Influence Of Purlins On Lateral-Torsional Buckling Of .
Influence of Purlins on Lateral-TorsionalBuckling of Steel Girders with Corrugated WebMaster of Science Thesis in the Master’s Programme Structural Engineering andBuilding TechnologyGUÐNI ELLERT EDVARDSSONBENGT LUNDQUISTDepartment of Civil and Environmental EngineeringDivision of Structural EngineeringSteel and Timber StructuresCHALMERS UNIVERSITY OF TECHNOLOGYGöteborg, Sweden 2014Master’s Thesis 2014:100
MASTER’S THESIS 2014:100Influence of Purlins on Lateral-Torsional Buckling ofSteel Girders with Corrugated WebMaster of Science Thesis in the Master’s Programme Structural Engineering andBuilding TechnologyGUÐNI ELLERT EDVARDSSONBENGT LUNDQUISTDepartment of Civil and Environmental EngineeringDivision of Structural EngineeringSteel and Timber StructuresCHALMERS UNIVERSITY OF TECHNOLOGYGöteborg, Sweden 2014
Influence of Purlins on Lateral-Torsional Buckling of Steel Girders with CorrugatedWebMaster of Science Thesis in the Master’s Programme Structural Engineering andBuilding TechnologyGUÐNI ELLERT EDVARDSSONBENGT LUNDQUIST GUÐNI ELLERT EDVARDSSON, BENGT LUNDQUIST, 2014Examensarbete / Institutionen för bygg- och miljöteknik,Chalmers tekniska högskola 2014:100Department of Civil and Environmental EngineeringDivision of Structural EngineeringSteel and Timber StructuresChalmers University of TechnologySE-412 96 GöteborgSwedenTelephone: 46 (0)31-772 1000Cover:Finite element model showing lateral-torsional buckling of a girder with trapezoidallycorrugated web with five restraints along its length.Chalmers Reproservice / Department of Civil and Environmental EngineeringGöteborg, Sweden 2014
Influence of Purlins on Lateral-Torsional Buckling of Steel Girders with CorrugatedWebMaster of Science Thesis in the Master’s Programme Structural Engineering andBuilding TechnologyGUÐNI ELLERT EDVARDSSONBENGT LUNDQUISTDepartment of Civil and Environmental EngineeringDivision of Structural EngineeringSteel and Timber StructuresChalmers University of TechnologyABSTRACTLateral-torsional buckling is a well-known stability problem in slender unrestrainedsteel girders with doubly symmetric I-section. A rather recent method to increase thestability of girders is to have a corrugated shape of the web instead of a flat web.Purlins connected to girders improve the stability due to their restraining effects. Thepurlins will restrain lateral movements of the girder at their connection point. Inaddition, rotation of the cross-section can be restrained by attaching the purlins rigidlyto the girder. In that way, the rotation is controlled by the stiffness of the purlins. Thisincrement in stability is of great interest from the designers’ point of view due toeconomic reasons.The aim of the thesis is to investigate the lateral-torsional buckling behaviour of steelgirders with corrugated web of trapezoidal shape, restrained by purlins. The increasedstability due to the torsional stiffness provided by rigidly attached purlins is of specialinterest. The thesis comprises a literature study and finite element analyses ofrestrained girders.In this thesis, the method stated by Horne & Ajmani (1968) to obtain the criticalmoment due to lateral-torsional buckling of restrained girders is investigated. Inaddition, the method presented by Lindner & Aschinger (1990) to include the effectsfrom a corrugated shape of the web is studied. The applicability of these expressionsis evaluated for restrained girders with corrugated web. This evaluation is carried outby comparing the critical buckling moment from these expressions with the resultsfrom linear buckling analyses using elastic material response. In addition, non-linearbuckling analyses are carried out in order to evaluate how restrained girders withcorrugated web fit with the design approach, for lateral-torsional buckling, suggestedin Eurocode 3.The thesis concludes that the critical buckling moment can accurately be calculatedwith Horne & Ajmani’s expression. This expression can be used with comparableaccuracy both for girders with flat web and girders with corrugated web, usingLindner’s method. In addition, it is concluded that the design approach for lateraltorsional buckling suggested in Eurocode 3 is applicable for restrained girders withcorrugated web in a conservative way.Key words: Lateral-torsional buckling, corrugated web, steel girder, purlins, discreterestraints, torsional stiffnessI
Takåsars inverkan på vippning av stålbalkar med korrugerade livExamensarbete inom Masterprogrammet Structural Engineering and BuildingTechnologyGUÐNI ELLERT EDVARDSSONBENGT LUNDQUISTInstitutionen för bygg- och miljöteknikAvdelningen för konstruktionsteknikStål- och träbyggnadChalmers tekniska högskolaSAMMANFATTNINGVippning är ett välkänt stabilitetsproblem i slanka ostagade stålbalkar meddubbelsymmetriskt I-tvärsnitt. En ny metod att öka stabiliteten i balkar är att ha ettkorrugerat liv istället för ett plant liv. Takåsar anslutna till balkar förbättrar stabilitetenpå grund av deras återhållande verkan. Takåsar hindrar rörelser i sidled i derasanslutningspunkter. Utöver detta, kan rotation av tvärsnittet hindras genom att fästatakåsarna styvt på balken. Då är tvärsnittets rotation styrd av takåsarnas styvhet.Denna ökning av stabilitet är, ur designers synpunkt, intressant på grund avekonomiska skäl.Syftet med denna rapport är att undersöka det kritiska vippningsmomentet avstålbalkar, med trapetskorrugerade liv, stagade av takåsar. Den ökade stabiliteten pågrund av vridstyvhet försedd av styvt anslutna takåsar är av särskilt intresse.Rapporten består av en litteraturstudie och finita element-analyser av stagade balkar.I denna rapport undersöks metoden föreslagen av Horne & Ajmani (1968) för atterhålla det kritiska vippningsmomentet av stagade balkar. Utöver detta, undersöksmetoden föreslagen av Lindner och Aschinger (1990) för att inkludera effekterna avatt ha ett korrugerat liv. Utvärderingen av dessa metoder utförs för stagade balkar medkorrugerat liv. Utvärderingen utförs genom att jämföra det kritiskavippningsmomentet från dessa uttryck med resultat från linjära bucklingsanalyser medelastisk materialrespons. Dessutom utförs icke-linjära bucklingsanalyser för attutvärdera hur stagade balkar med korrugarade liv passar in i designmetoden förvippning föreslagen i Eurocode 3.Rapporten fastslår att det kritiska vippningsmomentet med hög noggrannhet kanberäknas med Horne & Ajmanis metod. Uttrycket kan användas med jämförbarnoggrannhet både för balkar med plana liv och balkar med korrugerade liv,användandes Lindners metod. Utöver detta dras slutsatsen att designmetoden förvippning föreslagen i Eurocode 3 kan användas för stagade balkar med korrugeradeliv med konservativa resultat.Nyckelord: Vippning, korrugerat liv, stålbalk, takåsar, stagning, vridstyvhetII
round11.2Aim and objective11.3Method21.4Limitations21.5Outline of the thesis3LITERATURE REVIEW42.1Stability of steel members2.1.1Local and global behaviour3VI482.2Lateral-torsional buckling of I-beams2.2.1Coordinate system2.2.2Lateral buckling2.2.3Torsion2.2.4Lateral-torsional buckling991010132.3Restrained buckling2.3.1Lateral restraints2.3.2Torsional restraints1414152.4Critical moment for girders restrained by purlins2.4.1Girders restrained at the compression flange2.4.2Girders restrained at the tension flange1616172.5Girders with corrugated web2.5.1Lindner’s approach2.5.2Modified Lindner’s approach1919212.621Lateral-torsional buckling in Eurocode 3METHOD243.1Geometry of the studied girders243.2Parametric study253.3Evaluation of the design approach in Eurocode 326FINITE ELEMENT MODELLINGCHALMERS, Civil and Environmental Engineering, Master’s Thesis 2014:10028III
4.1Modelling procedure4.1.1Linear buckling analysis4.1.2Non-linear buckling analysis2828294.2Load application294.3End supports304.4Restraints304.5Convergence study314.6Verification of the model325RESULTS AND DISCUSSION345.1Parametric study5.1.1Torsional stiffness and spacing between purlins5.1.2Web thickness5.1.3Web height5.1.4Flange thickness5.1.5Flange width3434363739405.2Design approach in Eurocode 35.2.1Buckling curve d5.2.2Buckling curve c4041436CONCLUSIONS467SUGGESTIONS FOR FURTHER RESEARCH47REFERENCES48APPENDICESERROR! BOOKMARK NOT DEFINED.A – Analytical derivations of equations49B – Convergence study53C – Verification of the models57D – Results59E – Matlab code75IVCHALMERS, Civil and Environmental Engineering, Master’s Thesis 2014:100
PrefaceThis thesis was carried out at the Division of Structural Engineering, Department ofCivil and Environmental Engineering, of Chalmers University of Technology duringthe spring of 2014.In the thesis, the influence of purlins on the behaviour of steel girders, with corrugatedweb of trapezoidal shape, is investigated. Borga Steel Building produces steel portalframes with corrugated webs and they are interested in the possible advantages oftaking the torsional stiffness, provided by rigidly attached purlins, into account whensizing the frames. This thesis was initiated by Chalmers University of Technology andBorga Steel Buildings.We would like to thank our supervisor, Professor Emeritus Bo Edlund, and ourexaminer, Associate Professor Mohammad Al-Emrani, for their guidance throughoutthe project. We would also like to thank our opponent group, Hermann ÞórHauksson and Jón Björn Vilhjálmsson, for their feedback on the thesis. At last, wewould like to thank the other students carrying out their theses at the department formaking the work environment enjoyable.Guðni Ellert EdvardssonBengt LundquistCHALMERS, Civil and Environmental Engineering, Master’s Thesis 2014:100V
NotationsRoman upper case lettersCross-sectional areaModulus of elasticityTangent modulusForceShear modulusSecond moment of areaTorsion constantEquivalent torsion constant from Lindner’s approachEquivalent torsion constant from modified Lindner’s approachWarping constantEquivalent warping constant from Lindner’s approachEquivalent warping constant from modified Lindner’s approachSecond moment of area around the y-axisSecond moment of area around the x-axisSecond moment of area around the y-axis for the top flange of an I-beamSecond moment of area around the y-axis for the bottom flange of an IbeamSecond moment of area around the -axisSecond moment of area around the -axisBending momentDesign buckling resistance moment according to Eurocode 3Critical buckling momentCritical buckling moment caused by torsion along a restrained beamCritical buckling moment caused by lateral-torsional buckling betweenrestraintsDesign moment according to Eurocode 3Ultimate momentYield momentExternal bending moment applied at end supports of a beamBending moment around the -axisBending moment around the -axisBending moment around the -axisAxial loadCritical axial buckling loadSt. Venant torsional momentVlasov torsional momentStrain energyPotential energySectional modulusRoman lower case lettersLength of the longitudinal panel of a corrugated webDistance between the shear centre and the lateral restraint of a restrainedcross-sectionVICHALMERS, Civil and Environmental Engineering, Master’s Thesis 2014:100
Projected length of the inclined panel of a corrugated webWidth of the flange of an I-beamLength of the inclined panel of a corrugated webMaximum corrugation eccentricity of a corrugated webYield stressDistance between centroids of flanges of an I-beamHeight of the web of an I-beamBuckling coefficientReduction factor for initial imperfection in Eurocode 3Spring stiffnessTorsional stiffnessTotal length of a structural elementNumber of half-sine waves in the longitudinal direction of a memberNumber of restraints along the total length of a girderRadius of gyrationSpacing of purlinsWall thickness of a plate or shellThickness of the flange of an I-beamThickness of the web of an I-beamDeflection in the direction of the x-axisLateral component of lateral displacement due to lateral-torsionalbucklingTorsional component of lateral displacement due to lateral-torsionalbucklingDeflection in the direction of the y-axisWidth of a plateLateral axis of the global coordinate systemVertical axis of the global coordinate systemLongitudinal axis of the global coordinate systemGreek lower case letters̅Angle of the inclined panel in relation to the longitudinal axisImperfection factor in Eurocode 3Partial factor in Eurocode 3Maximum deflectionDeformation of a springElastic strainPlastic strainLongitudinal axis of the local coordinate systemVertical axis of the local coordinate systemNon-dimensional slenderness ratio of steel membersNon-dimensional slenderness ratio of beamsPoisson’s ratioLateral axis of the local coordinate systemCritical stressAngle of rotationReduction factorReduction factor according to Eurocode 3CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2014:100VII
DefinitionsBeam – Structural element that is capable of withstanding load primarily by resistingbending.Bifurcation load – The critical load for buckling to occur in a structure.Bifurcation point – The point on a load-deflection curve where the bifurcation loadhas been reached.Bracing – Lateral and/or torsional restraint on a structure.Critical buckling moment – Used to determine the load-carrying capacity againstelastic lateral-torsional buckling.Elastic strain, – Strain in which the deformed element returns to its original shapeand size when the deforming force is removed.Girder – Main structural member, often supporting smaller beams i.e. purlins.Lateral restraint – Full restraint against lateral movement at a specific point. In thisreport it is caused by purlins.Linear analysis – Assumes elastic material with no geometrical imperfections norresidual stresses in this study.Mode shape – Describes the shape of a buckled beam. The mode shape number tellsin how many half-sine waves the beam buckles.Non-dimensional slenderness ratio, – Used to describe the slenderness of steelmembers in a non-dimensional expression.Non-linear analysis – Assumes plastic material with strain hardening, geometricalimperfection and residual stresses in this study.Poisson’s ratio,– Negative ratio of transverse strain and axial strain.Plastic strain,– Strain in which the deformed element does not return to itsoriginal size and shape after the deforming force has been removed.Purlins – Secondary beams connected to girders to support the roof cladding. Providerestraint against lateral movement and torsion.Radius of gyration – Is used to describe the distribution of cross-sectional area in acolumn around its centroidal axis,Reduction factor,as:. .– Ratio between the critical stress and the yield stress, expressedTorsional restraint – Partial restraint against torsion at a specific point. In this reportit is caused by purlins.Ultimate moment,analysis.– The maximum moment obtained from non-linear plasticYield moment – Moment that causes the yielding in the cross-section, expressed as:VIIICHALMERS, Civil and Environmental Engineering, Master’s Thesis 2014:100
1 Introduction1.1 BackgroundSteel portal frames are commonly used in construction of industry halls. They aretypically low-rise structures that consist of rigidly connected columns and beams. Oneof the governing failure modes of such structures is often lateral-torsional buckling ofthe beam. The stability of the frame can be increased by designing a haunch in thebeams, by using a corrugated web or by adding stiffeners to the web.The top flanges of the beams in the steel portal frames are connected to purlins atcertain intervals which in turn are connected to the roof cladding. The top flange ofeach beam is usually in tension close to the eave of the frame and in compressionclose to the apex. In order to increase the lateral-torsional buckling capacity of thebeam, stiffeners are provided at certain intervals to prevent rotation. The connectionbetween the purlins and the frame in-between the stiffeners are usually not rigid, andcan be considered to only provide lateral restraint to the frame. If the connectionwould be made rigid, the torsional stiffness from the purlins could be taken intoaccount and consequently the distance between the stiffeners could be increased.Important parameters that affect the stability of the frames are the load distribution,the shape of the corrugation and the geometrical parameters of the cross-section. Theproblem can be simplified by only consider the beam segment in-between twostiffeners. The beam segment is considered to be subjected to pure bending andconnected to purlins at evenly spaced intervals.Figure 1.1: Beam segments extracted from a steel portal frame.1.2 Aim and objectiveThe aim of this thesis is to investigate the lateral-torsional stability of steel girderswith corrugated web of trapezoidal shape loaded in pure bending. The increasedstrength due to the torsional stiffness provided by rigidly attached purlins is of specialinterest. The objective of the thesis can be split into different parts:CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2014:1001
To carry out a literature study on how purlins affect the lateral-torsionalstability of steel girders and how existing knowledge is treating the problem.To investigate how corrugated shape of the web influence the lateral-torsionalstability of steel girders and how it is treated in analytical expressions.Evaluate the applicability of the analytical expressions for the critical momentdue to lateral-torsional buckling of restrained girders with corrugated web.Investigate if the existing design approach for lateral-torsional bucklingresistance from Eurocode 3 is applicable for laterally and torsionally restrainedgirders with corrugated web.1.3 MethodA literature review was performed in order to achieve comprehensive understandingof the behaviour of steel beams undergoing lateral-torsional buckling. The effectsfrom discrete lateral and torsional restraints along the beam were also investigated inorder to understand how purlins affect the lateral-torsional buckling phenomenon. Afurther investigation was also performed in order to find an effective method of takingthe increased stiffness, from the corrugated profile of the web, into account whencalculating the critical buckling moment of the beam with analytical solutions.A parametric study was carried out, with linear buckling analysis in ABAQUS CAE,in order to understand how the most influencing parameters affect the lateral-torsionalbuckling phenomenon of girders with flat and corrugated webs. The critical bucklingmoment from the parametric study was compared with an analytical solution, derivedby Horne & Ajmani (1968), in order to examine its applicability. The difference inbehaviour of girders with flat and corrugated webs was also investigated in order toevaluate the applicability of the analytical expression, proposed by Lindner (1990), totake the corrugated shape of the web into account.In order to evaluate the applicability of the design approach for lateral-torsionalbuckling resistance in Eurocode 3 for restrained girders, non-linear buckling analyseswith plastic material properties, including initial imperfections, were performed inABAQUS CAE. From the analyses, buckling curves were obtained and comparedwith equivalent buckling curves in Eurocode 3. This was performed for girders withflat web and girders with corrugated web in order to investigate the difference inbehaviour between the two girder types.1.4 LimitationsIn this project, only doubly symmetric, prismatic, I-sections with flat web andequivalent cross-sections with corrugated web will be considered. Fork supports arechosen as boundary conditions in order to imitate a beam segment between torsionalstiffeners in steel portal frames. All structural members mentioned in this report willbe considered to be of steel unless otherwise stated. Welds between the web and th
Influence of Purlins on Lateral-Torsional Buckling of Steel Girders with Corrugated . Wall thickness of a plate or shell Thickness of the flange of an I-beam Thickness of the web of an I-beam Deflection in the direction of the x-axis Lateral component of lateral
member of a truss bridge girder, principal rafter of a roof truss etc. 3.1.2.4 Purlins Purlins are beams provide over trusses to support the roofing between the adjacent trusses .these are placed in a tilted position over the principal rafters of the trusses. Channel and angle sections are commonly used purlins 3.1.2.5 Sag Tie:
2 Gable portal-frame rafter 3 Gable post 4 Separate strut Figure 1.3 Transmission with or without strut-purlin 1.3 Wind beam stanchion purlins Purlins may also be given the function of a stanchion of the roofing windward beam: see, in Figure 1.2, the stanchion purlins of the
purlins, FABRAL recommends a maximum spacing of 24” on-center (note that 5V requires solid decking). Pullout values decrease if the fasteners protrude completely through the purlins. Kiln-dried softwood is recommended for purlins or decking (pine, fir, hemlock, and spruce). Hardwoods are d
adjacent purlins, and only needs to be applied in alternating spaces between the purlins as shown in Figure 1a. The braces are co mmonly configured either with a channel connected to each purlin or diagonal angles as shown in Figure 1b. a) bay layout of torsion braces b) common torsion brace configurations Figure 1. Paired torsion braces
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to wall) Lateral Loads (parallel to wall) Lateral Load applied to end wall Lateral Load . the IRC gives no guidance on other aspects of the roof framing, such as: –Bracing design for high end of hip/valley rafters –Bracing design for rafter purlins –Non-symmetrical hip roofs –Roof diaphragms with plate height changes
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