Numerical And Experimental Investigation Of Loading Band On Buckling Of .
Research Journal of Recent Sciences ISSN 2277-2502Vol. 1(10), 63-71, October (2012)Res.J.Recent Sci.Numerical and Experimental Investigation of Loading Band on Bucklingof Perforated Rectangular Steel PlatesMahmoud Shariati1 and Ali Dadrasi2*1Fatigue and Fracture Research Laboratory, School of Mechanical Engineering, Shahrood University of Technology, Shahrood, IRAN2Department of Mechanics, Shahrood Branch, Islamic Azad University, Shahrood, IRANAvailable online at: www.isca.inReceived 21st August 2012, revised 27th August 2012, accepted 29th August 2012AbstractThe aim of this paper is to investigate the buckling behavior of the steel rectangular plates with circular and square cut outsunder uniaxial in-plane compressive loading in elasto-plastic range with various loading bands using the numerical and theexperimental methods. Some effective parameters on the buckling of plates have been studied separately and the requireddata for analysis have been gained through the experimental tests. The finite element Abaqus software has been used for thenumerical analysis and a set of servo hydraulic INSTRON8802 was applied in the experimental tests. Numerical andexperimental results show good agreement with each other.Keywords: Buckling, steel plates, cut out, finite element, loading band.IntroductionThin-walled members are the elements of many engineeringstructures. They become unstable and start to buckle if theysubjected under a compressive loads greater than their ultimatebuckling load. Moreover, some of these members usually havecut outs due to their applications and these discontinuities canaffect on their stability.The stability analysis of thin-walled structures under axialcompression has been investigated by some researchers1-3.Obviously, the stability of these structures is dependent on thetype of support and loading. The buckling and the geometricallynonlinear elasto-plastic collapse of perforated plates wereinvestigated using finite element solutions4. Elasto-plasticpostbuckling of damaged orthotropic plates based on the elastoplastic mechanics and continuum damage theory have beenstudied5. El-Sawy et al6 employed the FEM to determine theelasto-plastic buckling stress of uniaxially loaded square andrectangular plates with circular cutouts. Plates with simplysupported edges in the out-of-plane direction and subjected touniaxial end compression in their longitudinal direction wereconsidered.The nonlinear mathematical theory for initial and post localbuckling analysis of plates of abruptly varying stiffness havebeen established by Azhari et al7 using the principle of virtualwork. In another numerical investigation, the foundations of thedesign of perforated trapezoidal sheeting on effective stiffnessvalues for perforated sheeting with different arrays of holes areprovided by Kathage et al.8 Liu and Povlovic9 revisited thestability of simply supported rectangular plates under patchcompression using Ritz's energy method. Both single anddouble Fourier series are adopted as deflection series tocompute the values for buckling coefficients. An experimentalInternational Science Congress Associationand numerical study on buckling of thin-walled cylindricalshells under oblique Loading were done by Shariati et al.10 theyInvestigations on buckling and postbuckling behavior ofstainless steel 316ti cylindrical shells with cutout. Also inanother study Shariati et al11 investigate the buckling of tubularsteel shells with circular cutout subjected to combined loading.In their study the influence of shell length, shell diameter, shellangle and diameter of circular cutouts on the predicted bucklingvalues has been explored.The elastic buckling behaviors of rectangular perforated plateswere studied using the finite element method by Komur andSonmez12. To evaluate the effect of cutout location on thebuckling behavior of plates, they chose circular cutout atdifferent locations along the principal x-axis of plates subjectedto linearly varying loadings. Their results shown that the centerof a circular hole should not be placed at the end half of theouter panel for all loading patterns. A new approximateprocedure for buckling analysis of simply supported rectangularstepped or perforated plates subjected to uniform edge stresseswas formulated by Rahai et al13. The procedure uses energymethod based on modified buckling mode shapes. Eccher et al14have been presented the application of the isoparametric splinefinite strip method to the geometric nonlinear analysis ofperforated folded-plate structures. Paik et al15 has been studiedthe ultimate strength of perforated steel plates under axialcompressive loading along short edges using FEM. The platesare considered to be simply supported along all (four) edges,keeping them straight. The cutout was circular and located at thecenter of the plate.In this paper, the numerical and the experimental investigationon the buckling behavior of the rectangular plates with circularand square cut outs under uniaxial in-plane compressive loadingin elasto-plastic range with various loading bands are63
Research Journal of Recent SciencesISSN 2277-2502Vol. 1(10), 63-71, October (2012)Res. J. Recent Sci.performed. Moreover, the relation between stability of therectangular plates havingving square and circular cut out with thesame cross section has been studied. Several buckling tests wereperformed using an INSTRON 8802 servo hydraulic machine,and the results were compared with the results of the finiteelement method. A very good correlationrelation between experimentsand numerical simulations was observed. Finally, based on theexperimental and numerical results, formulas are presented forthe computation of the buckling load in such plates.Material and MethodsThe geometry and type of loadingading are shown in figure 1 in thisfigure, l is the loading band which varies in the range of(0 l a). The position l 0 is relates to the concentrated loadexerted on the middle of the width of the plate, a, and theposition l a represents to the distributed load exerted on theentire the width of the plate.In this investigation, the structural steel rectangular plates with100 x 150 x 2.07 mm dimensions are used. These plates havesquare or circular cut outs. The side of the square cut out isconsidered to be e 30 mm and for having the same area of twotypes of cut out, the diameter of the circle is considered to beD 33.84mm. The lower edge of the plates has been placed inthe support (both simply and clamped support) and oppoppositeedge has been exerted through a simply support with variousbands. The tests have been conducted for the width of loading ofl 15,30,50,75 and 100 mm. The right and left sides of platesaren't constrained. In these tests, the post buckling behavior ofthe plates has been fully studied, too.The mechanical properties of the tested structural steel plateshave been specified through the tensile test in accordance withthe ASTM-E8E8 standard using an INSTRON8802 servohydraulic machine. Based on the linearar portion of true stressstressstrain curve resulted of this test, the value of elasticity moduleand yield stress was obtained E 218Gpa and σy 349Mpa,respectively. Moreover the Poisson's ratio value is considered tobe ν 0.33.The data of the plastic region of the stress-strainstrain curve has beenused for analysis of the plastic behavior in ABAQUS software.Numerical analysis: In Abaqus software, after defining thegeometry, boundary conditions and the applied loading, wemust mesh the perforated plate to analyze. This is achievedusing the S8R5 quadrilateral non-linear elements. The S8R5element is very suitable for the element arrangement of the thinplates and shells3.After meshing specimens, a linear buckling analysis for gettinggetteigenvalues is performed in ABAQUS and the buckling modeshapes are obtained. Since in eigenvalue linear analysis, theplastic properties of the specimen are not taken into account,overestimates the real value for buckling load.Since buckling usuallyy occurs in smaller mode shapes, a linearanalysis should be performed first for all specimens, to find themode shapes with smaller eigenvalues. In this step, threeprimary mode shapes were obtained. We used all three primarymode shapes in analysis, becauseuse these mode shapes have moreeffects on buckling behavior.Effect of loading band on buckling behavior: In this part theeffect of the loading band on buckling behavior of perforatedplates has been numerically studied. Loading band is variedbetween 15 to 100 mm. The behavior of buckling load versusend shortening of plates (load-displacementdisplacement curves) undervarious load bands for plates with no cut out and plates withsquare and circular cut outs with the simply and clampedsupports have been displayed in figureure 2 through 4, respectively.Note that all diagrams were presented in dimensionless usingFref and height of plates, 150mm, for normalizing buckling loadsand end shortening, respectively. Fref is defined as follows:Fref at σ yWhere Fref is the reference load, the load required for theyielding of plates, a is the width of plates, t is the thickness,and σy is the yield stress. Therefore, the reference load of thespecimens is calculated in this wayFref 100 mm 2.07 mm 349 10 6 N/mm 2 72243 NLoad-displacementdisplacement curves for plates with circular and squarecutouts and perfect plate (with no cut out) for loading bandl 100 100 mm with simply and clamp support are compared infigure 5 (a) and (b).Loading band(a)Figure-1(b)The geometry of the plate and the type of the applied loading (a) The plate with circular cut out (b) The plate with square cut outoInternational Science Congress Association64
Research Journal of Recent Sciences ISSN 2277-2502Vol. 1(10), 63-71, October (2012)Res. J. Recent Sci.0.12Load Band 100mmLoad Band 75mm0.1Load Band 50mmBucklingload/FrefLoad Band 30mmLoad Band nd shortening/LFigure-2Load-displacement behavior of plate with no cutout and simply support at lower edge0.09Load Band 100mm0.08Load Band 75mmLoad Band 50mm0.07Load Band 30mmBucklingload/Fref0.06Load Band .03End shortening/LFigure-3The behavior of the load-displacement of the plate with square cut outs and lower edge with simply support0.2Load Band 100 mm0.18Load Band 75 mmLoad Band 50 mm0.16Load Band 30 mmBucklingload/Fref0.14Load Band 15 0.03Figure-4The behavior of load-displacement of the plate with circular cut out and lower edge with the clamped supportInternational Science Congress Association65
Research Journal of Recent Sciences ISSN 2277-2502Vol. 1(10), 63-71, October (2012)Res. J. Recent Sci.0.12PerfectCircle holeBuckling load/Fref0.1Square .0350.04End shortening/L(a)0.25PerfectCircle hole0.2Buckling load/FrefSquare hole0.150.10.05000.0050.010.0150.020.0250.03End shortening/L(b)Figure-5Comparison of load-displacement curves for plates with circular and square cutouts and perfect plate for loading bandl 100 mm (a)- Simply support. (b)- Clamped supportInternational Science Congress Association66
Research Journal of Recent Sciences ISSN 2277-2502Vol. 1(10), 63-71, October (2012)Res. J. Recent Sci.Also the critical buckling loads of plates with and without cutout for different loading bands are given in tables (1) and (2)Table-1Numerical results of buckling load for plates with no cut out(perfect) and plates with circular and square cutouts undersimply support at lower gbucklingbandload withload withload with no(mm)circularsquarecutout (kN)cutout (kN)cutout erical results of buckling load for plates with no cutout(perfect) and plates with circular and square cutouts underclamped support at lower gbucklingload withload withbandload with nocircular(mm)squarecutout (kN)cutout (kN)cutout 64100As shown in the curves, after the load of the plate reaches to thecritical value, the buckling phenomenon is occurred and theplate is bent by a low force. Moreover it is observed that, as theload band increases, the ultimate value of the buckling load isalso increased.The changes of the buckling load in proportion to the loadingband for two specimens with square and circular cut out havebeen numerically compared with two types of simply andclamped boundary conditions. These changes are also displayedin figure 6 The phrase "circle-simply support" indicates thespecimen with the circular cut out which its lower edge has beenplaced inside the simply support and the phrase "square–clampsupport" indicates the specimen with square cut out which loweredge has been placed inside the clamped support.As indicated in figure 6, for one specific loading band and underthe same boundary conditions, the buckling load of thespecimen with circular cut out is a little higher than the bucklingload of the specimen with square cut out. Owing to the lessdifficulties of manufacturing and creation of the circular cut outthan the square cut- out and due to the higher buckling load ofthe specimen with circular cut out than the specimen withInternational Science Congress Associationsquare cut out, it's recommended that use circular cut out forspecimens which don't have any limitations on type of thegeometry of cut out.Furthermore, in this diagram, it is also observed that, thebuckling load of each specimen with the clamped boundaryconditions is so higher than the buckling load with simplyboundary conditions. It’s about twice bigger for each specimen,for this specific case study. Different experimental tests wereconducted to confirm of the authenticity of the results obtainedfrom the numerical methodExperimental results: The lower edge of the plates was placedin the simply or clamped support and their upper edge wasplaced in a simply support with various loading bands (figure7). Also experiments are carried out under displacement controlwith speed of 0.01 mm/s.Some results of experimental tests which were conducted on therectangular plates with square and circular cut outs have beendisplayed in figure 8-9.Results and DiscussionAs shown in figure 8 and 9, the trend of the experimentaldiagrams is wholly similar to the trend of the numericaldiagrams and upon the increase loading band; the buckling loadis also increased. For better understanding of the results, thenumerical and experimental quantities of the buckling load arecompared in proportion to the boundary conditions and thevarious loading bands for the specimens having square andcircular cut out in tables 3 through 6.Table-3The numerical and experimental critical buckling loads forplate with square cut out and the lower edge, simply support(average error 8.05%)Numerical ExperimentalLoadingNumericalbucklingbuckling loadband(mm)error .4865.9258.65756.5905.9919.08Table-4The numerical and experimental critical buckling loads forplate with circular cut out and the lower edge, simplysupport (average error 8.92%)Numerical ExperimentalLoadingNumericalbucklingbuckling loadband(mm)error .5965.9939.14756.7026.0679.4767
Research Journal of Recent Sciences ISSN 2277-2502Vol. 1(10), 63-71, October (2012)Res. J. Recent Sci.Table-5The numerical and experimental critical buckling loads forplate with square cut out and the lower edge, clampedsupport (average error 6%)Numerical ExperimentalLoadingNumericalbucklingbuckling loadband(mm)error 075012.48611.7965.527512.91812.8210.75Table-6The numerical and experimental critical buckling loads forplate with circular cut out and the lower edge, clampedsupport (average error for plates with cut out g loadbucklingerror 8812.424306.0411.92012.687505.0413.04813.74175In figure 10, the ultimate value of the buckling load calculatedfrom the numerical and experimental analyses for some variousloading bands and the end of the clamped support have beencompared with each other for circular and square cut outs.Based on the experimental buckling loads of plates, formulasare presented here using Lagrangian polynomial for thecomputation of the buckling load of plates with circular andsquare cut outs subject to axial compression. To get theseformulas with using Lagrangian polynomial method19, thecurves were passed through the buckling load values. In theseformulas F and l are buckling load for plates with cutout andloading band, respectively. The general form of F is as follows:F A Bl Cl 2 Dl 3 .(1)The coefficients A, B, C, D, are computed using Lagrangianpolynomial. The formulas for computation of the buckling loadof plates with circular and square cutouts are represented inequations 2 and 3, respectively. For plates with circular cut outs:F 3E ( 0.05)l 3 0.004l 2 0.179l 8.82(2)And for plates with square cut outs:F 2 E ( 0.05)l 3 0.003l 2 0.163l 8.793(3)1412Circle-Simply SupportSquare-Simply SupportBuckling Load(kN)10Circle-Clamp SupportSquare-Clamp Support8642001020304050607080Loading Band(mm)Figure-6Changes of buckling load with loading band for various cut outs and supportsInternational Science Congress Association68
Research Journal of Recent Sciences ISSN 2277-2502Vol. 1(10), 63-71, October (2012)Res. J. Recent Sci.Figure-7The setup of experimental tests (in this case, both of the supports are simply)0.1Band Load 75mmBuckling load/Fref0.09Band Load 50mm0.08Band Load 30mm0.07Band Load 0250.03End shortening/LFigure-8The behavior of the load-displacement of the plate with circular cut outs and lower edge with simply support fromexperimental testsInternational Science Congress Association69
Research Journal of Recent Sciences ISSN 2277-2502Vol. 1(10), 63-71, October (2012)Res. J. Recent Sci.0.2Load Band 75mmBucklingload/Fref0.18Load Band 50mm0.16Load Band 30mm0.14Load Band 250.03End shortening/LFigure-9The behavior of load-displacement of the plate with square cut out and lower edge with the clamped support fromexperimental ental-Sq9.5901020304050607080Loading Band(mm)Figure-10Effect of loading band on buckling load in numerical and experimental analyses of plates with circular and square cut outsand clamped supportConclusionWe can conclude from the results obtained in this research that:when the buckling phenomenon occurs, the capacity of the loadtoleration is considerably decreased. The results show that, asloading band increases, the ultimate buckling load alsoincreases. The buckling load of the specimen with circle cut outis a little more than the specimens with the square cut out withInternational Science Congress Associationthe equal surface area, therefore it’s suggested that if there is nolimitation in manufacturing process, the specimens with circularcut out are appropriate, because the Production of circular cutouts are very easier than square cut outs. The buckling load ofthe plates with the clamped support is about twice bigger thanplates with simply support. Numerical and experimental resultsare in good agreement with each others.70
Research Journal of Recent Sciences ISSN 2277-2502Vol. 1(10), 63-71, October (2012)Res. J. Recent Sci.References11. Shariati M., Fereidoon A and Akbarpour A., Buckling ofSteel Cylindrical Shells with an Elliptical Cutout,International Journal of Steel Structures, 10(2), 193-205(2010)1.Timoshenko S.P. and Gere J.M., Theory of ElasticStability, 2nd ed., McGraw-Hill Book Company, New York(1961)2.Mignot F. and Puel J.P., Homogenization and Bifurcationof Perforated Plates, Engineering science, 18, 409-414(1980)12. Komur M., and Sonmez M., Elastic buckling of rectangularplates under linearly varying in-plane normal load with acircular cutout, Mechanics Research Communications,35(6), 361-371 (2008)3.Shariati M. and Mahdizadeh Rokhi M., Buckling of SteelCylindrical Shells with an Elliptical Cutout, InternationalJournal of Steel Structures, 10(2), 193-205 (2010)13. Rahai A.R., Alinia M.M. and Kazemi S., Buckling analysisof stepped plates using modified buckling mode shapes,Thin-Walled Structures, 46, 484–493 (2008)4.Roberts T.M. and Azizian Z.G., Strength of PerforatedPlates Subjected to In-Plane Loading, Thin-WalledStructures, 2, 153-164 (1984)14. Eccher G., Rasmussen K.J.R. and Zandonini R., Geometricnonlinear isoparametric spline finite strip analysis ofperforated, Thin-walled structures, 47(2), 219-232 (2009)5.Tian Y. and FU Y., Elasto-plastic postbuckling of damagedorthotropic plates, Applied Mathematics and Mechanics,29(7), 841-853 (2008)15. Paik Jeom Kee, Ultimate strength of perforated steel platesunder combined biaxial compression and edge shear loads,Thin-Walled Structures, 46, 207–213, (2008)6.El-Sawy, Khaled M., Nazmy Aly S., Martini and Ikbal M.,Elasto-plastic buckling of perforated plates under uniaxialcompression, Thin-Walled Structures, 42, 1083-1101(2004)16. Dadrasi A., An Investigation on Crashworthiness Design ofAluminium Columns with Damage Criteria, Res. J. RecentSci., 1(7), 19-24 (2012)7.Azhari M., Shahidi A.R. and Saadatpour M.M., Local andpost local buckling of stepped and perforated thin plates,Applied Mathematical Modeling, 29, 633–652 (2005)17. Murthy B.R.N., Lewlyn L.R. Rodrigues and AnjaiahDevineni, Process Parameters Optimization in GFRPDrilling through Integration of Taguchi and ResponseSurface Methodology, Res. J. Recent Sci., 1(6), 7-15 (2012)8.Kathagea K., Misiekb Th and Saal H., Stiffness and criticalbuckling load of perforated sheeting, Thin-WalledStructures, 44(12), 1223-1230 (2006)18. Purkar T. Sanjay and Pathak S., Aspect of finite ElementAnalysis Methods for Prediction of Fatigue Crack GrowthRate, Res. J. Recent Sci., 1(2), 85-91 (2012)9.Liu Y.G. and Pavlovic M.N., Elastic Stability of flatrectangular plates under patch compression, InternationalJournal of Mechanical Sciences, 49, 970-982 (2007)19. Gerald C.F. and Wheatley P.O., Applied numericalanalysis, Addison- Wesley, New York, (1999)10. Shariati M., Fereidoon A. and Akbarpour A., BucklingLoad Analysis of oblique Loaded Stainless Steel 316tiCylindrical Shells with Elliptical Cutout, Res. J. RecentSci., 1(2), 85-91 (2012)International Science Congress Association71
Some effective parameters on the buckling of plates have been studied separately and the required data for analysis have been gained through the experimental tests. The finite element Abaqus software has been used for the numerical analysis and a set of servo hydraulic INSTRON8802 was applied in the experimental tests. Numerical and
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