ME311 Machin Dsign Lctur 4: Strss Concntrations; Static Failur W Dornfld 8Sp017 Fairfild Univrsit School of Enginring Strss Concntration W saw that in a curvd bam, th strss was distortd from th uniform distribution of a straight bam, and highr at th insid radius. A similar distortion happns nar hols, notchs, and changs in sction: Th ffct varis with th loading tp: tnsion, bnding, or torsion. 1
Strss Concntration Strss Concntration Factors rlat th maimum strss to th avrag strss in th minimum cross sction of th part. d b 0.75.0 0.375 nom 300 (.0 0.75)0.5 300 0.315 960 psi ma Kc nom.5 960 160 psi Hamrock s plots tll ou th nominal strss. Hamrock Fig. 6.a Which has th biggr K c? r r 150 150 M M Us Fig.: H r h r K h c Us Fig.: H r h r K h c
Prdicting Failur W ar givn som matrial that is tnsil tstd and ilds at S ksi. It has a Mohr s circl that looks lik this: W want to us this matrial in an application whr it ss this loading: 40 80 50 60 0 It has a Mohr s circl that looks lik this: -40 80 Th Ma Principal strss is lss than ksi, but will it ild? Failur Thor Failur Thor addrsss how to translat a ral, multiaial stat of strss into somthing that can b compard with a simpl uniaial (tnsil) tst rsult. For ductil matrials, thr ar two prvailing thoris: 1. Ma Shar Strss Thor (MSST) or Trsca Thor. Distortion Enrg Thor (DET) or von Miss critrion. 3
Ma Shar Strss MSST sas "Yild occurs whnvr th maimum shar strss in an lmnt cds th ma shar at ild in a tnsil tst" In othr words, th Mohr's circl lis within th shar bounds (uppr and lowr) of th tnsil tst. 50 How did w do? Ma Shar Strss 60 50-40 80 Dosn t look good. MSST prdicts ilding whn 1-3 S. (~Eqn. 6.8) 1 is Ma principal and 3 is Min principal. In our cas, 1-3 [80 (-40)] 10 > S of. Anothr wa to sa is that ma S / 4
Ma Shar Strss Hr is a rprsntation of MSST in trms of limits to Ma and Min principal strsss: 1 1 & S t 1 1 1 1 S t 1 Insid th bo is saf. 1 This is th pur shar lin, whr 1 -. Distortion Enrg DET obsrvs that matrials can tak normous hdrostatic prssurs (.g., rocks dp in th arth that s prssurs wll abov thir comprssiv strngth) and not fractur suggsting that it must b distortion that causs failur. DET ssntiall computs th total strain nrg and subtracts th volum chang nrg to gt th distortion nrg. In trms of principal strsss: 1 ( 1 ) + ( 3) + ( 1 3) whr is th von Miss quivalnt strss. Not that for hdrostatic loading, 1 3, and 0! 5
Distortion Enrg In th mor common cas of biaial principal strsss ( 3 0), this rducs to: In our ampl, this is: + 1 1 80 (80)( 40) + ( 40) 6400 + 300 + 1600 11,00 105.8ksi This cds S ksi, so DET also prdicts ilding. Distortion Enrg - Simplifid If ou hav onl dirct biaial strss ( z 0), ou can calculat DET dirctl from,, and without gtting th principal strsss: + + 3 And, if ou hav onl bnding and torsion ( 0), this rducs to: + 3 (Hamrock buris this in E. 7.4 on pag 170.) 6
Th Failur Thoris, ovrlaid MSST is a littl mor consrvativ than DET. Brittl Thoris Brittl matrials can tak mor comprssion than tnsion, so hav diffrnt failur bounds. Ma Normal Strss Thor Intrnal Friction Thor & Modifid Mohr Thor Th Brittl thoris us Ultimat Strngths bcaus th fractur rathr than ild, and both Comprssiv and Tnsil strngths bcaus th ar diffrnt. 7
Rmmbr th Crank Arm? in lb Torqu B 300 lb Shar A 1800 in.lb Momnt C in lb Torqu z A At "A": Bnding M 1800 in.lb. Torsion T in.lb. Mc (1800)(0.375) 675 43,460 psi 4 I π (0.75) 64 0.01553 Tr ()(0.375) 450 z 14,487 psi 4 J π (0.75) 3 0.03106 4V (4)(300) BENDING 905 psi 3A 3π (0.75) 4 1.35 TOTAL z BENDING 13,58 psi Pur Shar 300 lb Shar B At B": Torsion T in.lb. Shar V 300 lb Not that at location C, th two shars would ADD to 15,39 psi. Strsss in a Crank Arm Th strsss at A occur at th sam point, so can us Mohr. For Xstrss 43.5, Ystrss 0.0, XYShar -14.5, Angl -16.85, Strss1 47.85, Strss 0.00, Strss3-4.39, SharMa 6.1 30 0 z A Mc 43,460 psi I Tr z 14,487 psi J Shar Strss Y,T 10 0-10 0 10 0 30 40 50 60-10 Th Mohr circl givs: 1 47.846 ksi 0.000 ksi 3-4.386 ksi Ma 6.116 ksi -0-30 Fac Strsss X,-T 8
What ar th DET strsss at A on th Crank Arm? At "A": Bnding M 1800 in.lb. Torsion T in.lb. A Mc 43,460 psi I Tr z 14,487 psi J Th Mohr circl for A gav: 1 47.846 ksi 0.000 ksi 3-4.386 ksi Ma 6.116 ksi Using th Principal strsss: 85 + 10 + 19 518 50.18ksi + + 3 1 3 + 3 1 1888.8 + 69.6 518.4 50.18ksi 47.8 (47.8)( 4.39) + ( 4.39) Using th Dirct strsss: 43.46 + 0 0 + 3(14.487) What ar th MSST strsss at A on th Crank Arm? At "A": Bnding M 1800 in.lb. Torsion T in.lb. A Mc 43,460 psi I Tr z 14,487 psi J Using th Principal strsss (our onl option with MSST): Th Mohr circl for A gav: 1 47.846 ksi 0.000 ksi 3-4.386 ksi Ma 6.116 ksi MSST 47.85 ( 4.39) 5. 4ksi 1 3 This numbr is slightl highr than th von Miss strss of 50. consistnt with th MSST bing mor consrvativ. 9
What ar th MSST & DET strsss at C on th Crank Arm? At C": Torsion T in.lb. Shar V 300 lb C 4V BENDING 905 psi 3A TOTAL z + BENDING 15,39 psi Pur Shar C ss pur shar, with & both zro. 15.4 So 1 15.4 & 3-15.4 MSST 1 3 15.4 ( 15.4) 30.8ksi DET using th Principal strsss: 1 1 3 + 3 15.4 (15.4)( 15.4) + ( 15.4) 3(15.4) 711.5 6.7ksi 15.4 DET using th Dirct strsss: + + 3 3(15.4) 0 + 0 0 + 3(15.4) 711.5 6.7ksi Factors of Saft Th Factor of Saft, n s, for a componnt is th ratio of th strss allowd b th componnt s matrial, dividd b th maimum strss prdictd (or masurd). W usuall us th ild strngth, S, as th allowabl strss for ductil matrials. S Hamrock s Appndi A for S. S n s ma Using th Failur Thoris, w will gnrall comput n s ithr b: S S ns ns ( 1 3) or or 0.5S n For DET (von Miss) s ma For MSST 10
Crank Arm Saft Factors n n MSST C DET C 3.5 30.8 3.75 6.7 C A n n MSST A DET A 1.91 5.4 1.99 50. This assums a matrial with S ksi. Can ou find a matrial with S ksi? Static Saft Factors A bar of AISI 1040 Stl ss a bnding and twisting load, rsulting in a bnding strss of 15 ksi and a shar strss of 6.5 ksi at its most highl strssd location. Find th MSS and th DET factors of saft against ilding. Calculat both FoS s using both Dirct strsss and Principal strsss. 11
Slcting Factors of Saft Hamrock givs guidlins for slcting Saft Factors in Sction 1.4.1. ns ns( A, B, C) ns ( D, E) Qualit of matrials, procssing, maintnanc, & inspction Environmnt (Load) control Qualit of analsis & tst Dangr to humans $$ impact 1.1 n s 3.95; 1.0 n s 1.6; so 1.1 n s 6.3 1