Unidad 4 Esfuerzos Combinados

 

]E@HAH 84 MVG]MZTNV FNOJ@EAHNV

 

8.1 F@ZF]CN HM ONKZ RAZA MVG]MZTN RCAEN (MF]AF@NEMV HM XZAEVGNZOAF@NE) Mc Fïrfuc Fïrfucn n hm Onkr ms uea tàfe`f tàfe`fa a usaha me `elme`mrïa  `elme`mrïa  y lmngïs`fa lmngïs`fa   para rmprmsmetar lrág`faometm   ue tmesn lrág`faometm tmesnrr s`o s`oàtr àtr`fn `fn (hm :x: n hm 6x6) y fac facfuc fucar ar fne mcc mcca a onometns hm `emrf`a,, hmgnroaf`nems `emrf`a hmgnroaf`nems   y tmes`nems, tmes`nems, ahaptaehn cns o`sons a cas faraftmrïst`fas hm uea f`rfuegmrmef`a f`rfuegmrmef`a   (rah (rah`n `n,, fm fmet etrn rn,, mt mtf.) f.).. Xa Xaoj oj`à `àe e ms pn pns` s`jcm jcm mc fá fácfu cfucn cn hm hmcc msgumrzn fnrtaetm   oáx`on ajsncutn y ca hmgnroaf`öe oáx`oa ajsncuta. fnrtaetm Mstm oàt Mstm oàtnhn nhn gum hms hmsarr arrncc nccahn ahn kaf kaf`a `a 1;;: 1;;:   pn pnrr mc `elme `elme`mrn `mrn f`v`c f`v`c  acmoáe acmoáe  Fkr`st Fkr`st`ae `ae Nttn Onkr   (1;6=-101;). Onkr

FAVN J@H@OMEV@NEAC

Me hns h`omes`nems, ca F`rfuegmrmef`a hm Onkr pmro`tm hmtmro`ear ca tmes`öe oáx`oa y oïe`oa, a part`r hm hns omh`f`nems hm ca tmes`öe enroac y taelmef`ac snjrm hns áelucns qum gnroae 02´4

ENXA4 Mc mdm vmrt`fac sm mefumetra `evmrt`hn, pnr cn qum msgumrzns pns`t`vns vae kaf`a ajadn y msgumrzns emlat`vns sm uj`fae me ca partm supmr`nr. ]saehn ]saeh n m mdms dms rmfta rmftaelucar elucarms, ms, hnehm mc mdm knr`znetac knr`znetac rmprm rmprmsmeta smeta ca ttmes`öe mes`öe enroa enroacc

y mc

mdm vmrt vmrt`fac `fac rmprm rmprmsmeta smeta ca tmes`ö tmes`öe e fnrta fnrtaetm etm n taelm taelmef`ac ef`ac para faha uen hm cns pcaen pcaens s aetm ae tmr` r`nr nrms ms.. Cn Cns s va vacn cnrm rms s hm ca f` f`rf rfue uegm gmrm rmef ef`a `a qum qumha hae e rm rmpr prms msme meta tahn hns s hm ca s`lu s`lu`m `met etm m oaemra4 

 

Fmetrn hmc fïrfucn hm Onkr4



 

Zah`n hm ca f`rfuegmrmef`a hm Onkr4

Cas tmes`nems oáx`oa y oïe`oa v`meme hahns me tàro`ens hm msas oale`tuhms s`opcmometm pnr4

 

Mstns vacnrms sm pumhme njtmemr taoj`àe facfucaehn cns vacnrms prnp`ns hmc tmesnr hm tmes`öe qum me mstm fasn v`mem hahn pnr4

  FAVN XZ@H@OMEF@NEAC Mc fa fasn sn hm hmcc ms msta tahn hn tm tmes es`n `nea eacc hm ue pu puet etn n R hm ue sö söc` c`hn hn tr`h tr`h`o `ome mes` s`ne neac ac ms oá oás s fnopc`fahn ya qum oatmoát`faometm sm rmprmsmeta pnr uea oatr`z hm 6x6 para ca qum mx`stme 6 vacnrms prnp`ns, en emfmsar`aometm h`gmrmetms.

Me mc fasn lmemrac, cas tmes`nems enroac (φ) y taelmef`ac (ψ), omh`has snjrm fuacqu`mr pcaen qum pasm pnr mc puetn R, rmprmsmetahas me mc h`alraoa (φ,ψ) fame s`moprm hmetrn hm uea rml`öe hmc`o`taha pnr 6 fïrfucns. Mstn ms oás fnopcmdn qum mc fasn j`h`omes`neac, hnehm mc mstahn tmes`neac faïa s`moprm snjrm uea ôe`fa f`rfuegmrmef`a. Faha uen hm cas 6 f`r f`rfu fueg egmr mrme mef`a f`as s qu qum m hm hmc` c`o`t o`tae ae ca rm rml`ö l`öe e hm pn pns`j s`jcm cms s pa parm rms s (φ (φ,ψ) ,ψ) sm fn fnen enfm fm fn fne e mc enojrm hm f`rfuegmrmef`a hm Onkr.

  F@ZF]GMZMEF@A HM ONKZ RAZA ONOMEXNV HM @EMZF@A

 

Rara söc`hns pcaens n fas`-pcaens, pumhm apc`farsm ca o`soa tàfe`fa hm ca f`rfuegmrmef`a hm On Onkr kr qu qum m sm us usö ö par para a tm tmes es`n `nem ems s me hn hns s h`om h`omes es`n `nem ems. s. Me ou oufk fkas as nf nfas` as`ne nems ms ms emfmsar`n facfucar mc onometn hm `emrf`a acrmhmhnr hm ue mdm qum sm mefumetra `efc`eahn, ca f`r f`rfue fuegmr gmrmef mef`a `a hm Onk Onkrr pum pumhm hm smr ut`c`z ut`c`zahn ahn par para a njt njtmem memrr mst mstm m vac vacnr. nr. Xao Xaoj`à j`àe e ms pns`jcm njtmemr cns onometns hm `emrf`a pr`ef`pacms. Me mstm fasn cas göroucas hm fácfucn hmc onometn hm `emrf`a omh`n y mc rah`n hm ca f`rfuegmrmef`a hm Onkr para onometns hm `emrf`a sne aeácnlas a cas hmc fácfucn hm msgumrzns4 

 

Fmetrn hm ca f`rfuegmrmef`a4



 

Zah`n hm ca f`rfuegmrmef`a4

 

 

8.: AEAC@V@V HM MVG]MZTN JADN FAZLAV FNOJ@EAHAV.  A omeuhn ms pns`jcm aeac`zar ue o`mojrn mstrufturac mstrufturac snomt`hn a farlas far las fnoj`eah fnoj`eahas as supmrpn supmrpne`me e`mehn hn cns msgumrz msgumrzns ns y hmgnroaf hmgnroaf`ne `nems ms fausahns pnr faha farla qum aftôa pnr smparahn. Aknra j`me, ca supmrpns`f`öe hm cns msgumrzns y cas hmgnroaf`nems ms pmro`s`jcm sncn me f`mrtas fneh`f`nems. ]e rmqu`s`tn ms qum cns msgumrzns y cas hmgnroaf`nems hmjme smr guef`ne hmgnroaf`nems guef`nems ms c`emacms hm cas farlas apc`fahas. Mstn rmqu`mrm a su vmz qum mc oatmr`ac njmhmzfa ca cmy hm Knnim y qum cns hmspcazao`metns smae pmqumñns. Ntrn rmqu`s`tn ms qum en hmjm mx`st`r `etmraff`öe metrm cas h`vmrsas farlas9 ms hmf`r, cns msgumrzns y hmgnroaf`nems fausahns pnr uea hm cas cas farla farlas s en hmjm hmjme e vmrsm vmrsm agmft agmftah ahns ns pnr pnr ca prms prmsmef mef`a `a hm ntras ntras farlas. Ca oaynr partm hm cas mstrufturas fnouems sat`sgafme mstas hns fneh`f`nems, pnr cn qum mc usn hm ca supmrpns`f`öe ms ouy fnoôe me mc trajadn `elme`mr`c. V` j`me kay oufkas oaemras hm aeac`zar uea mstruftura snomt`ha a oas hm ue t`pn hm farla, pnr cn lmemrac mc prnfmh`o`metn `efcuym cns s`lu`metms pasns4 1.- Vm mc`lm ue puetn me ca mstruftura para hmtmro`ear cns msgumrzns y cas hmgnroaf`nems (Rnr cn lmemrac sm msfnlm ue puetn me uea smff`öe traesvmrsac, hnehm cns msgumrzns sne lraehms9 pnr mdmopcn, me uea smff`ne smff `ne traesvm traesvmrsac rsac,, hnehm hnehm mc onometn onometn gcmx`nea gcmx`neaetm etm t`mem t`mem su vacnr  vacnr  oáx`on). :.- Rara faha farla snjrm ca mstruftura sm hmtmro`eae cas rmsuctaetms hm msgumrzn me ca smff`öe traesvmrsac qum fnetmela mc puetn smcmff`neahn (Cas pns`jcms rmsuctaetms hm cns msgumrzns sne uea gumrza ax`ac, ue onometn hm tnrs`öe, ue onometn gcmx`neaetm y uea gumrza fnrtaetm). 6.- Vm facfucae cns msgumrzns enroac y fnrtaetm me mc puetn smcmff`neahn hmj`hn a faha uea hm cas rmsu rmsucta ctaetm etms s hm ms msgu gumrz mrzns. ns. Ahmoá Ahmoás s s` ca mstruf mstruftu tura ra ms ue rmf`p rmf`p`m `metm etm a prms` prms`öe öe,, hmtm hmtmro` ro`ear ear cns msgumrzns hmj`hns a ca prms`öe `etmrea.

 

Mc prnfmh`o`metn hmsfr`tn para aeac`zar cns msgumrzns me cns puetns A y J, pumhm usarsm me ntrns puetns. Cns puetns hnehm cns msgumrzns facfucahns fne ca görouca hm gcmx`öe y cas göroucas hm cns fnrtaet fnr taetms ms t`meme t`meme vacnrms vacnrms oáx`ons oáx`ons y oïe`ons oïe`ons,, ccaoahns ccaoahns puetns puetns frït`fns frït`fns,, cns msgumrz msgumrzns ns enroacms enroacms hmj`hns a ca gcmx`öe sne oáx`ons me ca smff`öe traesvmrsac hm onometn gcmx`neaetm oáx`on qum sm prmsmeta me mc snpnrtm, pnr taetn, cns puetns me cas partms supmr`nr m `egmr`nr hm ca v`la me mc mxtrmon mopntrahn sne cns puetns frït`fns para mc fácfucn hm cns msgumrzns. Fnon pasn g`eac, cns msgumrzns pr`ef`pacms y cns msgumrzns fnrtaetms oáx`ons me cns puetns frït`fns pumhme fnopararsm fnopararsm metrm s` para hmtmro`ear hmtmro`ear cns msgumrzns msgumrzns enroacms y fnrtaetms fnrtaetms oáx`ons ajsncutns ajsncutns me ca jarra. Fne ca var`mhah hm s`tuaf`nems s`tuaf`nems praft`fas en parmfm tmemr c`o`tm, en vacm ca pmea njtmemr  gnroucas mspmfïg`fas para facfucar cns msgumrzns oáx`ons. Faha mstruftura sumcm tratarsm fnon fasn mspmf`ac.

Vmcmff`öe hm cns puetns frït`fns

V` mc njdmt`vn hmc aeác`s`s ms hmtmro`ear cns msgumrzns oáx`ons me fuacqu`mr partm hm ca mstruftura, metnefms kay qum msfnlmr cns puetns frït`fns me smff`nems traesvmrsacms hnehm cas rmsuctaetms hm msgumrzns acfaefme cns vacnrms oáx`ons. Pa me h`fkas smff`nems sm mcml`ráe cns puetns me qum cns msgumrzns enroacms n cns msgumrzns fnrtaetms tmelae sus vacnrms oáx`ons. V` ca smcmff`öe hm cns

 

puetns sm kafm fne jume du`f`n, pnhrmons mstar razneajcmometm smlurns hm kajmr njtme`hn cns msgumrzns oáx`ons me ca mstruftura. V`e mojarln, a vmfms ms h`gïf`c rmfnenfmr hm aetm oaen hnehm sm cnfac`zae cns msgumrzns oáx`ons me mc o`mojrn. Metnefms qu`zá sma emfmsar`n `evmst`lar cns msgumrzns msgumrz ns me ue lrae eôomrn hm puetns. Ntras mstratml`as taoj`àe pumhme rmsuctar rmsuctar ôt`cms, fnon njtm nj tmem emrr mfua mfuaf` f`ne nems ms mspm mspmfï fïg` g`fa fas s para para mc prnj prnjcm cmoa oa me fn fnes es`h `hmr mraf af`ö `öe e n mcaj mcajnr nrar ar k`pö k`pötm tms` s`s s s`opc`g`fahnras a g`e hm gaf`c`tar ue aeác`s`s qum pnhrïa rmsuctar suoaometm h`gïf`c s`e mccas. Mdmopcn4 ]e pnstm f`rfucar kumfn fne h`áomtrn mxtmr`nr hm ::2 oo y h`áomtrn `etmr`nr hm 1;2 oo snst`mem snst`me m ue cmtrmrn hm h`omes`nems hm :.2 o x 1.: o. Mc cmtrmrn msta hmspcazahn 2.= o hmc fmetrn hmc

pnstm y su jnrhm `egmr`nr msta 7.2 o snjrm mc tmrrmen.

Vncuf`öe. Ca prms`öe hmc v`metn fnetra mc cmtrmrn prnhufm uea gumrza rmsuctaetm \ qum aftôa me mc puetn omh`n hm mstm y ms `luac a ca prms`öe p ouct`pc`faha pnr mc árma A snjrm ca qum aftôa4  

Ca cïema hm aff`öe hm msta gumrza mstá a uea actura k ? 7.7 o snjrm mc sumcn y uea h`staef`a j ? 1.= o hm ca cïema cïema fmetr fmetrac ac hmc pnstm pnstm.. Ca gumrza gumrza hmc hmc v`metn v`metn qum qum af aftô tôa a snjrm snjrm mc cmtrm cmtrmrn rn ms mstát mstát`fa `faom ometm etm mqu`vacmetm mqu`vac metm a uea gumrza catmrac \ y a ue par hm tnrs`öe X qum aftôa snjrm mc pnstm. Mc par ms `luac a ca gumrza \ ouct`pc`faha pnr ca h`staef`a j4

 

Cas rmsuctaetms hm msgumrzns me ca jasm hmc pnstm sne ue onometn gcmx`neaetm O, ue par hm tnrs`öe X y uea gumrza fnrtaetm W. Vus oale`tuhms sne4

Mc mxaome hm msta rmsuctaetm hm msgumrzns oumstra qum cns msgumrzns hm gcmx`öe oáx`ons nfurrme fne mc puetn A y cns msgumrzns fnrtaetms oáx`ons fne mc puetn J9 Rnr taetn A y J sne puetns frït`fns hnehm hmjme hmtmro`earsm cns msgumrzns. Msgumrzns me mc cns puetns A y J. Mc onometn gcmx`neastm O prnhufm ue msgumrzn hm tmes`öe me mc puetn A pmrn e`elôe msgumrzn me mc puetn J. Mc msgumrzn hm tmes`öe me mc puetn A sm njt`mem fne ca görouca hm gcmx`öe4

Hnehm h: ms mc h`áomtrn mxtmr`nr (::2 oo) m @ ms mc onometn hm `emrf`a hm ca smff`öe traesvmrsac. Mc onometn hm `emrf`a ms4

 Hnehm h1 ms mc h`áomtrn `etmr`nr. Rnr ca taetn mc msgumrzn hm tmes`öe me mc puetn A ms.

 Mc par hm tnrs`öe X prnhufm msgumrzns fnrtaetms, me cns puetns A y J. Rnhmons facfucar h`fkns msgumrzns fne ca görouca hm tnrs`öe4

hnehm @p ms mc onometn pncar hm `emrf`a4

Metnefms4

 

Rnr ôct`on facfucaons cns msgumrzns fnrtaetms me cns puetns A y J hmj`hns a ca gumrza fnrtaetm W. mc msgumrzn fnrtaetm me mc puetn A ms fmrn y mc msgumrzn fnrtaetm me mc puetn J sm njt`mem fne ca görouca hmc fnrtaetm para ue tujn f`rfucar 4

mfu (d) Hnehm r: y r1 sne Hnehm sne cns ra rah`n h`ns s mxtmr` mxtmr`nr nr m `etmr `etmr`nr `nr,, rmspmf rmspmft`v t`vao aome metm, tm, y A ms mc árma árma hm ca smff smff`öe `öe traesvmrsac4

 Ac sust`tu`r sust`tu`r cns vacnrms vacnrms euoàr`fns euoàr`fns me ca mfu mfu (d), njtmemons4 njtmemons4

 Aknra kmons facfucahn facfucahn tnhns cns msgumrzns msgumrzns qum aftôae snjrm snjrm cns puetns A y J hm ca smff`öe traesvmrsac. Mcmometns hm msgumrzn. Mc s`lu`metm pasn ms onstrar mstns msgumrzns snjrm mcmometns hm msgumrzn. Rara aojns mcmometns, mc mdm "y" ms paracmcn ac mdm cnel`tuh`eac hmc pnstm y mc mdm x ms knr`znetac. Me mc puetn A, cns msgumrzns qum aftôae snjrm mc mcmometn sne4

 Me mc puetn J, cns msgumrzns sne

Rumstn qum en mx`stme msgumrzns enroacms qum mstàe aftuaehn snjrm mc mcmometn, me mc puetn J sm mefume mef umetra tra me mstahn mstahn hm fnrtaetm fnrtaetm purn. Aknra Aknra qum fnenfmons fnenfmons tnhns cns msgumrz msgumrzns ns qum aftôae

 

snjrm cns mcmometn snjrm mcmometns s hm msgumrz msgumrzn, n, pnhmons pnhmons usar cas mfuaf` mfuaf`nems nems para hmtmro` hmtmro`ear ear cns msgumrz msgumrzns ns pr`ef`pacms y cns msgumrzns fnrtaetms oáx`ons. Msgumrzns pr`ef`pacms y msgumrzns Msgumrzns msgumrzns fnrtaetms oáx`ons me mc puetn A. Cns msgumrzns msgumrzns pr`ef`pacms sm njt`meme fne ca mfuaf`öe4

Vust`tu`ons4

Cns msgumrzns oáx`ons me mc pcaen pumhme njtmemrsm fne ca mfuaf`öe

Mstm tàro`en sm mvacuö aetms, pnr cn qum vmons hm `eomh`atn qum

Rumstn qum cns msgumrzns pr`ef`pacms t`meme s`lens npumstns, cns msgumrzns fnrtaetms oáx`ons me mc pcaen sne oaynrms qum cns msgumrzns fnrtaetms oáx`ons me mc pcaen sne oaynrms qum cns msgumrzns fnrtaetms oáx`ons gumra hmc pcaen9 pnr taetn, mc msgumrzn fnrtaetm oáx`on me mc puetn A ms hm :;.:Opa. Msgumrzns pr`ef`pacms y msgumrzns fnrtaetms me mc puetn J. Cns msgumrzns me mstm puetn sne

Hahn qum mc mcmometn mstá me mstahn fnrtaetm purn, cns msgumrzns pr`ef`pacms sne

 

y mc msgumrzn fnrtaetm oáx`on me mc pcaen ms

Cns msgumrzns fnrtaetms oáx`ons gumra hmc pcaen t`meme ca o`tah hm mstm vacnr. Enta4 V` sm rmq rmqu`m `mrrme cns cns msgu sgumrzn rzns oáx` oáx`on ons s me fuacqu` cqu`mr mr par arttm hmc hmc pnstm, tm, kay qum hmtmro`ear taoj`àe cns msgumrzns me mc puetn frït`fn h`aomtracometm npumstn ac puetn A, pnrqum me h`fkn puetn mc msgumrzn hm fnoprms`öe hmj`hn a ca gcmx`öe acfaeza mc vacnr oáx`on. Cns msgumrzns pr`ef`pacms me msm puetn sne

y mc msgumrzn fnrtaetm oáx`on ms hm :;.: ORa9 pnr taetn, mc msgumrz msgumrzn n hm tmes`öe oáx`on me mc pnstm ms hm ==.< ORa, mc msgumrzn oáx`on hm fnoprms`öe ms hm -==.< ORa y mc msgumrzn fnrtaetm fnrtaetm oáx`on ms hm :;.: ORa (Zmfumrhm qum sncn sm kae fnes`hmrahn cns mgmftns hm ca prms`öe hmc v`metn me mc aeác`s`s. Ntras farlas, fnon mc pmsn hm ca mstruftura, taoj`àe prnhufme msgumrzns me ca jasm hmc pnstm).

8.6 F@ZF]CN HM ONKZ RAZA HMGNZOAF@NEMV