2 shear strain / L for small angle

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Molly Chase
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1 Sac quaons F F M al Sess omal sess foce coss-seconal aea eage Shea Sess shea sess shea foce coss-seconal aea llowable Sess Faco of Safe F. S San falue Shea San falue san change n lengh ognal lengh Hooke s aw falue shea san / fo small angle omal sess Moulus of lasc san Moulus of Reslence u pl pl pl / Moulus of oughness, u aea une he sess san cue osson s ao Hooke s aw of Shea sess la long Shea sess Moulus of Rg shea san al oas Consan oe a lengh n aal loa acng oe seeal segmens change n lengh loa n he segmen lengh of he segmen aea of he segmen moulus of elasc of he segmen numbe of segmens empeaue san an eflecons change n lengh hemal coeffcen change n he empeaue lengh of he segmen Remembe ha he empeaue can hae. San whou sess. Sess whou san Sacall neemnae aal loas. Case :. Case : geomec conons. Case : BgapB negaon of aal membes F q [ ] u F [ ] u whee, oson Shea Sess shea sess mamum ouse aus oque (F-) pola momen of nea

2 sol shaf - ouse aus - nse aus o o owe = f Foces () M() -M < a > - -M < a> on Foce Dsbue loa Ramp loa - F < a > -F < a> -w < a > -(w/) < a> -(w/)< a > -(w/6) < a> f powe (foce-lengh/me) oque angula eloc n a/s fequenc n Hz. of nea able seup o fn he ceno an momen of nea ngle of ws Consan oe a lengh osonal loas acng oe a numbe of segmens angle of ws n aans oque n he segmen lengh of he segmen pola momen of nea of he segmen moulus of g of he segmen numbe of segmens Sacall neemnae oque Compabl conons. Case :. Case : Shea an Benng Dagams. Compue he eacons. Fn (). Fn M(), whch s he aea une he shea agam w & M. Slope on he momen agam s he alue of he shea agam. w M & he sngula funcons ae efne fo aous foce loas. Foces Funcon F() < a > - -M < a> - on Foce Dsbue loa Ramp loa < a > - -F < a> - < a > -w < a> < a > -(w/) < a> n n n n ea of he secon cene of he secon momen of nea psmac secon, bh sol ccula secon, o hollow ccula secon, sance of he cene fom ceno n ao of moulus of lasc j Ceno -- whch s he locaon of he Fnng he momen of nea n n n n Fleual Sess Benng Sess M n M n sess ue o benng benng momen momen of nea sance fom he neual as ao of moul of elasc

3 Fleual Sess Secon Moulus zz ma S zz zz ma momen of nea sance fom he neual as Shea Sess (fleual) b Q Q Q Q b shea fom he shea agam momen of nea coss-secon hckness fs momen o -- hollow clne a neual as. osonal. Fleual Sess. Fleual Shea Sess, 5. essue, ncple of Supeposon M Q b a & h. Compue he sess fo each loa nepenen of he ohe sesses. Sum he sesses on he sess block o fnal sess loang quaons of lane Sess Sgn Conenon Shea Flow Q q q shea flow (foce / lengh ) shea foce fom shea agam Q fs momen (can nclue n*) momen of nea Conneco spacng s F q s essue essels F q spacng beween connecos numbe of connecos Foce /conneco shea flow. Sphecal essel. Clncal essel a h a & h aal o longunal sess hoop sess nenal pessue nne aus of he ank wall hckness ngle (+) CCW ( - ) CW Shea Sess (+) CCW on ( - ) CW on omal Sess (+) enson ( - ) Compesson cos cos sn ncpal Sesses cos sn sn p, p p o Combne oas. al loa ngles an, an p s

4 nomal sess acng n econ on he nomal sess acng n econ on he H shea sess acng on he,, ae he ansfome sesses p s p,p Moh s Ccle (oe s acng on he ) angle nee o oae he block o ge pncpal sesses. angle nee o oae he block o ge mamum nplane shea sess acng on he block pncpal sesses s a gaphcal meho o fn he sesses acng on an sess block. R Cene p = cene + R p = cene - R n = cene + Rcos( ) n = Rsn( ) ma n-plane = R Mamum Shea Sess mad p ma p mn, ae whee, o z s equal o zeo ecep on he nse of a pessue essel, whee = - essue. Consucon of Moh s ccle. lo he pons. Connec he wo pons wh a lne, whee cosses he -as s he cene of he ccle.. Daw he ccle.. he aus of he ccle s he mamum n-plane shea sess. he pncpal sesses ae he pons whee he ccle cosses he -as. lane Sess Fn he sans fom known sess olume laaon e z Deflecon Calculaons negae he momen equaon o fn eflecon () M () M Dsplacemen a locaon a locaon Moulus of elasc of nea Sole he poblems usng he bouna conons Usng he slope o eflecon. he sngula funcons ae efne fo aous foce loas o connue fom he eale sngulaes. Foces () M() -M < a > - -M < a> on Foce Dsbue loa Ramp loa - F < a > -F < a> -w < a > -(w/) < a> -(w/)< a > -(w/6) < a> Foces () () -M < a > -(M / ) < a> on Foce Dsbue loa Ramp loa -(F/) < a > -(F/ 6)< a> -(w/6)< a > -(w/)< a> -(w/)< a > -(w/) < a> 5

5 Sngula funcons o no un on unl a s eache. he bouna conons ae use o sole he poblem. () w () () M () S () w M M S w S Dsplacemen a locaon Foce a locaon Shea a locaon a locaon Slope a locaon ( * ) Moulus of elasc of nea / aus of gaon ffece lenghs sleneness ao he effece lengh s epenen on he bouna conons n he column Fe Fe anslaon e=. Supeposon fo eflecon of a beam ean o use he ables o fn he eflecon an slope a a pon on he beam. neemnae Beams wo mehos o sole he neemnae eflecon poblems fo he eflecon of beams.. negaon of he momen equaons (a) Sa fom he foce funcon (b) ppl bouna conons o sole fo he coeffcens of negaon.. Supeposon Meho (a) Beak he poblem no sacall eemnae poblems. (b) Sole he poblem usng he bouna conons fo splacemen an/o slope. Bucklng of Columns c e e c e c e c Ccal bucklng foce ffece lengh Ccal bucklng sess Coss-seconal aea Moulus of elasc of nea

Physics 201 Lecture 15

Physics 201 Lecture 15 Phscs 0 Lecue 5 l Goals Lecue 5 v Elo consevaon of oenu n D & D v Inouce oenu an Iulse Coens on oenu Consevaon l oe geneal han consevaon of echancal eneg l oenu Consevaon occus n sses wh no ne eenal foces