Stength of Mateals and Falue Theoes 010 State of Stess y xy z x Ths s a D state of stess only the ndependent stess components ae named. A sngle stess component z can exst on the z-axs and the state of stess s stll called D and the followng equatons apply. To elate falue to ths state of stess, thee mpotant stess ndcatos ae deved: Pncpal stess, maxmum shea stess, and VonMses stess. Pncpal stesses: x y x y 1, Gven o known 3 If y =0 (common case) then x 1, 3 x Gven o known xy If x = y =0 then 1 = xy. If y = xy = 0, then 1 = x and =0. xy 1
Maxmum shea stess Only the absolute values ae mpotant. Max(,, ) max,1 If 3 =0, the max,1 max,1 max,13 1 1 The Vom Mses stess: v max,3 max,1,3 1 3 max,1,3 1 max,3 3 max,3 ( 1 ) ( 3) ( 1 3) When 3 =0, the von Mses stess s: v 1 1 When only x, and xy ae pesent (as n combned toson and bendng/axal stess o pue toson), thee s no need to calculate the pncpal stesses, the Von Mses stess s: v x 3 xy Note that n pue shea o pue toson x =0. If x =0, then 3 3 v xy xy Accodng to dstoton enegy theoy, yeldng occus when v eached the yeld stength S y. Theefoe n pue shea, yeldng occus when xy eaches 58% of S y.
Common loadng applcatons and stesses (when oented popely) Dect Tenson/Compesson (only x ) Beam bendng (only x on top/bottom) Pue toson (only xy ) Rotatng shafts (bendng + toson) ( x and xy ) Poblem #S1 A membe unde load has a pont wth the followng state of stess: 10500 ps, Tensle 5500 ps, Compessve x xy 4000 ps 3 0 y Detemne 1,, max (Ans: 11444 tensle, 6444 Compessve, 8944 ps) y =5500 xy =4000 x =10500 3
Stan (one dmensonal) A ba changes length unde the nfluence of axal foces and tempeatue changes. Ognal Length Fnal Length L L Total stan defnton: total t L L Total stan s a combnaton of mechancal and themal stans: t M T F T EA Both the mechancal and the themal stans ae algebac values. T s postve fo an ncease n tempeatue. F s postve when t s a tensle foce. Poblem #S The end of the steel ba has a gap of 0.05 wth a gd wall. The length of the ba s 100 and ts coss-sectonal aea s 1 n. The tempeatue s ased by 100 degees F. Fnd the stess n the ba. ANS: 4500 Ps Comp. 100 4
Bendng of staght beams Bendng fomulas n ths secton apply when the beam depth (n the plane of bendng) s small (by at least a facto o 0) compaed to the beam adus of cuvatue. y F y y F z x z Bendng stess fo bendng about the Z-axs: M z y x I z M z F I z s aea moments of netas about the z and epesents esstance to otaton about z axs. Bendng stess fo bendng about the Y-axs: y L x M I y y z M y F L z I y s aea moments of netas about the y and epesents esstance to otaton about y axs. Use tables to look up moments of neta fo vaous coss-sectons. The paallel axs theoem can be used to fnd moment of neta w/ a paallel axs. 5
Poblem #S3 The sold ccula steel ba wth R= (damete 4 ) s unde two loads as shown. Detemne the nomal stess x at pont Q. Pont Q s on the suface closest to the obseve and the 000 lb goes nto the pape. 000 lb Q 6 ft 4.5 ft 4.5 ft 0000 lb [The most common stess analyss poblems n exams nvolve smple bendng, smple toson, o a combnaton of the two. Ths s an example of the combnaton the toson analyss would be teated late.] Answe: 15600 ps Poblem #S4 A beam wth the coss-secton shown s unde a bendng moment of FL=M z =10000 lb-n actng on ths coss-secton. The thcknesses of all webs ae 0.5 nches. Detemne: a) The locaton of the neutal axs (0.667 fom bottom) b) The moment of neta about the z-axs (0.158 n 4 ) c) Bendng stess at D (5700 ps) d) Solve pat b) f the coss-secton was H-shaped [Fndng aea moments of netas ae popula exam questons. Ths poblem s a lttle longe than typcal ones but t s a good pepaaton execse] 6
D 7/8 3/8 1.5 Bendng Stesses n Cuved Beams 0 M n Maxmum bendng stesses occu at and o - The magntude s lagest at M ( n ) ea The stess at the oute suface s smla but wth o eplacng. In ths expesson, M s the bendng moment at the secton, A s the secton aea and e s the dstance between the centodal axs and neutal axs. These two axes wee the same n staght beams. 7
e n The adus of the neutal axs fo a ectangula secton can be obtaned as: o n ln( / ) o Refe to Shgley o othe desgn handbooks fo othe coss-sectons: Ccula Tapezodal T-shaped Hollow Squae I-Shaped Note: When fndng bendng moment of foces, the exact moment am s n but the centodal adus s also close enough to be a good appoxmaton. Fo a ccula shape wth a adus of R, n s: n ( c R c R ) Whee c = R + Check Shgley fo othe coss-secton foms such as T-shaped beams. 8
Poblem #S5 Gven: = n o = 4 n b = 1 n F = 10000 lb b Fnd: Answe: maxmum bendng stess Maxmum total stess 57900 ps (bendng only) 6900 ps (total) F Toque, Powe, and Toson of Ccula Bas Relaton between toque, powe and speed of a otatng shaft: H Tn 63000 H s powe n Hp, T s toque n lb-n, and n s shaft speed n pm. In SI unts: H T H s powe n Watts, T s toque n N-m, and s shaft speed n ad/s. The shea stess n a sold o tubula ound shaft unde a toque: y x T 9
The shea stess s a maxmum on the suface of the ba. The state of stess can be epesented as a case of pue shea: xy The shea stess s: T J J s the aea pola moment of neta and fo a sold (d =0) o hollow secton, J 4 ( d o d 3 The Von Mses stess n pue shea s: 3 3 V xy 4 ) xy When the behavo s ductle, yeldng occus when v eaches the yeld stength of the mateal. Ths s based on the dstoton enegy theoy whch s the best pedcto of yeldng. Accodng to ths, yeldng occus when: V S y xy 1 3 S 3 y xy O S y xy 0.58S Ths pedcts that yeldng n pue shea occus when the shea stess eaches 58% of the yeld stength of the mateal. y 10
The angle of otaton of a ccula shaft unde toque TL GJ The angle of otaton s n adans, L s the length of the ba, and G s a constant called the shea modulus. The shea modulus can be obtaned fom the modulus of elastcty E, and the posson s aton : E G (1 ) Fo steels, ths value s 11.5*10 6 ps. Poblem #S6 Consde the loadng stuaton shown n Poblem #S3. Detemne: a) the tosonal shea stess fo an element on the shaft suface. b) The maxmum shea stess at pont Q. Use the gven (as answe n Poblem #S3) maxmum nomal stess at pont Q to estmate the maxmum shea stess. Answes: a) 11460, b)13860 000 lb Q 6 ft 4.5 ft 4.5 ft 0000 lb 11
Beam and Fame Deflecton - Castglano s Theoem When a body s elastcally deflected by any combnaton of loads, the deflecton at any pont and n any decton s equal to the ate of change of stan enegy wth espect to the load located at that pont and actng n that decton even a fcttous load. When toson o bendng s pesent, they domnate the stan enegy. The deflecton due to tosonal and bendng loads s: 0 L T T F GJ dx 0 L M M F EI dx Example: Sold steel tube wth ID=1.75 and OD=.75 nches. Detemne the deflecton of the end of the tube. P=100 lb 9 ft 0 0 L L M M F dx EI Px( x) dx EI whee M Px 3 PL 3EI 3 100(9*1) 6 3(30*10 )(.347) 0.6 n 1
Example: Sold steel tube wth ID=1.75 and OD=.75 nches. Detemne the deflecton of the end of the tube. P=100 lb 4 ft = L 1 9 ft = L x Deflecton fom bendng n the 9-ft span M M L F dx whee M Px 0 EI 3 3 L Px( x) PL 100(9*1) dx 0.596 0 6 EI 3EI 3(30*10 )(.347) Deflecton fom bendng n the 4-ft span M M L1 F dx1 whee M Px1 0 EI 0 L 1 Px1( x EI 1 ) dx 1 3 PL1 3EI 3 100(4*1) 0.157 6 3(30*10 )(.347) Deflecton fom toson n the 9-ft span T T L F dx whee T PL1 0 EI 0 L PL1 ( L1 ) dx EI 1 PL EI 100(4*1) (9*1) L 6 (30*10 )(.347) 0.353 Total Deflecton = 0.596 + 0.157 + 0.353 = 1.1 n 13
Deflectons, Spng Constants, Load Shang Axal deflecton of a ba due to axal loadng The spng constant s: K EA L Lateal deflecton of a beam unde bendng load A common cases s shown. The est can be looked up n deflecton tables. K 48EI L 3 Fo cantleveed beams of length L: 3EI K 3 L Tosonal stffness of a sold o tubula ba s: 14
K t GJ L The unts ae n-lbs pe adan. Load Dstbuton between paallel membes If a load (a foce o foce couple) s appled to two membes n paallel, each membe takes a load that s popotonal to ts stffness. F K 1 K K t1 T K t The foce F s dvded between the two membes as: F 1 K1 K K 1 F F K K K The toque T s dvded between the two bas as: T 1 K Poblem #S7 t1 K t1 K t T T K 1 t1 Kt K A one-pece ectangula alumnum ba wth 1 by ½ nch coss-secton s suppotng a total load of 800 lbs. Detemne the maxmum nomal stess n the ba. t F T 15
30 0 Answe: 960 ps Poblem #S8 A sold steel ba wth 1 damete s subjected to 1000 n-lb load as shown. Detemne the eacton toques at the two end suppots. 4 ft 6 ft Answe: 600 on the left, 400 on the ght. 16
Dect shea stess n pns Pns n double shea (as n tongue and clevs) s one of the most common method of axal connecton of pats. The shea stess n the pn and beang stesses ae appoxmately unfomly dstbuted and ae obtaned fom: F A b pn F td t The clevs s also unde tea-out shea stess as shown n the followng fgue (top vew): F F Tea-out shea stess s: F 4A clevs In ths fomula A clevs =t(r o -R ) s appoxmately and consevatvely the aea of the dotted coss-secton. R o and R ae the oute and nne ad of the clevs hole. Note that thee ae 4 such aeas. 17
Shea stesses n beams unde bendng foces When a beam s unde a bendng foce, ts layes lke to slde on oneanothe as a deck of cads would do f bent. Snce the beam layes can not slde elatve to each othe, a shea stess develops wthn the beam just as shea stesses develop between cad faces f they wee glued togethe. Ths s shown below. The shea stess n beams s elatvely small and can be gnoed fo one-pece beams. But fo composte beams that ae glued, welded, veted, bolted, o somehow attached togethe, ths shea stess can be sgnfcant enough to tea off the weldng o bolts. F V The value of the shea stess depends on the followng: The shea foce V actng on the coss-secton of nteest. In the above fgue, the shea foce s F n all coss-sectons. The lage the foce, the lage the stess. The wdth of the beam b at the coss-secton. The wde the beam, the lowe the stess. The aea moment of neta of the ente coss-secton w/ to neutal axs. The moe moment of neta, the less the stess. The last paamete s Q whch s the bendng stess balance facto. The moe Q, the moe bendng stess has to be balanced by shea. 18
VQ I b Z Q A 1 y 1 A 1 Y b y 1 y 1 A 1 s the aea of the coss-secton left hangng and y 1 s the dstance between the centod of A 1 and the neutal axs (whch s the same as the centodal axs of the ente coss-secton). The followng s anothe example. b A 1 y 1 19
Poblem # S9 : by 4 Pne wood boads have been glued togethe to ceate a composte beam as shown. Assume the dmensons ae by 4 (n ealty they ae less than the nomnal value). If the shea stength of the glue s 11 ps, detemne the lagest load P that the beam can cay w/o glue falue. Assume beam s long enough fo the classcal beam theoy to apply. Do not consde falue due to bendng stesses. Answe:90.4 lbs P Coss-secton Poblem #S10: A composte beam s glued as shown. Hozontal membes ae 1 by 6 nch and the vetcal membes ae ¼ by 10 nch. Tansvese load at ths coss-secton s F=50 lbs. Detemne the equed mnmum glue stength n shea. Answe: 11.8 ps Z Y 50 0
Shea Cente of a C-Channel S t V Shea Cente h t b Tansvese loads on non-symmetc sectons can ceate twstng toques and wap beam flanges. If such tansvese loads ae appled at an offset locaton, the shea foces balance and do not twst the beam. Ths locaton s called the Shea Cente. Fo the C-channel shown S h b 4I t Fo a sem-ccula coss-secton, the shea cente s at: s 4 ( 1) 1
Toson of Thn-walled Tubes T Shea stess n thn-walled tubes (left fo closed tubes ght fo open tubes) T 3T At St Whee T s the toque, t s the wall thckness, S s the pemete of the mdlne, and A s the coss-sectonal aea defned by the mdlne of the tube wall. Usng aea o pemete of the nne o oute bounday s also acceptable snce the wall thckness s small. Fo a membe of constant coss-secton, the angle of twst n adans s TSL 4A Gt Whee S s the pemete of the mdlne, L s the length of the beam, and G s shea modulus. Thee s a smla fomula fo open tubes. [Shgley] Poblem #S11: A squae tube of length 50 cm s fxed at one end and subjected to a toque of 00 Nm. The tube s 40 mm squae (outsde dmenson) and mm thck. Detemne the shea stess n the tube and the angle of ts otaton. Answe: Stess 34.6 Mpa Rotaton (twst of the beam end): 0.011 adans o 0.66 degees
Stess n Thn-Walled Cylndes If the thckness t s less than 1/0 th of the md adus of the pessue vessel, the stesses can be closely appoxmated usng the followng smple fomulas. The ctcal stess pont n pessue vessels s always on the nne suface. t a P The tangental o hoop stess s: t Pd t P s the ntenal pessue, t s the wall thckness, and d s the nne damete. The axal stess s: a Pd 4t The adal stess on the nne suface s P whch s gnoed as t s much smalle than the hoop stess. Stesses n Thck-walled Cylndes In thck-walled cylndes the tangental and adal stesses vay exponentally wth espect to the adal locaton wthn the cylnde and f the cylnde s closed the axal stess would be a constant. All the thee stesses ae pncpal stesses when stess element s cut as a pe pece they 3
4 occu on sufaces on whch shea stesses ae zeo. The ctcal stess pont s on the nne suface. The tangental stess: o o o o o t P P P P The adal stess s: o o o o o P P P P When the extenal pessue s zeo, the stesses on the nne suface ae: ) ( o o t P o o P P ) ( P o P t
5 When the ends ae closed, the extenal pessue s often zeo and the axal stess s o a P Poblem #S1: A steel cylnde wth a yeld stength of 57 ks s unde extenal pessue only. The dmensons ae: ID=1.5 and OD=1.75. If the extenal pessue s 1100 ps, what s the facto of safety guadng aganst yeldng. Use the dstoton enegy theoy. Answe: 1.5. Stesses n otatng dsks A otatng dsk develops substantal neta-caused stesses at hgh speeds. The tangental and adal stesses n a dsk otatng at ad/sec s as follows: ) 3 3 1 )( 8 3 ( o o t and ) )( 8 3 ( o o whee s the mass densty and s the Posson s ato. The dsk thckness s to be less than 1/10 of the oute adus.
Poblem #S13: A dsk s otatng at 069 pm. The dsk s OD=150 mm and ts ID s 5 mm. The Posson s ato s 0.4 and the dsk s mass densty s 330 kg/m 3. Detemne the maxmum tensle stess n the dsk as a esult of otaton. Answe: 0.715 Mpa. Inteface pessue as a esult of shnk o pess fts When the ntenal pessue s hgh, shnk-ft cylndes lowe the nduced stesses. When two cylndes wth a adal ntefeence of ae pess o shnk ftted, an nteface pessue develops as follows: The nteface pessue fo same mateal cylndes wth nteface nomnal adus of R and nne and oute ad of and o : E ( o P R R )( R R ( ) o Poblem #S14: A colla s pess-ftted on a sold shaft. Both pats ae made of steel. The shaft damete s 40.06 mm and the colla damete s 40 mm. The oute damete of the colla s 80 mm. Fnd the nteface pessue. Answe: 50 Mpa. ) When both shnk ft and ntenal pessue s combned, the method of supeposton must be used. 6
Impact Foces The equvalent statc load ceated by an object fallng and mpactng anothe object can be vey lage. Equatons of enegy n dynamcs can be used to detemne such loads. Two common cases nvolve an object fallng fom a heght and a speedng object mpactng a stuctue. In both cases the dampng s assumed to be small. w h k v w Fo a fallng weght (gnong the enegy loss dung mpact): hk F e 1 1 W W F e 1 h 1 W st If h=0, the equvalent load s W. Fo a movng body wth a velocty of V befoe mpact, the equvalent foce (gnong enegy losses) s: F e V mk These ae consevatve values as gnong the enegy loss leads to lage equvalent foces. 7
Poblem #S15: A 1000 lb weght dops a dstance of 1-n on a platfom suppoted by a 1 n steel ba of length 1 nches. What s the theoetcal tensle stess that would develop n the ba. Answe: 70.7 ks. 1 6 1000 6 1000 # S15 # S16 Poblem #S16: Ths s the same poblem as #S15 but the ba s made up of two segments. The uppe segment has an aea of n. Detemne the maxmum theoetcal stess developng n the ba as the esult of the weght doppng on the platfom. Answe: 81.6 ks. Execse Queston: You have made gocey shoppng and the cashe placed all you tems n a pape bag. The bag s dead weght s now 15 lbs. What foce would the bag handles expeence f you: a) Lft the bag gently and lowe t? b) Slde the bag off the countetop and suddenly esst the weght of the bag at a ate of 30 lbs/n of dop? c) Let the bag slde off and dop 5 befoe you suddenly esst t at a ate of 30 lbs pe/n of dop. d) Same as c) but ate of esstance s 60 lbs/n. 8
Falue of columns unde compessve load (Bucklng) A beam unde axal compessve load can become unstable and collapse. Ths occus when the beam s long and ts ntenal esstance to bendng moment s nsuffcent to keep t stable. The ntenal esstance s a functon of aea moment of neta, I, and the stffness of the mateal. Note that the longe the beam, the moe bendng moment s ceated at the cente and fo the beam to eman stable, t needs to be stffe o have moe bendng esstance aea. P Fo evey long beams thee s a ctcal load beyond whch even a tny nudge would esult s a collapse. Ths ctcal load can be found usng Eule fomula. In shote columns the ctcal load may cause stesses well above the yeld stength of the mateal befoe the Eule load s eached. Fo such cases, Johnson fomula s used whch elates the falue to yeldng athe than nstablty. The ctcal Eule load fo a beam that s long enough s: EI P c C L C s the end-condton numbe. The followng end-condton numbes should be used fo gven cases: When both ends ae fee to pvot use C=1. Fee to pvot means the end can otate but not move n lateal decton. Note that even f the ends ae fee to otate a lttle, such as n any beang, ths condton s applcable. When one end s fxed (pevented fom otaton) and the othe s fee, the beam buckles ease. Use C= 1/4. 9
When one end s fxed and the othe end can pvot, use C= when the fxed end s tuly fxed n concete. If the fxed end s attached to stuctues that mght flex unde load, use C=1. (ecommended). When both ends ae fxed (pevented fom otaton and lateal movement), use C=4. Agan, a value of C=1. s ecommended when thee s any chance fo pvotng. These condtons ae depcted below: Pvot - Pvot Fxed - Fee Fxed - Pvot Fxed - Fxed An altenate but common fom of the Eule fomula uses the slendeness ato whch s defned as follows: Slendenes s Rato L k whee k s the aea adus of gyaton of the coss-secton. k I A 30
Range of valdty of the Eule fomula Expementaton has shown that the Eule fomula s a good pedcto of column falue when: L k EC S y If the slendeness ato s less than the value n the fomula, then the bette pedcto of falue s the Johnson fomula: P c AS y S y L k 1 CE Altenatvely, we can calculate the ctcal load fom both the Eule and the Johnson fomulas and pck the one that s lowe. Poblem #S17: The axal load on a ound sold steel ba n compesson s 5655 lbs. The mateal s AISI 1030 HR. Assume the end condtons ae pn-pn o pvot-pvot. Detemne the facto of safety aganst falue fo the followng two condtons: a) L=60 and D=damete=1.5 b) L=18 and D= 7/8 Answes: a) 3.6 and b) 4.4 Note: When a beam s unde compesson, t would buckle about the axs wth smalle aea moment of neta. 31
Eccentcally loaded columns c The moe geneal case of column loadng s when the load s appled eccentcally. Ths eccentc load exacebates the stuaton as t nduces moe bendng moment due to ts eccentcty. The pedcton fomula s known as the Secant Fomula whch s essentally a classcal bendng stess fomula although t may not look lke t. The secant fomula s: P c ec 1 k AS sec y L Ck P c 4EA whee e s the eccentcty, c s the dstance fom the oute laye to the neutal axs, and the est of the symbols have aleady been defned. A slght techncal dffculty wth ths fomula s that P c appeas on both sdes of the equaton esultng n the need to use tal-and-eo o use a nonlnea equaton solve. Howeve, usually the load s gven and you would calculate the stess (n place of S y n the fomula). 3
Example: A column has a fxed end and the othe end s fee and unsuppoted. The column length s 8 feet long. The beam coss-secton s a squae tube wth oute dmensons of 4 by 4 nches. The aea of the cosssecton s calculated to be 3.54 n and ts smallest aea moment of neta s 8 n 4. Detemne the maxmum compessve stess when the beam s suppotng 31.1 kps at an eccentcty of 0.75 nches off the beam axs. Soluton We fnd the stess fom the secant fomula. The aea adus of gyaton s: k I 8 1. n A 3.54 5 The fomula s AS y Pc ec L Pc 1 sec k Ck 4EA Fo ths poblem, P=31100 lbs s known and S y becomes the unknown max. Substtutng the numbes: 3.54( ) 31100 max 0.75() 8(1) 31100 1 sec (1.5) 6 (0.5)(1.5) 4(9)(10 )(3.54) Calculatng fo max we get: max = 000 ps Notes: 1. The end condton s C=0.5 (some books do not apply C but nstead they use an equvalent length L eq whch s L dvded by squae oot of C.. The agument of the secant functon s n adans. Convet to degees fst befoe takng cosnes. 3. The angle n degees n secant functon must be between 0 and 90 degees (0 and /4 n adans). Add o subtact multples of 90 degees untl the angle s between 0 and 90 degees. In ths poblem the angle s 16 degees. 33
Falue Theoes Falue unde load can occu due to excessve elastc deflectons o due to excessve stesses. Falue pedcton theoes due to excessve stesses fall nto two classes: Falue when the loadng s statc o the numbe of load cycles s one o qute small, and falue due to cyclc loadng when the numbe of cycles s lage often n thousands of cycles. Falue unde statc load Pats unde statc loadng may fal due to: a) Ductle behavo: Falue s due to bulk yeldng causng pemanent defomatons that ae objectonable. These falues may cause nose, loss of accuacy, excessve vbatons, and eventual factue. In machney, bulk yeldng s the ctea fo falue. Tny aeas of yeldng ae OK n ductle behavo n statc loadng. b) Bttle behavo: Falue s due to factue. Ths occus when the mateals (o condtons) do not allow much yeldng such as ceamcs, gey cast on, o heavly cold-woked pats. Theoes of ductle falue (yeldng) Yeldng s a shea stess phenomenon. That means mateals yeld because the shea stesses on some planes causes the lattce cystals to slde lke a deck of cads. In pue tenson o compesson, maxmum shea stesses occu on 45-degee planes these stesses ae esponsble fo yeldng and not the lage nomal stesses. The best pedcto of yeldng s the maxmum dstoton enegy theoy (DET). Ths theoy states that yeldng occus when the Von Mses stess eaches the yeld stength. The moe consevatve pedcto s the maxmum shea stess theoy (MST), whch pedcts yeldng to occu when the shea stesses each S y /. Fo example n a pue toson stuaton, the DET pedcts the yeldng to stat when eaches 58% of S y. But the MST pedcts yeldng to stat when eaches 50% of S y. Use of DET s moe common n desgn wok. Note that n statc loadng and ductle behavo, stess concentatons ae hamless as they only ceate small localzed yeldng whch do not lead to 34
any objectonable dmensonal changes. The mateal yeldng pe se s not hamful to mateals as long as t s not epeated too many tmes. Poblem # S18: A damete steel ba wth Sy=50 ks s unde pue toson of a 0,000 n-lb. Fnd the facto of safety guadng aganst yeldng based on: a) Dstoton enegy theoy, and b) Max shea stess theoy. Rounded answes:.3 and. Theoes of bttle falue Thee ae two types of theoes fo bttle falue. The classcal theoes assume that the mateal stuctue s unfom. If the mateal stuctue s non-unfom, such as n many thck-secton castngs, and that the pobablty of lage flaws exst, then the theoy of factue mechancs pedcts the falue much moe accuately. Many old shp hulls have splt nto two whle the exstng classcal theoes pedcted that they should not. We wll only look at the classcal bttle falue theoes. An mpotant pont to emembe s that bttle mateals often show much hghe ultmate stength n compesson than n tenson. One eason s that, unlke yeldng, factue of bttle mateals when loaded n tenson s a nomal stess phenomenon. The mateal fals because eventually nomal tensle stesses factue o sepaate the pat n the decton nomal to the plane of maxmum nomal stess (o pncpal stess see Page 1). In compesson the stoy s qute dffeent. When a bttle mateal s loaded n compesson, the nomal stess cannot sepaate the pat along the decton nomal to the plane of maxmum nomal stess. In the absence of sepaatng nomal stesses, shea stesses would have to do the job and sepaate o factue the mateal along the decton whee the shea stesses ae maxmum. In pue compesson, ths decton s at 45 degees to the plane of loadng. Bttle mateals, howeve, ae vey stong n shea. The bottom lne s that t takes a lot moe compessve nomal stess to ceate a factue. We only dscuss these theoes fo a D state of stess 3D s smla but s moe fomula-based. Theoes of falue n bttle factue dvde the 1 - egon nto 4 quadants. In the fst quadant, both pncpal stesses ae postve. 35
S ut II I S uc IV S ut 1 III When both 1 and ae postve (tensle), the factue s pedcted to occu when one of the two pncpal stesses eaches S ut. When both 1 and ae negatve (compessve), the factue occus when the magntude of one of the two pncpal stesses eaches S uc. The magntude of S uc s often moe than S ut as the po dscusson ndcated. In the othe two quadants, whee one pncpal stess s postve and the othe s negatve, the Columb-Moh theoy s a consevatve theoy fo falue pedcton. It s also easy to use. The Columb-Moh theoy falue lne smply connects the falue ponts as shown n the fgue as double lnes. Usng only the magntudes of the stesses, n Quadant II o IV: S 1 ut S uc 1 n In ths fomula ( 1, ) s the load pont (two pncpal stesses), and n s the facto of safety assocated wth that load pont. The postve pncpal stess s assocated wth S ut and the negatve pncpal stess s assocated wth S uc. Poblem #S19: A flywheel made of Gade 30 cast on has the followng dmensons: ID=6, OD=10 and thckness=0.5. What s the speed that would lead to the flywheel s factue? Answe: 13600 pm S uc 36
Summay of Falue Theoes Ductle Falue Defnton Macoscopc and measuable bulk defomaton Slght change n geomety Condtons fo ductle falue Metals (Except cast ons and P/M pats) At least % stan befoe factue Cause of falue (defomaton) Excessve SHEAR stesses Pedcton Theoes Maxmum DET o Yeldng occus when Maxmum Shea Stess Theoy V S y o Yeldng occus when max S y What to do wth stess concentaton? IGNORE them They cause small aeas of yeldng and do not cause macoscopc and measuable bulk defomaton. 37
Bttle Falue Defnton Factue Condtons fo Bttle falue Gay cast ons and P/M pats [I], ceamcs [II] Othe metals n specal condtons: o Exteme cold o exteme mpact o Exteme cold-wokng o exteme heat teatment Cause of falue (factue) Excessve nomal stesses n tenson, shea n compesson Pedcton Theoes Columb-Moh theoy S ut S uc III II I IV S ut 1 S uc S ut S 1 uc 1 n What to do wth stess concentaton? Ignoe fo [I] the stength s aleady educed, Apply fo [II] 38
Fatgue Falue Repeated loadng can lead to fatgue falue at loads much less than those leadng to statc falue. Fatgue falue s senstve to the magntude of the stess egadless of how localzed and small the stess aea s. Theefoe, stess concentatons play an mpotant ole n fatgue falue. Note: If the mateal bulk tself s full of unseen stess ases (such as n gey cast on), the geometc stess ases must be gnoed. Desgn fo nfnte lfe stats wth test esults of the mateal n otatng bendng test (known as Mooe test). The Mooe test stess lmt s called the otatng bendng enduance lmt, S n. Ths s the stess fo whch no falue occus egadless of the numbe of cycles. In the absence of dect expemental data, Mooe test enduance lmt s 50% of the ultmate stess fo steels. S 10 3 S f S n 10 3 10 6 10 5 Numbe of cycles - N The otatng bendng o Mooe test enduance lmt has to be coected fo the actual pat loadng and condtons. Ths ncludes coectons fo suface oughness, gadent effect, and sze of the pat (n Mooe test the specmens ae polshed, unde otatng bendng, and ae 0.3 n damete). The esult of these coectons s the enduance lmt S n. Anothe notaton fo enduance lmt s S e 39
Puely Altenatng Load a Combned Altenatng Loadng When the state of stess s known, the Von Mses stesses can be analyzed. In the case of ths fgue all stesses ae puely altenatng. y,a xy,a x,a V,a Most common loadngs n shafts nvolves x, xy, o both. 40
S n Enduance Lmt Von Mses Stess V,a The ndex a n the above fomula emphaszes that the loadng s puely altenatng. Poblem #S1 The steel shaft shown below s unde puely altenatng toque of 56 N-m. The toque fluctuates between 56 Nm CW and 56 Nm CCW. Assume S ut =518 MPa, and the coecton factos of 0.9 and 0.78 apply fo gadent and suface fnsh. Also assume a fatgue stess concentaton facto of 1.48 fo the shoulde fllets. Answe: About 0 mm 41
Fluctuatng and Steady Loads (optonal) Mean Stess Altenatng Stess When both mean and fluctuatng loads ae pesent, the Goodman cteon s used to detemne how much the mean loadng affects (educes) the enduance lmt. To begn the analyss, detemne the mean and altenatng Von Mses stesses. These ae actual maxmum stesses and they do nclude the fatgue stess concentaton factos. As a esult we should be able to calculate the followng: V, m V, a The mean Von Mses s only due to mean loads and the altenatng Von Mses s only due to altenatng loads. In powe tansmsson shafts the loadng ncludes a steady shea (powe toque) and an altenatng bendng stess (due to shaft flexue and otatng just lke Mooe test set up). The load ponts plot n the Goodman dagam as shown below: 4
Load Pont S n v,a v,m S u S S a m 1 e u n To detemne the facto of safety guadng aganst fatgue falue, we must consde the oveload mechansm. If both the steady and altenatng components of stess ae subject to ncease as shown, the magn of safety s detemned by the Goodman lne. Fatgue Falue Defnton Factue Condtons fo Fatgue falue Repeated loadng All metals Cause of falue (factue) Excessve LOCALIZED SHEAR stesses causng epeated yeldng Local bttle factue Cack gowth 43
Pedcton Theoes Falue occus when the local VonMses stess eaches the Enduance Lmt. What to do wth stess concentaton? Apply to all (mean and altenatng stesses) except gay cast on o othe mateals wth type-i ntenal stuctue Enduance Lmt S 10 3 S f S n 10 3 10 6 10 5 Numbe of cycles - N Cumulatve Fatgue Damage (Mne s o Palmgen Rule) If a pat s stessed to a load fo whch the fatgue lfe s 10 3 cycles, then each cycle takes 0.001 of the lfe of the pat. If stessed to a load fo whch the fatgue lfe s 10 4 cycles, then each cycle takes 0.0001 of the lfe of the pat and so on. Ths nfeence leads to the followng cumulatve fatgue damage fomula: n1 n nk... 1 N N N 1 k 44
In ths elaton, n 1 s the numbe of cycles n a loadng that would have a fatgue lfe of N 1 cycles, etc. Example: A ctcal pont of a landng gea s analyzed fo fatgue falue. Expements show that n each landng a compound load cycle s appled to the membe consstng of 5 cycles of 80 ks stess, cycles of 90 ks, and 1 cycle at 100 ks stess. All stess cycles ae fully evesed (no mean component). An expemental S-N cuve s also avalable fo ths pat (ths cuve can also be constucted usng Mooe test but fo ctcal pats t s always best to spend the money and ceate a tue S-N cuve). The S-N cuve shows the fatgue lves of the component at the loadng stesses to be as follows: Stess Level Numbe of Fatgue lfe cycles 80 Ks 5 10 5 cycles 90 Ks 38000 cyc 100 Ks 1 16000 cyc Detemne the lfe of ths pat n the numbe of compound cycles. Soluton: Each compound cycle takes the followng facton of lfe out of the pat: 5 10 1 38000 16000 5 0.0001651 The numbe of cycles s ecpocal of ths value whch s 6059 cycles. Unt Convesons Poblem #S11: Length: 1.640 feet Toque: 147.4 ft-lb OD: 1.575 n Thckness: 0.07874 n Answe (Stess): 5 Ks Poblem #S14: Shaft Damete: 1.5758 Colla damete: 1.5748 OD of colla: 3.1496 Answe (Pessue): 7.5 Ks 45