eco 7. 7. Aalyss of Three Dmesoal ress ad ra The cocep of raco ad sress was roduced ad dscussed Par I.-.5. For he mos par he dscusso was cofed o wo-dmesoal saes of sress. Here he fully hree dmesoal sress sae s eamed. There wll be some repeo of he earler aalyses. 7.. The Traco Vecor ad ress Compoes Cosder a raco vecor acg o a surface eleme Fg. 7... Iroduce a Caresa coordae sysem wh base vecors e so ha oe of he base vecors s a ormal o he surface ad he org of he coordae sysem s posoed a he po a whch he raco acs. For eample Fg. 7.. he e dreco s ake o be ormal o he plae ad a superscrp o deoes hs ormal: ( e e e e (7.. Each of hese compoes s represeed by where he frs subscrp deoes he dreco of he ormal ad he secod deoes he dreco of he compoe o he plae. Thus he hree compoes of he raco vecor show Fg. 7.. are : ( e e e e (7.. The frs wo sresses he compoes acg ageal o he surface are shear sresses whereas acg ormal o he plae s a ormal sress. (ˆ ( e e e e Fgure 7..: compoes of he raco vecor ( e ( e ( e Cosder he hree raco vecors acg o he surface elemes whose ouward ormals are alged wh he hree base vecors e j Fg. 7..a. The hree (or s surfaces ca be amalgamaed o oe dagram as Fg. 7..b. I erms of sresses he raco vecors are old Mechacs Par II
eco 7. ( e ( e ( e e e e e e e e e e or ( e e (7.. j (a e ( e ( e e ( e e e (b ( e e ( e ( e e Fgure 7..: he hree raco vecors acg a a po; (a o muually orhogoal plaes (b he raco vecors llusraed o a bo eleme The compoes of he hree raco vecors.e. he sress compoes ca ow be dsplayed o a bo eleme as Fg. 7... oe ha he sress compoes wll vary slghly over he surfaces of a elemeal bo of fe sze. However s assumed ha he eleme Fg. 7.. s small eough ha he sresses ca be reaed as cosa so ha hey are he sresses acg a he org. Fgure 7..: he e sress compoes wh respec o a Caresa coordae sysem The e sresses ca be coveely dsplayed mar form: old Mechacs Par II
eco 7. [ ] (7..4 I s mpora o realse ha f oe were o ake a eleme a some dffere oreao o he eleme Fg. 7.. bu a he same maeral parcle for eample alged wh he aes show Fg. 7..4 oe would he have dffere racos acg ad he e sresses would be dffere also. The sresses acg hs ew oreao ca be represeed by a ew mar: (7..5 [ ] Fgure 7..4: he sress compoes wh respec o a Caresa coordae sysem dffere o ha Fg. 7.. 7.. Cauchy s Law Cauchy s Law whch wll be proved below saes ha he ormal o a surface ( s relaed o he raco vecor e acg o ha surface accordg o e (7..6 j j Wrg he raco ad ormal vecor form ad he sress mar form [ ] [ ] [ ] (7..7 ad Cauchy s law mar oao reads old Mechacs Par II
eco 7. (7..8 oe ha s he raspose sress mar whch s used Cauchy s law. ce he sress mar s symmerc oe ca epress Cauchy s law he form Cauchy s Law (7..9 j Cauchy s law s llusraed Fg. 7..5; hs fgure posve sresses are show. ( Fgure 7..5: Cauchy s Law; gve he sresses ad he ormal o a plae he raco vecor acg o he plae ca be deermed ormal ad hear ress I s useful o be able o evaluae he ormal sress ad shear sress acg o ay plae Fg. 7..6. For hs purpose oe ha he sress acg ormal o a plae s he ( projeco of he dreco of ( (7.. The magude of he shear sress acg o he surface ca he be obaed from (7.. old Mechacs Par II 4
eco 7. Fgure 7..6: he ormal ad shear sress acg o a arbrary plae hrough a po Eample The sae of sress a a po wh respec o a Caresa coordaes sysem s gve by: [ ] Deerme: (a he raco vecor acg o a plae hrough he po whose u ormal s ( / e ( / e ( / e (b he compoe of hs raco acg perpedcular o he plae (c he shear compoe of raco o he plae oluo (a From Cauchy s law ( so ha ( / e ˆ e e. 9 (b The compoe ormal o he plae s ( ( / (/ ( / ( / / 9.4. (c The shearg compoe of raco s {[ ( ( ( ] [( ]} /. 9 old Mechacs Par II 5
eco 7. Proof of Cauchy s Law Cauchy s law ca be proved usg force equlbrum of maeral elemes. Frs cosder a erahedral free-body wh vere a he org Fg. 7..7. I s requred o deerme he raco erms of he e sress compoes (whch are all show posve he dagram. Δ Δ e α Δ Fgure 7..7: proof of Cauchy s Law The compoes of he u ormal are he dreco coses of he ormal vecor.e. he coses of he agles bewee he ormal ad each of he coordae drecos: ( e e cos (7.. Le he area of he base of he erahedra wh ormal be Δ. The area Δ s he Δ cosα where α s he agle bewee he plaes as show o he rgh of Fg. 7..7; hs agle s he same as ha bewee he vecors ad e so Δ Δ ad smlarly for he oher surfaces: Δ Δ (7.. The resula surface force o he body acg he dreco s he F Δ Δ Δ Δ (7..4 j j j j For equlbrum hs epresso mus be zero ad oe arrves a Cauchy s law. oe: As proved Par III hs resul holds also he geeral case of accelerag maeral elemes he preseces of body forces. old Mechacs Par II 6
eco 7. 7.. The ress Tesor Cauchy s law 7..9 s of he same form as 7..4 ad so by defo he sress s a esor. Deoe he sress esor symbolc oao by. Cauchy s law symbolc form he reads (7..5 Furher he rasformao rule for sress follows he geeral esor rasformao rule 7..: Q p p Q jq Q Q qj pq pq K K T [ ] [ Q][ ][ Q ] T [ ] [ Q ][ ][ Q] ress Trasformao Rule (7..6 As wh he ormal ad raco vecors he compoes ad hece mar represeao of he sress chages wh coordae sysem as wh he wo dffere mar represeaos 7..4 ad 7..5. However here s oly oe sress esor a a po. Aoher way of lookg a hs s o oe ha a fe umber of plaes pass hrough a po ad o each of hese plaes acs a raco vecor ad each of hese raco vecors has hree (sress compoes. All of hese raco vecors ake ogeher defe he complee sae of sress a a po. Eample The sae of sress a a po wh respec o a coordae sysem s gve by [ ] (a Wha are he sress compoes wh respec o aes whch are obaed o from he frs by a 45 roao (posve couerclockwse abou he as Fg. 7..8? (b Use Cauchy s law o evaluae he ormal ad shear sress o a plae wh ormal / e / e ad relae your resul wh ha from (a ( ( old Mechacs Par II 7
eco 7. old Mechacs Par II 8 Fgure 7..8: wo dffere coordae sysems a a po oluo (a The rasformao mar s [ ] ( ( ( ( ( ( ( ( ( cos cos cos cos cos cos cos cos cos Q ad I QQ T as epeced. The roaed sress compoes are herefore ad he ew sress mar s symmerc as epeced. (b From Cauchy s law he raco vecor s ( ( ( so ha ( ( ( ( ˆ / / e e e. The ormal ad shear sress o he plae are / ( ad / (/ ( The ormal o he plae s equal o e ad so should be he same as ad s. The sress should be equal o ( ( ad s. The resuls are e e e e e e o 45 o 45
eco 7. dsplayed Fg. 7..9 whch he raco s represeed dffere ways wh. compoes ( ad ( ( e Fgure 7..9: raco ad sresses acg o a plae Isoropc ae of ress uppose he sae of sress a body s [ ] δ (7..7 Oe fds ha he applcao of he sress esor rasformao rule yelds he very same compoes o maer wha he ew coordae sysem{ Problem }. I oher words o shear sresses ac o maer wha he oreao of he plae hrough he po. Ths s ermed a soropc sae of sress or a sphercal sae of sress. Oe eample of soropc sress s he sress arsg a flud a res whch cao suppor shear sress whch case [ ] p[ I] (7..8 where he scalar p s he flud hydrosac pressure. For hs reaso a soropc sae of sress s also referred o as a hydrosac sae of sress. 7..4 Prcpal resses For cera plaes hrough a maeral parcle here are raco vecors whch ac ormal o he plae as Fg. 7... I hs case he raco ca be epressed as a scalar ( mulple of he ormal vecor. old Mechacs Par II 9
eco 7. ( o shear sress oly a ormal compoe o he raco Fgure 7..: a purely ormal raco vecor From Cauchy s law he for hese plaes j (7..9 Ths s a sadard egevalue problem from Lear Algebra: gve a mar [ ] fd he egevalues ad assocaed egevecors such ha Eq. 7..9 holds. To solve he problem frs re-wre he equao he form j (7.. ( I ( δ or (7.. Ths s a se of hree homogeeous equaos hree ukows (f oe reas as kow. From basc lear algebra hs sysem has a soluo (apar from f ad oly f he deerma of he coeffce mar s zero.e. f de( I de (7.. Evaluag he deerma oe has he followg cubc characersc equao of he sress esor I I I Characersc Equao (7.. ad he prcpal scalar varas of he sress esor are old Mechacs Par II
eco 7. I I I (7..4 ( I s he deerma of he sress mar. The characersc equao 7.. ca ow be solved for he egevalues ad he Eq. 7.. ca be used o solve for he egevecors. ow aoher heorem of lear algebra saes ha he egevalues of a real (ha s he compoes are real symmerc mar (such as he sress mar are all real ad furher ha he assocaed egevecors are muually orhogoal. Ths meas ha he hree roos of he characersc equao are real ad ha he hree assocaed egevecors form a muually orhogoal sysem. Ths s llusraed Fg. 7..; he egevalues are called prcpal sresses ad are labelled ad he hree correspodg egevecors are called prcpal drecos he drecos whch he prcpal sresses ac. The plaes o whch he prcpal sresses ac (o whch he prcpal drecos are ormal are called he prcpal plaes. Fgure 7..: he hree prcpal sresses acg a a po ad he hree assocaed prcpal drecos ad Oce he prcpal sresses are foud as meoed he prcpal drecos ca be foud by solvg Eq. 7.. whch ca be epressed as ( ( ( (7..5 Each prcpal sress value hs equao gves rse o he hree compoes of he assocaed prcpal dreco vecor. The soluo also requres ha he magude of he ormal be specfed: for a u vecor. The drecos of he ormals are also chose so ha hey form a rgh-haded se. old Mechacs Par II
eco 7. Eample The sress a a po s gve wh respec o he aes O by he values 5 [ ] 6. Deerme (a he prcpal values (b he prcpal drecos (ad skech hem. oluo: (a The prcpal values are he soluo o he characersc equao 5 6 ( (5 (5 whch yelds he hree prcpal values 5 5. (b The egevecors are ow obaed from Eq. 7..5. Frs for 5 6 9 ad usg also he equao leads o ( / 5 e (4 / 5 e. mlarly for 5 ad 5 oe has respecvely ad 9 4 6 whch yeld e ad ( 4 / 5 e (/ 5 e. The prcpal drecos are skeched Fg. 7... oe ha he hree compoes of each prcpal dreco are he dreco coses: he coses of he agles bewee ha prcpal dreco ad he hree coordae aes. For eample for wh / 5 4 / 5 he agles made wh he coordae aes are respecvely 7 o ad 7 o. ˆ ˆ o 7 ˆ Fgure 7..: prcpal drecos old Mechacs Par II
eco 7. Ivaras The prcpal sresses are depede of ay coordae sysem; he aes o whch he sress mar Eq. 7..9 s referred ca have ay oreao he same prcpal sresses wll be foud from he egevalue aalyss. Ths s epressed by usg he symbolc oao for he problem: whch s depede of ay coordae sysem. Thus he prcpal sresses are rsc properes of he sress sae a a po. I follows ha he fucos I I I he characersc equao Eq. 7.. are also depede of ay coordae sysem ad hece he ame prcpal scalar varas (or smply varas of he sress. The sress varas ca also be wre ealy erms of he prcpal sresses: I I I (7..6 Also f oe chooses a coordae sysem o cocde wh he prcpal drecos Fg. 7.. he sress mar akes he smple form [ ] (7..7 oe ha whe wo of he prcpal sresses are equal oe of he prcpal drecos wll be uque bu he oher wo wll be arbrary oe ca choose ay wo prcpal drecos he plae perpedcular o he uquely deermed dreco so ha he hree form a orhoormal se. Ths sress sae s called a-symmerc. Whe all hree prcpal sresses are equal oe has a soropc sae of sress ad all drecos are prcpal drecos he sress mar has he form 7..7 o maer wha oreao he plaes hrough he po. Eample The wo sress marces from he Eample of 7.. descrbg he sress sae a a po wh respec o dffere coordae sysems are / / / [ ] [ ] / / / / / The frs vara s he sum of he ormal sresses he dagoal erms ad s he same for boh as epeced: I 6 The oher varas ca also be obaed from eher mar ad are I 6 I old Mechacs Par II
eco 7. 7..5 Mamum ad Mmum ress Values ormal resses The hree prcpal sresses clude he mamum ad mmum ormal sress compoes acg a a po. To prove hs frs le e e e be u vecors he prcpal drecos. Cosder e a arbrary u ormal vecor e. From Cauchy s law (see Fg. 7.. he sress mar Cauchy s law s ow wh respec o he prcpal drecos ad he ormal sress acg o he plae wh ormal s ( j ( (7..8 prcpal drecos Fgure 7..: ormal sress acg o a plae defed by he u ormal Thus (7..9 ce ad whou loss of geeraly akg oe has mlarly ( (7.. ( (7.. Thus he mamum ormal sress acg a a po s he mamum prcpal sress ad he mmum ormal sress acg a a po s he mmum prcpal sress. old Mechacs Par II 4
eco 7. hear resses e wll be show ha he mamum shearg sresses a a po ac o plaes oreed a 45 o o he prcpal plaes ad ha hey have magude equal o half he dfferece bewee he prcpal sresses. Frs aga le e e e be u vecors he prcpal drecos ad cosder a arbrary u ormal vecor e. The ormal sress s gve by Eq. 7..9 (7.. Cauchy s law gves he compoes of he raco vecor as (7.. ad so he shear sress o he plae s from Eq. 7.. ( ( (7..4 Usg he codo o elmae leads o ( ( [( ( ] (7..5 The saoary pos are ow obaed by equag he paral dervaves wh respec o he wo varables ad o zero: ( ( { [ ]} ( ( ( ( ( ( { [ ]} (7..6 Oe sees mmedaely ha (so ha ± s a soluo; hs s he prcpal dreco e ad he shear sress s by defo zero o he plae wh hs ormal. I hs calculao he compoe was elmaed ad was reaed as a fuco of he varables (. mlarly ca be elmaed wh ( reaed as he varables leadg o he soluo e ad ca be elmaed wh ( reaed as he varables leadg o he soluo e. Thus hese soluos lead o he mmum shear sress value. A secod soluo o Eq. 7..6 ca be see o be / (so ha ± ±/ wh correspodg shear sress values (. Two oher 4 old Mechacs Par II 5
eco 7. soluos ca be obaed as descrbed earler by elmag ad by elmag. The full soluo s lsed below ad hese are evdely he mamum (absolue value of he shear sresses acg a a po: ± ± ± ± ± ± (7..7 Takg he mamum shear sress a a po s τ ma ( (7..8 ad acs o a plae wh ormal oreed a 45 o o he ad prcpal drecos. Ths s llusraed Fg. 7..4. τ ma prcpal drecos τ ma Fgure 7..4: mamum shear sress a a po Eample Cosder he sress sae eamed he Eample of 7..4: 5 [ ] 6 The prcpal sresses were foud o be 5 5 ad so he mamum shear sress s 5 τ ma ( Oe of he plaes upo whch hey ac s show Fg. 7..5 (see Fg. 7.. old Mechacs Par II 6
eco 7. old Mechacs Par II 7 Fgure 7..5: mamum shear sress 7..6 Mohr s Crcles of ress The Mohr s crcle for D sress saes was dscussed Par I.5.4. For he D case followg o from seco 7..5 oe has he codos (7..9 olvg hese equaos gves ( ( ( ( ( ( ( ( ( ( ( ( (7..4 Takg ad og ha he squares of he ormal compoes mus be posve oe has ha ( ( ( ( ( ( (7..4 ad hese ca be re-wre as ( [ ] ( [ ] ( [ ] ( [ ] ( [ ] ( [ ] (7..4 ˆ ˆ ˆ o 7 τ ma
eco 7. If oe akes coordaes ( sress space Fg. 7..6. Each po ( he equaly sgs here represe crcles ( hs sress space represes he sress o a parcular plae hrough he maeral parcle queso. Admssble ( gve by he codos Eqs. 7..4; hey mus le sde a crcle of cere ( ( ad radus ( he crcle wh cere ( ( ad radus ( wh cere ( ( ad radus ( pars are. Ths s he large crcle Fg. 7..6. The pos mus le ousde ad also ousde he crcle ; hese are he wo smaller crcles he fgure. Thus he admssble pos sress space le he shaded rego of Fg. 7..6. Fgure 7..6: admssble pos sress space 7..7 Three Dmesoal ra The sra ε symbolc form ε s a esor ad as such follows he same rules as for he sress esor. I parcular follows he geeral esor rasformao rule 7..6; has prcpal values ε whch sasfy he characersc equao 7.. ad hese clude he mamum ad mmum ormal sra a a po. There are hree prcpal sra varas gve by 7..4 or 7..6 ad he mamum shear sra occurs o plaes oreed a 45 o o he prcpal drecos. 7..8 Problems. The sae of sress a a po wh respec o a coordae sysem s gve by [ ] Use Cauchy s law o deerme he raco vecor acg o a plae rough hs po whose u ormal s ( e / e e. Wha s he ormal sress acg o he plae? Wha s he shear sress acg o he plae?. The sae of sress a a po wh respec o a coordae sysem s gve by [ ] old Mechacs Par II 8
eco 7. Wha are he sress compoes wh respec o aes whch are obaed o from he frs by a 45 roao (posve couerclockwse abou he as. how boh he de ad mar oao ha he compoes of a soropc sress sae rema uchaged uder a coordae rasformao. 4. Cosder a wo-dmesoal problem. The sress rasformao formulae are he full cosθ sθ cosθ sθ sθ cosθ sθ cosθ Mulply he rgh had sde ou ad use he fac ha he sress esor s symmerc - o rue for all esors. Wha do you ge? Look famlar? ( 5. The sae of sress a a po wh respec o a coordae sysem s gve by 5 / / [ ] / 5 / Evaluae he prcpal sresses ad he prcpal drecos. Wha s he mamum shear sress acg a he po? old Mechacs Par II 9