Bolted Timber Joints Citation Gayer, Martin Roger (1944) Bolted Timber Joints. Master's thesis, California Institute of Technology. doi:10.7907/nnqs-1w59. https://resolver.caltech.edu/CaltechTHESIS:03312025-212000717 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Civil Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Civil Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 19 June 1944 Record Number: CaltechTHESIS:03312025-212000717 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:03312025-212000717 DOI: 10.7907/nnqs-1w59 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17119 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 01 Apr 2025 23:29 Last Modified: 01 Apr 2025 23:29 Thesis Files PDF - Final Version See Usage Policy. 15MB Repository Staff Only: item control page

BOLTED TJJ'IBZR JOilfrS :P:c·osentocl a s 1-,a rti-etl fulfillrnent Qf the 1"equ1~8lllents for the degree ot :Master of . Scionoe in Civil Engineering ~ Marth R. Ga:;rer J"une 19, 1944 l Defleetd011 Ou.I've of o. l3eam cm an Elastic foundation l Bolted Joint with Steel S:plioe Plates 6 Bolted loint ·with Wood Spliee Plates 9 Detem.ination of- the Mod\ilus of the Wood 16 Determin.ation ot Allowable ATarage Bearing Str0as 20 rension 1n the Bolt 22 1 BOLTED 'ril'IDER JO.U:I'TS Ie Intr•ocluct,ion . In a bolted timber joint the stress in the wood umier the bolt is not uniform; on tho contrary, high etresa concentrations occur at the edges of the timbers. The more slender the bolt 'the gr eater is the stress concentration and the smaller the portiQn of' ·the length of bolt over, which the a.o tual bearing is distributed. Tests conducted at the Forest Products Laboratory have revealed the existenoe of a relationship between the length to diameter ratio of the bolt and the proportional limit of the average bearing value of the wood. The tests indicated that the average :proportional limit stress (stress at which i~he slip •of t~ie joint ceases to be proportional to the applied. load) deereases with increa::;ing length to diameter rs.ties• Therefore sate u.nifo:rr, c1.esign stresses, based on the average proportional limit ztress, r.rust decrerrn e with increasing length to diameter :r.·c,tios. It ts the :purpose of tho following cliscussio11 to determine • analytically the relationship between the stress distribution in bolted joints and the elastic properties of the bolt and the r1ooa.. IIo Deflection Curve of a Beam. on an Elastio Foundation ' The ac"'cion of a bolt in a timber joint is that of a beam on an elas tic foundation. That is 9 both the bending s ·trength of the bolt. and the elar:rtio :p1~operties of the vmod mmit be corn; i dared a Consicle1· a beam on an elastic 'foundation, with coordinate ax@s as shovma - The bending moment ld is ta.ken as positiv0 vmen ['''''''''''''''''''' y it produces upward concavity, and fo; such bending:;, is negative when the eoordins.te axes are directed as indicated c.boveo Tience the differential equa:tion of the deflect.ion d2.Y Differentiating b,oth members of the d4 eq_u.:::,tion tv;iae, there results E.I d;. =-~ , in ·which q denates CUl"Ve i s · £ Id x' =~M. the intensity of the load acting on the beam. Now consider the berun to be ir1qedded along its entire length in a material cn.pable of exerting both downvrard and upward ferees on 1t ~ It v-rill be nssurr1ec1 that when the b~am is defleoted the intensity of the contil).uous reaction a.t every section. is :pr.oporti.onal to t he defleation at that section. Hence the intensity of the· · l·oact acting on the beam raay be expressed as ky, where k ia •the modulus of the foundation. k denotes the_:magnitude ot the oon- -'Ginuous. rea-ct,ion per unit of length of the beairt per unit of deflection. For an unloaded portion of the beam, the only force acting· on the beam is the continuous reacti'on. Therefore the a.ifferential equation o!' the defleet;i-on cm:."Ve of a 'beillll on an , elastic foundation is 'I'he gone:cal solution of this differential equation may be represented a s folloYrs: y= sinlix{C, .sinh/.ix + C2 co.sh,<Jx} + cos/.1x(C3 nnht3x. + C_, co.rhA1x) lrx.. .vrhere J3'" Vff! . to be deter-mined from the knovm. eonditions at certain points ot, ~he beam. n1e d.efl ~ction cu:r-ve vrill ftrst be cmtahltshed for t-vro fundanont i1l types of loading: 1) a ooncentratea. load applied at the end of a be1:1Iil ~ and 2)' a bending moment npplied at the end of a . rJeamo For othe:t,. 0@'.11.<U.tions of loading the deflection curve will be found by superposition.• Gase l .s p L y Cl (CO .r ,<JL co .t/24t. - ,f//1 J/31.. ,un/2/.il..) -C.i (.s,n/JJ. .unh/.li +c<J,U'Jt. C1J1A4J.) - C4 (.fih4L. co.r/,,<JJ. + ~o.r.,(Jt. .:i/1/,.41.) =0 tho b em-n : V = 2 tp ./ sinh/.JL cos/.3x, cosh/J (L-x) - s1n/3L cosh/.Jx cos /.J (L -,,t} 2 s in/2 /J L - sin 2/.i '- L )' ,6. C. I =0 .st.!L.~ d,¥. x=1. C, ~o.s/.lt- cos/,/JL. + C1 c.o.s/31. .f/n/2/.J/. -C3 J'I/J4L co.s/241- - c~ s,n/.Jt. .,;,,,1, /.J '- ,. o d".YJ 3 =o x=o dx 3 =o d. dd3Y1 X ~:J. C, (co t/Jt. .rin/2.it. - ,un/Jt. eo,1A41.)+ C2 (co.r,<Jt: uuh/Jt. -uA4t..s,n/,IJt) - C 3 ( 11n/Jt. unh /.U. +cou3J. to.1l>.dt.)-t;(.s1114/.cosAIJt. +co ~.ii.. .s/nAa1..) = 0 2 /J zMo [ • /, /.J • ( ) • ~ • • 1 / . L zA'L . 1 ) .tin ,JI. cot /Jx .r1nh/J J. -x - JI/J.n/.Jt. .r111.ix co.rn.ict.-x Y = - I<( Sinn -s111 "/.II. ) /,J, + ll;,l3L ,osl>l!J~ sin 13(L -,,:). - s1hi0L s/n/,/Jx eoslJ(i.-X)] p p L y oxp;:·ocrn i on Le: obto. L.n.cC. for ·i.:,he c1e1'lect ion ourvo of ·the be irn1: y= 2.iP I( . co.sh ,0x coJl.?(1.. -.t) +co~lix coJ/,~(t. -x) S1/Jh /3L + sin /.JL.. L .Y Y = - k (.r,:f~~sin1.u} [ .r1.nh'8~ coS/4?(L-x) - ~ o.sh~x J//J/.J (L -x) +cost:Jx s1nh/.J(L-x) -.sin,<Jx cosh/.i(L-x)] .IIIe Bolted J"oint with Steel Splice Plates Ansur2ing tha t the bolt fits tightly in the hole so that bcfth an u.pur:.rd· and a dovmv:ru:rd reaction may be developed, the above cqu o.-'~iorn.1 can be applied t.o "i;he deflection elll'Ve of the bolt in o. • ·i~_ted ti:c,1ber joint., f1, ~ y~ _ _j ____ l .... f - · - J X L .Sinoe the thiekness of the splice . plates is small relative • to tha t of the ma.in tLr.iber, a...'11.d sinoe the deformn.tion of the splice pla ten under the bolt is negligible ill oom.pariE~on to that of t.he wood, this case mny be oonsi,clE-)red to b e that of a bemu. ttith ooncen·trated- loads applied at its extremities. · If P d.enote o ·t.he tens ion force in the oentral member of the joint, a load. of P/2. vdll be transmi tte<l to the extremities of the bolt in the eantrn.1 member by the · splice ple.te:1.,, rrhe defl~eticn of Jche bolt a t o.ny section at a distr-m.oe x f!'om the edge of the cent,:ral ne:ml,er is co J h ti x. co, /J ( J. - x,J + c o s /.J x co .rI, /.J (t. - x) sinh/3L + .sin/.3/... The rJ.nr·;nitu.dc of the continuous reaation per unit of length at any section is equal to k times ·the d.ef'leotion a t tha t section .. Renee the unit stress in bearing under the bolt is e(1Ual to kyid, where d is the : dirunet.er of the bolt. If the load we:r:e unlfo:r.,:,-Jly distributed over the length of the bolt, the untt s·tress in bea:ring would be P/Ld. It follqvrs that the ratio of the a ctual bearing stress · at any section to the average bearing stress 1s equal to kLy/P,, • a ctual stress ,average stress l~ Since /3 = V:f£"j , where E and I are the modulus of elas·:jici ty and momsnt of inertia of the bolt, respectively, for o. given length and diameter of bolt I.JL is a constant. In a follov.ririg pa ragraph 1t is shovvn that k is a funotion of the n odulus of eln.stioi ty of the wood and is indeJ.)end~nt of -'che d" /rii L cliar:1eter of t,he bolt. :B urther-.ro.ore, since I= 4 , /JJ. =l.50V'£ -;r 7J 1 That is, 13L is directly proportional to the L/d. ratio of: the bolto ' ' Hence tho follovrlng discussion will be· in tar.ms of /JL , , since this terr:1 is the more genera l and may be applied to a11y speoies of wood and to any strer.1:gth bolt. On sheet 10 a re plotted ou.rves shovdng the dis.t ribution of stress along the length of the belt for various ·.values or /JL ~ ~che.. cUi·ves a1--e symmetrica l a bout the oenter line of the timber mo:'.1ber.. For small values c-f /Jl (henoe sm.a ll va lues of ' . the L/d ra ti~) the stress is substantially unif'o11n4 for }1igher vrJ.lues of/JI. ·there are high stress conoer:rbra tions at -the edges of t ho timb-e-1"« Fer val·ues of /Ji. 4 and ebov~ a .. crve~ only ,1 ~m.B.1 1 re1rtlon of' the len.gth of the bolt. r.l:h@f:'. e "u.rves o.lso represen.t the relative deflections of the bol te On sheet· 11 in })letted o. curve shovring the var;_ati'-tn of tb.e nw:.:xirn:ran str~ss ,. oeQur-ring at the edge of the timber, '\-Yi t h /Ji. 11 Jlor values ot BL of 7 and higher, the maximum. stress inore2ses in. di re et- proport,ion i,\1, th BL"' It is of inte1•0f.1 t to note haw the bending moment i n the bolt varies ·wit.h IJL. · .An OXJ?1·ession -for the bending moment at an~r section is us follews: M·= _EI d2.Y = _ _E_ .rin/2/.3x .sin/.J(L-x) + .sin/.ix ui/)/2/.J(J..-x) . • dx. 2 2/J ,t1p/2/.JL+s1n/.J/.. The mi nus s ign results from the ohosen direction. of the coordina te axes " The ratio of the bending moment to PL~ neglecting the minus sign, is expressed in ter-111s ot I.JI. G.loneo M I -~ Pi. 2/.U.. lt/>h /.J X- .sin /J(J.. -x) + .rinA x s1Al,.i(J. -.x) .sin/J/.JL + .rin/.JJ. On r;11.cc·0 12 are plotted. ourves shovrl.ng the variation in b0miing moment along the length of the bolt tor various v a.1 1.J.es of IJL o As /JL increases II the point of ma.xim.u.m moment aprirne.,ehes the edge et t he timber . For high va.l ues of /J L (10 and above) a reversal. of bendi ng mcusnt occurs, but the nngn,it-uo.e of" ·the n.ege. tive msment i s m!llill and cannot be nho\m l1o·v; t~l10 na:d.rr.iun bonc.1ing xa0Bent cLeoroases v;rit h inoreasi:r1g vaJ.uen of /JL o rv o Dolt;ed J oint t-dth Woec'. Spl ioe Plates ~ _L _ _ 7-L p ~ .Y Mo X Mo M{ X y .~ e; Since the bending of t110 bolt in tho side mara.lier must be corn3id.0red, cu1d since tlw foundation is disoontinuous at the ;:. -•_-3 _·_:+'~:7 :_.~:_-.! -1-~+----+---'+--+"'--. _.,, , ., • - !:... :· 1; •- .L:-:--:-!c- - - ' • -'-~':----+---~-+-----+ ·-· . • ~:- -:..h~:-1--,.-._-+---+--+-----J __ , • • 1 - - -• 1• -••I •• .. ,..... - ,·.,-~t.-· I • ' ! :- , : -; .: . ; '.. -~ ~ t~ -i i :. ; :-~ .- • • - ,.-- 7--- i l ~~~L --~--- ,. HLJ •• : ::·i: ·: _. ·1 •. ! . . • ... . . :·• •wJ _T-~1~~~-: ' •• I ,- . :: j \~: \ '1·" 1•··· :=-:r:~.~ :· ,' ::~J::J . ,. . . · 1-- .,. · 1 !- ,·; r~: . . :-1: ... ·t-t-:--;-r:-:::-n:'. :r :-:·: :::::l=. · · ·:;~: Hiir • : _;- ~ I • : .:-r :\ n~:~ - -~.. . > __:·. I -- - , _:.:r>: . .. f ·, ·· - .:.:..:.L:.: +---::~T-:::::!::: : T :• ::.1:-· • •• • l • -C\;rt:- · ~ ~ ~ IB}J.~ ---l-:. _1 .. - ! . : : ·-t••·•-· • I' ' I - I • , eci.gea oi' t he ee~tre.l tira.ber • this case must b~ broken down into that of tv'TO sepa:i.~uta beam.s aa shown above • . '?ae bending monont in the bolt, M0 , a t the edge of the oentral timber, n c:1 :/ he dot,ei~ined by the 1,rinoiple of eo1l'tinui ty. That is» n::.mu.1ning rote.tion of ·the side mer'l'oers is prevented. 5 t....'1.e slope of' t~ho tlo:£".leotion ou.1--ve 01' t,he bolt a t the point of disoonti~- Fo1., the center ti..T11.ber, YM0 = 2/.JzMo sinlt/J.x. co, tJ(L-x)-co,,/,Qx11hAl1.•x) •c1u I(!.¥ ~ln.4/J(L-x) -sin.ix cos/,R(t.-~) I< .sil'1h /JJ.. + ,1in/JJ.. . "";,4/:,04/JX. co.rh/.3(;--x)- s,'nl.lJ:e61l>/.Jxco.r/J(-t-x) /JP V: JL I 6 /JL - sin 1~ ~~2 I< sinh 2 2 diseonti,nuityj ·the value of M0 is as follaws: For small values of/.il, the, above equation should result in the same expression fer Mo aa for the case of unif'orxn stress distribution. For very sm....9.ll angles the sine .an.d sinh are equal to the angle, and the cosine and cosh are equa.1 to 1. · Mald.ng tJi.is subst~tution, the above equation red,oes to M0 ' • FL/8, ?rhioh is correct :for uniform stress distribution. On sheet 17 is plotted a curve sho'Wing the variation of TuJ.o ,;Ji th valuas of /JL • Fer values of /JL above 6, Mo is eque.l. to z$:ro. Taking a free body diagram ot the side membett and the portion of the bolt within the side :member, if the only forces acting on the free body are the tensi0?1 force P/2,. the shear in the bolt P/2 1 :md M0 , then Mo must be equal to . FL/8,. Assurdng no rotatio:n of the side member, Hi0 is less than J(L/8 for the higher values of /JL ; hence there must be crthErro to1"ces acting on the. sido •member,;. This problem 1s discussed in a later pax-a.gra:Ph• As befol'e, the ratio of the.. actual bearing stress to the average bearing stress is equal to k!.q/P for both the center and 's ide timbe:r~ This r atio for the two members is siln.1.la:r to that for the ease Qt steel aplioe plates, being in terms of /Ji.. i The curTes 6n sheet 18 show 'tbe va:tiation ot stress along ·tha bolt for various values of' /JL.. r:i:heee curves clearly incl ioa te t he high stress ocncentration at the edges of the member8 and demonstrate tha t the actual bearing is distributed over only a szn.a.ll :portion of' the bolt length for higher values of /J.L • V. Datar..nina•cion of k, the Modul.~~ o'f the Wood The modulus of the v,rood may be approximated analytioally b y considering tha defor1nation ot rm elastio bod:y under direct boa.ring s"'u:ress. Consider the oas e in \\1:lich a imi:fom pressure of p lbs. per sq. inoh is applied over a vridth d. a.cross the surfa ce of a s emi-infinit.-e pla te, 1"or w:hioh ease an expression for the deflection of -the :plate has been developed: ){. I Th:1.s eas e ma y be a:pplioa. to the bearing ot the bolt in a t:l.ra.ber • if it be oons i dored that the bee.rlll-S stress i s distributed to the vmod en a pl ane through the center of the bolt. Let d r ep1. , es ent the di ameter 01· the bolt and b the width ef' the timber through w:hioh t he bolt is inserted. * 1:ormul r:s lor Stress_ and Str.._-a,in, by Ra~nd. · J.· Roar · k , p.. (-.'"'4t! . . J _.., ,.u r--.- ' I . : 1·· -~ ---,--. -,-- . . I I - . I • ___,: f? -· ,i _: ___ _ . I . - ! ' . . •- ~ ' ' ~! ---- _ f ~+,:-:-=~+;-:::.:~+_::_:_~-+--,~HT"--fJ-""-fr,J.+:-...,..+---:-J---4-~-+~=--+~~f-:-:-,-:--l-r.-:-:t:---,-r.;::-:+-r,-t-~t , ;tf?j - l~ , J ~ :-:r~ -• ~l~ ;J 1-- ~ - ~ v = deflection relative to a remote point A a distance m from " tho edge of the loaded s.rea V :::: Pois3on' s ratio a ) At any. point 01 outside of loaded -area, b-d At the eclge of the tiraber, x,:. - r - , and . b) At any point inside the loaded area, At the oenter of the bolt, X =.Ji. , and 2 2 _ 2pd/. 2,n • J- nE ".Jed+ j:,d c) ( J-Y) n£ y, =d.efleotion a-'.; eenter o:r bolt relative to the edge of the rn.etnber Letting b • ndt ovei- the ordinary range of values ot n, from 5 to 15, the -20- va1~1e.tion ot y 1 is small, and the average val~e ever this 2J>d re.nr;e 1.s y, = "c . From the definition et' the modulus of the vrood, 1 t f ollov-.rs that . _.£ /( -_,od y, - 2 Since the equ.atiens employed in this derivation are based upon a.~ homogeneous, elastically isotropic body, and sinae wood is neither homogeneous nor isotro:r;.i1¢, it is understood that the above expression for the modulu.s of the \rood is merely an ·approxi111ation. VI. Dete:rraination of Allowable Average Bearing Stress For the case of a bolted joint vdth steel splice plates, the maximum deflection, which occurs at the edge of the timber, . is given by /JP Yma.i..= K co.sh /JI..+ to,t /J'.sinhl'JJ. + -1/n/JJ. Also,. t,hc rimximum bearing stress under the bolt is given by HenceJ if the rnaximum bearing stress be f'ixed 1 the allowable load on the joint is '· s/nhl3L. + .sin/Jt. p =fsllfa)(-f} eo.rh,(JL + e.o.s /.JI.. Since the average bearing stress is equal to P/Ld, it fo1lows that th~ allovrable · average bearing stress, s, is ll'or design purposes it is necessary to. express the allowable bearing stress in terms of the L/d ratio of the bolt. Consider the ease of a bolted joint ,_with steel splice plates loaded :parallel to tho grain, using Douglas Fir (Qoast region). ~v c 1. 2 x 10 6 lbs. ~er sq. inch, smax m 1300 lbs. per sq. . ineh, and E8 = 29 :z:. 10 6 lbs. per sq,, inch .. . . Hence k, the modulus of the vmod, is aqua.l tQ Ew/2 = o._;f x 10 6 lbs., per sq. inch. !3L ~1.soVf-{~) L :;(). S77 .. .s Stnd¥,. = This ratio of t he allovra.ble uniform stress to the basic stress i s I>lotted against the L/'d ratio on sheet 25. It may be seen tha t the allovmble st1·ess drops off rapidly vvith increasing L/d r atios. The redu°'~Jon of the basic stress recommended by the Forest Products Laboratory is less than that indicated by this theoretical curve. r.rhts diserepanoy ~ be due to three f a cr1;ors: l) _'fflle bolt a ctually does not fit tightly in the tir~ber ; 2) the deternined modulus of the wood is incorrect; and 3 } the xn.axj.mura edge str13as at ·wl11oh the fslip of' th$ joint oonn os to be proportio_nE.1 to the a.]!pliecl loe.a. cl.ON~ not ~o-rres pona. to the prop0rtional iirnit stress of the ,med :ltselt. Ho-r1eve1.. , i f the theoretical curve be ad.justed. at one :point, the rema ining portion of the curve closely follows the Forest :Produots Labor~.tory reommnend.a.tions. If the joint· is ~oaded perpendicrular to ·the grain of the r,rood, . this theo:ret1ea.l appX.a.oh f'ails . In this case the wood :t'ihers under tht3 bolt a~e pla.eed in tension, thus affording greater bearing ca:paoity than normal • . The bearing capn•c ity .i s ralat~d to th~ vddth of the bolt, henoe the modulus of the wood is not independent of the diameter of the bolt. VII. Tenoion in the Belt ' z 2 "'> j' "'> t,.. - ·- - _l__ - - --- - I--- - -=--r- --- ~ L✓.? L L/2 I 'P It 'f:'jas stated that .under the action of the tans ion in the u1 the member, tht shE:Jar in the bolt, and 110 , the side member t ... '. .. : . . . . .. ·- · ·-+~-,----c-t---r -~ h,•<--1- ,--+- 7+ - - + - --+- J.n not in eov.ilihriu:m for t i s h L/d :ratios. This i s due to ·~.he i'o.Gt tho.t :t'oi· hi?;b. L/d r atios r-,!o i s not ~qu.al to PL/o. A po::tion of ·the excrnss norn.ent is t ~'llten u:p by btmdlng of' th& rdde •meniber, sJld_ the remainder i s ·~aken by the tans ion in t he bolt and compression betvreen the side and oentrul 1-aeTJ.bo!". If it be sssu.n.ed that bending of the side member is negligible, tho t'lagni tudG. of the 1;en.sion in ·che bolt may be <1etei-dtl:n.ed .. ,.o~ is the t. es ign :practice, let the distance from the holt te the end ot the . eeimtral mam.ber- be equal to 7d. Ass-ume thnt th~ intensity of the compressive 3tress varies lin~arly from the bolt to t he end ef the central member. T(-o/) = f (f)-Mo T{ 14 L) = el:._ -Br'L JR 8 · T =f, (f-B)RP , The ou...~e on sheet 25 shev,JU hm1v the tension in the bott varies It is 'Wlderntced that ·due to 09ding of t.he l'.!lQ'.ln.ber itself the tension. in the bolt is less than that ind.ica·i,eu by the above expression._ I . .. . --+-~~+