c. F. Kollbrunner.K. Basler Torsion in Structures An Engineering Approach Translated from the German Edition by E. C. Glauser With Annotations and an Appendix by B. G. Johnston Springer-Verlag Berlin Heidelberg GmbH 1969 CURTF.KOLLBRUNNEEKONBADBASLEB Senator h.c., Dr.h. c., Dr.sc.techn. Dipl.Bau-Ing. ETH, SIA Zollikon/Zurich, Switzerland Presidentofthe InstituteforEngineeringResearch Zurich, Switzerland Dr.Ing. (Ph.D.) Dipl. Bau-Ing. ETH, SIA, M. ASCE Partner, Basler&Hofmann Consulting Engineers, Zurich, Switzerland EENSTC. GLAUSER Dr.Ing.(Ph.D.) Dipl. Bau-Ing. ETH, SIA, A.M. ASCE ProjectEngineerin Basler&Hofmann, ConsultingEngineers Zurich, Switzerland BRUCEG. JOHNSTON Ph.D., M.S., F. ASCE Professorof CivilEngineering UniversityofArizona, Tucson, Arizona, U.S.A. Title of theOriginal Edition: Torsion (1966) ISBN 978-3-662-22559-2ISBN 978-3-662-22557-8 (eBook) DOI10.1007/978-3-662-22557-8 With112figures and 22 tables This work is subject to copyright. All rightsare reserved, whether the whole or part ofthe material is concerned,specificallythose of translation, reprinting, re-use of illustrations, broadcasting,reproduction by photocopying machine or similarmeans, and storage indatabanks. Under 54 of the GermanCopyrightLawwherecopiesare madefor otherthan private use, a fee is payable to the publisher, the amount of the fee to be determined byagreement with the publisher. bySpringer-VerlagBerlinHeidelberg1969. Originallypublished by Springer-Verlag Berlin/Heidelberg i n1969 Softcoverreprint of the hardcover1st edition1969 Library of Congress Catalog Card Number:79-88817 Thereproductionofregisteredtrade-marksin thisbookdoesnot warrant the assumption, evenwithoutany specialmarking, thatsuchnames are to be consid-ered free under the trade-marklawand may be used by anyone TitleNo. 01599 Preface of the Authors ThisEnglisheditionhasbeentranslatedfromtheGermaneditionbyDr. E. GLAUSER,Dipl.Ing.ETH,intheofficeofBasler&Hofmann,Consulting Engineers,Zurich,Switzerland.Thetranslatormadealargenumberofminor improvements and alterations. Dr.E. GLAUSERreceivedhisPh.D.from theUniversityofMichigan,Ann Arbor,underthesupervisionofProfessorBRUCEG.JOHNSTON,thenProfessor ofStructural Engineeringat that university.Dr.JOHNSTONkindly reviewedthe draft ofthis translation and made annotations and corrections.He alsoenriched thebookbytheappendixwhichgivesapreciseevaluationofthetorsioncon-stantsofstandardrolledshapesandoffersinformationonstressconcentration factors. OurspecialthanksareduetobothDrs.E. GLAUSERandB. G.JOHNSTON fortheir careful translation and editing. Zollikon and Zurich, July 1969 Curt F. KollbrunnerKonrad Basler Preface of the Translator Thistextbookrepresentsatranslationof thecorrespondingGermanversion which was published by the same publishing company in 1966. The good reception ofthe Germanedition induced the authors to make this fundamentally newand comprehensivetreatmentoftorsionaccessibletoalargergroupofinterested engineers.Thetranslatordidnotfollowtheoriginalversionliterally.Hetried to eliminate the fewmistakes in the original and made other changesand modi-ficationswhich promised to facilitate the understanding of the text. It istheaimofthisbooktodevisemethodsfortheanalysisoftorsionin structuresusingtoolsfamiliartomoststructuralengineers.Themethodsyield thestressdistributionintwisted single-span or continuous memberswithsolid, thin-walled,open or closed cross sections resting on either regular or skew supports. Aprismaticmemberexhibitstwowaystoresisttwist.Thefirstresultsina circulatoryshear flowin thecrosssectionwhilethesecondyieldsshear stresses resultingfromthechangein axialstresses.The first0ontribution isdenoted by Saint-Venant torsion,the second by warping torsion. The discussionofeachoftheseeffectsmaybedividedupintoastructural mechanicssectionandintoastructuralanalysissection.Whilethefirstiscon-cernedwiththe analysisofstressesinagivencrosssectiontakingintoaccount onlytheconditionsoftheconsideredcrosssection,theseconddealswiththe structure as awhole and determines the distribution of the torsional moments for agiven load. The sum ofthe Saint-Venant torsionalmomentTsand the warpingtorsional momentT wisin equilibriumwith the total torsionalmomentT.It isafurther taskofthestructuralanalysistodeterminethedecompositionofthetorsional moment Tinto the twocomponentsT8and T W.Mixed torsion issaid to occur if neither of the componentsTsand T w Anumberofadvantages result fromthe introduction and discussion of these twocomponentsseparately,i. e.,relativelyunknowntermssuchasthesec-torial coordinatewor the warping moment M wmay be explained within aclosed context.Amuchbetter justification,however,isgivenby the practical applica-tionsthemselves.Chapter10providesademonstrationthatformostpractical cases,theeffectofonecomponentmaybe neglectedascompared to theeffect ofthe other.Warping torsion isusually negligibleor may at bestbeconsidered bylocalcorrectionsin slendermemberswith compact solidor hollowcrosssec-tions.Saint-Venanttorsion,ontheother hand,maybeneglected in thin-walled open cross sections such as in cold formed profiles or open orthotropic plate brid-ges.Thestructural mechanics aspect of the analysis,however,isindependent of Prefaceofthe Translatorv whether the Saint-Venant torsional momentTsor the warping torsional moment T OJ occur separately or together. It was forthose reasonsthat it wasdecidedtopresent the subject oftorsion in structures in the following13chapters: Parts ISaint-Venant torsion IIWarping torsion IIIMixed torsion IVFolded Plates StructuralMechanics Chapter 1 and 2 Chapters 5and 6 Chapters 11,12 and 13 StructuralAnalysis Chapters 3and 4 Chapters 7 and 8 Chapters 9and 10 Both the theoriesofbendingand warpingtorsionassume that thecrosssec-tionsofaloadedmembermaintaintheirshape.Thereareofcoursestructural members which do not satisfy thiscondition. Chapter IV demonstrates foracertain classof structures(folded plates) what happensifthecondition ofconstant shapeiscompletelydropped.Themethods presented therein,however,assume that the plates are hinged to each other.The resultofthisfoldedplatetheorytogetherwiththeresultoftheordinaryben-dingand warping torsion theory represent therefore bounds for the true behaviour of aplate member whose crosssections are only partially maintained.Chapter 13 showsthatthesetwoboundsareformostpracticalapplicationsclosetoeach other soastoprovideapowerfultoolforthe designer.The applicationofmore sophisticated methods (applying Fourier Series, finitedifferences, finiteelements, etc.)doesnot seem to be necessary as long as the distribution of the plate stresses remains linear. If nototherwisementioned,allchaptersassumeprismaticmembersofan elastic material whose shear deformations may be neglected as compared to those causedby axial stresses.Theinternalforcesarecalculatedfortheundeformed state of the structure and thus do not permit conclusions as to its stability. There is hardly any subject in the theory of structural mechanics in which the signconventions,thedefinitionofcoordinatesystemsandthepresentationof thetheoryisofasgreatimportanceasinthetheoryofwarpingtorsion.The systematology and clearness of presentation is lost if,just to name a few examples, the x-axis isforsome discussions in the plane of the cross sections and forothers pointing along the length of the member, if the y-axis points upwards fordiscus-sionsofcrosssectionsand downwards forthecalculation ofdeflections,ifloads are positive when pointing in the direction of apositive axis and equally directed stressesnegative,etc.Inordertoarriveataconsistentsystemofdefinitions, the authors wereforcedtointroduce notationsand signconventionswhichmay notbeveryfamiliartosomereaders.The system isdefinedbeforeits firstuse, i. e.with respect to coordinates in Section 5.1. Uptotheturnof the last century, the torsionphenomenonwasthought to becompletelycoveredbythetheorydevelopedbyDESAINT-VENANT[1].It wasnot realizedatthattimethat the shear stressesin the crosssectionsmay not be solelyapart ofaclosed shear flowbut may also be caused by achange in the axial stresses (warping torsion). VIPrefaceof the Translator ImportantcontributionstotheapplicationofdeSaint-Venant'stheorywere madebyR. BREDT[2]towhomweoweformula(2.5),byL.PRAND'fL[3]who discoveredtheanalogybetweenthetorsionandthemembraneproblem,by A.FOPPLbecauseofhiscontributions to the evaluation ofthe torsionconstants of rolled sections[4,5]and finallyby C.WEBERand W. GUNTHER[6]. Ageneralproblemof mixedtorsionseemstohavebeensolvedforthefirst timeby S.TIMOSllENKO.After L. PRANDTLtreated in hisdissertation the lateral bucklingofbeamswith rectangularcrosssection,S.TIMOSHENKOinvestigated the same problem for the case of I-sections.This investigation led TIMOSHENKOto thediscoveryofwhathecalled"torsionwithflangebending."Hisfamousre-sults,thesolutionsofan eigenvalueproblem,werefirstpublishedinRussianin 1905and in German in 1910. In1909,C.VONBACH[8]reportedtheresultsofexperimentsconductedon beamswith[-typecrosssectionwhichclearlyshowedanonplanarstraindistri-bution(sincethestraingagesindicatedthesumofthebendingandwarping strains). It was more than 10 years until R. MAILLART[9]and A.EGGENSCHWYLER [to] gave acorrect interpretation to this suspected contradiction of the Bernoullic Navierhypothesis.Theyintroducedtheshearcenterofthecrosssectionas a point common to all shear forcescausingbending without torsion. Thegeneraltheoryforwarpingtorsionofopen,thin-walledcrosssections wasinitiatedbyH. WAGNER[11]andR. KAPPUS[12].Asaircraftengineers, theywereconcernedwiththetorsionalbucklingofthin-walledmembers.More than20yearslater,these fundamentalideasappearedin the structural enginee-ringliteratureaswell.F. W. BORNSCHEUER[13]contributedasystematologyof cross-sectional quantities forboth bending and warping.The application of these theories to the designof steel structures wasoutstandingly described by F. WANS-LEBEN[14].R. HEILIG [15]investigated the influence of shear deformations which turned out tobe negligiblein mostcases. J. N. GOODIER[16],S.TIMOSHENKO[17]andF.BLEICH[18]presentedex-cellentsummariesofthetheoriesforopen,thin-walledmemberswithspecial emphasisonstabilityproblems.Aspectsapplicabletoaircraftengineeringwere presented by P. KUHN[19]. Very recently, the German and English translations of abook by V. Z. VLASOV [20]appeared.Thisbook must have existed in Russia since1940.It presents the subjectofmixedtorsioninamostoutstandingmanneranditseemsthatits author applied the term"sectorial coordinate"forthe firsttime. Without special mention, the authors of this book utilized the results of many otherscientistsandengineerswhichcontributedtothepresentknowledgeof torsion,e. g.[21]. Thisbookrepresentsacondensationoftheresearchwhichwascarriedout from1961to 1965.Apart of this work was published by the Swiss Association of Steel Manufacturers [22,23,24,25]andby theSwiss construction magazine [26]. Zurich,July 1969 Ernst Glauser References tothe Preface [1]SAINT-VENANT,B.DE:Memoires des savants etrangers,Vol.14,1855. [2]BREDT,R.: Kritische Bemerkungen zur Drehungselastizitat.Z.VDl40 (1968)785. [3]PRANDTL,L.: Zur Torsion von prismatischen Staben. Phys.Z.4(1903)758. [4]FOPPL,A.:Der Drillingswiderstand von Walzeisentragern.Z.VDl61(1917)694. [5]FOPPL,A.,and L. FOPPL:Drang und Zwang,Miinchen/Berlin:R. Oldenbourg1928. [6]WEBER,C.,and W.GUNTHER:Torsionstheorie,Braunschweig:Vieweg1958. [7]TIMOSHENKO,S.:EinigeStabilitatsproblemederElastizitatstheorie.Z.Math.Phys.58 (1910). [8]BACH,C.VON:VersucheiiberdietatsachlicheWiderstandsfahigkeitvonBalkenmit [-fOrmigemQuerschnitt.Z.VDI 1909,1910. [9]MAILLART,R.:Zur Frage der Biegung.Schweiz.Bauztg. 77(1921)195. [10]EGGENSCHWYLER,A.:tiberdieFestigkeitsberechnungvonSchiebetoren und ahnlichen Bauwerken.Diss.E. T. H.,1921,Borna beiLeipzig:Robert Noske [11]WAGNER,H.: Verdrehung und Knickung von offenen Profilen. Festschrift 25 Jahre T. H. Danzig,1929,or Luftf.-Forschg.11(1934)329. [12]KAPPUS,R.:DrillknickenzentrischgedriickterStabe mit offenemProfilim elastischen Bereich.Luftf.-Forschg.13(1937)444. [13]BORNSCHEUER,F. W.:SystematischeDarstellungdesBiege- undVerdrehvorganges unter besonderer Beriicksichtigung der W6lbkrafttorsion.Stahlbau 21(1952)1. (14)WANSLEBEN,F.:DieTheoriederDrillfestigkeitvonStahlbauteilen,K6ln:Stahlbau-Verlag1956. [15]HEILIG,R.:DerSchubverformungseinfluBaufdieW6lbkrafttorsionvonStabenmit offenem Profil.Stahlbau 30(1961)67. [16]GOODIER,J. N.:TheBucklingofCompressedBarsbyTorsionandFlexure.Cornell University,Engineering Experiment Station, Bulletin 27,1941. [17]TIMOSHENKO,S.:Theory ofBending,Torsion and Buckling ofThin-Walled Membersof Open Cross-Section.J. Franklin lnst. 239(1945). [18]BLEICH,F.: Buckling Strength of Metal Structures, New York:McGraw-Hill1952. [19]KUHN,P.:Stresses in Aircraft andShellStructures,New York:McGraw-Hill1956. [20]VLASOV,V.Z.:Thin-WalledElasticBeams.EnglishTranslation,NationalScience Foundation,WashingtonD.C.,U.S.Dept.Commerce1961,also,London:Oldbourne Press 1961. [21]THURLIMANN,B.: Lecture Notes C.E. 453 and C.E. 411.Lehigh University, Bethlehem, Pa., U.S.A.,1958,1959. [22]KOLLBRUNNER,C.F.,andK. BASLER:TorsionskonstantenundSchubspannungenbei Saint-VenantscherTorsion.Mitt.TKSSV.,Vol.23,VerlagSchweizerStahlbau-Ver-einigung,July 1962. [23]KOLLBRUNNER,C.F., and K. BASLER:Torsionsmomente und Stabverdrehung bei Saint-VenantscherTorsion.Mitt.TKSSV.,Vol. 27,VerlagSchweizerStahlbau-Vereinigung, Oktober 1963. [24]KOLLBRUNNER,C.F.,and K. BASLER:SektorielleGri:iBenund Spannungen beioffenen, diinnwandigenQuerschnitten.Mitt.TKSSV.,Vol.28,VerlagSchweizerStahlbau-Vereinigung,January 1964. [25]KOLLBRUNNER,C.F.,andK. BASLER:StatikderW6lbtorsionunddergemischten Torsion.Mitt.TKSSV.,Vol.31,VerlagSchweizer Stahlbau-Vereinigung,May1965. [26]BASLER,K.: Zur Statik schief gelagerter Trager.Schweiz.Bauztg. 82(1964)269. Contents Nomenclature..XIV I.Saint-VenantTorsion1 1Solid Cross Sections1 1.1Introduction.1 a)Rotation of the Shaft.1 b)Shear Stress Distribution2 1.2Applications.......3 a)Circular Cross Section3 b)Narrow Rectangular Cross Section4 c)Equilateral Triangular Cross Section5 1.3Saint-Venant's Approximation for the Torsion Constant7 1.4Built-Up Solid Cross Sections9 2Closed, Thin-Walled Cross Sections10 2.1Introduction....10 a)Prandtl's Analogy11 b)Bredt's Formula.12 2.2Applications....13 a)Radially Symmetric Hollow Cross Sections13 b)Validity ofthe Assumption "Thin-Walled"15 c)Mixed Open-Closed Cross Sections16 d)Composite Cross Sections...17 e)Lattice Walls........19 2.3Multicellular Box Section Members21 a)General Remarks......21 b)Representation of the Resulting Shear Flows(Differences in Elevation)23 c)Evaluation of the Shear Flows qi(Lid Elevations)24 d)Computational Scheme....27 e)Examples..........29 2.4Special Cases and Limiting Values.31 a)Separated Cells.......31 b)Limiting Values for the Torsion Constant32 c)Several Cells in Two-Dimensional Configuration34 2.5Cells Connected by aCommon Base Cell35 a)Introduction.......35 b)Analysis.........36 c)Interpretation of the Results38 d)Special Cases.......40 2.6Shear Stresses Caused by aResultant Shearing Force Acting on the Clo-sed,Hollow CrossSection.........42 a)Problem...............42 b)Shear Stresses Caused by a Shearing Force.43 c)Shear Center.............44 ContentsIX 3Torsional Moments and Member Rotation for Saint-Venant Torsion46 3.1Introduction......46 a)Saint-Venant Torsion46 b)Differential Equation46 c)Notations......47 3.2The Single Member...47 a)Statically Determinate Support47 b)Member With No End-Rotations.48 3.3The Continuous Rod on Elastically Rotating Supports50 a)The "Three-Rotation Equations"50 b)The "Three-Torsional-Moment Equations"52 3.4Examples........53 a)Systems.......53 b)Selection of the Method.55 c)Influence Lines....57 4Skew Supported Members With Saint-Venant Torsion58 4.1Introduction.........58 a)Assumptions.......58 b)Notations and Abbreviations58 c)Torsional Moments and their Side-Effects62 d)Example: Iterative Procedure63 4.2Analytical Solution....65 a)Slope at Skew Supports.65 b)Three-Moment Equations66 c)Forces and Deformations68 d)Example: Three-Span Continuous Beam71 4.3Eccentric Loads............75 a)Decomposition of Loads.......75 b)System Acted Upon by aConcentrated Moment76 c)System Acted Upon by aGeneral Eccentric Load79 d)Example: Influence Surfaces81 4.4General Considerations85 a)Member Constant C85 b)Conditions at the Supports86 c)Maximum Shearing Stresses Caused by Skewed Supports93 d)Summary.94 II.Warping Torsion96 5Stress Analysis96 5.1Notation96 a)Sign Convention96 b)Coordinate System.96 c)Quantities Depending on the Coordinates xandy96 d)Correlation of Signs for Quantities Depending on the Coordinatez97 Mkand Mk+l ortheloaddependentelementsmaybeputseparatelyontherightsideofthe 4.2Analytical Solution67 equalsign.Therelationsgivenhereinafterutilizethe secondwayofpresenting thethree-momentequationsfortheskewsupported,continuousmember.The three-momentequationswillbegivenfourtimesinorderofincreasingsystem regularity.Thegeneralcaseisfollowedbythecasewhichassumesconstant flexuralandtorsionalrigiditywithinanyonespan(SpecialCase 1),thecase assuminginadditionconstantskewangleofthesupportsisSpecialCase 2and finally the case with both constant rigidities and constant skew angleofsupports over the entire system isSpecial Case 3. General case {Variable cross section:EI(z), GK(z), Different skew angles of supports: bk-1 01=15k 01=t5k+1 + Mk [(,8i-lk+ ,8i-lk Oi-lok-l- iXi-lkOi-1Pk--l)Di+ + (iXik+ iXikOiPk+1- ,8ik Oi.k+1) Di - 1]+ fordisplacements iX,,8:v. Table 4.1, firstcolumn, member constants 0, D: v. Table 4.2, first column, b .kF42(k=1,3,5, ...) suscnpts ~ : v.Ig..i=2,4,6, .... S.l 01{Cross section constant within one span: Eli, G K i, pectaaseDifferent skew angles of supports: t5k-1 01=15k 01=t5k+1. (4.9) Thevaluesfromthesecondcolumnoftable4.1and4.2areintroducedinto formula (4.9). + Mk[_li=L- (2+ 0 Ki-1 tg2 15k-I)D;+(2+ 0 Kitg2 t5k+1)Di-1]+ 6 Eli-lEli- 126EliEli2 + M ~ (1_ OKitgt5k tgt5k+1)D._= k+16EliEli2,I (4.10) + [,8- OKi2tgt5k+ tgt5k+1t15-. (1+ OK,tgt5k+ 2tgt5k+t tt5)] D. ,0E [.6gk+1iX,OEL6gk+1,-1, ,t 5'" 684 Skew Supported Members with Saint-Venant Torsion where: and:~ =1,3,5,...). ~ =2,4,6,.. . TheratioGK;/EIiisquiteoftenapproximatelyconstant.Thisleadstoa furthersimplification,as follows: {Cross section: E I=Eli =constant within span, Special Gase2G KjE I=constant for entire system, Same skew angle at each support: {jk={jk+l={j. Mk-l ~ (1- G)+ Mk .~ + ---.!:L)(2+ G)+ Mk+l ---.!:L(1- G) 6 E ~ - 1 6E1i-l6EI;6Elj where: =- [(Pi-l 0+ lXiO)(1+ G)- (lXHO+ PiO) G], G =GKtg2t5 EI2' (k=1,3,5,...) i=2,4,6,.... (4.11) {Constant cross section along the entire beam, Special Gase3Same skew angle at each support. where: Mk-lli-l(l- G)+ Mk(li-l + li) (2+ G)+ Mk+lld1- G) =-6EI[ -lXi-lOG + (Pi-lO+ lXiO)(1+ G)- P;oG],(4.12) G=GKtg2t5 EI2' ~ =1,3,5,...). ,=2,4,6,.. . c)Forces andDeformations ThetorsionalmomentinducedbyskewedsupportsisgivenbyEq.(4.2) whereinthesupportdisplacementslXiandPiaredeterminedbytheexpressions (4.7).Considering that and Gi{Jk+ltg {jk=O;{Jktg {jk+l Gik tg {jk+l=O;d+l tg {jk, the torsionalmoment in span iforthe most generalcase(crosssectionvariable and skew angle different from support to support) may be given by the expression: -1 ~ =-- [lXiOtg {jk+ {JiOtg {jk+l+ Mk(lXiktg {jk+ Pik tg {jk+l)+ YiDi + Mk+l(lXik+ltg {jk+ Pik+l tg {jk+l)](4.13) 4.2Analytical Solution69 The determinant Diis list.ed in the firstcolumn of table 4.2. Expression(4.13)maybegiveninamoreexplicitmannerforthespecial cases defined above. SpecialCase1(E 1;,G Kiconstant within onespan,Ok=1=Ok+1): SpecialCase2and3(Eli,G Kiconstantwithinonespanandsameskew angle at each support,Ok=Ok+\=0) : Ti=_tg 6[EI;(IX;O+ fJiO)+Mk~ Mk+l] Elit26li GK+ g , (4.15a) or withthesubst.itutiona.=G Kitg26 ,Eli2 (4.15b) The interpretation of the expression in brackets is as follows: Eli lXiOrepresents (accordingtotheconjugatebeammethod)thefictitioussupport,reaction at the leftendofmemberiwhichisloadedbythemomentareaMiOElif3iOisthe corresponding reaction at the opposite member end. The sum of these two reactions isthusequaltothemomentarea.Thefirstterm inthebracketsmaytherefore be written as: (4.16) It isequaltotheaveragemomentduetotheappliedloadsinthemember~ For aconcentrated loadP,it becomes Eli (lXiO+ fJiO)=-.!. M=aibiP li2max2li (4.16a) and foraunifonnly distributed load P, (4.16b) The evaluation of the moment at any point alongspan irequires aknowledge oftheendmomentsofthemember.TheseendmomentsaredesignatedasMki and Mk+1.iand may be expressed by means of Eq.(4.3)in the form: Mki=Mk+ LlMki=Mk+ Ti tg Ok,} Mk+1i=Mk+1+ LlMk+1i=Mk+l+ T;tg 0k+1' (4.17) 704 Skew Supported Members with Saint-Venant Torsion Themomentatany. pointalongthespanisconvenientlyexpressedasthe superpositionofsimplebeam momentand theproportionatecontributionofthe endmomentswhich,writtenintermsofthenondimensionalcoordinates, ,Ci=zdliandCi=zilli'is: (4.18) In order to plot the moment diagramonemay advantageouslyplot firstthe baselinebetweenendmomentsuponwhichthesimplebeam momentsMiOare superposed. Based on the end moments in Eq. (4.17),both the end shear Q and the support reactions Fmay be expressed as follows: (4.19) (4.20) ThelineofactionofthereactionFactssomewherealongthebearingedge ofthesupportbutnolongerintersectswiththememberaxiseventhoughthe appliedloadsactwithouteccentricity.Thepointofapplicationofthisreaction has an x-coordinate;7:j'kas determined by Eq.(4.21): Ti- Ti-} XFk=----''------''-''-Fk (4.21) Theverticaldeflectionyofthememberaxismay likewisebedeterminedas -thesuperpositionofthedifferentdeflectionstatesofthesimplysupportedbase system. Herein are: YiO=Deflection due to applied loads, Yik=Deflection due to M ki=1, YikH=Deflection due to MkHi =1. (4.22) Thelasttwodeflectioncoefficientsatlocationzare,according to the Reci-procalTheorem,equaltotheangularrotationsatthecorrespondingsupports causedby aunit loadP=1at z,i.e.Yik=(XiOandYikH=PiO.Both (XiOand PiOare listed in table4.1(inthe section valid forthe concentrated loadP=1). Expressionsfortheverticaldeflectionofmemberlocationswhichdonot coincidewiththememberaxismustincludethetorsionalrotationp.If this rotationismeasuredfromahorizontallineandispositivewhendirectedas shown in Fig. 4.2,it can be givenby the expression: z, P(Zi)=P(Zi=0)+ J p'dz. o (4.23) SinceinskewsupportedmembersP(Zi=0)=(Xitg lJk(v.Fig.4.3)and since,inSaint-Venanttorsionp'=T/GK,therelationgivenabovemaybe 4.2Analytical Solution71 rewritten as Zi ,.T (l)=q>"(l)=O. 5.1 99 In thisanalogy,thetorsionalmomentTwcorrespondsto theshearwhilethe warpingmomentjlfw1correspondstothebendingmomentinthesimplebeam. Apositive,e.g.clockwisetorsionalloadmDcorrespondsthereforetoat",ist with the samesenseofrotation. ThetorsionalmomentT wactingonacrosssectionwithpositiveoutward normalissaidtobepositivewhenpointingin+ go-direction(Fig.5.2).The equilibriumconditionforthememberelementoflengthdzacteduponbythe distributed torsional load mDmay thus bewritten as follows: This leads to the relation: whichisequivalent to the connectionbetweenshearand load.In analogyto the relation=+ Qyit isthus apparent that adominating function,the warping moment Mw,maybe defined as=+Tw' ThedefinitionforthespecificangleoftwistOw,go'=+ Ow,completesthis newchainoffunctionswhichissummarizedinthesecondcolumnofTable 5.1. Table 5.1.Sign Connection Between thez-Dependent Quantities at aMemberUnder Pure Bending and Warping Torsion Displacement Slope Bending moment and warping m. Shearing forces and torsional moments Load to lateral load py 'YJOy=+ 'YJ' My=-ElyyO; =-Elyy'YJ" Qy=+ = =-(Elyy'YJ")' py===+ (ElyyO;,)" =+ (Elyy'YJ")" Subjected to torsional load mD rp Ow=+ rp' Mw=-ElwwO:, =-Elwwrp" T(,)=+ M;) =-(ElwwO;Y =-(Elww'P")' 1nn=-T;o =+ =+ (Elww'P")" Subjected to lateral load Px Mx=-ElxxO.;. =-Elxxf;" Qx=+ M; =-(ElxxO;Y =-(Elxxf;"), Px= =+ =+ (Elxxf;")" y)Bending'inx-Direction.Thesignconventionsetdownforthey-direction willbevalid forthe x-direction as well(v.column 3ofTable 5.1).This statement seemstrivialsinceit isreally the prerequisite forasystematic treatment andthe analogiesofproblems in flexure and torsion.Note that thesesignconventionsare different fromthose which assume moments to be positive whenever their vectors point in thepositivedirectionofacoordinateaxis. d)Axial1in aMember cnderthe End Warping Moments X. Awayfromthememberends(forwhichtheordinarystaticalanalysisisnot abletogiveareliabl esolution)itmaybeobservedthatthewarpingstresses vanishalongthemember forlargevaluesofx(foraboutx>10).Onlyin such casesisitpermissibletoresolvetheeffectsoftheaxialforcePwithrespectto thecentroidalmemberaxesandtocalculatethestressesbymeansofthewell known stress formulas. Fig.7.9showsthestressdistributioninthecrosssectionat mid-spanforthe two limitingcases;pure warping torsion(x=0)and pure Saint-Venant torsion (x=(0).Even though these two limiting cases cannot occur in anactual member, theyareneverthelessvaluableinestablishinglimitsfortheactualstressdistri-bution aswasshown inFig.7.9forthe parametersx=1,3 and 10. 19010Miscellaneous Problems Thetwolimitingcaseshaveadditionalsignificanceinprovidingabasisfor thedevelopmentofapproximateexpressionswhenlimitingconditionsareap-proached.Approximaterelationsforbothlargeandsmallvaluesfor",willbe derived in the next two sections. MembersacteduponbyauniformlydistributedtorsionalloadmDandbya concentrated torsional momentMiJwillserveasexamples.If the localinfluence of warping moments at the member ends is also considered, the basis for the appro-ximate analysisofcontinuous systems willbecomplete. c)Large x-Values, Limiting Caseand Approximation for maxMCD If'"isverylarge,approximateexpressionsforasimplysupportedmember undertheuniformlydistributedtorsionalloadmDmaybederivedfromthe solutions (9.27). mDl2 (1-4';:2) GSK~ GKq/ =-mDl' =Ts=T,(10.16) M",=T", =0. These expressionscoincide exactly with those for the member with only Saint-Venant torsion (Section 3.2). If,however,an approximation for the warping moment M",forlarge ",-values is desired,one may read fromthe corresponding relation ofEqs.(9.27) M=mD l2 =mElm", ",",2DGK which,according to Eq.(5.33),yields the followingwarping stresses: EmD(J) (1=---",GK. (10.17 a) (1O.17b) TheEqs.(9.32)leadtocorrespondingexpressionsforthe memberunderthe concentratedtorsionalmomentMD'Forextremelylargevaluesof"',these expressions again coincide with those for pure Saint-Venant torsion in Table 3.2a. Theprevioussectionmentionsthatanordinarystructuralanalysiswillnot determinethetruestateofstressinthecloseneighbourhoodofconcentrated loads.With thislimitationkept in mind,thewarpingmomentat thefixedend ofamembermay be derived fromthecorrespondingexpressionofthesolutions (9.10). (10.18) T(J)refersto the total torsionalmoment at the fixedend and dis the charac-teristic length as defined by Eq.(9.7). IfaconcentratedtorsionalmomentMDactsatalargedistancefromthe nearest support,warping restrainteffectsare produced on eachsideofthe point 10.2 Approximate Solutions191 of load application.The warping moment (10.19a) (10.19b) Boththewarpingmomentandthewarpingstressesdecreasetoeitherside of the disturbance accordingtothelawe-z/d [v.curveat thebottomofFig. 9.1 andcorrespondingexpressionofthesolutions(9.10)].Thesameholdstrue fora warpingmomentXappliedat amemberend.For largevaluesof"itsdecrease with increasing distancez fromthemember end isgivenby the expression: (10.20) Thisexpressionandthecorrespondingrelationforthewarpingstressare veryunreliableif thecharacteristiclengthd issmallerthanthebreadthofthe crosssection.Since the warping stressesare usually small in this case,the result-ing error is not significant. d)Small x-Values, Limiting Case and Approximation Thewarpingmomentinasimplysupportedmemberwhichisactedupon byawarpingmomentXattherightendofthemember(Fig. 9.2)isgivenby Eqs.(9.22). M=XSin,,1; '"Sin" If bothnumeratoranddenominatorareexpandedinaTaylorseries,this expression becomes: 11 "I; + - (u 1;)3+ - (,,1;)5+... M=X3!5! '"11 u+-u3+--x5+ ... 3!5! (10.21 ) ThisexpressionleadsimmediatelytothelimitingsolutionM",=XCfor vanishingvalues"(purewarpingtorsion)whichmighthavebeenderivedfrom the analogy of warping torsion and bending. In additiontothislimitingsolution,approximatesolutionsfor,,> 0with anydesireddegreeofaccuracymaybederived.Theaccuracydependsonlyon the number of terms considered in the series. 19210 Miscellaneous Problems Setsofapproximate solutions willnow be derived forallcases treated in Sec-tion 9.2.All terms of higher than second order in uare neglected.The subsequent expressionsareimmediatelytransformedintothoseforpurewarpingtorsion if u2 isput equal tozero.They represent furthermoreapproximatesolutionsfor mixed torsion if theparameter udoesnot exceed acertain valuewhichdepends ontherequiredaccuracy(e.g.u< 2).ThewarpingtorsionalmomentTwis given instead ofthe total torsional moment T. For awarping moment Xacting at therightend of amember[derived fromthe solutions (9.22) for small values u]: 2 ~ ~ ~ 3Mw=X6 2' 1 ~6 2 1 ~6 (10.22) For uniformlydistributedtorsionalloadmD[derivedfromthesolutions(9.27) for small values u] : (10.23) 10.2Approximate Solutions193 ForconcentratedtorsionalloadJYI DatC =iXand(= f3[derivedfromthe solutions(9.32)forsmall valuesu\: rp=MD!3 [1_f32_C2 + u2 (1_ f34_C4_10 f32C2)]. 6EI"""1 +203 6 (p'= [1_f32- 3C2+(1- f34-1Of32C2 - 5C4)]. 6Elww1 + u20 6 Mw=MDl1:1; u2 [1+(f32+ C2)]. (10.24) G Tw= _f3_ [1+ u2 (f32+ 3C2)]. u2 6 1 +-6 Thesolutionsintheregion0:;::;;(10).It alsoisnoted,thatthesetheoreticalwarpingstressesoccuratthe point ofload application and are therefore to be accepted with some reservation. As mentioned in Section 10.2, members with large values of xshow such abrupt changesintheaxialstressesthattheshearstressesandshear resultingtherefrommaynolongerbeneglected.Theactualwarpingrestraint isthereforesmallerthanthatcalculatedwithoutconsiderationofsheardefor-mations.Thetheoryaspresentedhereinthereforeoverestimatesthewarping momentsundertheload.Fig. 10.5presentsthereforethesequestionedresults by abroken line. Theapproximatesolutionsforsmallandlargex-valuesareindicatedbya dash-and-dotline.Theirdeviationsfromthecorrespondingexactsolutionsare accentedbythecross-hatchedregions.It isseenthattheapproximationboth forwarping torsion and Saint-Venant torsion isexcellent.One might wellassume therefore that the twoapproximate solutions would in themselvesbesatisfactory forthe entire rangeofmixed torsion(0'"fO 2G J' 7 0 Fig.11.S. Bridge ('ross Section. thicknesswhiletheintermediateplatestretchesbeyondthetwoconnecti ng hinges. The three platesarecharacterizedby the followingvalues: Plate2: 13 a2 = -ab2 =-a 2'2' F2=2Fo , 12 5 =- a2Fo' 6 Plate 4: a4=2a,b4 =2a, F4=SFo, (11.33) 14=24a2Fo' Plate6: 31 a6 = -a b6 ="2 a, 2' F6=2Fo, 16 = a2Fo' 6 IfthebendingmomentfortheloadshowninFig.l1.SisdenotedbyM 0' thebending moments in theplates in terms ofthis referencemoment are: (11.34} 21811General Analysis of Folded Plates Since there are only two unknown edge forces(R3and R5),the system (11.30) reduces to two equations with two unknowns: Thesolutionsofthissystemfortheplateproperties(11.33)andtheload definition(11.34)are as follows: R_ __1_5_ Mo 3- 2.23.53a' R5=+3 . 419Mo. 22353a (11.35) Thesesolutionsdeterminetheaxialstressesintheedges1,3,5and'7by means ofEq.(11.29) 3Mo 4 - [degrees]. - c (13.10a) This condition is quite conservative since it was derived for quite unfavourable conditionswhichrarelyoccurinpractice,aconcentratedloadat mid-spanand platesconnected by longitudinal hinges. Thecondition(13.10a),ontheotherhand,doesnotconsiderthefactthat the ordinary folded plate theory, which determines the stresses for the undeformed shape of the crosssection,overestimates the capacity ofafolded-platestructure. Sincetheeffectsofthenonlinearbehaviorbecomeconsiderablebothfor extremelyobtuseandacuteangles,it isadvisabletosatisfythefollowingrule when selecting the angleXkbetween adjacent plates: ll 4 - < ()/;< 180- 4- [degrees]. c- - c (13.10 b) If, forinstance,the span l isfivetimesaslargeasthe averagewidth c ofthe plates,the anglexl:is restricted by the followinginequality: Exercise 13.1.TheSrw,p-ThroughProblemof aFoldedPlate a) Amodel of the structure shown in Fig.H.l is to be built in ordertoverifyexperimen-tallytheload-deformationcurvesofFig. 13.3.Devisetheconditionswhichensurethat these curves may be produced without exceeding the proportional limit lipof the model mate-rial. 13.2 Equivalence of Both Theories243 (Hint.: CalC'ltlatethc maximumstl"{'SSin the region0.:::cx< cxo.) b)Which regionofcxmaybeeoveredby theexperiment if CXoisselectedsnchthat the proportionallimit cpisreachedbut not exceeded in the range0< cx.-:- IXo? Solutiolls: a)The maximum stl"{'SSisa.as given by Eq.(13.Sa).If the load isintroducedas given by Eqs.(13.3)and (13.4),this expression becomes: 05 _ ~ __ ~ ~ E ~ ) 2 cx (cxo- cx) 5l1+(."( The stress ao as!mmesthe followingextreme valueatcx~ CXo: 2 9E(C)2., I max 0:;I =5TCXii (a) (b) Thegivenconditionfor the proportional limit ep imposes therefore the followingrestric-tion on the initial slope 0