1 ADHESIVE BONDED JOINTS

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1 Verson... Theorecl Bcgrond of SAComp Anlyses V.. ADHSIV BODD JOITS Flemmng orensen nd Ole Thyo Thomsen (Alorg Unersy Inse of echncl ngneerng Denmr ) The mehod sed n SAComp for engneerng nlyss of dhese onded jons of ros complees s presened. The jons consdered re dded n o ypes: sndrd nd dnced. The sndrd jons conss of o or hree dherends onded ogeher h srgh connos dhese lyer prllel o he n-plne drecon of he dherends. The dnced jons conss of o dherends onded ogeher h eher sngle or dole-sded scrfed dhese nerfce. The dherends re modelled s ems or ples n cylndrcl endng. They re formed from lmnes h rrry ly-ps sng he clsscl lmnon heory (CT). The dhese lyer s modelled y o-prmeer elsc fondon model here he dhese lyer s ssmed composed of connos lyer of lner enson/compresson nd sher sprngs. Snce non-lner effecs n he form of dhese plscy ply n mporn role n he lod rnsfer he nlyss llos nclson of non-lner dhese properes y n ere mehod sed pon he lner-elsc pproch. The lod nd ondry condons cn e chosen rrrly. Approches for predcng he cohese flre n he dhese lyers nd lmne flre n he jon re re lso presened. SYBOS A j lemen of he dherend n-plne sffness mr B j lemen of he dherend coplng sffness mr c f C S C V Consn sed n sysem eqons Consn sed n effece sress formlon Consn sed n effece sress formlon D j lemen of he dherend endng sffness mr e e h f I I D J Adhese elsc modls ffece srn ffece srn n he h eron sep Adhese elsc sher modls Consn sed n sysem eqons Frs nrn of he generl srn ensor Second nrn of he deorc srn ensor Frs nrn of he generl sress ensor V Jons Adhese Bonded Jons

2 V.. Verson. Theorecl Bcgrond of SAComp Anlyses.. J D g m f RF dh y y y RF dhprop RF FPF s * s s s prop s () y Second nrn of he deorc sress ensor Consn sed n sysem eqons engh of he oerlp zone n he dhese jon engh of dherends osde he oerlp zone yy Adherend momen reslns Consn sed n sysem eqons yy Adherend n-plne sress reslns yy Adherend sher force reslns Resere fcor for cohese flre of dhese (lner or non-lner dhese model) Resere fcor for proporonl lm of dhese (non-lner dhese model) Resere fcor for dherend (lmne) frs ply flre n he cny of he jon ffece sress Clcled sress n he h eron sep permenl sress n he h eron sep Sress proporonl lm Dfference eeen clcled nd epermenl sress Adherend hcness Adherend hcness s fncon of Adhese lyer hcness Adherend n-plne coordne sysem n he longdnl drecon ongdnl dsplcemen of he dherend md-plne (-drecon) ongdnl dsplcemen of he dherend (-drecon) Dsplcemen of he dherend md-plne n he dh drecon (ydrecon) Dsplcemen of he dherend n he dh drecon (y-drecon) Trnserse dsplcemen of he dherend (z-drecon) Trnson ngles of scrfed dherend Roon of md-plne norml o he dherend Wegh fcor for he chnge n elsc modls V Jons Adhese Bonded Jons

3 Verson... Theorecl Bcgrond of SAComp Anlyses V.. f Prncpl srns (f = ) Ro eeen compresse nd ensle yeld sress n n c Adhese lyer o-of-plne norml sress Adhese lyer o-of-plne norml sress Adhese lyer o-of-plne norml sress Compresse yeld sress f Prncpl sresses (f = ) y n Tensle yeld sress Adhese lyer sher sress Adhese lyer sher sress Adhese lyer sher sress Sscrps Adhese lyer Adherend ( = ) Ieron nmer for non-lner ngen modls Dfferenon h respec o he -coordne y Dfferenon h respec o he y-coordne l Ulme Sperscrps Adherend ( = ) end Adherend end hcness he oerlp zone end Adherend hcness he lef end of he oerlp zone endr Adherend hcness he rgh end of he oerlp zone Idenfer sed for non-lner ngen modls V Jons Adhese Bonded Jons

4 V.. Verson. Theorecl Bcgrond of SAComp Anlyses... ITRODUCTIO Jonng of compose srcres cn e cheed hrogh se of oled reed or dhese onded jons. The performnces of he menoned jon ypes re seerely nflenced y he chrcerscs of he lyered compose merls dhese onded jons prode mch more effcen lod rnsfer hn mechnclly fsened jons. Accre nlyss of dhese onded jons for nsnce y sng he fne elemen mehod s n elore nd componlly demndng s s descred y Crocrome e l. [] Hrrs e l. [] nd Frosg e l. [5]. Hence here s n oos need for nlyss nd desgn ools h cn prode ccre resls for prelmnry desgn prposes. Ths chper nrodces he nlyss pproch sed n SAComp for deermnng he sress nd dsplcemen felds n commonly sed dhese onded jon confgrons. The ls secons del h he hndlng of plscy effecs n he dhese lyers nd flre predcon of onded jons. The onded jon ypes consdered n SAComp re: Sngle lp jon (S) Sngle srp jon (SS) Bonded doler (BD) Dole lp jon (D) Dole srp jon (DS) Sngle sded scrfed lp jon (SSC) Dole sded scrfed lp jon (DSC) These jon ypes re llsred n Fgre.. All he jon confgrons cn e composed of smlr or dssmlr lmnes h n rrry ly-p. The jons re sjeced o generl lodng condon s shon n Fgre.. V Jons Adhese Bonded Jons

5 Verson... Theorecl Bcgrond of SAComp Anlyses V..5 Fgre. Schemc llsron of n dhese sngle lp jon sjeced o generl lodng condon. Accordng o he compley of he jons he lp nd srp jons nd onded dolers cn e referred o s sndrd jons. In hese jons he dhese lyer or lyers re prllel o he nplne drecon of he dherends. Correspondngly he scrfed jons cn e referred o s dnced jons. The dnced jons re more effcen de o he redced eccenrcy of he lod ph he dnced jons re lso mch more epense o mnfcre nd hey re herefore only sed for hgh-performnce pplcons.. STRUCTURA ODI The srcrl modellng s crred o y dopng se of sc resrce ssmpons for he ehor of onded jons. Bsed on hese resrcons he conse nd nemc relons for he dherends re dered nd he conse relons for he dhese lyers re doped. Fnlly he eqlrm eqons for he jons re dered nd y comnng ll hese eqons nd relons he se of goernng eqons s oned... odel dmensons The dhese onded jon confgrons ere nrodced n Secon.. The dherend hcnesses re gen y nd for ll he jons osde he oerlp zone. For he dole lp jon he hcness of he hrd dherend (loer dherend) s. The dherend lengh osde he oerlp s nd nd he lengh of he dhese lyer s s llsred n Fgre.. V Jons Adhese Bonded Jons

6 V..6 Verson. Theorecl Bcgrond of SAComp Anlyses.. Fgre. Illsron of dherend lenghs nd dhese lyer lengh nd hcness. Insde he oerlp zone ( ) he hcnesses re: Sngle lp jon sngle srp jon nd onded doler: (..) Dole lp nd dole srp jon: (..) Sngle sded scrfed lp jon: end end (..) end Where he sperscrp end n end ends of he oerlp see Fgre.. nd end refers o he hcnesses of he dherends he free Fgre. Thcnesses nd scrf ngle for sngle sded scrfed lp jon. V Jons Adhese Bonded Jons

7 Verson... Theorecl Bcgrond of SAComp Anlyses V..7 Dole sded scrfed lp jon: end end R end end end R end end R end end end R (..) end end R Where he sscrps nd nd he sperscrp end n nd refer o he hcness of dherend he lef () nd rgh (R) ends of he oerlp oe nd elo dherend (Fgre.5). Fgre.5 Thcnesses for dole sded scrfed lp jon... Bsc ssmpons for he srcrl modellng The sc resrce ssmpons for he srcrl modellng re he follong: Adherends The dherends re modelled s ems or ples n cylndrcl endng sng ordnry Krchhoff ple heory ( oe-krchhoff ssmpons). The conse ehor of he dherends s oned sng he clsscl lmnon heory (CT). o resrcons re se on he lmne ly-p.e. nsymmerc nd nlnced lmnes cn e nclded n he nlyss. The lmnes re ssmed o oey lner-elsc conse ls. The srns re smll nd he roons re ery smll. Adhese lyers V Jons Adhese Bonded Jons

8 V..8 Verson. Theorecl Bcgrond of SAComp Anlyses.. The dhese lyers re modelled s connosly dsred lner enson/compresson nd sher sprngs. on-lner dhese properes re consdered y sng secn modls pproch for he non-lner ensle sress-srn relonshp n conjncon h modfed Von ses yeld creron. ods nd ondry condons The srcrl model llos ondry condons o e chosen rrrly s long s he sysem s n eqlrm. Ses of prescred eernl lods (n-plne nd o-of-plne forces nd endng momens) nd geomerc ondry condons re defned n SAComp o od selecons of nconssen ondry condons hch cn led o snglry prolems n he sysem of eqons. The sysem of goernng eqons s se p for o dfferen cses.e. he dherends re modelled s ples n cylndrcl endng or s de ems. In he follong he cse here he dherends re modelled s ples n cylndrcl endng s prmrly consdered snce he modellng of he dherends s ems s redced cse of hs... Conse relons for dherends modelled s ples For he prposes of he presen nesgon nd h references o Fgres.6 nd. cylndrcl endng cn e defned s de ple (n he y-drecon) here he dsplcemen feld cn e descred s fncon of he longdnl coordne only. As conseqence he dsplcemen feld n he dh drecon s nform. Ths he dsplcemen feld cn e descred s (..5) here s he md-plne dsplcemen n he longdnl drecon (-drecon) s he md-plne dsplcemen n he dh drecon (y-drecon) nd s he dsplcemen n he rnserse drecon (z-drecon). The dsplcemen componens nd re ll defned rele o he md-plne of he lmnes nd = corresponds o he lmnes nd respecely. Bsed on he erler ssmpons he follong holds lso re: (..6) y y y yy V Jons Adhese Bonded Jons

9 Verson... Theorecl Bcgrond of SAComp Anlyses V..9 Fgre.6 Schemc llsron of dhese sngle lp jon clmped eeen o ercl lmnes hch preen he dherends of he sngle lp jon from mong nd rong freely n he dh drecon. Ths represens he concepl nerpreon of cylndrcl endng s defned n he presen formlon. In he concep of cylndrcl endng he ondry condons he ondres n he dh drecon re no ell defned. Hoeer s ssmed h here re some resrce consrns on he ondres sch h he ondres re no cple of mong freely. I shold e noed h he concep of cylndrcl endng s no nqe nd h oher defnons hn he one sed n he presen formlon cn e doped see Whney []. Sson of he qnes n q. (..5) no he conse relons for lmned compose merl ges he conse relons for lmne () n cylndrcl endng []: yy y A A A 6 A A A B -B -B 6 yy y B B B 6 B B B D -D -D 6 (..7) here A B j j nd D j (j = 6) re he eensonl coplng nd flerl rgdes sed on he clsscl lmnon heory (see Pr III Chper ). sress reslns dherends he rgdes yy nd A B j j y nd yy nd y re he n-plne re he momen reslns. For he jons h scrfed D j (j = 6) hn he oerlp zone re chnged s fncon of he longdnl coordne n ccordnce h her defnon.e. A j s chnged V Jons Adhese Bonded Jons

10 V.. Verson. Theorecl Bcgrond of SAComp Anlyses.. lnerly B j s chnged prolclly nd D j s chnged cclly (j = 6) eeen her les he ends of he oerlp zone. Ths s of corse n ppromon snce he cl sffnesses of he lmnes re chngng y chnges hn he lyers s fncon of he longdnl drecon... Conse relons for dherends modelled s ems odellng of he dherends s de ems cn e consdered s specl cse of cylndrcl endng. When he dherends re modelled s ems he dh drecon dsplcemens re no consdered nd only he longdnl nd ercl dsplcemens re nclded. Ths he dsplcemen feld n q. (..5) s redced o (..8) For hs cse he conse relons for compose em re redced o A -B B -D (..9)..5 Knemc relons From he oe-krchhoff ssmpons he follong nemc relons for he lmnes n cylndrcl endng re dered: y z (..) here s he longdnl dsplcemen s he longdnl dsplcemen of he md-plne nd s he ercl dsplcemen of he h lmne. The nemc relons of q. (..) re he sme for he em cse s for he cylndrcl endng cse ecep h ll he rles ssoced h he dh drecon re nl...6 Conse relons for he dhese lyer The coplng eeen he dherends s eslshed hrogh he conse relons for he dhese lyer hch s frs ppromon s ssmed homogeneos soropc nd lner elsc. The conse relons for he dhese lyer re eslshed y se of oprmeer elsc fondon pproch here he dhese lyer s ssmed o e composed of connosly dsred sher nd enson/compresson sprngs. The conse relons of he dhese lyer re sggesed n ccordnce h Thomsen [6 7] Thomsen e l. [8] nd Tong [9]: V Jons Adhese Bonded Jons

11 Verson. V.... Theorecl Bcgrond of SAComp Anlyses V Jons Adhese Bonded Jons j j j j j y j j j j (..) here nd j re he nmers of he dherends s he sher modls nd s he elsc modls of he dhese lyer. The conseqence of sng he smple sprng model pproch for he modellng of he dhese lyers s h s no possle o ssfy he eqlrm condons he (free) edges of he dhese. Hoeer n rel dhese jons no free edges re presen he ends of he oerlp snce flle of srpls dhese so-clled spe-flle s formed he ends of he oerlp zone. Ths spe flle llos for he rnsfer of sher sresses he oerlp ends. odellng of he dhese lyer y sprng models hs een compred h oher non nlyss mehods sch s fne elemen nlyss (Crocrome e l. [] nd Frosg e l. [5]) nd hgh-order heory pproch ncldng spe flles (Frosg e l. [5]). The resls sho h he oerll sress dsron nd he predced les re n ery good greemen.. UIIBRIU UATIOS The eqlrm eqons re dered sed on eqlrm elemens nsde nd osde he oerlp zone for ech of he consdered jon ypes... Adherends osde he oerlp zone The eqlrm eqons re dered for ples n cylndrcl endng snce he eqlrm eqons for he em modellng cn e consdered s redced cse of hs. The eqlrm eqons osde he oerlp zone for ech of he dherends.e. n he regons - nd + re ll he sme (see Fgre.) nd re dered sed on Fgre.7: nd y y y (..) here correspond o he dherends =.

12 V.. Verson. Theorecl Bcgrond of SAComp Anlyses.. V Jons Adhese Bonded Jons Fgre.7 qlrm elemens of dherend osde he oerlp zone; - nd +... Sngle lp nd sngle srp jons The eqlrm eqons nsde he oerlp zone for he sngle lp jon nd he sngle srp re dered sed on Fgre.8. For he sngle lp jon he dherend hcness ll remn he sme n he enre oerlp zone s specfed y qs. (..) hs gng he eqons: y y y y y y y y y y (..)

13 Verson... Theorecl Bcgrond of SAComp Anlyses V.. Fgre.8 qlrm elemen of dherends nsde he oerlp zone for jons h one dhese lyer nd srgh dherends;... Bonded doler Insde he oerlp zone for onded doler jon he eqlrm eqons re dered sed on Fgre.8. nd qs. (..) nd yelds ecly he sme eqons s for he sngle lp jon (see he preos secon)... Dole lp nd dole srp jon The eqlrm eqons nsde he oerlp zone for he dole lp jon nd he dole srp re dered sed on Fgre.9: V Jons Adhese Bonded Jons

14 V.. Verson. Theorecl Bcgrond of SAComp Anlyses.. Fgre.9 qlrm elemen of dherends nsde he oerlp zone for jons h o dhese lyers nd srgh dherends;. V Jons Adhese Bonded Jons

15 Verson. V..5.. Theorecl Bcgrond of SAComp Anlyses V Jons Adhese Bonded Jons y y y y y y y y y y y y y y y y y (..)..5 Sngle sded scrfed lp jon The eqlrm eqons nsde he oerlp zone for he sngle sded scrfed lp jon re dered sed on Fgre.. They re dfferen from he erler ones de o he lner chnge of he dherend hcnesses nd he slopng ond lne: end y y y y end y y y y end end y y y y (..)

16 V..6 Verson. Theorecl Bcgrond of SAComp Anlyses.. here he relonshp eeen n q. (..) nd n n shon n Fgre. s eslshed hrogh eqlrm: n cos sn cos sn cos cos (..5) n here s he scrf ngle of he dherends n he oerlp zone (see Fgre.). Fgre. qlrm elemens n he oerlp zone for sngle sded scrfed lp jon (scrf ngles );. The dherend hcnesses () () ry lnerly hrogh he oerlp lengh s specfed y q. (..). V Jons Adhese Bonded Jons

17 Verson. V..7.. Theorecl Bcgrond of SAComp Anlyses V Jons Adhese Bonded Jons..6 Dole sded scrfed lp jon Fnlly he eqlrm eqons nsde he oerlp zone for he dole-sded scrfed lp jon re dered sed on Fgre.: end R end y y y end R end y y end R end y y y y end R end y y y y y y y y y (..6) here () () nd () re he dherend hcnesses ccordng o q. (..) nd nd re he dhese lyer hcnesses.

18 V..8 Verson. Theorecl Bcgrond of SAComp Anlyses.. Fgre. qlrm elemen of dherends nsde he oerlp zone for dole sded scrfed lp jons;. The relonshp eeen n q. (..6) nd n n shon n Fgre. s ell s he relonshp eeen n q. (..6) nd n n shon n Fgre. s eslshed hrogh eqlrm: cos sn cos sn cos cos (..7) n n V Jons Adhese Bonded Jons

19 Verson... Theorecl Bcgrond of SAComp Anlyses V..9 here (=) s he scrf ngles of he dherends n he oerlp zone (see Fgre.5).. TH COPT ST OF SYST UATIOS From he eqons dered s possle o form he complee se of sysem eqons for ech of he onded jon confgrons. Ths comnon of he conse nd nemc relons.e. qs. (..7) nd (..) ogeher h he conse relons for he dhese lyers.e. qs. (..) nd he eqlrm eqons led o se of 8 lner copled frsorder ordnry dfferenl eqons descrng he sysem ehor of ech of he dherends. The ol nmer of copled frs-order ordnry dfferenl eqons hn he oerlp zone s herefore 6 for jons h o dherends nsde he oerlp zone nd for he jons h hree dherends nsde he oerlp zone. Osde he oerlp zones he sysem ehor for ll he jons s descred y 8 lner copled frs-order ordnry dfferenl eqons ecep for he dole lp jon hch hs o dherends n he regon + nd herefore s descred y se of 6 lner copled frs-order ordnry dfferenl eqons n hs regon. The se of goernng eqons for ll he consdered dhese onded jon ypes re presened n hs secon. The goernng eqons presened re hose here he dherends re modelled s ples n cylndrcl endng. The cse here he dherends re modelled s de ems cn e consdered s specl cse of cylndrcl endng nd resls n redced se of he goernng eqons. Hoeer o demonsre h hs s re he goernng eqons for he cse of dherends modelled s ems re lso shon for he sngle lp jon... Sngle lp nd sngle srp jons From he eqons dered s possle o form he complee se of sysem eqons for he prolem. Ths comnon of qs. (..7) (..) nd (..) yelds for he lmne nd n he res - nd + (osde of oerlp): y 7 y 5 y 8 y 6 9 (..) V Jons Adhese Bonded Jons

20 V.. Verson. Theorecl Bcgrond of SAComp Anlyses.. V Jons Adhese Bonded Jons qs. (..) conse se of egh lner copled frs-order ordnry dfferenl eqons. The coeffcens I 9 ( = ) conn lmne sffness prmeers nd re resl of solng nd from y nd n qs. (..7): c c c c c c c c c h h h h h m m m m m (..) here he coeffcens c j h j nd m j (j=) re: h B c A m h B c c A m c c A h B A m c D B h D c D c B h c D c B D B h D B c D B B A c D B B A c (..) Whn he oerlp zone.e. for comnon of qs. (..7) (..) (..) nd (..) yelds for he lmne nd :

21 Verson. V.... Theorecl Bcgrond of SAComp Anlyses V Jons Adhese Bonded Jons y y y y y y y y (..) qs. (..) conse se of 6 lner copled frs-order ordnry dfferenl eqons... Sngle lp jon h dherends modelled s em For he lmnes nd n he res - nd + (osde of oerlp) comnng qs. (..) (..9) (..) ogeher h he eqlrm eqons yelds: 6 5 y y (..5)

22 V.. Verson. Theorecl Bcgrond of SAComp Anlyses.. V Jons Adhese Bonded Jons qs. (..5) conse se of s lner copled frs-order ordnry dfferenl eqons. The coeffcens 9 ( = ) conn he lmne sffness prmeers nd re deermned y solon of nd from nd n q. (..9): D B B B B A D D B B D B B A D B A (..6) By comprson h he coeffcens for he cylndrcl endng cse s seen h he coeffcens for he em cse re srongly redced nd only conn fe of he lmne sffness prmeers. Whn he oerlp zone.e. for he goernng for lmne nd re: (..7) qs. (..7) conse se of lner copled frs-order ordnry dfferenl eqons. By comprson h he eqons for he cylndrcl endng cse s seen h he eqons

23 Verson... Theorecl Bcgrond of SAComp Anlyses V.. dsply he sme oerll ppernce ecep h ll rles ssoced h he dh drecon re nl n qs. (..7)... Bonded doler The goernng eqons for he onded doler jon re ecly he sme s for he sngle lp jon n he oerlp zone nd osde he oerlp zone n he regon +... Dole lp jon The goernng eqons for he dole lp jon re ecly he sme s for he sngle lp jon n he regon. The goernng eqons for lmne nd hn he oerlp zone.e. for re dered y comnng qs. (..7) (..) (..) nd (..): V Jons Adhese Bonded Jons

24 V.. Verson. Theorecl Bcgrond of SAComp Anlyses.. V Jons Adhese Bonded Jons y y y y y y y y (..8) y y y y

25 Verson... Theorecl Bcgrond of SAComp Anlyses V..5 qs. (..8) conse se of lner copled frs-order ordnry dfferenl eqons hn he oerlp zone. The goernng eqon for he lmnes nd n he regon + (osde of oerlp) re dered y comnng qs. (..7) (..) nd (..): y 7 y 5 y 8 y 9 6 y 7 y 5 y 8 y 9 6 (..9) qs. (..9) conse se of 6 lner copled frs-order ordnry dfferenl eqons...5 Sngle sded scrfed lp jon The goernng eqons osde he oerlp zone re he sme s for he sngle lp jon.e. qs. (..). Whn he oerlp zone.e. for he goernng eqons for he lmnes nd re dered y comnng of qs. (..7) (..) (..) nd (..): V Jons Adhese Bonded Jons

26 V..6 Verson. Theorecl Bcgrond of SAComp Anlyses.. V Jons Adhese Bonded Jons end y y y y end y y y y (..) qs. (..) conse se of 6 lner copled frs-order ordnry dfferenl eqons...6 Dole sded scrfed lp jon The goernng eqons for he dole-sded scrfed lp jon re he sme s for he sngle lp jon osde he oerlp zone. The goernng eqons for he lmnes hn he oerlp zone.e. for re dered y comnng of qs. (..7) (..) (..) nd (..6):

27 Verson. V..7.. Theorecl Bcgrond of SAComp Anlyses V Jons Adhese Bonded Jons end y y y y y y y y (..)

28 V..8 Verson. Theorecl Bcgrond of SAComp Anlyses.. V Jons Adhese Bonded Jons end y y y y qs. (..) conse se of lner copled frs-order ordnry dfferenl eqons hn he oerlp zone..5 BOUDARY CODITIOS To sole he dhese onded jon prolems he ondry condons nd conny condons he o e sed. The conny condons ms e sed he ends of he regons n hch he jon s dded s shon n Fgre.. In he follong he ondry condons nd conny condons re sed for he dfferen jon ypes..5. Sngle lp jon The ondry condons for sngle lp jon re jncon cross Conny dherend dherend jncon cross Conny dherend dherend or or or or prescred y y y o : : : : : : : : (.5.) The ondry condons for he dherend = nd for dherend = re dered from he ssmpon h he dherend edges re free see Fgre..

29 Verson... Theorecl Bcgrond of SAComp Anlyses V Sngle srp jon The ondry condons for sngle srp jon re : : : prescred : Symmery : dherend ( f dherend ( f dherend ( f dherend : dherend : dherend ( f y or or or or y or o ) : or ) : y Conny cross jncon ) : ) : y y y y y (.5.) The ondry condons = re dered from he ssmpons h here s symmery rond he cenerlne of he srp jon. For dherend = s ssmed h he dherend edge s free..5. Bonded doler Smlr ssmpons s for sngle srp jons re sed onded dolers hch yelds he follong ondry condons: : : : Symmery : dherend ( f dherend ( f dherend : dherend : dherend ( f dherend ( f y ) y ) dherend : Conny cross jncon y prescred : or or o or or y : ) : : ) : y y y y y y (.5.) V Jons Adhese Bonded Jons

30 V.. Verson. Theorecl Bcgrond of SAComp Anlyses...5. Dole lp jon The ondry condons for dole lp jon re : : : dherend : dherend : dherend : dherend : dherend : dherend : prescred : y y y or or or o Conny cross jncon Conny cross jncon Conny cross jncon or y (.5.) For he dherend nd = nd for he dherend = he ondry condons re dered from he ssmpon h he dherend edges re free..5.5 Sngle sded scrfed lp jon The ondry nd conny condons he ends of he jon nd he ends of he oerlp zone re he sme s for he sngle lp jon n q. (.5.)..5.6 Dole sded scrfed lp jon The ondry nd conny condons re he sme s for he dole sded sepped lp jon n q. (.5.) ecep h no conny condons hn he oerlp zone s reqred for he dole sded scrfed lp jon..6 UTI-ST THOD ch se of goernng eqons ogeher h he ppropre ondry condons for he prclr onded jon prolem consdered conses mlple-pon ondry le prolem o hch no generl closed-form solon s onle. Ths nmercl solon procedre ms e sed o sole he onded jon prolems. In generl hs cn e done y sng mehods sch s fne dfference mehods or drec negron mehods. The se of norml drec negron pproch ll nole some dsdnges from hch he mos mporn s h complee loss of ccrcy nrly ll occr f he lengh of he negron nerl s ncresed eyond cern le. The loss of ccrcy s csed y srcon of lmos eql nd ery lrge nmers n he process of deermnon of he nnon ondry les. Hoeer he se of modfed drec negron mehod clled he ml-segmen mehod of negron cn oercome he loss of ccrcy eperenced h he norml drec negron mehods. V Jons Adhese Bonded Jons

31 Verson... Theorecl Bcgrond of SAComp Anlyses V.. The mehod s sed on rnsformon of he orgnl mlple-pon ondry le prolem no seres of nl le prolems. The prncple ehnd he mehod s o dde he orgnl prolem no fne nmer of segmens here he solon hn ech segmen cn e ccomplshed y mens of drec negron. Flflmen of he ondry condons s ell s flflmen of conny reqremens cross he segmen jncons s ssred y formlon nd solng se of lner lgerc eqons. As n emple he sngle lp jon confgron shon n Fgre. s dded no hree regons. Fgre. Schemc llsron of sngle lp jon dded no + + segmens. Accordng o Fgre. he hree regons re: The regon o he lef sde of he oerlp zone.e. - The oerlp zone.e. The regon o he rgh sde of he oerlp zone.e. +. ch of he regons r (r = n r ) re hen dded no fne nmer of segmens r see Fgre.. The segmens hn regon re denoed y S (j = r ) nd he j'h segmen eends from r j o r j. The solon procedre doped n he ml-segmen mehod of negron ncldes for seps: Solon of he goernng eqon hn ech segmen r j r S j n ech regon r. Specfcon of conny condons eeen ech segmen hn ech regon r. Specfcon of ondry nd conny condons he ends of he regons. V Jons Adhese Bonded Jons

32 V.. Verson. Theorecl Bcgrond of SAComp Anlyses.. Formlon nd solon of se of lner lgerc eqons connng he nnon rles. I s eyond he scope of hs docmen o go no frher dels of he solon mehod nd he mplemenon n he SAComp sofre (see orensen [] for dels). I s essenl hogh o emphsse h he drec negron of he nl le prolems s performed y n emedded Rnge-K mehod h dpe sep sze conrol sed on prescred ccrcy hch enles SAComp o conrol he nmer of segmens sed..7 PASTICITY FFCTS I TH ADHSIV AYR.7. Inrodcon The srcrl modellng descred n Secon. s sed on he ssmpon h he dhese lyers ehe s lner elsc merl. Ths s good ppromon for mos rle dheses especlly lo lod leels nd he pproch s sefl o predc he sress dsron nd he locon of pe sress les. Hoeer mos polymerc srcrl dheses eh nelsc ehor n he sense h plsc resdl srns re ndced een lo leels of eernl lodng nd plsc yeldng ll pper n mos dhese onded jons s he lod s ncresed o flre see Hr-Smh [8 ] Pce [ 5] Adms [] l [6 7] nd Thomsen [6 7]. Ths he ssmpon on lner elscy of he dhese s clerly n ppromon. Bsed on he srcrl nlyss descred n Secon. he onded jon nlyss hs een eended o nclde dhese plscy. Hoeer non-lner me nd emperre dependen effecs ncldng sco-elscy creep nd herml srnng re no consdered..7. on-lner formlon nd solon procedre The concep of effece sress/srn s one y of pprochng he non-lner prolem. In hs pproch s ssmed h for dcle merl he plsc resdl srns re lrge compred h he creep srns norml lodng res. Therefore plsc yeld hypohess cn e ppled nd he mldreconl se of sress cn e reled o smple ndreconl sress se hrogh fncon smlr o h of Von ses. Hoeer s dely cceped h he yeld ehor of polymerc srcrl dheses s dependen on oh deorc nd hydrosc sress componens. A conseqence of hs phenomenon s dfference eeen he yeld sresses n nl enson nd compresson see Adms e l. [] [] l e l. [6 7] Hrrs e l. [] nd Thomsen [6 7]. Ths ehor hs een ncorpored no he nlyss y he pplcon of modfed Von ses creron sggesed y l e l. [7]: s C J S c D CV J CS CV (.7.) V Jons Adhese Bonded Jons

33 Verson... Theorecl Bcgrond of SAComp Anlyses V.. here s s he effece sress J D s he second nrn of he deorc sress ensor J s he frs nrn of he generl sress ensor nd s he ro eeen he compresse nd ensle yeld sresses. J D nd J re defned y: J D 6 (.7.) J For = qs. (.7.) re redced o he ordnry Von ses creron. A he flre lod leel he frs of qs. (.7.) s rnsformed no he epresson: J J l s l C S l D l C V l (.7.) here he sscrp l denoes lme. q. (.7.) descres he flre enelope for he generl cse of dcle merl. In hree-dmensonl sress spce q. (.7.) represens prolod h s s concden h he lne = =. The effece srn e s gen y l e l. [7]: e CS I D CV I (.7.) here s he Posson's ro I D s he second nrn of he deorc srn ensor nd I s he frs nrn of he generl srn ensor. I D nd I re defned y: I D 6 (.7.5) I The non-lner dhese properes re nclded y mplemenng n effece sress-srn relonshp dered epermenlly from ess on dhese l specmens Thomsen [6 7] nd Tong [9]. Ths s ssmed h he l nd n-s mechncl properes of he srcrl dhese re closely correled s dscsses y l e l. [7] nd shon y lleheden []. V Jons Adhese Bonded Jons

34 V.. Verson. Theorecl Bcgrond of SAComp Anlyses.. Fgre. () ffece sress-srn relonshp oned from ensle es on l specmen. () Illsron of pece-se lner ppromon o he cre nd he solon procedre for he sress nlyss n he non-lner rnge. Bsed on secn modls pproch for he non-lner effece sress-srn relonshp for he dhese s shon n Fgre. he solon procedre for deermnng he sress dsron n he dhese lyer cn e descred y he follong seps: * () Clcle he effece srns e nd sresses s (qs. (.7.) nd (.7.)) for ech pon of he dhese lyer sng he lner elsc solon procedre nd ssmng nform elsc modls for he dhese. * () If he clcled effece sresses s re oe he proporonl lm denoed y s prop deermne he effece sresses s for ech pon of he dhese lyer ccordng o he correspondng effece srns e (sng he epermenl relonshp gen y Fgre.) clcled n sep (). () Clcle he dfference s = effece sress nd deermne he specfc secn-modls s -s eeen he clcled nd he epermenl * } defned y: s { s (.7.6) here s egh-fcor hch deermnes he chnge of he modls n ech eron. () Rern he procedre (seps ()-()) h he elsc modls for ech dhese pon modfed s per sep (). (5) Compre he clcled effece sresses s * for ech dhese pon h he epermenl les s oned from he effece sress-srn cre (Fgre.). (6) Repe seps ()-(5) nl he dfference eeen he clcled nd epermenl sresses (s) drops elo specfed frcon (%) of he epermenl sress le. Conergence s slly cheed hn fe erons. The non-lner sress-srn relonshp oned from ensle es on l specmen s llsred n Fgre. s n V Jons Adhese Bonded Jons

35 Verson... Theorecl Bcgrond of SAComp Anlyses V..5 SAComp defned y pece-se lner ppromon o he cre s llsred n Fgre.. The procedre descred oe hs preosly een sed for he nlyss of non-lner dhese ehor n lr lp jons y Thomsen [6 7]. The mmm effece sress nd srn crer he een nesged y [] nd [] y ncorporng he o crer n fne elemen nlyss of dole nd sngle lp jons. Ther nesgons shoed h for rle dheses here s ery close correlon h epermenl resls y sng he mmm effece sress creron. For oghened dcle dheses hey fond h he mmm effece srn creron ge he es predcon of he jon srengh. From he fne elemen nlyses s lso possle o predc he flre mode frly ccrely..8 VAIDATIO OF TH ODS The ldon of he dhese lyer model y comprson h hgh-order heory pproch nd fne elemen models hs een presened y orensen n reference []..9 FAIUR AAYSIS Flre n dhese onded jons cn e dded no he follong for ypes:. The dhese my fl de o hgh sher nd rnserse norml sresses (cohese flre).. The dhese/dherend nerfces my fl de o hgh sher nd rnserse norml sresses (nerfce flre).. The dherends my fl de o he eernl lods copled h he lrge endng momen concenrons ndced n he regons ner o he ends of he oerlp.. If he dherends re mde of compose merl hey my fl de o ply-flre csed y hgh nerlmnr sher sresses. The flre ypes nd re consdered n he SAComp mplemenon s descred n he follong ssecons. The flre ype slly ppers de o nsffcen ondng or srfce prepron nd s herefore prmrly qeson of proper mnfcrng. The flre ype s no predced n he crren SAComp mplemenon he dhese sher sresses nd dheren resln sher forces oned form he jon nlyss cn e sed s he ss for ssessng he crcly of hs mode..9. Cohese flre nlyss The cohese flre nlyses re dded no o ypes of nlyss lner nd non-lner dhese flre nlyss. The procedre for deermne he o flre leels re descred n hs secon. V Jons Adhese Bonded Jons

36 V..6 Verson. Theorecl Bcgrond of SAComp Anlyses.. ner cohese flre nlyss The lner cohese flre leel s reched hen he effece dhese sresses o he proporonl lm (s prop ) for he dhese l d. The procedre for deermnng he resere fcor (RF dh ) s s follos: * s re eql. Clcle he dhese lyer effece sresses correspondng o he ppled lod ecor {F} nd deermne he mmm le ( s * m ).. Clcle he resere fcor s RF dh = s prop / s.. If he resere fcor RF dh. he resls from lod response nlyss eql o he ppled lod ecor re shon.. If he resere fcor RF dh <. he lod ecor s mlpled h he resere fcor.e. RF dh {F} nd he lod response nlyss s performed gn h he redced lod ecor. The resls from he nlyss re shon.e. he resls flre lod leel. 5. If he jon poses o dhese lyers sep nd re performed for ech of he dhese lyers. The loes resere fcor of he o dhese lyers s dsplyed s he resere fcor for he jon (RF dh ). on-lner cohese flre nlyss In he non-lner cohese flre nlyss o resere fcors re dsplyed he resere fcor hch he proporonl lm s reched.e. here plscy srs (RF dhprop ) nd he resere fcor o flre leel (RF dh ). The resere fcor o he proporonl lm (RF dhprop ) s deermned s for he lner cohese flre nlyss descred oe. The resere fcor o flre leel or he lme lod-erng cply of he onded jons re deermned y n ere se of he non-lner solon procedre descred n Ssecon.7. here he eernl lods re modfed eeen ech eron. The eron scheme s repeed nl he clcled mmm effece srn reches he lme le e l. The procedre for deermnng he proporonl dhese resere fcor (RF dhprop ) nd he flre resere fcor (RF dh ) s s follos:. Clcle he dhese lyer effece sresses correspondng o he ppled lod ecor {F} nd deermne he mmm le ( s * m ). * m. Clcle he resere fcor s RF dhprop = s prop / * s m.. Cll he non-lner solon procedre descred n Ssecon.7... Increse or decrese he lod ecor y mlplyng h he frcon of he lme effece dhese srn nd he mmm effece dhese lyer srn (e l / e ).e. {F} + = {F} * e l / * e m here s he eron nmer srng from o n flre. 5. Cll he non-lner solon procedre descred n Ssecon.7. h he modfed lod ecor np {F} Repe sep 5 nd 6 nl he clcled mmm effece srn reches he lme le e l.e. nl e m = e l. * m V Jons Adhese Bonded Jons

37 Verson... Theorecl Bcgrond of SAComp Anlyses V Clcle he resere fcor o flre RF dh = {F} flre /{F} ppled. 8. If he resere fcor RF dh. cll he non-lner solon procedre h he ppled lod ecor {F} ppled o dsply op from he lod response eql o he ppled lod ecor ogeher h he resere fcors. De o he smple y of modellng he dhese lyer (he dhese s no modelled s connm) s no possle o predc he flre mode h hs pproch. Hoeer shold e possle o predc he jon srengh h resonle ccrcy y pplyng he mmm effece sress or srn crer snce eqlly smple models of he dhese lyer he een sed sccessflly for he predcon of he jon srengh y [8 ]. Hoeer he predcons shold e sed for compre prposes only. For relsc elon of he predced resls hey shold e compred h epermenl resls..9. mne flre The flre of he dherends de o eernl lods comned h jon ndced endng momens s predced sng he lmne frs ply flre (FPF) nlyss of SAComp (Pr III Chper 5). Poenlly crcl locons n he cny of he jon re consdered s llsred n Fgre X. The n-plne forces nd endng momens cng hese locons re oned from he jon nlyss. For comprson he FPF resere fcors he end sppors re lso comped. As resl he resere fcor for dherend (lmne) FPF re V Jons Adhese Bonded Jons

38 V..8 Verson. Theorecl Bcgrond of SAComp Anlyses.. RFRCS. Adms Roer D. nd We Wllm C. Srcrl Adhese Jons n ngneerng lseer Appled Scence Plshers 98. ed.. Adms R. D. nd Coppendle J. nd Pepp. A. Flre Anlyss of lmnm-lmnm onded jons Adheson 978 ol. pp Crocome A. D. nd Adms R. D. An effece sress/srn concep n mechncl chrcerzon of srcrl dhese ondng Jornl of Adheson 98 ol. pp Adms R. D. Sress nlyss: fne elemen nlyss pproch Deelopmens n Adheses Appled Scence Plshers ondon 98 ed. 5. Frosg Y. nd Thomsen O. T. nd orensen F. Anlyss of Adhese Bonded Jons Sqre-end nd Spe-Flle: Closed-Form Hgher-Order Theory Approch Repor o. 8 Inse of echncl ngneerng Alorg Unersy Denmr 997 smed. 6. l S. nd Ish O. Inerlmnr sress dsron hn n dhese lyer n he nonlner rnge Jornl of Adheson 978 ol. 9 pp l S. nd Dole. nd Ish O. An effece sress/srn concep n mechncl chrcerzon of srcrl dhese ondng Inernonl Jornl of Adheson nd Adheses 98 ol. pp Hr-Smh. J. Adhese onded sngle lp jons Techncl repor ASA CR 6 Dogls Arcrf Compny cdonnell Dogls Corporon USA Hr-Smh. J. Adhese onded dole lp jons Techncl repor ASA CR 7 Dogls Arcrf Compny cdonnell Dogls Corporon USA 97.. Hr-Smh. J. Adhese onded scrf nd sepped-lp jons Techncl repor ASA CR 5 Dogls Arcrf Compny cdonnell Dogls Corporon USA 97.. Hrrs J. A. nd Adms R. D. Srengh predcon of onded sngle lp jons y non-lner fne elemen mehods Inernonl Jornl of Adheson nd Adheses 98 ol. pp lleheden. Properes of dhese n s nd n l Inernonl Jornl of Adheson nd Adhese 99 ol. pp. 7.. orensen F. Deelopmen of Tools for ngneerng Anlyss nd Desgn of Hgh-Performnce FRP-Compose Srcrl lemens Ph.D.-hess Inse of echncl ngneerng Alorg Unersy Denmr 998 Specl Repor o. 7. (hp://..c.d/phd/). Pce A. K. Sress nlyss of dhese onded lp jons Ph.D. hess Unersy of Srrey 98. V Jons Adhese Bonded Jons

39 Verson... Theorecl Bcgrond of SAComp Anlyses V Pce A. K. nd Holly. The nlyss of elso-plsc dhese sress n dhese onded lp jons Compose Srcres 985 ol. pp Thomsen O. T. Anlyss of dhese onded generlly orhoropc crclr cylndrcl shells Ph.D.-hess Inse of echncl ngneerng Alorg Unersy Denmr 989 Specl Repor o.. 7. Thomsen O. T. lso-sc nd elso-plsc sress nlyss of dhese onded lr lp jons Compose Srcres 99 ol. pp Thomsen O. T. nd Rs W. nd on D. C.. nd Bron S. Ply Drop-off effecs n sndch pnels - heory Composes Scence nd Technology 996 ol. 56 pp Tong. Bond srengh for dhese-onded sngle-lp jons. Ac echnc Sprnger-Verlg 996 ol. 7 pp... Whney J.. Srcrl Anlyss of mned Ansoropc Ples Technomc Plshng Compny. Inc. ncser 987. V Jons Adhese Bonded Jons

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41 Verson... Theorecl Bcgrond of SAComp Anlyses V.. CHAICA JOITS Tmo Brnder (HUT/S ) The procedre for nlyzng nl n-plne lod ndced sresses n mechnclly fsened sngle lp nd dole lp jons s presened. The eernl lod cn e eher ensle or compresse nl lod n he jon lengh drecon. Frs he fsener lods re clcled. The sresses of n nfne dherend on he fsener hole re clcled from he fsener lod nd from he y-pss lod. The jon flre lod nd he flre mode re clcled. The procedre s prmrly nended for nlyzng oled jons lso reed jons h sold res cn e nlyzed sng hs procedre prodng h proper les for cern prmeers descrng fsener flely re defned. SYBOS A A j Fsener cross-seconl re In-plne sffness mr of lmne A A Cross-seconl re of dherend (= W h) A s [A] [B] {B} C D d f h I JF JS ffece re of ple oer hch he fsener sher cs Adherend eenson mr Adherend coplng sffness mr Fsener/hole eenson ecor Fsener hed roonl sffness Fsener dmeer Chrcersc dsnce Yong's modls Fsener/hole flely Sher modls Adherend hcness Sffness momen of ner Jon flely Jon sffness ffece sffness per n hcness of dherend spporng fsener V Jons echncl Jons

42 V.. Verson. Theorecl Bcgrond of SAComp Anlyses.. e l e r P {P} q r S W X Oerll jon lengh engh of dherend efore fsener (lef) nd fsener (rgh) respecely Inernl momen n fsener mer of fseners long jon Sher lod n fsener Fsener lod ecor Sher lod Recon of ple spporng fsener per n lengh Rds of fsener hole or fsener Sffness of fsener n sngle dherend Wdh of sngle lne of fseners Tol n-plne longdnl lod of sngle lne of fseners of dh W Roon of fsener s de o sher = /( A ) In-plne longdnl eenson; Tol deformon of fsener nd dherend Srn Shpe fcor for crclr em = 6(+ )/(7+6 ) Coeffcen of frcon Posson's ro Drecon ngle from he -s Sress Sher sress Roon of fsener s de o endng Sscrps AB p c f Sngle lp jons composng dole-lp jon erng Fsener By-pss Compresson chrcersc Inde h fsener or pch long jon Flre V Jons echncl Jons

43 Verson... Theorecl Bcgrond of SAComp Anlyses V.. p Adherend (ple) pch r Rdl drecon Tenson y Orhoropc n-plne coordne sysem Tngenl drecon on he hole ondry Adherends n sngle lp jon V Jons echncl Jons

44 V.. Verson. Theorecl Bcgrond of SAComp Anlyses... ITRODUCTIO Prcclly ll rel lfe srcres conss of seerl s-srcres or re conneced o oher srcres. Ths srcres lmos nely conn jons. The o mos commonly sed jonng mehods of compose srcres re mechncl jonng nd ondng. Hghly loded mechncl jons se re or ol fseners. The SAComp nlyss procedres re sclly smlr n jons sng eher fsener. When he fsener lods re clcled s ssmed h he fseners re ols. I s lso ssmed y defl h he ols re ghened o orqe h ges deqe roonl sffness o he ol hed does no dmge he dherends y hrogh-he-hcness lods. The frcon eeen dherends s no consdered. The fsener lod ndced sresses re clcled ssmng pn ype fsener. Ths mens h no clmpng n he lmne hcness drecon s consdered hs sge of nlyss. Ths n hs respec he procedre ges consere jon flre lods h respec o oled jons here les some clmpng or consrn s presen.. AAYSIS APPROACH The nlyss of mechncl jons s sed on he follong ssmpons:. Adherend hcnesses re consn. ffecs of dherend endng re negleced. Adherend srns re ssmed consn hrogh he hcness of he dherend. od from fsener o he dherend s rnsferred prely y erng (pn jon). The nlyss procedre cn e olned s follos:. The fsener lods re soled ccordng o he heory presened n SDU 855 [] nd 85 [] n hch he flely of he componens (dherends) nd fseners s nclded. A ne SDU shold replce he foremenoned SDUs. A drf erson of h SDU S68D [] s lso sed. For nsymmerc lmnes zero-crre modl re sed.. The sress feld he fsener hole s soled sng he heory of nsoropc ples. Hoeer he presen solon s lmed o orhoropc dherends here A 6 = A 6 = nd [B] =. The solon ncldes he effecs of he fsener lod nd y-pss lod. The ppled heory pples for nfne dherends only. The procedre s s follos: ) The sresses ndced y he fsener re soled ccordng o presenon of Zhng nd Ueng []. The solon s sed on he heory of nsoropc ples y ehns [5]. ) The y-pss lod ndced sresses re soled sng he crclr open hole solon for nfne ple (see Pr IV Chper ). c) In oh cses he solon s sed on lner elsc ehor of merls. Ths he sress felds cn e smmed.. The flre modes nd mrgns of sfey re eled ros pons rond he fsener hole o ssess he lod crryng cply nd he poenl flre mode of he V Jons echncl Jons

45 Verson... Theorecl Bcgrond of SAComp Anlyses V..5 jon.. I s lso possle o nlyze he cse of fne dh dherends. Hoeer hs procedre cn e ppled only o ngenl sress = /.e. locons deng 9 from he fsener lod drecon.. CHAICA JOIT OAD RSPOS.. Fsener nd y-pss lods The fsener lods re deermned ccordng o prncples presened n references [ ]. The pproch s ld only for sngle ro sngle lp jons sjeced o n-plne enson lods. Hoeer he sme nlyss cn lso e ppled o dole lp jons nd mlple ro jons proded he ros re dencl. The nlyss prodes he lods crred y ech fsener ypss lods nd he oerll n-plne flely of he jon. The pproch s n prncple ld only for soropc merls. Hoeer soropy hs sgnfcn effec only on he fsener/hole flely. Ths s eleed h hs pproch cn e ppled lso o orhoropc merls h deqe confdence. The nlyss pproch s sed on smll dsplcemen elsc heory nd does no consder he momen effecs de o he eccenrcy of he lodng. The jon geomery nd noon s shon n Fgre.. I Fsener Fsener Fsener Fsener + Fsener - Fsener I e l p p p - e r X X-P X-P P P P X-P X Fgre. Sngle lp jon geomery nd noon V Jons echncl Jons

46 V..6 Verson. Theorecl Bcgrond of SAComp Anlyses.. ch pch eeen he fseners s consdered seprely nd he effecs re smmed. For comply he pch eensons for dherends nd fsener posons re eql = (..) The pch eensons re mde p of o componens ) dherend eensons ( p p ) nd ) fsener/hole deformons ( ). The dherend eensons for he dherend nd for he dherend re p p= X P (..) A = p p= P (..) A = here P s he fsener lod of h fsener nd X he ol lod of sngle lne of fseners. Adherend sffnesses re zero crre sffnesses. The fsener flely f s reled o oh dherend nd dherend. The pch eenson de o he fsener nd fsener hole flely s = P + f P f + (..) Comnng qons (.. - ) he comply eqon for ech pch ecomes A p X = P p A = P P + f + P f (..5) here f s he sm of he nddl fsener/hole flely n ech dherend s eplned n he ne ssecon. Oerll lod comply yelds X = P (..6) = qons (..5) nd (..6) represen smlneos eqons hch my e ren n he mr form V Jons echncl Jons

47 Verson... Theorecl Bcgrond of SAComp Anlyses V..7 P = B A (..7) The fsener lods P cn e soled from hs eqon. The y-pss lod for ech pch s P X p P (..8) The jon eenson s fond y ddng () he dherend eensons of he op dherend () he eensons de o he flely of he end fsener nd (c) he free dherend eensons n dherends nd ech end eeen fseners nd nd pons I nd I respecely: p X el X er = X P P f + + (..9) = A = AF AF The jon flely s defned s JF (..) X nd he correspondng jon sffness s JS (..) JF The flely of dole lp jon s oned y nlyzng o sngle lp jons hch re proned from he dole lp jon o he mdplne of he cener dherend s shon n Fgre.. The flely of he dole lp jon s J F = A B J F A J F B J F + J F A B (..) here A nd B refer o sngle lp jons composng he dole lp jon. The correspondng sffness s JSAB JSA JSB (..) V Jons echncl Jons

48 V..8 Verson. Theorecl Bcgrond of SAComp Anlyses.. Adherend Sngle lp jon A Adherend h X/ Cener lne of h cener ple X/ X Sngle lp jon B Fgre. Dole lp jon modeled s o sngle lp jons The erng sress for ech fsener s P B (..) D h.. Fsener flely In qon (..5) he only nnon prmeer s he fsener/hole flely f. I cn e soled y consderng sngle fsener n sher n sngle fl ple (dherend) s shon n Fgre.. The fsener s resrned gns roon. The locl deformons of he ple re nclded n he nlyss. The fsener flely s defned s f j = P j h z (..5) If o neql dherends re conneced s sngle lp jon he fsener flely s oned y smmng he flely of he fsener n he nddl dherends nd he sffness s f = f + f (..6) S = f + f (..7) V Jons echncl Jons

49 Verson... Theorecl Bcgrond of SAComp Anlyses V..9 Unloded ol h z Cener lne of nloded ol lsc sppor Cener lne of loded ol od P Fgre. Fsener/hole flely The sc goernng dfferenl eqon for he fsener cng s em on n soropc elsc fondon s z + As A d d z A + z As d + I d z I y = (..8) here s he shpe fcor for crclr em A s s he effece re of ple oer hch he ol sher cs s he effece sffness of he ple spporng he ol nd I s he sffness momen of ner of he ol. The recon of he ple fondon s q pz d z As (..9) dz The lner smll dsplcemen ssmpon s mde hch ges V Jons echncl Jons

50 V.. Verson. Theorecl Bcgrond of SAComp Anlyses.. z = I d dz (..) The follong for ondry condons re ppled: z d h P z As z (..) dz z h z (..) z d h z As z (..c) dz z h C z (..d) qon (..8) cn e ren s '''' '' (..) here he prmes refer o he dfferenon h respec o z. The eqon s soled sng he ondry condons of (.. - d). The eqon for he roon of fsener s de o endng reqred n qon (..d) s = d dz d = + dz = d dz z A (..) From qons (..) nd (..) follos V Jons echncl Jons

51 Verson... Theorecl Bcgrond of SAComp Anlyses V.. z d q pz = I + d z A (..) o hch he fondon recon qon (..9) s plced. The sher force s oned y deron from qon (..) z = d dz z = I d + d z A dq dz pz (..5) qon (..8) ncldes hree qnes A s C nd hch re ypclly no lle. To prode he hghes possle ccrcy he les for hese prmeers shold e deermned hrogh ess. The follong esmes cn mnly e sed for mellc dherends. If he fsener hed s effecely resrned gns roon he fsener hed roonl sffness C my e consdered s nfne. The effece re of dherend oer hch sher cs A s my e esmed o e A s =. D. The effece dherend sffness shold e deermned epermenlly ccordng o he mehod descred n SDU 85 Append B []. The le =.8 p gen n [] shold only e sed hen nm lloy fseners re sed n n lmnm lloy dherend... Fsener lod ndced sresses he pn-loded hole The fsener lod ndced sresses he fsener hole re deermned ccordng o he presenon of Zhng nd Ueng []. The nlycl solon s sed on he heory of nsoropc ples n he presen solon hs een resrced o orhoropc ples o on compc nlycl solons. The epresson of rdl sress s ( c ) c r n( ) cos 5 ( c )( y n) n( ) cos cgr cgr P r ( c ) ( c ) ( y n n) ( y n) cos (..6) cgr cgr nd for sher sress ( c ) c r n ( ) sn 5 ( c )( y n n) n( ) sn cgr cgr V Jons echncl Jons

52 V.. Verson. Theorecl Bcgrond of SAComp Anlyses.. P r ( c ) ( c ) ( y n n) ( y n) sn (..7) cgr cgr When he effec of frcon s consdered he follong condon pples / r r d / r d r (..8) In qons (..9) nd (..) follong prmeers re sed / (..) y / n( y ) (..) y y g c y y B A A y (..) (..) g P A B A n B n y y (..5) here A 9 n n n 6n 5 5 (..6) y n n B n y (..7) y The ngenl sress cn e epressed s (..8) 5 V Jons echncl Jons

53 Verson... Theorecl Bcgrond of SAComp Anlyses V.. here P y cos ( y ) cos sn r cos ( n y )sn (..9) ( c) n cos (sn cos ) ( )( y n) cos cgr ( y n y n)sn sn sn cos (sn cos ) sn n ( n n ) cos ( ) ( n ) sn cos (..) y y ( c ) n cos (sn cos ) n ( ) cos cgr n y sn sn sn cos sn cos ( n )cos ( )( ) sn cos n sn y y (..) ( c ) n cos cos (sn cos ) ( )( y n) cos cgr ( y n y n) sn sn sn cos (sn cos ) sn sn n ( n n ) cos ( ) ( n ) sn cos (..) y y y y c 5 sn (..) cr n he preos eqons s he dherend Yong s modls n he -drecon... By-pss lod ndced sresses The y-pss lod ndced sresses s soled ccordng o Pr IV Chper of hs docmen. The sresses re clcled eher on he fsener hole or on chrcersc cre (Pon Sress Creron) here he pon of clclon s deermned s V Jons echncl Jons

54 V.. Verson. Theorecl Bcgrond of SAComp Anlyses.. r c r d (..) here d s he chrcersc dsnce. The sresses clcled oe pply o hole n n nfne ple...5 Flre of he jon The flre of he jon s deermned prmrly on he fsener hole ccordng o he follong procedre sed on reference [6]:. The fsener lod ndced lmne sresses r nd r re clcled.. The y-pss lod ndced sresses re dded o he fsener lod ndced sresses.. The FPF nlyss pproch (Pr III Chper 5) h he seleced flre creron s ppled o predc he flre lod n erms of resere fcor or mrgn of sfey. In enson lodng he flre mode s deermned s follos -5 f 5 < f < 6 75 < f < 9 erng flre mode sher-o flre mode enson flre mode f s he ngle of he pon here he comned sress reches he crcl le. A nermede les of f flre my e csed y comnon of he modes. In compresson lodng he flre mode s -5 f 5 75 < f < 9 erng flre mode compresson flre mode A nermede ngles he flre mode s ndefned. Alernely f he chrcersc dsnces re non he flre cn e clcled s follos:. The fsener lod ndced lmne sresses r nd r re clcled long he chrcersc cre r r c r c r r = r c r = r c rc rc = (..5). The y-pss lod ndced sresses re dded o he fsener lod ndced sresses long he chrcersc cre. The fnl sep s dencl o he sep oe. V Jons echncl Jons

55 Verson... Theorecl Bcgrond of SAComp Anlyses V..5 Some commens concernng he pproch presened oe re:. Chrcersc dsnces shold e deermned epermenlly for he lmne f relen les re no fond n lerre.. The pproch does no e no ccon he sresses ndced y he oher closely sed fseners/holes.. The pproch here he sresses re clcled on he hole ondry ges consere flre lods for he jon compred o he cse here he les re clcled on he chrcersc cre...6 Fne dh jon The heory presened oe for fsener ndced sresses nd y-pss lod ndced sresses s ld for nfne ples only. Hoeer n ypcl lmnes h relsc end nd sde dsnces nd pches he heory s pplcle h resonle ccrcy. The fne dh correcon s ppled only o ngenl norml sress = ± /. o oher sress componens re correced nor nclded n he flre nlyss. The correced sress for he fsener lod nd for he y-pss lod s K K T T K K T T p p (..6) here s he ngenl sress n n nfne dherend nd sscrps nd p refer o fsener nd y-pss lod respecely. K T nd K T denoe he sress concenron on he hole ondry on he s norml o he ppled lod for nfne ple nd fne ple respecely [7]. The ron of K T nd K T s clled Fne Wdh Correcon (FWC) fcor [7]. The FWC for he fsener lod s [8] K K T T D D D W W W (..7) here W s he dh of ple connng cenrl openng or fsener. The forml s ld for soropc merls s sed here de o he lc of correspondng forml for orhoropc merls. In ddon ypcl lmne srcres for mechncl jons he close o qs-soropc properes. V Jons echncl Jons

56 V..6 Verson. Theorecl Bcgrond of SAComp Anlyses.. V Jons echncl Jons The FWC for he y-pss lod s [7] 6 / / W D K W D W D W D K K T p T T (..8) here A A A A A A A A K T (..9) nd / / / 8 W D W D W D (..5) qons..6 8 pply for orhoropc ples (lmnes).