDL and ANSYS commands, and executed te serial linear programming optimization procedure to optimize te plate structure. II. MATHEMATICAL MODEL Wit te researc on te structure optimization deepening, structure optimization softwares ave developed greatly. Moreover, te ANSYS software as been te most outstanding finite element analysis software of structure optimization [8], te wole parameterized options of wic can be cosen as te optimize parameters. And APDL (ANSYS Parametric Design Language) is an indispensable important tecnology of ANSYS software, wic can realize te parameterized finite element analysis, batced analysis, secondary development and optimization design. Te process of optimization design based on APDL is as follows. And te parameter analysis file is built for optimal circulation based on APDL and ANSYS commands, ten te analysis processing is executed in OPT and te serials of optimal design is cecked up. doi:.44/jcp

2 8 JOURNAL OF COMPUTERS, VOL. 9, NO., MARCH 4 value of te buckling load; C -C are te undetermined coefficients wic can be obtained by fitting te software analysis results; V is te volume of rectangular plate before optimization; K is te quantity of transverse stiffeners; J is te quantity of longitudinal stiffeners. Fig. Stiffened plate under uniform normal compression In tis paper, finite element analysis software ANSYS was employed and secondarily te optimization model was set up wit APDL. Fig. sown tat te rectangular plate wit J longitudinal stiffeners and K transverse stiffeners under uniform normal compression q(t), were a is te lengt of plate, b is te widt of plate, is te tickness of plate, is te widt of stiffeners and is te eigt of stiffeners. In addition, te low alloy steel (Q4R) was adopted in te rectangular plate structure model, wic could be assumed as perfect elastic material. In te optimization analysis, it was assumed tat te rectangular plate was supported simply on four sides. And te lengt a was cosen as mm and te widt was cosen as 6 mm. According to engineering experiment, te tickness was cosen as mm. Te sections of longitudinal stiffeners and transverse stiffeners were bot rectangle and te same to eac oter, wose widt and eigt were mm and mm respectively. Taking te tickness of rectangular plate, te widt of stiffeners and te eigt of stiffeners as optimization design variables, te optimization model was built as following forms: max qcr = q + C, ) + C, ) + C, ) s.t. abx + Jbx x + Kax x V were x, -x, are te initial value of,,, respectively; is optimization object; q is te initial () III. OPTIMIZATION RESULT ANALYSIS Tere are two instability patterns of tin stiffened rectangular plate under normal compression, wic are global instability and local instability surrounded by stiffeners. For te rectangular plate wit overcrowded stiffeners, because te area surrounded by stiffeners is small, te global instability takes place easily rater tan te local instability firstly. For te rectangular plate wit sparse stiffeners, due to te bigger area surrounded by stiffeners, te local instability of tin plate appens more easily tan te global instability. In tis situation, te reinforce effect of stiffeners is not obvious [7]. In tis paper, te optimization limits of,, were.mm mm,.mm t 6mm,.mm 6mm, respectively; Te initial optimization value of,, was mm, mm, mm, respectively. A. Longitudinal Stiffeners, Transverse Stiffeners Te longitudinal stiffeners and transverse stiffeners were arranged on te back of rectangular plate. Te geometrical properties before and after optimization were sown in Table I and te variation processes of variables and objective were sown in Fig.. Based on te optimization results analysis, te buckling capacity of stiffened rectangular plate under uniform normal compression was improved rarely. Te variables, converged to te lower limits of te dimensional constraints, wile te variable increased correspondingly. Te local instability took place in te tin plate firstly due to te too sparse stiffeners. Furtermore, te result was to make te structure as a no-stiffener plate. TABLE I. COMPARISON OF MECHANICS PROPERTY OF RECTANGULAR PLATE WITH LONGITUDINAL STIFFENERS AND TRANSVERSE STIFFENERS BEFORE AND Tickness of plate Widt of stiffener Heigt of stiffener / MPa / mm / mm / mm Before optimization After optimization Teoretical value* *Te teoretical value of no-stiffener plate in te same volume wic can be looked up in te tables of te Reference. Te same as in te following tables Fig. Variation process of variables and objective of rectangular plate wit longitudinal stiffeners and transverse stiffeners

3 JOURNAL OF COMPUTERS, VOL. 9, NO., MARCH 4 8 B. Longitudinal Stiffeners, Transverse Stiffeners Te longitudinal stiffeners and transverse stiffeners were arranged uniformly on te back of rectangular plate, and te variation processes of variables and objective were presented in Fig., and te geometrical properties before and after optimization were sown in Table II. Troug analysis of te optimization results, te buckling normal compression was improved evidently. Te variables,, converged to certain value witin te dimensional constraints, and te density of longitudinal stiffeners was bigger tan tat of transverse stiffeners. And te ratio of te eigt to te widt of te stiffener was about.. Meanwile, te global instability took place under te normal compression and te reinforce effect of stiffeners was obvious. TABLE II. COMPARISON OF MECHANICS PROPERTY OF RECTANGULAR PLATE OF LONGITUDINAL STIFFENERS AND TRANSVERSE STIFFENERS BEFORE AND Tickness of plate Heigt of stiffener Widt of stiffener / mm / MPa / mm / mm Before optimization After optimization Teoretical value Fig. Variation process of variables and objective of rectangular plate wit longitudinal stiffeners and transverse stiffeners C. 7 Longitudinal Stiffeners, Transverse Stiffeners Te 7 longitudinal stiffeners and transverse stiffeners were arranged uniformly on te back of rectangular plate, and te variation processes of variables and objective were presented in Fig.4. Troug analysis of te optimization results sown in Table Ⅲ, te buckling normal compression was improved greatly. Te variables,, converged to certain value witin te dimensional constraints, wile te variable got muc smaller. And te densities of longitudinal stiffeners and transverse stiffeners were similar. In te meantime, te stiffener frame was nearly square, wic made te vertical and orizontal non-deformability corresponding and was supportive to improve buckling capacity under normal compression. And te ratio of te eigt to te widt of te stiffener was about..terefore, te reinforce effect was remarkable. TABLE III. COMPARISON OF MECHANICS PROPERTY OF RECTANGULAR PLATE OF 7 LONGITUDINAL STIFFENERS AND TRANSVERSE STIFFENERS BEFORE AND Tickness of plate Heigt of stiffener Widt of stiffener t / mm / mm / mm / MPa Before optimization After optimization Teoretical value Fig.4 Variation process of variables and objective of rectangular plate wit 7 longitudinal stiffeners and transverse stiffeners

4 84 JOURNAL OF COMPUTERS, VOL. 9, NO., MARCH 4 D. 7 Longitudinal Stiffeners, Transverse Stiffeners Te 7 longitudinal stiffeners and transverse stiffeners were arranged uniformly on te back of rectangular plate, and te variation processes of variables and objective were presented in Fig.. Troug analysis of te optimization results sown in Table Ⅳ, te buckling normal compression was also improved obviously. Te variables,, converged to certain value witin te dimensional constraints, wile also got muc smaller. And te densities of longitudinal stiffeners and transverse stiffeners were similar and suitable. Hence, te reinforce effect of stiffeners was remarkable. TABLE IV. COMPARISON OF MECHANICS PROPERTY OF RECTANGULAR PLATE OF 7 LONGITUDINAL STIFFENERS AND TRANSVERSE STIFFENERS BEFORE AND Tickness of plate Heigt of stiffener Widt of stiffener t / mm / mm / mm / MPa Before optimization After optimization Teoretical value Fig. Variation process of variables and objective of rectangular plate wit 7 longitudinal stiffeners and transverse stiffeners E. Longitudinal Stiffeners, Transverse Stiffeners Te longitudinal stiffeners and transverse stiffeners were arranged uniformly on te back of rectangular plate, and te variation processes of variables and objective were presented in Fig.6. Troug analysis of te optimization results sown in Table Ⅴ, te buckling capacity was improved a little. Te variables, converged to te lower limits of te dimensional constraints, wile increased correspondingly. And te result was to make te volume of stiffeners decrease but te tickness of te plate increase, leading te structure to be a rectangular plate witout stiffeners. Consequently, for te too overcrowded stiffeners, te reinforce effect of stiffeners was not remarkable. TABLE V. COMPARISON OF MECHANICS PROPERTY OF RECTANGULAR PLATE OF LONGITUDINAL STIFFENERS AND TRANSVERSE STIFFENERS BEFORE AND Tickness of plate Heigt of stiffener Widt of stiffener / mm / MPa / mm / mm Before optimization After optimization Teoretical value Fig.6 Variation process of variables and objective of rectangular plate wi longitudinal stiffeners and transverse stiffeners IV. CONCLUSION Based on APDL and ANSYS commands, we set up te optimization model of stiffened rectangular plate, and executed te serial linear programming optimization procedure, obtained te following conclusions. Weter te stiffeners are too sparse or too overcrowded, te widt and te eigt of stiffeners converge to te lower limits of te dimensional constraints, leading te structure to be a rectangular nostiffener plate. Only wen te plate as an appropriate density, te normal buckling capacity of stiffened

5 JOURNAL OF COMPUTERS, VOL. 9, NO., MARCH 4 8 rectangular plate will be significantly improved after optimization. Wile te stiffener frame is nearly square, it will make te vertical and orizontal non-deformability corresponding, wic will be supportive to improve buckling capacity under normal compression. As a result, for te rectangular plate wit determined structure sizes, te density of stiffeners sould be regulated in a reasonable scope. Moreover, canging te dimensional constraints of stiffeners as no influence on optimization results. And te ratio of te eigt to te widt of te stiffener sould be about., wic could improve te normal buckling capacity of stiffened rectangular plate. ACKNOWLEDGEMENTS Te financial supports from Yout Science Fund of Jiangsu University of Science & Tecnology and Yout Science Fund of Zangjiagang Campus are greatly appreciated. REFERENCES [] HUANG Ke-zi, Limit equilibrium of tin cylindrical sell wit stiffeners, Cinese Journal of Teoretical and Applied Mecanics, 7, pp. 9-8, (964). [] C. R. Calladine, Toery of Sell Structures, Cambridge: Cambridge University Press, pp. 47-4(98). [] ZHU En-cun, C. R. Calladine, Buckling of tin cylindrical sells under locally normal compression, Engineering Mecanics,, pp. 68-7, (). [4] ZHU En-cun, C. R. Calladine, Buckling of tin cylindrical sells under uniform normal compression, Journal of Harbin University of Civil Engineering and Arcitecture,, pp. -, (). [] ZHU En-cun, P. Mandal, C. R. Calladine, Analysis of buckling of tin cylindrical sells under normal compression, Cina Civil Engineering Journal, 4, pp. 8-, (). [6] L. G. Pilippe, L. V. An, Elastoplastic bifurcation and collapse of normally loaded cylindrical sells, International Journal of Solids and Structures, 4, pp , (8). [7] N. E. Sanmugam, M. Arockiasamy, Local buckling of stiffened sells in offsore structures, Construct Steel Res, 8, pp. 4-9, (996). [8] I. A. Seik, A. E. Elwi, G. Y. Grondin, Stiffened steel sells under combined compression and bending, Journal of Constrctional Steel Researc, 9, pp. 9-9, () [9] SUI Yun-kang, ZHANG Wei, DU Jia-zeng, Sectional area optimization subjected to strengt constraints for structures combined by multiple elements, Journal of Beijing University of Tecnology,, pp. -8, (7). [] WANG De-yu, LI Dong-seng, Optimum design of reinforced cylindrical sell on buckling, Cina Offsore Platform, 4, pp. -7, (999). [] LIANG Bin, YUE Jin-cao, Optimum design of cylindrical sell on stability, Journal of Mecanical Strengt, 4, pp , (). [] M. M. Alinia, A study into optimization of stiffeners in sells subjected to sear loading, Tin-Walled Structures, 4, pp , (). [] Li Xue-jun, Jiang Sou-bo, Zeng Qing-liang, Optimization of Two-Stage Cylindrical Gear Reducer wit Adaptive Boundary Constraints, Journal of Software, 8, pp.-7,(). [4] Yang Yong, Zang Wei-min, Xie Hong-lei, A Selfadaptive Optimization Removal Algoritm for Continuum Evolutionary Structural Topology, Journal of Software, 7, pp.9-,(). [] V. L. Krasovsky, V. V. Kostyrko, Experimental studying of buckling of stringer cylindrical sells under normal compression, Tin-Walled Structures, 4, pp , (7) [6] N. D. Lagaros, F. Micalis, P. Manolis, Optimum design of sell structures wit stiffened beams, AIAA Journal, 4, pp. 7-84, (4). [7] FANG Hai-feng, GE Si-rong, CAI Li-ua, et al. Buckling capacity optimization of coal mine refuge camber s sell under uniform axial compression, ICMTMA., 6, pp , (). [8] Li Xue-jun, Wang Ke, Jiang Ling-li, Rotor Crack Detection by Using Multi-vibration Signal from Te Basement,Journal of Software, 7, pp.99-96,(). Haifeng Fang was born in 984, e received te P.D. degree from te Scool of Mecanical Engineering at Cina University of Mining and Tecnology, Xuzou, Cina, in. He as been a faculty member of Jiangsu University of Science & Tecnology.He is actively engaged in te area of tecnology and equipment of coal mine safety. Liua Cai was born in 984. Se received te P.D. degree from Cina University of Mining and Tecnology at Xuzou city in Jiangsu Province Cina. Her main researc interests include automated reasoning and intelligent planning. Se as been a faculty member of Jiangsu University of Science & Tecnology