A PREDICTION METHOD FOR LOAD DISTRIBUTION IN THREADED CONNECTIONS

Transcription

1 JOURNAL OF THEORETICAL AND APPLIED MECHANICS 56,, pp , Warsaw 208 DOI: /jtam-pl A PREDICTION METHOD FOR LOAD DISTRIBUTION IN THREADED CONNECTIONS Dongmei Zhang School of Mechanical and Electrical Engineering, North University of China, Taiyuan, China dongmei zhang@63.com Shiqiao Gao, Shaohua Niu, Haipeng Liu State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing, China A new method has been developed for predicting the load distribution along the thread portion of a bolt and nut connection. The calculated results were validated by comparison with three-dimensional finite element analysis and Yamamoto s method. It was shown that theloaddistributionpredictedbythemodelinthispaperwasingoodagreementwiththe resultsfromfiniteelementmodel,andtheloadratioonthefirstthreadbytheprediction model and finite element model was slightly larger than the results from Yamamoto s method. In addition, the results of calculation and finite element analysis indicated that the decreasing of the lead angle could improve the load distribution, the increasing of the length of thread engaged could significantly improve the load bearing capacity of the first thread, andtheadoptingofamaterialwithlowstiffnessforthenutwithrespecttotheboltcould improve the load distribution slightly. Keywords: load distribution, threaded connections, finite element analysis. Introduction The bolted joint is a typical connection that is widely used for construction of structures from components. In order to ensure functionality of the joint, the load distribution in threaded connections is of concern. The load distribution in threaded connections has been studied since the beginning of the century, but only a few of the most essential papers are referenced here(goodier, 940; Hetenyi, 943; Motosh, 975; Kenny and Patterson, 985; Brutti and D Eramo, 987; Patterson and Kenny, 986). The Sopwith(948) theory for predicting the load distribution of thread fasteners isthemostwellknownanalyticalmodel.theactionofanumberofstrainsisformulatedbythe axial extension of the bolt and compression of the nut. These strains include bending deflection of the thread, axial recession due to radial compression of the threads, and axial recession duetoaxialcontractionoftheboltandexpansionofthenutcausedbyradialpressureofthe joints. Alternatively, Yamamoto(980) proposed a procedure for calculating the deflection due to bending moment, shear loading and radial contraction and expansion on the bolt and the nut. Due to the progress of the modern finite element method, the solution of contact problem becomes possible by using finite element software. Grosse and Mitchell(990), Wileman et al. (99), Lehnhoff and Wistehuff(996), Chaaban and Muzzo(99) and Chaaban and Jutras (992) studied stresses in threaded connections by axisymmetrical finite element models by ASME Code. In order to investigate the helical effect on the thread connection, Chen and Shih (999) built a three-dimensional bolt-nut assembly by ABAQUS. But there was a small hole on

2 58 D.Zhangetal. the center of the bolt, which slightly reduced the area of the applied load. Moreover, Eraslan and Inan(200) built 3D finite element models of screws by Solidworks. In order to achieve a more convenient prediction for the load distribution in screw threads, we developed a new analytical model to calculate the load distribution in the thread connection. Comparing the previous methods, especially Yamamoto s method which includes five deflections, there are only two main thread deflections in our method. In addition, we fully consider the ununiformity of load distribution on thread and use unit deflection per unit width under unit force. Meanwhile, a three-dimensional finite element model of the bolt-nut assembly is built to validate the prediction method. 2. The new analytical model According to Sopwith s theory, the ISO triangle thread can be simplified into a cantilever beam with a variable cross section. The strain in the thickness direction is considered to be zero, namely, it is a plane strain problem. A coordinate system for the thread under axial concentrated load is established. The length direction of the thread is the y-axis, the depth direction is the x-axis,andtheoriginislocatedattherootofthethread,asshowninfig.a.theoriginalcross sectionofthethreadisatriangle.thesemi-angleofthethreadisα,thethreadlengthis2a, andthepitchdiameterisd,asshowninfig.b. Fig..(a) Thread structure and coordinate system,(b) bolt structure Thegeometricequationoftheupperhalfofthethreadis y=a kx (2.) wherek=tanαandα=30. Thegeometricequationofthelowerhalfofthethreadis y=kx a (2.2) Themomentofinertiaofonethreadis I(x)= y 2 (x)ds= 2πD 3 (a kx)3 (2.3) S(x)

3 A prediction method for load distribution in threaded connections 59 wherea=kh,hisdepthofthethread(asshowninfig.a),therefore I(x)= S(x) y 2 (x)ds= 2πD 3 k3 (h x) 3 (2.4) 2.. The elastic deflection of thread According to Sopwith s theory, the thread under axial loading will deform due to several reasons. Owing to the above assumption, these deflections mainly include bending and shearing deflectionofthethread.anisotrianglethreadcanbeseenasacantileverbeamundera concentratedloadpatx=l,whichisshowninfig.2.thebendingdeflectionδ andshearing deflectionδ 2 alongtheaxialdirectionatx=l canbeobtainedbythefollowingfunctions. Fig. 2. Bending deflection and shearing deflection 2... The deflection due to bending moment ForathreadshowninFig.2,accordingtothetheoryofmechanicsofmaterials,thestatic bending equation is EI(x) 2 w x 2=P (l x) x<l (2.5) wherep istheaxialforceononethread,i(x)isthemomentofinertia,l isthedistancefrom theactionpointoftheforcetotherootofthethread,eisyoung smodulusandwisthe bending deflection. Substituting Eq.(2.4) into(2.5) leads to 2 w x 2= 3P 2πDEk 3 l x (x l) 3 x<l (2.6) wherelisheightofthethreadasshowninfig.2. Primary integration of Eq.(2.6) is conducted in accordance with the assumption that the deflectionangleattherootofthethreadiszero,i.e.,theboundarycondition w/ x=0at x=0andthedeflectionanglecanbeexpressedas w x = 3P ( 2πDEk 3 x l + l l ) 2(x l) 2+ l l l 2l 2 x<l (2.7)

4 60 D.Zhangetal. Primary integration of Eq.(2.7) is conducted on the assumption that the displacement at therootofthethreadiszero,i.e.,theboundaryconditionw=0atx=0is w= 3P [ 2πDEk 3 ln l x l l ( ) l 2(x l) + l l l 2l 2 x l l ] 2l Undertheunitforce,thebendingdeflectionofonethreadatx=l is 3 [ δ = 2πDEk 3 ln l l + ( ) l 2 + l l l 2l 2 l l l ] 2l x<l (2.8) Because the deformation of thread is treated as a plane deformation, under the unit force, the unit deflection due to bending moment can be expressed as δ = 3( ν2 )[ 2πDEk 3 ln l l + ( ) l 2 + l l l 2l 2 l l l ] 2l The deflection due to shear loading Atx=l,theshearstressforthethreadcanbeexpressedas τ(x)= P πd2(a kx) Therefore, the shearing deflection of one thread should be (x)= τ(x) G L wherethelistotallengthofthethreadasshowninfig.b. Theshearingdeflectionoftheunitlength(L )is δ U (x)= τ(x) G (2.9) (2.0) (2.) (2.2) (2.3) wheregisshearmodulus,andg=e[2(+ν)].therefore,theunitshearingdeflectionofone threadundertheunitforce(p )canbeexpressedas δ(x)= +ν EπDa kx (2.4) Accordingtotheboundaryconditionδ(x)=0forx=0andδ(x)=δ 2 forx=l,integration ofeq.(2.4)leadsto δ 2 = +ν l dx EπD a kx = +ν keπd lna b 0 (2.5) AccordingtoYamamoto(980),theloadPislocatedatthecenteroftheoriginaltriangle of the thread, namely l = 2 l b= 2 a (2.6) Therefore, for the bolt and nut under the unit force, substituting of Eqs.(2.6) into Eqs. (2.0) and(2.5), respectively, the elastic deformation of unit length thread in the direction of thehelixis δ b =δ b +δ 2b = 0.5( ν2 b )+.2(+ν b) πde b δ n =δ n +δ 2n = 0.5( ν2 n)+.2(+ν n ) πde n (2.7)

5 A prediction method for load distribution in threaded connections 6 whereδ b andδ 2b representthebendingandshearingdeflectionofbolt,respectively,δ n and δ 2n representthebendingandshearingdeflectionofthenut.fortheisotrianglethread,there isk=tan30.forthethreadedconnectionstructurecomposedoftheboltandnut,thetotal elasticdeformationoftheunitlengththreadinthedirectionofthehelixis δ y =δ b +δ n (2.8) whereδ b andδ n representthetotaldeflectionoftheboltandnut,respectively Threadstiffness Forthethreadonthebolt,thestiffnessoftheunitlengththreadundertheunitforceinthe direction of the helix is k bu = δ b (2.9) BecausethetotallengthofthethreadisπD,thetotalaxialstiffnessofthethreadonthe boltundertheunitforceis K b =πdk bu = πd δ b (2.20) Similarly,forthethreadonthenut,thestiffnessoftheunitlengthundertheunitforcein thedirectionofthehelixis k nu = δ n Thethreadstiffnessofthethreadonthenutundertheunitforceis K n =πdk nu = πd δ n (2.2) (2.22) If one thread is considered to be a spring, the threaded connection that consists of several threadsisaparallelspring.thus,forthebolt,thetotalstiffnessofnthreadisnk b.ifthepitch isp,thenthestiffnessoftheunitaxiallengthundertheunitforceisregardedas k by = K b p =πdk bu p (2.23) Iftheleadangleofthehelixisβ,thentanβ=p/(πD).Fortheboltandnut,thestiffness of the unit axial length under the unit force is respectively k by = k bu sinβ = δ b sinβ k ny = k nu sinβ = δ n sinβ (2.24) For the threaded connection structure composed of the bolt and nut, the total stiffness of theunitaxiallengthis k y = k by + = k ny (δ b +δ n )sinβ = δ y sinβ (2.25) Becausethestiffnessistheratiooftheforcetodeflection,theaxialdeflectionofthethread body is y = k y P y (2.26) where P/ y is the axial force on the unit length thread. The preceding equation indicates that the axial deflection of the thread body varies directly with load distribution.

6 62 D.Zhangetal Axial load distribution in threaded connection Figure3showsthatthethreadedconnectionstructureincludesafixednutbodyandabolt body.intheboltbodyunderacompressingforcef B,theforcewillbetransferredtothenut bodythroughthethreads.thestructureiselastic.thus,theboltbody,thenutbodyorthe thread body will exhibit deflection under the force. However, the forms of forces on the bolt body and the nut body differ from those on the thread body. Furthermore, their load distributions alsovary.thus,thedeformationmodesoftheboltbodyandthenutbodydifferfromthoseof thethreadbody.however,theboltbodyandthenutbodyarejoinedbythethreadbody.thus, their deformations are compatible. Fig. 3. Threaded connection structure Theregionofthecompressingforceonthethreadbodyrangesfrom0tol,wherelisthe engagedlengthoftheboltedjoint.thus,thestrainε b generatedbytheboltbodyunderthe compressingforcef(y)atthelocationyis ε b = f(y) S b E b wheres b isthecross-sectionalareaoftheboltbody. Similarly,strainε n generatedbytheboltbodyunderthecompressingforcef(y)is (2.27) ε n = f(y) S n E n (2.28) wheres n isthecross-sectionalareaofthenutbody. Forthethreadbody,theloadfisdistributedalongthedirectionofthehelixandalongthe axial direction y. The thread deformation analysis indicates that the deflection and displacement ofthreadsontheboltbodyisrelatedtothegradientofforcedistribution,i.e., δ b = k by f y Similarly,thedeflectionofthethreadbodyonthenutbodyis (2.29) δ n = k ny f y (2.30)

7 A prediction method for load distribution in threaded connections 63 Their displacement gradients are respectively δ b y = 2 f k by y 2 δ n y = k ny 2 f y 2 (2.3) Inordertomakethedeformationcompatible,ε b,ε n,δ b andδ n mustmeetthefollowing relationship ε b +ε n = δ b y + δ n y (2.32) Substitution of Eqs.(2.27),(2.28) and(2.3) into Eq.(2.32) leads to λ 2 f= 2 f y 2 (2.33) where λ= S b E b + S ne n k ny + k by ThesolutiontoEq.(2.32)is f=c sinhλy+c 2 coshλy (2.34) Bytheboundaryconditionsf(y)=f B fory=0andf(y)=0fory=l,wecanobtain C 2 =f B C = coshλl sinhλl f B SubstitutingC andc 2 intoeq.(2.34)leadsto f(y)=f B ( coshλl sinhλl sinhλy+coshλy ) (2.35) wheref B istheloadonthefirstthreadofthebolt.theloadalongtheaxialdirectionycanbe calculated by Eq.(2.35). This distribution is more precise than Yamamoto s method because we considered the load distribution along the helix of the screw. 3. Finite element model AgroupofISOmetricscrewthreadsM36areusedtostudytheloaddistributionandtheeffect of pitch, material and length of the thread engagement. The parameters of all screws are listed in Table. Commercial finite element software ANSYS is used for modeling and analysis. The model geometry is meshed by 8 node hexahedron elements(solid85). The contact and target elements are TARGE69 and CONTA72, respectively. The friction coefficient changes between accordingtothematerialofthethread.Figure4shows3-Dfiniteelementmodelsofthe standard bolt assembly and the stress distribution on the screw under the external load. There is no sliding between the bolt and nut threads because friction coefficients are large enough to preventsliding.inthemodels,theaxialloadingisappliedtothetopsurfaceofthebolt,and theouterbottomsurfaceofnutisassumedfixed.convergencestudyiscarriedoutontheinitial finite element model by decreasing the element size near the threads. The smallest element size is0.25mmby0.25mmandthetotalnumberofelementisabout80000.thereisnosignificant improvement in the accuracy by using smallest elements.

8 64 D.Zhangetal. Table. Parameters of the thread Speci- Nominal diameter Pitch Engaged length Materials men of thread p of bolted joints No. d[mm] [mm] l[mm] bolt nut steel steel steel copper steel brass steel aluminum aluminum steel Fig D model mesh of standard threaded connection:(a) bolted joint assembly,(b) bolt,(c) contour plotofthescrewstress

9 A prediction method for load distribution in threaded connections 65 Table 2. Parameters of material Material Young s modulus Poisson s ratio Density E[GPa] ν[ ] ρ[kg/m 3 ] Steel Copper Brass Aluminum An elastic material is used throughout this work. A uniform pressure loading p is applied to thetoprootsurfaceofthebolt.so,thetotalforceduetotheappliedpressurecanbeexpressed as f B =p πd2 4 (3.) wheredisthenominaldiameterofthebolt. 4. Result and discussion ForthreadNo.,theresultoftheloaddistributiononeachthreadbyYamamoto smethod, finiteelementmodelandthepredictionmodeldevelopedinthispapercanbeshowninfig.5. Fig.5.TheloadratiooneachthreadforboltedjointsNo. Figure5showsthattheloaddistributionobtainedfromthemodelinthispaperareinaclose agreement with the finite element analysis. It is a load corresponding to the pitch section of the thread. The load distribution with prediction from the analytical model and finite element model is shown to be slightly larger than the results of Yamamoto s method. For the FEM calculation, theaxialcomponentofstressisutilizedfortheloadonthreadandtheloadratioistheratioof theloadonthreadtotheloadonthesurfaceofthebolt.theloadonthreadcanbecalculated byequation(2.32)forthepredictionmodel,anditistheaveragevalueofsomeelement(all theelementsalongthethreadedge)forthefemcalculation.fromthefigure,wecanseethat theloadingratioisquitesimilaroneachthreadinthecaseoftheanalyticalmodelandfinite element model for all specimens.

10 66 D.Zhangetal. TherearefourpitcheswhichhavebeenusedinthisSection,namely4mm,3mm,2mm,.5mm,andthecorrespondingleadanglesare2.804,.6068,.05,0.780,respectively. TheloadratioofthepredictionmodelinthispaperwithdifferentpitchinFig.6showsthat the decreasing of the pitch can improve the load distribution and reduces the load bearing on the first thread. Fig.6.TheloadratiooneachthreadforboltedjointsNo.,2,3,and4 Figure7istheloadratioofthepredictionmodelandfiniteelementmodelwithlengthof threadengagement28mm,24mm,20mm,6mm,2mm,respectively.thefiguresshowthat the load ratio is modified greatly with the increasing of thread engagement length. And the longer the length of thread engagement, the smaller the load bearing on the first thread. But whenthelengthismorethan20mm,theeffectisveryslight. Fig.7.TheloadratiooneachthreadforboltedjointsNo.,5,6,7and8 Inaddition,whenYoung smodulustheofbolte b islargerthanthatofthenute n,theload distribution is improved slightly. The calculation result by the prediction model for specimens No.,9,0,,2whoseboltsandnutsaremadeofdifferentmaterials,areshowninFig.8.

11 A prediction method for load distribution in threaded connections 67 Fig.8.TheloadratiooneachthreadforboltedjointsNo.,9,0,and3 5. Conclusions The load distributions on the thread connection by the analytical model and finite element model have been studied. The results can be concluded as follows..theloaddistributionsobtainedfromthemodelinthispaperareingoodagreementwith finite element analysis and Yamamoto s method. 2.Theresultsoftheanalyticalmodelinthispaperarenearertothefiniteelementanalysis than Yamamoto s method, because we considered the load distribution along the helix of the screw. 3.Theeffectofpitchontheloaddistributionofthethreadedassemblyisveryobvious.A decreaseinthepitchnotonlycanimprovetheloaddistribution,butalsocanreducethe loadratiointhefirstthread. 4. The load distribution of the first thread is significantly decreased by a properly increased length of the thread engagement. 5.Adoptingamaterialwithlowstiffnessforthenutwithrespecttotheboltcanimprove the load distribution slightly. References. Brutti C., D Eramo M., 987, Methods to evaluate load distribution in screw threads, Proceedings of 2th Canadian Conference of Applied Mechanics, Edmonton, Chaaban A., Jutras M., 992, Static analysis of buttress threads using the finite element method, Journal of Pressure Vessel Technology, 4, Chaaban A., Muzzo U., 99, Finite element analysis of residual stresses in threaded end closure, Journal of Pressure Vessel Technology, 3, ChenJ.J.,ShihY.S.,999,Astudyofthehelicaleffectonthethreadconnectionbythree dimensional finite element analysis, Nuclear Engineering and Design, 9, 2, EraslanO.,InanO.,200,Theeffectofdesignonstressdistributioninasolidscrewimplant: a 3D finite element analysis, Clinical Oral Investigations, 4, Goodier J.N., 940, The distribution of load on the threads of screws, Transactions ASME, 62, 0-6

12 68 D.Zhangetal. 7. Grosse I.R., Mitchell L.D., 990, Non-linear axial stiffness characteristic of axisymmetric bolted joint, Journal of Mechanical Design, 2, Hetenyi M., 943, A photoelastic study of bolt and nut fastenings, Transactions ASME, 65, Kenny B., Patterson E.A., 985, Load and stress distribution in screw threads, Experimental Mechanics, 25, Lehnhoff T.F., Wistehuff W.E., 996, Non-linear effects on the stress and deformations of bolted joints, Journal of Mechanical Design, 8, Motosh N., 975, Load distribution on threads of titanium tension nut and steel bolts, Transactions ASME, 97, Patterson E.A., Kenny B., 986, A modification to the theory for the load distribution in conventional nuts and bolts, Proceedings of the Institution of Mechanical Engineers, 2,, Sopwith D.G., 948, The distribution of load in screw threads, Proceedings of the Institution of Mechanical Engineers, 59, Wileman J., Choudhury M., Green I., 99, Computation of member stiffness in bolted connections, Journal of Mechanical Design, 3, Yamamoto A., 980, The Theory and Computation of Threads Connection(in Japanese), Yokendo, Tokyo Manuscript received June 2, 206; accepted for print August 7, 207