MECHANICS OF MATERIALS Poisson s Ratio

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1 Fouh diion MCHANICS OF MATRIALS Poisson s Raio Bee Johnson DeWolf Fo a slende ba subjeced o aial loading: 0 The elongaion in he -diecion is accompanied b a conacion in he ohe diecions. Assuming ha he maeial is isoopic (no diecional dependence), 0 Poisson s aio is defined as laeal sain ν aial sain 2006 The McGaw-Hill Companies, Inc. All ighs eseved. 2-1

2 Fouh diion MCHANICS OF MATRIALS Genealied Hooke s Law Bee Johnson DeWolf Fo an elemen subjeced o muli-aial loading, he nomal sain componens esuling fom he sess componens ma be deemined fom he pinciple of supeposiion. This equies: 1) sain is lineal elaed o sess 2) defomaions ae small Wih hese esicions: ν ν + ν ν + ν ν The McGaw-Hill Companies, Inc. All ighs eseved. 2-2

3 Fouh diion MCHANICS OF MATRIALS Dilaaion: Bulk Modulus Bee Johnson DeWolf Relaive o he unsessed sae, he change in volume is e 1 ( 1+ )( 1+ )( 1+ ) ν [ ] [ ] + + ( + + ) dilaaion (change in volume pe uni volume) Fo elemen subjeced o unifom hdosaic pessue, 3( 1 2ν ) p e p k 3 1 ( 2ν ) k bulk modulus Subjeced o unifom pessue, dilaaion mus be negaive, heefoe 0 <ν < The McGaw-Hill Companies, Inc. All ighs eseved.

4 Fouh diion MCHANICS OF MATRIALS Sheaing Sain Bee Johnson DeWolf A cubic elemen subjeced o a shea sess will defom ino a homboid. The coesponding shea sain is quanified in ems of he change in angle beween he sides, τ ( ) f γ A plo of shea sess vs. shea sain is simila o he pevious plos of nomal sess vs. nomal sain ecep ha he sengh values ae appoimael half. Fo small sains, τ Gγ τ Gγ τ Gγ whee G is he modulus of igidi o shea modulus The McGaw-Hill Companies, Inc. All ighs eseved. 2-4

5 Fouh diion MCHANICS OF MATRIALS ample 2. A ecangula block of maeial wih modulus of igidi G 90 ksi is bonded o wo igid hoional plaes. The lowe plae is fied, while he uppe plae is subjeced o a hoional foce P. Knowing ha he uppe plae moves hough 0.04 in. unde he acion of he foce, deemine a) he aveage sheaing sain in he maeial, and b) he foce P eeed on he plae. SOLUTION: Bee Johnson DeWolf Deemine he aveage angula defomaion o sheaing sain of he block. Appl Hooke s law fo sheaing sess and sain o find he coesponding sheaing sess. Use he definiion of sheaing sess o find he foce P The McGaw-Hill Companies, Inc. All ighs eseved. 2-5

Fouh diion MCHANICS OF MATRIALS Bee Johnson DeWolf Deemine he aveage angula defomaion o sheaing sain of he block. γ 0.04in. an γ γ 0.020 ad 2in.

6 Fouh diion MCHANICS OF MATRIALS Bee Johnson DeWolf Deemine he aveage angula defomaion o sheaing sain of he block. γ 0.04in. an γ γ ad 2in. Appl Hooke s law fo sheaing sess and sain o find he coesponding sheaing sess. τ Gγ 3 ( 90 psi)( ad) 1800psi Use he definiion of sheaing sess o find he foce P. 3 P τ A ( 1800 psi)( 8in. )( 2.5in. ) 36 lb P 36.0kips 2006 The McGaw-Hill Companies, Inc. All ighs eseved. 2-6

Fouh diion MCHANICS OF MATRIALS Relaion Among, ν, and G Bee Johnson DeWolf An aiall loaded slende ba will elongae in he aial diecion and conac in he ansvese diecions.

7 Fouh diion MCHANICS OF MATRIALS Relaion Among, ν, and G Bee Johnson DeWolf An aiall loaded slende ba will elongae in he aial diecion and conac in he ansvese diecions. An iniiall cubic elemen oiened as in op figue will defom ino a ecangula paallelepiped. The aial load poduces a nomal sain. If he cubic elemen is oiened as in he boom figue, i will defom ino a hombus. Aial load also esuls in a shea sain. Componens of nomal and shea sain ae elaed, 2G ( 1+ν ) 2006 The McGaw-Hill Companies, Inc. All ighs eseved. 2-7

Fouh diion MCHANICS OF MATRIALS Sample Poblem 2.5 Bee Johnson DeWolf A cicle of diamee d 9 in. is scibed on an unsessed aluminum plae of hickness 3/4 in.

8 Fouh diion MCHANICS OF MATRIALS Sample Poblem 2.5 Bee Johnson DeWolf A cicle of diamee d 9 in. is scibed on an unsessed aluminum plae of hickness 3/4 in. Foces acing in he plane of he plae lae cause nomal sesses 12 ksi and 20 ksi. Fo 6 psi and ν 1/3, deemine he change in: a) he lengh of diamee AB, b) he lengh of diamee CD, c) he hickness of he plae, and d) he volume of he plae The McGaw-Hill Companies, Inc. All ighs eseved. 2-8

9 Fouh diion MCHANICS OF MATRIALS SOLUTION: Appl he genealied Hooke s Law o find he hee componens of nomal sain psi ν ν ν + ν ν ( 12ksi) 0 ( 20ksi) in./in. ν in./in. + in./in. 1 3 Bee Johnson DeWolf valuae he defomaion componens. δ B d ( in./in. )( 9in. ) A δ C d D δ δ B A in. ( in./in. )( 9in. ) δ C D in. ( in./in. )( 0.75in. ) δ Find he change in volume e V + ev V in 3 /in in. 3 3 ( ) in in 2006 The McGaw-Hill Companies, Inc. All ighs eseved. 2-9

Fouh diion MCHANICS OF MATRIALS Composie Maeials Bee Johnson DeWolf Fibe-einfoced composie maeials ae fomed fom lamina of fibes of gaphie, glass, o polmes embedded in a esin mai.

10 Fouh diion MCHANICS OF MATRIALS Composie Maeials Bee Johnson DeWolf Fibe-einfoced composie maeials ae fomed fom lamina of fibes of gaphie, glass, o polmes embedded in a esin mai. Nomal sesses and sains ae elaed b Hooke s Law bu wih diecionall dependen moduli of elasici, ν ν Tansvese conacions ae elaed b diecionall dependen values of Poisson s aio, e.g., Maeials wih diecionall dependen mechanical popeies ae anisoopic The McGaw-Hill Companies, Inc. All ighs eseved. 2 -

Sess and sain disibuions become unifom a a elaivel sho disance fom he load applicaion poins.

11 Fouh diion MCHANICS OF MATRIALS Sain-Venan s Pinciple Bee Johnson DeWolf Loads ansmied hough igid plaes esul in unifom disibuion of sess and sain. Concenaed loads esul in lage sesses in he vicini of he load applicaion poin. Sess and sain disibuions become unifom a a elaivel sho disance fom he load applicaion poins. Sain-Venan s Pinciple: Sess disibuion ma be assumed independen of he mode of load applicaion ecep in he immediae vicini of load applicaion poins The McGaw-Hill Companies, Inc. All ighs eseved. 2-11

12 Fouh diion MCHANICS OF MATRIALS Sess Concenaion: Hole Bee Johnson DeWolf Disconinuiies of coss secion ma esul in high localied o concenaed sesses. K ma ave 2006 The McGaw-Hill Companies, Inc. All ighs eseved. 2-12

13 Fouh diion MCHANICS OF MATRIALS Sess Concenaion: Fille Bee Johnson DeWolf 2006 The McGaw-Hill Companies, Inc. All ighs eseved. 2-13

14 Fouh diion MCHANICS OF MATRIALS ample 2.12 Bee Johnson DeWolf SOLUTION: Deemine he lages aial load P ha can be safel suppoed b a fla seel ba consising of wo poions, boh mm hick, and especivel 40 and 60 mm wide, conneced b filles of adius 8 mm. Assume an allowable nomal sess of 165 MPa. Deemine he geomeic aios and find he sess concenaion faco fom Fig. 2.64b. Find he allowable aveage nomal sess using he maeial allowable nomal sess and he sess concenaion faco. Appl he definiion of nomal sess o find he allowable load The McGaw-Hill Companies, Inc. All ighs eseved. 2-14

15 Fouh diion MCHANICS OF MATRIALS Bee Johnson DeWolf Deemine he geomeic aios and find he sess concenaion faco fom Fig. 2.64b. D d 60mm mm K 1.82 d 8mm 40mm 0.20 Find he allowable aveage nomal sess using he maeial allowable nomal sess and he sess concenaion faco. K 165 MPa 1.82 ma ave 90.7 MPa Appl he definiion of nomal sess o find he allowable load. P A ( 40mm)( mm)( 90.7 MPa) ave N P 36.3kN 2006 The McGaw-Hill Companies, Inc. All ighs eseved. 2-15

16 Fouh diion MCHANICS OF MATRIALS lasoplasic Maeials Bee Johnson DeWolf Pevious analses based on assumpion of linea sess-sain elaionship, i.e., sesses below he ield sess Assumpion is good fo bile maeial which upue wihou ielding If he ield sess of ducile maeials is eceeded, hen plasic defomaions occu Analsis of plasic defomaions is simplified b assuming an idealied elasoplasic maeial Defomaions of an elasoplasic maeial ae divided ino elasic and plasic anges Pemanen defomaions esul fom loading beond he ield sess 2006 The McGaw-Hill Companies, Inc. All ighs eseved. 2-16

loadings above he maimum elasic load, a egion of plasic defomaions develop nea he hole As he loading inceases, he

17 Fouh diion MCHANICS OF MATRIALS Plasic Defomaions Bee Johnson DeWolf A P ave A ma K P A K lasic defomaion while maimum sess is less han ield sess Maimum sess is equal o he ield sess a he maimum elasic loading P U A K P A loadings above he maimum elasic load, a egion of plasic defomaions develop nea he hole As he loading inceases, he plasic egion epands unil he secion is a a unifom sess equal o he ield sess 2006 The McGaw-Hill Companies, Inc. All ighs eseved. 2-17

18 Fouh diion MCHANICS OF MATRIALS Residual Sesses Bee Johnson DeWolf When a single sucual elemen is loaded unifoml beond is ield sess and hen unloaded, i is pemanenl defomed bu all sesses disappea. This is no he geneal esul. Residual sesses will emain in a sucue afe loading and unloading if - onl pa of he sucue undegoes plasic defomaion - diffeen pas of he sucue undego diffeen plasic defomaions Residual sesses also esul fom he uneven heaing o cooling of sucues o sucual elemens 2006 The McGaw-Hill Companies, Inc. All ighs eseved. 2-18

19 Fouh diion MCHANICS OF MATRIALS ample 2.14, 2.15, 2.16 Bee Johnson DeWolf A clindical od is placed inside a ube of he same lengh. The ends of he od and ube ae aached o a igid suppo on one side and a igid plae on he ohe. The load on he od-ube assembl is inceased fom eo o 5.7 kips and deceased back o eo. a) daw a load-deflecion diagam fo he od-ube assembl b) deemine he maimum elongaion c) deemine he pemanen se d) calculae he esidual sesses in he od and ube. A 0.075in ( ) 36ksi 2 psi A 0.0in ( ) 45ksi 2 psi 2006 The McGaw-Hill Companies, Inc. All ighs eseved. 2-19

20 Fouh diion MCHANICS OF MATRIALS ample 2.14, 2.15, 2.16 Bee Johnson DeWolf a) Daw a load-deflecion diagam fo he od-ube assembl 2 ( P ) ( ) A ( 36ksi)( 0.075in ) ( δ ) ( ) L 36-3 ( ) in L 30 2 ( P ) ( ) A ( 45ksi)( 0.0in ) ( δ ) ( ) L 90-3 ( ) in. 45 L kips psi psi psi psi ( 30in. ) 4.5kips ( 30in. ) P P + P δ δ δ 2006 The McGaw-Hill Companies, Inc. All ighs eseved. 2-20

21 Fouh diion MCHANICS OF MATRIALS ample 2.14, 2.15, 2.16 Bee Johnson DeWolf b,c) deemine he maimum elongaion and pemanen se A a load of P 5.7 kips, he od has eached he plasic ange while he ube is sill in he elasic ange P P ( P ) P P P A δ L 2.7 kips ( ) 3.0 kips 30ksi 2 0.1in 30 L 15 kips 3.0 kips 3 6 psi 30in. psi δ ma δ 60 in. The od-ube assembl unloads along a line paallel o 0 4.5kips m 125kips in. slope in. P δ m ma 5.7 kips kips in. in. δ p δ ma + δ ( ) in. δ p 14.4 in The McGaw-Hill Companies, Inc. All ighs eseved. 2-21

22 Fouh diion MCHANICS OF MATRIALS ample 2.14, 2.15, 2.16 Bee Johnson DeWolf Calculae he esidual sesses in he od and ube. Calculae he evese sesses in he od and ube caused b unloading and add hem o he maimum sesses. δ L in. in in. in. 6 ( 1.52 )( 30 psi) 6 ( 1.52 )( 15 psi) 45.6 ksi 22.8ksi esidual, esidual, + + ( ) ksi 9. 6ksi ( ) ksi 7.2ksi 2006 The McGaw-Hill Companies, Inc. All ighs eseved. 2-22

MECHANICS OF MATERIALS Poisson s Ratio

MECHANICS OF MATERIALS Poisson s Ratio Poisson s Raio For a slender bar subjeced o axial loading: ε x x y 0 The elongaion in he x-direcion i is accompanied by a conracion in he oher direcions. Assuming ha he maerial is isoropic (no direcional