ME/BioE C117 April 13, 2006 Professor Lisa Pruitt Exam #2. This is a closed notes/closed book exam. Show all work. Write your name on every page.

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1 ME/BoE C7 Aprl 3, 00 Proessor Ls Prutt Ex # Ths s closed otes/closed book ex. Show ll work. Wrte your e o every pge. AME: (0 pts) You re egeer workg or edcl devce copy, d surgeo proposes desg or spl dsc replceet tht s show below. D R Rg Core h C Fgure. Spl Dsc Replceet (ot to scle) The devce s de up o core tht s surrouded by rgs. The surgeo tells you tht the resog behd ther desg s to c the bsc structure o the turl dsc. The core supports lod by pressurzto, d the rgs wll keep the core ro over expdg. Ths s logous to the ucleus pulposus d ulus brosus o the turl tervertebrl dsc. The core s de ro sot d copressble terl, d the rgs re de o Coblt-Chroe wre. Ths tl desg hs 3 totl rgs. Desos d terl propertes o the devce re gve below: Core Icopressble Deter, D C 3.00 c Heght, h C.00 c Rg Modulus, E 00 GP Yeld Stregth, yld 00 MP Deter, D R totl rgs (tl desg) D C Your job s to detere ths s se d esble desg. Cosder the uor xl lodg scero show below.

2 Uor lod (ot to scle) Sttc Alyss Fgure. Illustrto o Uor Uxl Lodg. Clculte the stress the rgs or worst-cse lod equl to 4X body weght. Assue 70 kg or ss.. Clculte the pressure the dsc. Sce ths s copressble terl, the pressure wll be the se ALL drectos the core. Aswer The pressure, p, the dsc s the xl orce dvded by the xl crosssectol re. ( )( )( Force 4 BW 4 70kg 9.8 ) p s 3.9MP () Are AXIAL π π DC ( 0.03) 4 4 b. Peror equlbru o the ollowg cut secto o the devce to detere the hoop stress, hoop, the rgs. The orce due to the pressure ust be blced by the orce the rgs. Assue uor stress, hoop, the rgs. (ote: Solve or hoop sybolclly becuse you wll be usg tht equto g.)

3 Hoop stress, hoop (out o pge) y Pressure (ll drectos) (ot to scle) z x Fgure 3. Secto o Devce or Equlbru Alyss The ethod to clculte the hoop stress the rgs s slr to tht used to derve the hoop stress or cyldrcl th wlled pressure vessel. Sttc equlbru s eorced to clculte the hoop stress. F Z 0 () Assue the z-xs s out o the pge d s postve. ote tht the oly et orce, due to pressure, s the z-drecto. The x d y copoets or pressure ccel out. The oly et orces re show below. The orce due to pressure, Fp, s blced by the orce due to hoop stress, F. F + F 0 (3) p Fp, s equl to the pressure ultpled by the projected re, d F s equl to the hoop stress ultpled by the totl re o the rgs. padisc + ARIGS 0 (4) pa A (4b) DISC RIGS

4 Equto 4b s llustrted the ext gure. A DISC A RIGS pressure, p, cts o ths shded re, A DISC Hoop stress,, cts o these shded res, A RIGS Equto 4b sttes tht the pressure ctg o the shded re o the let ust be blced by the stress ctg o the shded re o the rght. The re o the dsc s gve by the ollowg equto. A D DISC C h C () It wll be coveet to put the re o the rgs ters o rbtrry uber o rgs,. π ARIGS DR ( ) () 4 (the ctor o s there due to the ct tht ech rg hs two ces o the cross secto s show Fgure 3.) π pdc hc DR ( ) (4c) 4 Solve or Aswer b pdc hc (4d) πdr (For 3 rgs s show the tl desg, the stress s equl to 989 MP.) c. Wht s the u uber o rgs eeded or sety ctor or yeld o the rgs equl to t lest?

5 The swer ro Proble b (Eq. 4d) c be used to solve or. pd h C C (4e) πdr The gve sety ctor llows you to clculte the llowble hoop stress. Allowble YLD YLD 00MP Sety _ Fctor 300MP Appled All quttes re kow so you c solve or drectly usg Eq. 4e. ( )( 3.9 P)( 0.03)( 0.0) π ( 0.00) ( 300 P) Aswer c 9.9 rgs. The uber o rgs ust be rouded up to the ext whole uber. The resultg hoop stress or rgs s equl to 97 MP.. The spe expereces pproxtely llo totl lodg cycles per yer d you wt the devce to lst yers. Assue tht the stress cycles vry ro zero stress to xu vlues s ollows: 98% o totl cycles experece stress equl to 0. x hoop.% o totl cycles experece stress equl to 0.8 x hoop 0.% o totl cycles experece stress equl to.0 x hoop ( hoop s the vlue resultg ro the desg you clculted c.) You hve ctul tgue dt or the terl used or the rgs s show the S- plot Fgure 4. The dt ws geerted ro tests perored wth stress rge ro 0 to the xu vlue, S.. Wht s the edurce lt o ths terl? Aswer The edurce lt s deed s the xu cyclc stress vlue, S, where the terl does ot l. Accordg to the dt, ths terl does ot l whe subjected to cyclc stress vlues s hgh s 00 MP. b. Wll the rgs survve or the requred uber o cycles? Mer s Rule ust be used to detere the rgs wll l. The rgs wll survve the ollowg equlty s stsed.

6 (7) The vlues or c be oud ro the proble stteet where ech s percetge o the uber o totl requred cycles. A totl o 0 llo cycles re requred The vlues or ust be oud ro the S- plot. A best t le c be used to detere vlues or ech vlue o S. S S S MP 37MP 48MP The vlues or correspodg to the vlues or S re drw o the S- plot. Approxte vlues re gve below Vlues or d c be plugged to Equto 7. Aswer b FAIL A desg wth rgs wll ot stsy tgue requreets. Sce redg vlues ro the S- plot s subjectve, t s ore portt or you to uderstd the procedure rther th get the exct ubers show here. c. Wht s the u uber o rgs ecessry or the prt to survve the requred uber o cycles? There re y wys to pproch ths proble. Oe ethod s to observe tht ddg just eough rgs to drop S below the edurce lt wll lkely result cceptble desg. Vlues or stress c be clculted

7 usg Eq. 4d. Rgs () Stress S S S ew vlues or c be oud ro the S- plot d plugged to Equto 7. Aswer c PASS The resultg u uber o rgs s pproxtely. However, sce redg vlues ro the S- plot s subjectve, the techque s ore portt th the l uber o rgs. S- Dt or Rg Mterl S (MP) S * Test speces rked wth rrows dd ot l. S 00 0 S 3 3 (to ) 0.0E+03.0E+04.0E+0.0E+0.0E+07.0E+08 (cycles) Fgure 4. S- Dt

8 3. Durg pltto trls, t s see tht occsolly the surgeo kes scrtch o oe o the wres, org surce lw. Ths ppers to be uvodble, so you ust desg the prt to tolerte cert tl lw sze cused ter qulty cotrol specto. D R 0.0 Fgure. Flwed Wre Crtcl stress testy (or ths spece sze) s 90 MP* / Prs coecets re C9e- d.9.(or d/d /cycle, K MP* / ) Threshold stress testy (ths cse) s MP* /.. Gve the equto or the stress testy ctor: K I Y ) ( where π Y(). 0<<0. Y(). 0.<<0.4 Y().0 0.4<<0..

9 Seprte d tegrte the Prs Equto to d the uber o cycles requred to grow crck ro tl sze,, to l sze. Assue here tht 0.<<0.4. Show sybolclly. Igore y possble threshold eects. We strt wth the Prs equto, d d CΔK where we kow tht K s ucto o the crck legth, so we collect stu tht s ucto o oto the let sde, d d oto the rght sde. K I Y ) ( π d CΔK d [ ( Δ ) ] π C Y ( ) d d Thus, we tegrte ro the ttl crck stte to the l oe. d C Y [ ( ) ] Δ π ( ) d The proble sttes tht we re cosderg Y() s costt, sce we re oly oe o the pecewse deed reges by presupto. Thereore we c pull out ll costt ters ro the tegrls. C d [ ( ) ] Δ Y π d Thus, we eed oly tegrte -/ betwee the lts d the rest s just uercl. d d + + Thus, we hve

10 d C Y [ ( Δ ) π ] + + whch s the sybolc or o the uber o cycles requred to propgte crck betwee two speced crck legths. b.. Oly you solved 3(): uerclly clculte result ro 3() bove, wth ll preters s gve bove ths proble d 0. d 0., except leve the ppled stress s vrble. Ths gves the le s ucto o the ppled stress. Igore threshold d crtcl eects. Clculte coecet (wth C or /cy or splcty): [ ( Δ ) π ] ( 9e 3)[. π ] C Y.9.9 ( ).48e ( ) Q.9 Uts here re bguous. ow, Q Q.48e Thus, or.e-4 d.e ( ) (.08)(.9 ( ) cycles 3.3e, where the stress ust be put MP!. Oly you dd ot solve 3(): Estte the le o the prt by coputg d/d or ve eqully spced crck legths betwee 0. d 0., d ssug tht d/d s costt betwee these crck legth sttes, lke the hoework. Fd the estted le s ucto o oly the stress, s descrbed 3(b ). Igore threshold d crtcl eects. Crck legths: 0., 0.4, 3 0.8, 4 0., 0.. (Δ0.03) d d Δ Δ C [ ] ( )[ ].9 π 9e 3. π ( ). 9 ( ΔK ) C Y ( Δ ) ( )

11 Thus, we rerrge Δ.9 ( 9e 3)[. π ] ( ). 9 Δ Thus or to (usg verge crck legth) Δ Δ ( ) ( 9e 3). π ( 0.8e 3).9. 9 ( ).e ( ) 0.03e ( ), Δ.4 e ( ), Δ.7 e ( ) e 34 4 Thus, ddg ll these creets up we get Δ.9. 9 ( ) 3.3e ( ).4e Ths s very good estte (whe copred to the closed or soluto)! c. Fd the ppled stress requred or the crck to tke 0 llo cycles to propgte the lyss prt 3(b). How y reorceet rgs does ths requre? We ust reeber (t ll tes) tht the Prs equto ths cse uses stress MP. I you put stress o 7MP to t s 7e P, you wll get out osese. Usg the swer ro bove, we hve 0e cycles: 3.3e. 9 ( ) 0e.9 ( MP) 4. MP (or 0 llo cycles). 3.3e Thus or our le, we eed the rgs to experece o ore th 4. MP servce or sety ctor o oe, or or lure rght t 0 llo cycles. We re ot cosderg sety ctor ths proble, whch y be dubous cocept here ywy. Fro the soluto to the rst proble we hve the uber o rgs requred or gve hoop stress o the rg. Here we ssue tht orce o the plt s the ol weght o the ptet. Other ssuptos or servce orce y lso be pplcble, so stted. ( 0.0)( 70kg) 9.8( ) hmg e s rgs π π Dd ( 0.03)( 0.000) 4 4

12 For stress 4.e (/ ) we requre rgs >7.4 rgs, or 8 dscrete rgs or plt to survve 0 llo cycles ths codto. ote tht ths s er xu, s oly 0 c be stcked wth the uloded heght o the plt (c), d there wll be soe copresso o the devce servce. Ths y requre desg terto! d. Clculte the crtcl crck legth or the stress coputed 3(c). Wll the crck rech ths legth servce? I so, whe? I ot, whe wll lure occur? We re gve tht K c 90MP d we kow K Y ) I ( π so we bck out the crck legth tht gves ths stress testy uder the stress coputed bove. 90MP. MP π c ( 4. ) 90MP MP c ( 4. ) π 0.3 (!) Ths uses rst guess o Y. or < (!) Sce c >0.4, but ths hs lttle eg. Ths s very lrge, d thus we do ot ever expect st rcture the copoet. Icdetlly, the crtcl crck sze or the yeld stress s.8, whch s stll uch lrger th the wre. Also, or ultte stress o 800 MP, the crtcl crck sze s, both ssug Y. Sce the crck wll ot spoteously propgte to lure, the t sees tht the crck ust propgte through the etre wre thckess or t to brek. Ths s ot the cse. As the crck propgtes, the cross sectol re reduces, d hece the stress over the secto hed o the crck creses. Whe the stress throughout ths crtcl secto reches the ultte stregth o the terl, the wre sps. Ths hppes whe bout hl o the thckess o the wre s coprosed, d hece why 0. ws chose.