TECHNICALSCIENCESAbbrev.: Techn. Sc., No 12, Y 2009DOI10.2478/v10022-009-0008-5DETERMINATION OF INTERNAL FORCESIN END PLATES OF SIMPLE END PLATE JOINTSStefan Dominikowski, Piotr BogaczDepartment of Civil Engineering and Building StructuresUniversity of Warmia and Mazury in OlsztynK e y w o r d s: end plate joints, internal forces, end plate.A b s t r a c tIn the paper the authors propose computer simulation ofsimple end plate joints. Acomputermodel has been elaborated using the Autodesk Robot Structural Analysis Professional 2010 softwareprogramme. Maps of internal forces and stresses on end plates ofa joint have been analysed. Theanalysis of the maps enabled the authors to determine the course of variation of bending moments inany cross-section of the joint and to compare the obtained diagram with a diagram of moments citedin the literature (TOWSKI et al. 2000, BIEGUS 1997), which serves the purpose of determination ofthe minimum thickness of end plates needed due to the lever effect. The paper also determines thedependencebetweentheloadingforceactingonthejointandthethicknessofendplatesontheassumption of maximum stresses damaging the steel of the end plates, as this is one of the criteria fordimensioning end plate joints. In addition, effort of joint bolts was determined in relation to a value ofthe loading force.WYZNACZENIE WARTOCI SI WEWNTRZNYCH W BLACHACH CZOOWYCHPOCZE DOCZOOWYCH ZWYKYCHStefan Dominikowski, Piotr BogaczKatedra Budownictwa i Konstrukcji BudowlanychUniwersytet Warmisko-Mazurski w OlsztynieS o w a k l u c z o w e: poczenie doczoowe, siy wewntrzne, blacha czoowa.A b s t r a k tWpracy zaproponowano symulacj komputerowpoczenia doczoowego zwykego. Modelkomputerowy opracowano na podstawie programu Autodesk Robot Structural Analysis Professional2010. Przeanalizowano mapy si wewntrznych i napre na blachach czoowych poczenia. Analizamap pozwolia okreli przebieg zmiennoci momentw zginajcych w dowolnymprzekrojupoczenia oraz porwna uzyskanywykres momentw zwykresem momentw podanym w litera-turze (TOWSKI i in. 2000, BIEGUS 1997), sucymdo okrelenia minimalnej gruboci blachczoowychzewzgldunaefekt dwigni. Wpracyokrelonorwniezaleno siyobciajcejpoczenie od gruboci blach czoowych, z zaoeniemmaksymalnych napre niszczcych stal blachczoowych. Jest to jedno z kryteriw wymiarowania pocze doczoowych.Wewzorcowympoczeniuwyznaczonorwniewytenierubpoczeniawzalenoci odwartoci siy obciajcej.IntroductionIn contemporary steel constructions, bolt joints are mainly used for joiningshipping or assemblage elements. End plate joints, either non-bolted or boltedwithhighstrengthbolts, are amodernwayof joiningstretchedor bentelements of metal constructions, in which the main component of the load isparallel to the axis of the bolts. These joints can be made easily and quickly ata construction site under any atmospheric conditions and without any heavy orspecialist equipment. With such joints,it is also possible toassemble and,ifnecessary, dismantle a steel construction easily.This paper discussesthe problem ofthe effort ofboltsandendplatesinsimple, strechtedendplatejoints. Amongthemostimportantissueswhichhave not beensolved satisfactorily until present day are distributionofinternal forces in a joint (in end plates and in bolts), the way the edges of endplates interact with one another, the lever effect on the actual bearing capacityof a joint and the minimumthickness of end plates. Moreover, the nature of thework performed by bolts and the effect of the thickness of end plates on thecharacterof workperformedonendplatesinendplatejoints still awaitcomplete clarification.Description of a computer modelIn order to improve our understanding of the above problems, a computer-basedsimulationofasimpleendplatejointsubjecttostretchinghasbeencompletedusingthesoftwareprogrammecalledAutodeskRobotStructuralAnalysis Professional 2010licence 3251. Astretchedendplate joint wasdesigned in the formoftwoplate girder double-tee barsjoinedthrough endplates with M16 bolts fixed in 18 holes (Fig. 1).The connection of the end plates was modelled as a joint of platesrestrained at the sites where bolts joining the plates were fitted. The end platejoint was made using a fixed connection with a 16 rod, which simulated theproperties of a grade 8.8 M16 bolt with all degrees of freedom being blocked(Fig. 2).Stefan Dominikowski, Piotr Bogacz 8450050 40020030010f18505050100300 1010Fig. 1. A model of an end plate in a joint. Thickness of the end plates in the end plate jointis gb = 16 mm. Thickness of the plates of the connected rods is gp = 10 mmSource: own elaboration.Fig. 2. Model of the bolted joint of end plates: rod 16. Steel in the bolt simulating grade 8.8 steel.Source: Autodesk Structural Analysis Professional 2010 licence 3251.It was assumed that the end plates and connected rods were made of S235grade steel. The bolts were modelled from steel of mechanical properties closeto those of grade 8.8 bolt steel. The bottom part of the joint was supported byarticulated linesupports. Totheupperone, theconnectedplategirderrod,concentrated transmitted load was applied. The load was applied to the middleDetermination of Internal Forces... 85node of a FEM grid at the upper edge of the plate girder web. A Finite ElementMethodgrid was establishedaccording to Coonses method, allowing foradjustment of FEM grids. Around the joints there were visible codes of blockeddegrees of freedom.In the subsequent stage of the simulation, thicknesses of end plates werechanged, thus determining the dependences between the force causing normaldamaging stresses and the thicknes of the end plates. In addition, the value ofthe load on the joint was determined at which the efforts of the bolt steel wereclose to one.Analysis of the resultsThe followingwere obtained: amapof normal stresses onthe uppersurfaces of end plates (Fig. 3), a map of bending moments My (Fig. 4) and a mapof translocations of the FEM grid centres (Fig. 5).At thesametime, inthevicinityof theeffect producedbythebolts,a diagram of variation of bending moments was made, which verified diagramsof bending moments accepted in the literature (TOWSKI et al. 2000)(cross-section AA) (Fig. 6). towski presents a diagram of bending momentsat the contact edge of end plates assuming a zero value of bending moments atthe longitudinal edges of the end plates (Fig. 7).Fig. 3. Map of normal stresses in upper fibers of end plates in the jointSource: Autodesk Structural Analysis Professional 2010 licence 3251.Stefan Dominikowski, Piotr Bogacz 86Fig. 4. Map of bending moments in upper fibers of end plates in the jointSource: Autodesk Structural Analysis Professional 2010 licence 3251.Fig. 5. Map of translocations of the FEM grid nodes in end plates of the jointSource: Autodesk Structural Analysis Professional 2010 licence 3251.Determination of Internal Forces... 87Fig. 6. Diagrams of bending moments M on end plates of the jointSource: Autodesk Structural Analysis Professional 2010 licence 3251.M Zcz=~0.66 1wQ+ZQ+ZtQQtQQ+ZQQ+ZM1c2M1c1c1c2M2N Z =2N Z =2Fig. 7. The lever effect and diagram of moments on the surface of an end plate in an end plate jointSource: own elaboration.Takingadvantageof suchadiagramof moments, hederivesaknownstandard dependence for the minimumthickness of end plates. It assumes thatmoment Mz = ~ 0.66 Z c1 (Fig. 8, 9). By distinguishing mentally a strip of theendplate of the width bs (Fig. 10), one can determine the value of moment Mz(DOMINIKOWSKI et al. 2005).M = 0.66 c Z = W fd M fd(1)WStefan Dominikowski, Piotr Bogacz 88W = 1 bz t2(2)60.66c Z = 0.167 bz t2 fd(3)t =0.66 c SRt= c SRt(4)0.167 bz fdbz fdFig. 8. Deformation of end plates in the jointSource: Autodesk Structural Analysis Professional 2010 licence 3251.Due to the limited increase in the value of the force in a bolt caused by thelever effect, it is assumed that the minimum thickness of the plate in the jointis tmin = 0.6 t.Then, the minimum thickness of the plate is:tmin = 0.6 t = 1.2c SRt(5)bz fdDetermination of Internal Forces... 89Fig. 9. Computer model of the analysed jointSource: Autodesk Structural Analysis Professional 2010 licence 3251.The above dependence is valid as long as the bending moment at the edgesof the endplates is assumedas zero (PN-90/B-03200). Inreality, at theendplate edges are there are non-zero, residual bending moments and normalstresses (Fig. 3, 9). The assumptionthat bending moments at endplateedges are equal zero is quite a good approximation in the light of the analysis ofthe diagram of moments (Fig. 3, 9). The diagram of moments (Fig. 7) is close tothe diagram of moments (Fig. 6), but the assumed values M2 are higher thanthe actual ones (Fig. 6). The flattening of the diagram (Fig. 6) is caused by thestiffening of the plate along the xx axis. Paradoxically, the presence of factorbz in the denominator of the fraction (Fig. 10) should diminish the thickness ofendplates as the spacing of bolts in the joint increases (PN-90/B-03200).The cases described in (TOWSKI et al. 2000, BIEGUS 1997, MAREK et al.2003) are applicable to relatively thin end plates stiffened along one direction.The computer-basedsimulationpresentedinthis paper accounts for thebracing ofendplatesalongbothdirections.Theyyaxisisbracedwiththewebs of the connecting rods, whereas the xx direction is braced with shelves ofconnected rods.Stefan Dominikowski, Piotr Bogacz 90bscbsbsbsFig. 10. Width of interaction of plate bs per one bolt and distance c from the edge of the hole to theouter edge of the weldSource: own elaboration.MAREK et al. (2003) describes a computer-based simulation of a joint basedontheSBRA(Simulation-BasedReliabilityAssessment) method, assumingthat supports at bolts are elastic and articulated at edges (the edge of one endplate is supportedon the edge ofthe other plate). Both methods attain zeromoments at edges of endplates andassumethat plates havearticulatedsupports. At the same time, Marek compares the SBRA results with the EC3recommendations (prEN1993-1-1:2002), repeatedinlater publications ofeurocodes (PN-EN 1993-1-8:2006).The results of the SBRAsimulations are close to the results of thesimulation reported in this paper and cited in (TOWSKI et al. 2000, BIEGUS1997). They are also approximately the sameasthe ones given in EC3. Theformula for the minimumthickness of anend plate should also includenon-zero valeus of moments at edges of end plates (Fig. 4).Moreovertheappliedcomputer-basedsimulationallowedfordetermina-tion of the dependence of the load on the joint on the thickness of the end platesinthejoint, assumingthatthereappearnormal stressesintheendplateswhich damage the joint. It was assumed that the steel in the end plates wasS235gradesteel. Fordifferentthicknessesoftheendplates, thejointwasloadedwithaforcewhichcausedtheappearanceintheendplatesteel ofnormal stresses close to damaging ones. The assumed values of the axial forcewere such as to attain the value of extreme stresses in end plates close to theDetermination of Internal Forces... 91values causing damage to their steel. For S235 steel, the value was fd = 375MPa(PN-90/B-03200). Thesedependencesareillustrated in figure10.Withsuch dependences, it is possible to determine the value of loads damaging theend plates in a joint. For a given thickness of an end plate, one can read thevalue of a force at which the values of stresses in end plates are nearly Rm.The computations were performed for the thickness of end plates between10 and 24 mm. The regression curve represents foreseen results of thickness ofend plates up to 30 mm. At the same time, the value of a force was measured atwhich the efforts of the steel in the bolts are close to 1. The force which causesthe break of grade 8.8 M16 bolts in the modelled joint is P = 320 kN. The loadper 1 bolt is P1 = 80 kN SRt 81.3 kN (PN-90/B-03200). The load capacity ofa bolt in a joint also depends on the thickness of plates (the lever effect) and islower for less thick plates (e.g. for a plate 16 mm in thickness, the load capacityof bolts is~ 55 kN).ConclusionsThe currently available computer methods applied to analysis of construc-tions enable us to analyse more precisely how joints, including semi-rigid ones,work. Analysis of maps of bending moments (Fig. 4) makes it possible to verifytheacceptedstaticmodelsof workof endplates. Theassumptionof zerobendingmomentsatedgesof platesisclosetothedistributionof bendingmoments (vector of moments alongaxis y) obtainedinacomputer-basedsimulation. At the edges of end plates there are moments whose values can beneglected while analysing how a joint works. They are residual in nature.Inboltswhichjoinendplates, apartfromtensileforces, therearealsobending moments (Fig. 4). Values of bending moments on joining boltsdecrease as the thickness of end plates of the joints rise. Diagrams of bendingmoments in a cross-sections in the vicinity of bolts obtained in our computersimulationareveryclosetothediagramsofmomentspresentedinarticles(TOWSKI et al. 2000, BIEGUS 1997, DOMINIKOWSKI et al. 2005, MAREK et al.2003). Littlechanges inthe character of thedistributionof stresses andbending moments on end plates were observed when different types of supportgiven to edges of plates were assumed.Foranyloadonajoint,thequestionofmutualsupportofedgesofendplates seems to be negligible because under the influence of loads thereappears a gap between the end plates, as a result of which there is no mutualsupport of their edges.In the simulation discussedin this paper, the maximum value of normalstressesonsurfacesof endplateswasachievedinthevicinityof thejointStefan Dominikowski, Piotr Bogacz 92connectingendsofshelvesofplategirdersjoined(welded)withendplates.This confirms the principle of concentration of stresses in these places.The computer-based simulation presented in this article enabled theauthors to determine the dependence of the load on a joint on the thickness ofend plates, assuming that there are normal stresses on end plates close to Rm(Fig. 11).0501001502002503003504004505 10 15 20 25 30 35platethickness [mm] daxialforce[kN]PFig. 11. Dependence of the axial force in the joint on the thickness of the end plates in the joint, onassumption of attaining stresses in the end plates that damage the jointSource: own elaboration.Thesimulationalsomadeit possibletodeterminethevaluesof forceswhich damage the joint by breaking the bolts. It turned out that on assumptionthat the joint would be braced with a rod modelled from steel whose mechan-icalpropertieswereclosetothesteelfromwhichtheboltsweremade, theobtained damaging forces were nearly identical to values SRt (PN-90/B-03200).At the same time, the authors observed a change in the value of axial forcesPintheboltsdependingonthethicknessoftheendplates. Therefore,thecommonlyacceptedassumptionabout areverselyproportional dependencebetweenthicknessofendplatesandthenormmultiplierbs(Fig. 10)seemsquestionable. Paradoxically, an increase in value bz diminished the minimumthickness of end plates.The authors of this paper are fully aware of how fragmentary this analysisof semi-rigid joints is and are now working on expanding their analysis so as tocover more complex issues (greater density of bolts, analysis of welds) and onanalysis of bent end plate joints.Accepted for print 12.10.2009Determination of Internal Forces... 93ReferencesBIEGUS A. 1997. Poczenia rubowe. PWN, Warszawa.DOMINIKOWSKI S., BOGACZ P. 2005. Konstrukcje stalowe.Konstrukcje stalowe. Obliczenia statyczne i projektowanie. PN-90/B-03200,Eurocode 3 Design of steel structures, part 11: General rules and rules for buildings CEN may 2002.prEN 1993-1-1:2002,Eurokod 3: Projektowanie konstrukcji stalowych Cz 18: Projektowanie wzw. PN-EN 1993-1-8:2006.MAREKP., KRIVYV. 2003. Reliabilityassessment of semi-rigidpartial- strengthsteel joints andstructures, Konstrukcje stalowe, 5 (63).TOWSKI W., FILIPOWICZA., UBISKI M. 2000. Konstrukcjemetalowe. Cz1. Podstawyprojek-towania. Arkady, Warszawa.Stefan Dominikowski, Piotr Bogacz 94