Mechanics and strength of materials

Size: px

Start display at page:

Download "Mechanics and strength of materials"

Virginia Miles
6 years ago
Views:

Lecue pogam D ż. Po Szulc Wocław Uves of Techlog Facul of Mechacal ad Powe Egeeg 00 Mechacs ad segh of maeals. Kemacs of a po.. Moo of a gd bod 3. Damcs of fee ad cosaed moo of a po 4.

1 Lecue pogam D ż. Po Szulc Wocław Uves of Techlog Facul of Mechacal ad Powe Egeeg 00 Mechacs ad segh of maeals. Kemacs of a po.. Moo of a gd bod 3. Damcs of fee ad cosaed moo of a po 4. Damcs of a gd bod 5. Cosevao laws. 6. Wok, powe ad kec eeg 7. Mass geome ad mpac heo 8. Teso ad compesso. Hooke s law. 9. Bedg. 0. Bedg le of beam.. Shea, oso ad bucklg.. Hpohess of exeo. Combed sess. 3. Eeg mehods Leaue Ioduco. TYLOR J., Classcal mechacs, Uves Scece Books, 005. SCHECK F., Mechacs- Fom Newo s Laws o Deemsc Chaos, Spge, SINGH U.K., DWIVEDI M., Poblems ad soluos mechacal egeeg, New ge Ieaoal, Mo R., ppledseghofmaeals, UppeSaddleRve, Peaso/PeceHall, SIUT W., Mechaka Techcza, Wdawcwa Szkole Pedagogcze, Waszawa MISIK J., Mechaka Ogóla, WNT, Waszawa ŻUCHOWSKI R., Wzmałośćmaeałów, Ofca Wdawcza PW., Wocław 998. KINEMTICS seco descbg he mechacs of moo of a po o block, whou akg o accou he wegh ad causes of chage moo - The geome of moo MOTION descbed as chagg bod poso elave o he efeece bod whch emas a es

2 Tack ofa po Ths s a sold le lfomed b he subseque locao of a movg po. Po pah ma be a sagh le o a cuve Kemac equaos of moo of a maeal po Recagula coodaes x f (), f (), z f 3 () z Radus veco () x z O z x x(), (), z z() x x Veloc of a po cceleao of a po v v + B v v v v O veage veloc v s Isaaeous veloc d v lm & 0 Medum v cceleao - v v a s - Isaaeous cceleao v dv a lm v & && 0

3 The equaos of ufom lea moo Equaos of lea moo vaable ufoml v v acceleao dv a cos s v veloc v vo + a veloc dsace ds v cos s so + v s g α g αv dsace a s so + vo + a > 0 ufoml acceleaed moo a < 0 ufoml eaded moo Lea moo vaable ufoml dv a cos v vo + a a s so + vo + Cuvlea moo a a s a v Tageal acceleao a a dv v& Nomal cceleao v ρ a a + a value a a +a

4 Ufom moo a ccle dsace s lea veloc agula veloc α ( ) ds d α v ω ω dα ω π 60 gula acceleao Tageal acceleao Nomal cceleao ε a dω ( ω) dv d dω as ε v ω ( ) ( ) a a + a ω + ε ω + ε 4 s Moo of a gd bod Moo of a gd bod z O B C C B B C B C Rgd bod dsaces bewee pos ae uchaged Moo of a gd bod ca be deemed b veco equaos of hee pos, B, C z O C B C B u() u() u() C B C () ( ) + u(), o () ( ) + u(), B B o () ( ) + u(), C C o x () () B B () C C x u() -Shf s equal fo all pos of he bod

5 Moo of a gd bod Dffeeag he above equaos of moo vecos wh espec o me we ge veloc ad acceleao of pos,b,c du() v vb vc d u() a ab ac Vecos of veloc ad acceleao of all pos of a gd bod, movg wh a advacg moo aehesamea he same me The oa moo of he sold aoud a fxed axs lump ca be oaed aoud he axs ol (passg hough wo pos), called he axs of oao ϕ ϕ() ω ϕ& ε ω & ϕ&& v ω v ω cceleao oao Newo's Secod Rule cceleao of ageal ad omal acceleao of a po of a gd bod lg a a dsace fom he axs of oao wh espec o me we ge b dffeeag he fomula fo lea veloc eldg: dω a v& ε a v ω ω a a + a ε + ω 4 The chage of moo s popooal o he appled foce ad akes place he deco of he sagh le alog whch ha foce acs m cos d F (mv) dv m ma F

6 Iea a F ma Ca acceleaes wh he acceleao. We mus heefoe wok foce. ccodg o he pcple of aco ad eaco ou hads, wok he same foce fom he ca, bu eued o he coa. D ma Ths s he foce of ea (D'lembe) D F The damcs of fee ad cosaed moo of a po. D'lembe's Pcple I he case of fee moo of a po ssem of acve foces balaces he foce of ea F + ( ma) 0 I he case of moo cosaed po foce espose of acvead balaced es wh he foce of ea F + R + ( ma) 0 The momeum of he maeal po We henewo's secod law he fom of: d (mv) dp The veco s called he momeum o qua of moo of a po. mv p F The pcple of cosevao of momeum of a po I he eve ha a maeal po does o wok foce o foces balace, he momeum of he maeal po s cosa. d dp (mv) p cos. 0

7 The pcple of mass momeum ad mpulse foces The Newo s secod law: F d(mv) F o dπ Elemea mpulse foce acg o a maeal po s equal o he ga momeum of he elemea po. Π The pcple of mass momeum ad mpulse foces Iegag boh sdes of pevous equao, we oba F d m v m v ( ) F -mpulse s he oal foce F he me eval -, We oba v Π p p The gowh momeum of a movg mass po s equal o he oal mpulse foces gula momeum of maeal po Ko mv Thsshemome of momeumb he chose pole K& o Mo o o M 0 K cos Damc equaos of moo of a maeal po Damc equao of moo veco fom ca be eplaced b hee aalcal equaos: Fx F ma mx &&, F m &&, Fz mz &&. The fs ask (smple) -hese ae he paamec equaos of he ack, whch moves he maeal po, fd he foce acg o po. The secod ask (evese) - o deeme he equao of moo, wh a pacula segh

8 Relave mooof a maeal po Moo elave o he fxed po s defed b he equao m ab ad I whch Du mau called he lfg foce of ea. I s equal o mass mulpled b acceleao of floag po ad s oppose ha a u F a a + a b w u I a movg ssem he equao of moo s deemed ma F ma w u Wok ofa cosa foce Pemae wok ofa foce o a sagh moveme he deco of foce s called he poduc of hs foce b he legh of shfs Fs The u of wok he SI ssem sj (dżul): kgm J Nm m s Wok ofa cosa foce F If he veco of foce s cled o he deco ems of a shf, he wok s calculaed fom he fomula: Fs Fs cos α The wok makes ol a ageal foce compoe o he ajeco of F. Jobs omal compoe o he ajeco of F s equal o zeo F F Wok ofa cosa foce α 0 Fs > 0 0 < α < 90 Fs cos α > 0 α < α < 80 Fs cos α < 0 α 80 Fs < 0 I geeal: a)wok s a scala, b)wok ma ake posve oegavevalues ad zeo, c)wok s doe ol b compoe of foce ageal o he pah.

9 Wokofa vaable foce The elemea wok of he vaable foce o a shf he do poduc of foce F b a elemea shf δ F ds because F ds Fds cos F, ds Fds so δ F ds ( ) ds s called Wokofa foceo a shf s equal o he oal wok foces he especve cosue compoes of dsplacemes Wokofa vaable foce Toal wok of he poso o poso o he ack ae obaed b egag he expesso of a elemea wok. F X + jy + kz s δ s ds dx + jd + kdz x z Xdx + Yd + Zdz x z Wok o ccula pah Whe he foce F acg o a po movg alog a ccula pah (bel eso bel dve), we oba Wok o ccula pah Expesso F deemes he mome of foce F elave measue O (eg, cee of he dsc). We call oque ds F d F O δ F ds Fds ds dϕ δ F dϕ M o F The fomula fo he elemea wok akes he fom: δ M dϕ o

10 Wok o ccula pah ϕ Toal wok o he oad o ϕ deeme he agula egal ϕ o ϕ M dϕ Powe I pacce, we ae ofe eesed he volume of wok he moo o he mache ca pefom pe u me. The wok s elaed ohe u of me s called powe. I pacula, he saaeous powe of he emplome elaoshp s called he elemea o he me a whch hs wok was pefomed δ P expesso fo he saaeous powe s peseed he followg fom: Fds P o P F v Powe oao Powe P Powe Modϕ Powe oao P M o ω gula veloc dϕ ω Whe he oao velocsepo s usg he ege speed, pm -he he agula veloc s calculaed fom he fomula: ω π 60 Powe π P Mo 30 The fudameal u of powe s W J/s Nm/s Ths echcal us : kw MW

11 Effcec Pcple ofwok ad kec eeg Effcecs he ao of eceved wok(powe)o opeae he pu wok(powe) dv F ma m δ F ds Mechacal effcec ao deemed b: u η P P u 0 0 The effcec s a dmesoless umbe ad ca be egaded as a chaacesc measue of compaso eges ad maches, as fa as he ecoomcal wa o wok ulzao s o loaded dv δ m ds mvdv The gh sde of hs equao s a fuco of he oal dffeeal called he kec eeg of a movg maeal po. E k mv ds v Pcple ofwok ad kec eeg Based o he above depedece we oba δ de Ths equao shows he pcple of wok ad kec eeg, expessed he fom of dffeeal equao. fe egao we oba E E The kec eeg of a movg maeal po ceases o deceases he volume of wok doe b foces acg o he maeal po. Fuco of feld foces Toal dffeeal feld fuco s equal: Φ Φ Φ dφ dx + d + dz x z So he dffeeal was equal o he elemea wok X δ Xdx + Yd + Zdz mus be me depedg o Φ Y x Φ Z Φ z

12 Poeal of feld foces Veco offeldfoce ca be we he fom Φ Φ Φ F + j + k x z The gh sde s he gade of he fuco, Φ so F gad Φ ma Pogessve moo of a gd bod c F whee: m mass of a gd bod a C - acceleao of cee of mass mx && c Fx x m && z O c F c F C F C zc a C xc mz && F F c Fz The heoem o he devave of agula momeum The devave of agula momeum of he bod elave o s cee of mass s equal o he geomec momes of all foces exeal o hs measue d Kc M c Pogessve moo of a gd bod The pogessve moo of all pos of a gd bod have he same veloc, such as he cee of mass of he bod. Thus, gula momeum of a gd bod elave o he cee of mass s equal o zeo K 0 The equao shows ha whe he bod moves wh a advacg moo s he sum of he geomec momes of exeal foces o he cee ofbod mass mus be zeo M c 0 The exeal foces mus ceae a laou ha hashe esul W of a le of aco passg hough he cee of mass C. c

13 Plae moo of gd bod Show he dawg seco of he bod obaed b eseco of he plae paallel o he plae of he decg ad passg hough he cee of mass C xc F F C c F F x Damc equaos of moo of a gd bod To oba he damc equaos of moo of a gd bod we use a fla: damc equaos of Pogessvemoo he pcple of agula momeum a oag moo I z ε M z O x Damc equaos of moo of a gd bod The equao of pogessvemoo he x deco The equao of pogessvemoo he deco The pcple of cosevao of agula momeum a oag moo && x,&& hadwae acceleao of he cee of mass C c Iz ε c mx && z c m && c I ε mome of ea wh espec o he axs z of he bod agula acceleao of he axs z of oao of he bod F F M x z Mome of ea Mome of ea of a maeal po elave o he plae, a axs o polespoduc of he mass b he squae of he dsace of hs po fom he plae, a axs o pole: I m U s [ I ] kg m

14 Mome of ea of maeal pos ssem Mome of eaofmaealposssemhe plae, a axs o polescallehe sum of he momes of ea of all he maeal pos of hs plae, axs o pole. Mome of ea of he cosa Mome of ea of he cosa (les, sufaces o sold maeal) of vew of he plae, called he axs o pole of he egal I dm I m seched o he whole mass of he ssem Mome of ea of he plae Momes of ea wh espec o he coodae plaes defe he fomulas: I I I x z zx m z, mx, m I Mome of ea axs ad pole I I I x ( ) ( ) ( ) m + z, m x + z, m x + z Mome of ea axs Mome of ea pole ( ) m x + + z o

15 Mome of devao Mome of devao The mome of devao of a po muuall pepedcula plaes, called he poduc of he mass b he dsace fom he po of plaes D m ρ αβ Momes of devao ca be posve, egave ad, pacula, equal o zeo Mome ofdevaoof maeal pos elave o he wo muuall pepedcula plaes, s he sum of he momes of devao of dvdual pos of he maeal ems of hese plaes. D D αβ Fo he cosa αβ m ρ ρ dm exeded o he whole mass. m m Mome of devao The spaal coodaes of he ssem of maeal pos s hee momes of devao D D D m x, x m z x, zx m z z I fla coodaes he maeal ssem has oe mome of devao Paallel asfomao of he momes of ea Mome of ea wh espec o a axs s equal o he momeum paallel o he axs passg hough he cee of gav plus he poduc of he oal wegh b he squae of he dsace of he wo axes. I l I + md c D D m x x

16 Ceal smple collso Ceal smple collso v v w w mv + mv mw + mw he same me w adw showshe velocof boh masses afehe collso. The appome also use he equao esulg fom eeg cosdeaos. Veloc w ad w wll deped o whehe he loss of kec eeg a) eeg eued 00% (pefecl elasc collso of bodes) b) eeg absobed a 00% (pefecl plasc bod collso), c) eeg absobed pa (he acual collso of bodes.) Fo he deemao of hese losses wll oduce he so-called eeg. collso ae, callg hm a model k w w 0 k v v The lm values coespod o he ao k k k 0 fo he bod pefecl elasc fo pefecl plasc bod Ceal smple collso Dagoal ceal collso The acual loss of kec eeg s E mv mv mw mw + Spead he veloc vecos v ad v o compoes omal ad ageal o he plae of coac v v v cos α v s α v v v cos α v s α v w v w ad fe subsuo of equaos fo w ad w eceve m m E v v k m + m ( ) ( ) v v v v v v Fall, afe collso w w w w + w + w w w + v, + v

17 Sess Le's cosde he foce F pe eleme of aea Sess pa a po whee s called a sold bouda, whch seeks he ao of eal foce F b elemea feld of hs seco, whee he feld eds o zeo. F df N p lm, d m 0 T F N Tageal ad omal sess fe speadg he foce Fo he omal compoe N ad ageal T wll eceve he omal ad ageal τ sesses: N dn N lm, d m 0 τ T dt N lm, d m 0 Lea Defomao Smple sechg SegmeB l -chaged afe loadg: B l+ l. The aveage elogao of he segme B wll be : ε s l l Local exeso: ε lm l 0 l l Smple sechg occus whe he esul of educo of he eal foces fom he cee of he coss seco of he bodwll eceve ol he pcpal veco, omal o hs seco N N axal foce, coss-secoal aea of ba omal sess

18 Smple sechg The codo of balace -he sum of he pojecos of all foces he deco of he axs of he od Fx d N 0 () N d () Whe he sesses ae he same fo he ee coss-seco: Ths meas ha he ma veco s equal o he segh of he aggavag ad he absece of ohe sesses o he coss-secoal aea. N Smple sechg Effo of he maeal degee of appoxmao of he load maeal o a ccal sae The codo of he ba segh: llowable sess dop safe faco( ), eb Dageous Sess N dop eb dop Hoocke slaw Laeal defomao F ε l l F l l E F Hooke s law akes he fom: ε E The dffeece of al ad fal hckess s called oal seoss, h h h. Rao of scue of he oal hckess of he al call o he aowg of he u ε ε h h

19 Posso sumbe Gaph dawg low cabo seel The absolue value of he ao of seoss (swellg) of he u ε o u ε elogao (shoeg), s called he coeffce of asvese sa ad Posso's umbe ν ε ν ε d sp p Rm Posso s umbe assumes values wh he 0 0,5 Gaph dawg low cabo seel Idvdual pos o he gaph meas: lm of popooal (he lm of applcabl of Hooke's law) B elasc lm - pacce s assumed ha le ea each ohe, pos ad B ae of equal value: p C, D - eld Re, - cleal vsble o he gaph dawg ad eas o se ol fo cea maeals, such as low cabo seel. R e Fe sp 0 Gaph dawg low cabo seel K -eld esle segh R m (Emegec segh of he maeal). Tesle segh R m s he ao of he maxmum esle segh F max obaed he pocess of dawg he sample hough he box seco of he al sample 0 : R m F fe eachg he sess o he sample ases R m local cosco (he eck). I place of he sample upues - segme KL sechg fom he cha. max 0

20 Dawg gaphs of dffee maeals low cabo seel, coppe cas alumum allos Beam subjeced o pue bedg Pue bedgca be obseved he case of bedg of a psmac beam loaded as show R R x F F R B The codo of bedg segh The sesses he bedgod ae he lages he fbes, hus M g max dop W Beams of equal segh The codo of cosa-segh beam bedg segh alog he ee legh has he fom: M g (x) W (x) dop s we appoach he fbe axs of he beam load deceases

21 Defleco of he beam le The equao fo beam defleco agle Iegag he pevous equao, we oba he depedece of he vaable x s used o desgae he agle of defleco j he fom: Dffeeal equao of defleco of he beam le d dx M g (x) E I ϕ d(x) M (x) ( ) dx E I + g x dx C The equao fo defleco of he beam le fe e-egao wh espec o he vaable x we oba: ϕ (x) dx + D Cosas of egao C ad D we deeme he bouda codos. We have wo pes of hese codos ad he esul fom: sffess of he suppos (bouda codos), cou of he beam (cou codos). Shea sess I coss-seco of a esle od, led a a agle s addo o omal sesses ae also shea sess o shea : τ α s α

22 Pue shea Techcal Shea Sae of sess hese secos, whch hee ae ol a shea sess s called he pue shea Sae of pue shea s dffcul o be pepaed b dec ode of he bod hemselves shea sess, wheeas such a effec ca be obaed b callg eg, esle ad compessve same, absolue value sess, acg wo muuall pepedcula decos whee: τ dop allowable shea sess, T sheafoce, codo of shea segh: τ T τ dop coss-seco subjeced o shea. Cases of he echcal shea Toque Elemea oque eso od axs s: ves weld dm τ dρ ρ

23 gle oso ba Toso agle oud ba of dmesos d, ladm s s he: Codo of he osoal segh Sess he scew od ca o exceed he allowable osoal sessk s ϕ M s G I l 0 M s τ max W0 k s Sesses he maeal Maeal effo dτ zx τ zx + dz dz τ x τ z d x x + dx dx τ x dτ xz τ xz + dx dx τ xz dτ x τ + dτ x x dx dx τ x + d d τ z d z z + dz dz dτ z τ z + dz dz x τ zx dτ z τ z + d d d + d d Maeal effo degee of appoxmao of he load maeal o he bouda codo. Hpoheses ofmaeal effo desgs fo he coveso of he load o he load-dmesoal spaal. z

24 Hpoheses effo Basc coceps Hpoheses effo depedg o he adoped measue of effo ca be dvded o: Sess Defomao Eeg Mxed The pupose of effo hpohess s o deeme he coelaobewee he compoes of he effo a sae of sess: ( x z x z zx ) W F,,, τ, τ, τ, C Ccal values of effo ca be deemed b cag ou he expeece fo oe specfc sae of sess 0, s bes fo he uaxal eso ( ) W F,0,0,0,0,0, C 0 The hpohess of he lages esle sess The auhos of hs hpohess ae Galleusz (63) ad Lebz(864) ccodg o hs hpohess, a measue of he effo s he lages esle sess Ths s he hpohess of sess ad s occasoall used facue mechacs The hpohess of he geaes elogao The auhos of hs hpohess ae E. Maoe, B. S Vea ad J.V. Pocele. The measue of effo accodg o hs hpohess s he geaes elogao. ε εk ε εk ε3 εk Fo ε k adoped elogao ε Z coespodg esle segh Rm

25 The hpohess of he geaes elogao Whe he plae sae of sess s expessed b meas of sa x,, τ x he ma sess s deemed b he fomula: + ± + τ ( ) ( ) 4, x x x ad he educed sess s deemed fom oe of he hee equaos ν k k ν ν ( + ) k Wh educed sess should be he lef sde of he equao, whch s hghe ha he lef sde of each of he ohe wo equaos The hpohess of he lages shea sess Is auhos aec.. Culomb, H. Tesca J.J. Gues. The measue of effo hs hpohess s he geaes ageal sess τ max max m I he case of a smple esle sess s educed equao has he fom: ed max m k ed 0 The hpohess of he lages shea sess Whee max ad m 3, ha: ed 3 k I he case of plae sae of sess, whe: x, 0, τ x τ, educed sess ca be deemed: ed + 4τ The hpohess of eal eeg shea (Hube s hpohess) s a measue of effo b hs hpohessshe specfc eeg shea, whch a sae of sess s spaall: φ + ν ( x ) + ( z ) + ( z x ) + 6( τ x + τ z + τzx ) 6E I he uaxal sae of sess esposble shea eeg wll be: + ν φ 6E 0

26 The hpohess of eal eeg shea (Hube s hpohess) Reduced sess : τ + τ + τ ( ) ( ) ( ) 6( ) ed x z z x x z zx Reduced sess he ssem of pcpal sesses ca be deemed fom he equao: + + ( ) ( ) ( ) ed 3 3 The hpohess of eal eeg shea (Hube s hpohess) Fo a plae sae of sess x,, x educed sess expessed he equao: ( ) ed x + 3τ If x, 0, τ x τ hs equaoof educed sess ake he followg fom: ed + 3τ Eeg mehods Casglao s Theoem geealzed coodaes f coespodg o he foce F s equal o he paal devave of elasc eeg V f δv δ F Meabe s Theoem paal devave of he elasc eegvof he ssem wh espec o sacall deemae eaco X s equal o zeo δv 0 δx

EMA5001 Lecture 3 Steady State & Nonsteady State Diffusion - Fick s 2 nd Law & Solutions

EMA5001 Lecture 3 Steady State & Nonsteady State Diffusion - Fick s 2 nd Law & Solutions EMA5 Lecue 3 Seady Sae & Noseady Sae ffuso - Fck s d Law & Soluos EMA 5 Physcal Popees of Maeals Zhe heg (6) 3 Noseady Sae ff Fck s d Law Seady-Sae ffuso Seady Sae Seady Sae = Equlbum? No! Smlay: Sae fuco