MECHANICS OF MATERIALS

Transcription

1 MEHNS OF MTERS UNX STRESS-STRN Stess-Sti ue fo Mild Steel Te slope of te lie potio of te ue equls te modulus of elstiit. DEFNTONS Egieeig Sti ε / o, wee ε egieeig sti (uits pe uit) STRESS, S ge i legt (uits) of meme o oigil legt (uits) of meme eet Elogtio % Elogtio D m # eet Redutio i e (R) Te % edutio i e fom iitil e, i, to fil e, f, is: i - f %R e o # i Se Stess-Sti γ τ/g, wee γ se sti τ se stess G se modulus (ostt i lie tosio-ottio eltiosip) G E, wee ^ E modulus of elstiit (Youg's modulus) oisso's tio (ltel sti)/(logitudil sti) o STRESS, M Uiil odig d Defomtio σ /, wee σ stess o te oss setio lodig oss-setiol e ε δ/, wee δ elsti logitudil defomtio legt of meme E f d d E Tue stess is lod diided tul oss-setiol e wees egieeig stess is lod diided te iitil e. THERM DEFORMTONS δ t α(t T o ), wee δ t defomtio used ge i tempetue α tempetue oeffiiet of epsio legt of meme T fil tempetue T o iitil tempetue YNDR RESSURE VESSE lidil essue Vessel Fo itel pessue ol, te stesses t te iside wll e: t i o i o - i d - Fo etel pessue ol, te stesses t te outside wll e: t -o o i o - i d σ t tgetil (oop) stess σ dil stess i itel pessue o etel pessue i iside dius o outside dius i -, wee Fo essels wit ed ps, te il stess is: i i - o i σ t, σ, d σ e piipl stesses. o Fli, Rid., d ul K. Toj, Egieeig Mteils & Tei pplitios, t ed., Hougto Miffli o., Bosto, MEHNS OF MTERS

2 We te tikess of te lide wll is out oe-tet o less of iside dius, te lide e osideed s tiwlled. wi se, te itel pessue is esisted te oop stess d te il stess. i i t t d t i o wee t wll tikess d. STRESS ND STRN iipl Stesses Fo te speil se of two-dimesiol stess stte, te equtios fo piipl stess edue to, -! d Te ile dw wit te ete o te oml stess (oizotl) is wit ete,, d dius, R, wee, R - d Te two ozeo piipl stesses e te: σ R σ R w, R i R, Te two ozeo lues lulted fom tis equtio e tempoil leled σ d σ d te tid lue σ is lws zeo i tis se. Depedig o tei lues, te tee oots e te leled odig to te oetio: lgeill lgest σ, lgeill smllest σ, ote σ. tpil D stess elemet is sow elow wit ll idited ompoets sow i tei positie sese. Mo's ile Stess, D To ostut Mo's ile, te followig sig oetios e used.. Tesile oml stess ompoets e plotted o te oizotl is d e osideed positie. ompessie oml stess ompoets e egtie.. Fo ostutig Mo's ile ol, seig stesses e plotted oe te oml stess is we te pi of seig stesses, tig o opposite d pllel fes of elemet, foms lokwise ouple. Seig stesses e plotted elow te oml is we te se stesses fom outelokwise ouple. w Te mimum iple se stess is τ i R. Howee, te mimum se stess osideig tee dimesios is lws - m. Hooke's w Tee-dimesiol se: ε (/E)[σ (σ σ z )] γ τ /G ε (/E)[σ (σ z σ )] γ z τ z /G ε z (/E)[σ z (σ σ )] γ z τ z /G le stess se (σ z ): ε (/E)(σ σ ) ε (/E)(σ σ ) ε z (/E)(σ σ ) Uiil se (σ σ z ): ε, ε, ε z oml sti σ, σ, σ z oml stess γ, γ z, γ z se sti τ, τ z, τ z se stess E modulus of elstiit G se modulus oisso's tio R V S W E S W - - S W T X σ Eε o σ Eε, wee f * * f dll, S.H., d N.. Dl, todutio to Meis of Solids, MGw-Hill, New Yok, MEHNS OF MTERS

3 TORSON Tosio stess i iul solid o tik-wlled (t >. ) sfts: T J wee J pol momet of ieti TORSON STRN z z limit ^Dz/ Dz d ^ z/ dz Dz " Te se sti ies i diet popotio to te dius, fom zeo sti t te ete to te getest sti t te outside of te sft. dφ/dz is te twist pe uit legt o te te of twist. zz Gzz Gd ^ z/ dz T G^dz/ dz # d GJ^dz/ dz z # GJ T dz o φ totl gle (dis) of twist T toque legt of sft T, wee GJ T/φ gies te twistig momet pe di of twist. Tis is lled te tosiol stiffess d is ofte deoted te smol k o. Fo Hollow, Ti-Wlled Sfts T, wee t t m m tikess of sft wll te totl me e elosed te sft mesued to te midpoit of te wll. BEMS Seig Foe d Bedig Momet Sig oetios. Te edig momet is positie if it podues edig of te em oe upwd (ompessio i top fies d tesio i ottom fies).. Te seig foe is positie if te igt potio of te em teds to se dowwd wit espet to te left. OSTVE BENDNG NEGTVE BENDNG Te eltiosip etwee te lod (w), se (V), d momet (M) equtios e: dv^ w^ - d dm^ V d V - V # 7- w^d M - M # V^d Stesses i Bems Te oml stess i em due to edig: σ M/, wee M te momet t te setio te momet of ieti of te oss setio te diste fom te eutl is to te fie lotio oe o elow te eutl is Te mimum oml stesses i em due to edig: σ ± M/, wee diste fom te eutl is to te outemost fie of smmetil em setio. σ M/s, wee s /: te elsti setio modulus of te em. Tsese se stess: τ VQ/(), wee V se foe Q l l, wee e oe te le (o ple) upo wi te desied tsese se stess ts l diste fom eutl is to e etoid B widt o tikess o te oss-setio Tsese se flow: q VQ/ Timoseko, S., d Gleso H. Mulloug, Elemets of Stegts of Mteils, K. V Nostd o./wdswot ulisig o., 99. OSTVE SHER NEGTVE SHER 78 MEHNS OF MTERS

4 Defletio of Bems Usig /ρ M/(E), d E M, diffeetilequtio of defletioue d d E / d dm^ d V d E dv / d w d ^ - Detemie te defletio ue equtio doule itegtio (ppl oud oditios pplile to te defletio d/o slope). E (d/d) M() d E [ M() d] d Te ostts of itegtio e detemied fom te psil geomet of te em. omposite Setios Te edig stesses i em omposed of dissimil mteils (mteil d mteil ) wee E > E e: σ -M/ T σ -M/ T, wee T te momet of iteti of te tsfomed setio te modul tio E /E E elsti modulus of mteil E elsti modulus of mteil Te omposite setio is tsfomed ito setio omposed of sigle mteil. Te etoid d te te momet of ieti e foud o te tsfomed setio fo use i te edig stess equtios. OUMNS itil il lod fo log olum sujet to uklig: Eule's Fomul E, wee _ K, i, ued olum legt K effetie-legt fto to out fo ed suppots Teoetil effetie-legt ftos fo olums ilude: ied-pied, K. Fied-fied, K. Fied-pied, K.7 Fied-fee, K. itil uklig stess fo log olums: E, ^ K, / wee dius of gtio / K, / effetie sledeess tio fo te olum EST STRN ENERGY f te sti emis witi te elsti limit, te wok doe duig defletio (etesio) of meme will e tsfomed ito potetil eeg d e eoeed. f te fil lod is d te oespodig elogtio of tesio meme is δ, te te totl eeg U stoed is equl to te wok W doe duig lodig. U W δ/ OMOSTE SETON TRNSFORMED SETON MTER MTER E, E, E, E, NEUTR XS Te sti eeg pe uit olume is u U/ σ /E (fo tesio) 79 MEHNS OF MTERS

5 MTER ROERTES Tle - Tpil Mteil opeties (Use tese lues if te speifi llo d tempe e ot listed o Tle elow) Mteil Modulus of Elstiit, E [Mpsi (G)] Modulus of Rigit, G [Mpsi (G)] oisso's Rtio, oeffiiet of Teml Epsio, α [ 6 /ºF ( 6 /º)] Desit, ρ [l/i (Mg/m )] Steel lumium st o Wood (Fi) Bss oppe Boze Mgesium Glss olstee olil loide (V) lumi Fie mide Fie Boo Fie Bellium Fie BeO Fie o Fie Silio ide Fie 9. (.). (69.). (.).6 (.).88. () 7 (7).97. (96) 6. (). (7). () <.6 (<) 8 () 8. () 8 (). () 8 (). (7) 8 (). (8.).8 (6.) 6. (.).6 (.).8 () 6. () 6. (). (6.) (.7). (.6) 6.7 (.).7 (.). (8.7) 9. (6.6). (8.) (). (9.) 8.9 (7.) 8. (.).8 (7.8).98 (.7).6.8 (6.87.8).. (8.8.7). (8.9).78. (7.78.7).6 (.7).9 (.).8 (.).7 (.). (.9).7 (.).8 (.).69 (.9).8 (.).8 (.).6 (.) Tle - ege Meil opeties of Tpil Egieeig Mteils (U.S. ustom Uits) (Use tese lues fo te speifi llos d tempetue listed. Fo ll ote mteils efe to Tle oe.) Mteils Metlli Speifi Weigt γ (l/i ) Modulus of Elstiit E ( ksi) Modulus of Rigidit G ( ksi) Yield Stegt (ksi) Ultimte Stegt (ksi) σ σ u Tes. omp. Se Tes. omp. Se % Elogtio i i. speime oisso's Rtio oef. of Tem. Epsio α ( 6 )/ F lumium Wougt llos st o llos oppe llos -T6. 66-T6.98 G STM.6 Mllele STM Red Bss 8.6 Boze Mgesium llo [ m -T6] Steel llos Stutul 6 Stiless Tool Titium llo [ Ti-6l-V] Nometlli oete ow Stegt Hig Stegt lsti Reifoed Kel 9 % Glss Wood Selet Stutul Gde Dougls Fi Wite Spue d.8 d.9 d.97 d.9. SEF VUES MY VRY FOR RTUR MTER DUE TO OY OR MNER OMOSTON, MEHN WORKNG OF THE SEMEN, OR HET TRETMENT. FOR MORE EXT VUE REFERENE BOOKS FOR THE MTER SHOUD BE ONSUTED. THE YED ND UTMTE STRENGTHS FOR DUTE MTERS N BE SSUMED EQU FOR BOTH TENSON ND OMRESSON. MESURED ERENDUR TO THE GRN. d MESURED RE TO THE GRN. e DEFORMTON MESURED ERENDUR TO THE GRN WHEN THE OD S ED ONG THE GRN. Hiele, R.., Meis of Mteils, t ed., etie Hll,. 8 MEHNS OF MTERS

6 Simpl Suppoted Bem Slopes d Defletios M m BEM SOE DEFETON EST URVE m ( ) 6E 8E 8E m m () 6E ( 6E ) ( 6E ) () 6E M E M 6E m / M M ( E 6E ) w m w w E m 8E w ( E ) m w m w 8E 7w 8E / m w 768E.66 w E t.98 w ( ) E / w ( 8E 8 7 ) / < w 7w m w.6 6E E w t.9 E w 6E ( 7 ) Hiele, R.., Meis of Mteils, t ed., etie Hll,. 8 MEHNS OF MTERS

7 tileeed Bem Slopes d Defletios BEM SOE DEFETON EST URVE m m ( E E 6E m ) m m m m 8E m 8E 6E ( ( E _ / _ ( ( / w m m 6E w m w 8E w ( E 6 ) m m M m m M M M m E E E w m m m w 8E m 7w 8E w E ( w ( /) 9E / / ( w m m w E w E m w E ( ) m Hiele, R.., Meis of Mteils, t ed., etie Hll,. 8 MEHNS OF MTERS

8 DESGN OF STEE OMONENTS (NS/S 6-) RFD, E 9, ksi BEMS Fo doul smmeti ompt -sped memes et out tei mjo is, te desig fleul stegt φ M is detemied wit φ.9 s follows: Yieldig M M p F Z wee F speified miimum ield stess Z plsti setio modulus out te -is tel-tosiol Buklig Bsed o ig wee is te legt etwee poits tt e eite ed gist ltel displemet of te ompessio flge o ed gist twist of te oss setio wit espet to te legt limits p d : We p, te limit stte of ltel-tosiol uklig does ot ppl. We p < -p M > Mp -_ Mp - 7. FS ie oh# Mp - p wee. Mm. M M M M m B M m solute lue of mimum momet i te ued segmet M solute lue of mimum momet t qute poit of te ued segmet M B solute lue of mimum momet t etelie of te ued segmet M solute lue of mimum momet t tee-qute of te ued segmet Se Te desig se stegt φ V is detemied wit φ. fo wes of olled -sped memes d is detemied s follows: V.6 F (d t w ) OUMNS Te desig ompessie stegt φ is detemied wit φ.9 fo fleul uklig of memes witout slede elemets d is detemied s follows: F g wee te itil stess F is detemied s follows: K () We. F E F # 7, F : 68. Fe DF K () We > 7. F E, F 877. Fe wee K/ is te effetie sledeess tio sed o te olum effetie legt (K) d dius of gtio () K is detemied fom S Tle --7. o S Figues --7. d --7. o p. 8. F e is te elsti uklig stess E/(K/) OD w VUES OF FOR SMY SUORTED BEMS TER BRNG ONG SN NONE OD T MDONT T OD ONT NONE ODS T THRD ONTS T OD ONTS ODS SYMMETRY ED NONE ODS T QURTER ONTS T OD ONTS ODS T QURTER ONTS NONE T MDONT T THRD ONTS T QURTER ONTS T FFTH ONTS NOTE: TER BRNG MUST WYS BE ROVDED T ONTS OF SUORT ER S SEFTON HTER F. dpted fom Steel ostutio Mul, t ed., S,. V ENGNEERNG

9 d t f X Y X Tle -: WSpes Dimesios d opeties t w f e Dept We Flge is X-X is Y-Y Spe d t w f t f..... WX WX WX WX WX WX WX WX WX WX WX W8X W8X W8X W8X W8X W8X W8X W6X W6X W6X W6X W6X W6X WX WX WX WX WX WX WX WX WX WX WX WX WX W W W W W dpted fom Steel ostutio Mul, t ed., S,. S Z V ENGNEERNG

10 TBE --7. ROXMTE VUES OF EFFETVE ENGTH FTOR, K BUKED SHE OF OUMN S SHOWN BY DSHED NE. () () () (d) (e) (f) THEORET K VUE REOMMENDED DESGN VUE WHEN DE ONDTONS RE ROXMTED END ONDTON ODE ROTTON FXED ND TRNSTON FXED ROTTON FREE ND TRNSTON FXED ROTTON FXED ND TRNSTON FREE ROTTON FREE ND TRNSTON FREE FOR OUMN ENDS SUORTED BY, BUT NOT RGDY ONNETED TO, FOOTNG OR FOUNDTON, G S THEORETY NFNTY BUT UNESS DESGNED S TRUE FRTON-FREE N, MY BE TKEN S FOR RT DESGNS. F THE OUMN END S RGDY TTHED TO ROERY DESGNED FOOTNG, G MY BE TKEN S.. SMER VUES MY BE USED F JUSTFED BY NYSS. S Figue --7. ligmet t, sidesw iiited (ed fme) S Figue --7. ligmet t, sidesw uiiited (momet fme) G K G B G K G B Steel ostutio Mul, t ed., S,. 8 V ENGNEERNG

11 Figue e & etoid e Momet of eti (Rdius of Gtio) odut of eti 6 STTS / / / / / / / ( )/ / / / ( ) ( ) ( ) si ( os )/ ( si )/ /6 /6 / / /6 /6 / / 6 [ ( )] 6 [ ( )] [ ( )] J ( ) p ( ) 6 8 ( ) ( ) ( ) ( ) ( ) 6( ) [ si( os ) ] ( si ) [ si( os) ] ( sios) 6 6 8( ) ( ) ( ) si si ( os ) ( si) ( os) ( os) 6 House, Geoge W., d Dold E. Hudso, pplied Meis Dmis, D. V Nostd omp,., ieto, NJ, 99. Tle epited pemissio of G.W. House & D.E. Hudso. [ ] ( ) [ ( ) ] ( ) [ ] 6 7 [ ] ( si os) 7

12 66 STTS J p ( ) J p / /() si ( si os )/ ( si os )/ os si os si os si si si os si os si os si os si os si os si os si os si House, Geoge W., d Dold E. Hudso, pplied Meis Dmis, D. V Nostd omp,., ieto, NJ, 99. Tle epited pemissio of G.W. House & D.E. Hudso. [ ] [ ] [ ] [ ] [ ] Figue e & etoid e Momet of eti (Rdius of Gtio) odut of eti

13 67 STTS / / / / /8 / /7 7 / House, Geoge W., d Dold E. Hudso, pplied Meis Dmis, D. V Nostd omp,., ieto, NJ, 99. Tle epited pemissio of G.W. House & D.E. Hudso. (/ ) (/ / ) / Figue e & etoid e Momet of eti (Rdius of Gtio) odut of eti