CH.7. PLANE LINEAR ELASTICITY Coninuum Mechanics Course (MMC) - ETSECCPB - UPC
Overview Plane Linear Elasici Theor Plane Sress Simplifing Hpohesis Srain Field Consiuive Equaion Displacemen Field The Linear Elasic Problem in Plane Sress Eamples Plane Srain Simplifing Hpohesis Srain Field Consiuive Equaion Sress Field The Linear Elasic Problem in Plane Sress Eamples
Overview (con d) The Plane Linear Elasic Problem Governing Equaions Represenaive Curves Isosaics or sress rajecories Isoclines Isobars Shear lines Ohers: isochromaics and isopachs Phooelasici 3
7.1 Plane Linear Elasici Theor Ch.7. Plane Linear Elasici 4
Inroducion A lineal elasic solid is subjeced o bod forces and prescribed racion: Iniial acions: 0 Acions hrough ime: b,0,0 b,, The Linear Elasic problem is he se of equaions ha allow obaining he evoluion hrough ime of he corresponding displacemens u,, srains, and sresses,. 5
Governing Equaions The Linear Elasic Problem is governed b he equaions: 1. Cauch s Equaion of Moion. Linear Momenum Balance Equaion., b, 0 0 u,. Consiuive Equaion. Isoropic Linear Elasic Consiuive Equaion. Tr, 1 3. Geomerical Equaion. Kinemaic Compaibili. 1 S, u, u u This is a PDE ssem of 15 eqns -15 unknowns: u,, 3 unknowns 6 unknowns, 6 unknowns Which mus be solved in 3 he space. R R 6
Plane Linear Elasici For some problems, one of he principal sress direcions is known a priori: Due o paricular geomeries, loading and boundar condiions involved. The elasic problem can be solved independenl for his direcion. Seing he known direcion as z, he elasic problem analsis is reduced o he - plane There are wo main classes of plane linear elasic problems: Plane sress Plane srain PLANE ELASTICITY REMARK The isohermal case will be sudied here for he sake of simplici. Generalizaion of he resuls obained o hermo-elasici is sraigh-forward. 7
7. Plane Sress Ch.7. Plane Linear Elasici 8
Hpohesis on he Sress Tensor Simplifing hpohesis of a plane sress linear elasic problem: 1. Onl sresses conained in he - plane are no null z 0 0 0 0 0. The sress are independen of he z direcion.,,,,,, REMARK The name plane sress arises from he fac ha all (no null) sress are conained in he - plane. 9
Geomer and Acions in Plane Sress These hpohesis are valid when: The hickness is much smaller han he pical dimension associaed o he plane of analsis: e L b u * * The acions,,, and, are conained in he plane of analsis (in-plane acions) and independen of he hird dimension, z. *, is onl non-zero on he conour of he bod s hickness: 10
Srain Field in Plane Sress The srain field is obained from he inverse Hooke s Law: 1 Tr 1 E E As z0 z0 z 0,,,,,, 1 1 (1 ) E E 1 1 z z E 0 1 z z E 0 11 And he srain ensor for plane sress is: 1 0 1,, 0 0 0 z 1 wih z
Consiuive equaion in Plane Sress Operaing on he resul ields: 1 (1 ) E E 1 z z E 0 z z E 0 plane sress C E 1 E 1 E 1 1 0 E 1 0 1 1 0 0 1 Consiuive equaion in plane sress (Voig s noaion) C plane sress
Displacemen Field in Plane Sress The displacemen field is obained from he geomeric equaions,, S u,. These are spli ino: Those which do no affec he displacemen : u,, u,, u u,, Inegraion in. u z u u,, u u,, u z Those in which appears: uz z,, (,, ) uz(,, z, ) 1 z u(, ) uz uz z,, z 0 z 0 uz(,) z u (, ) uz uz z,, z 0 z 0 13 Conradicion!!!
The Lineal Elasic Problem in Plane Sress The problem can be reduced o he wo dimensions of he plane of analsis. The unknowns are: The addiional unknowns (wih respec o he general problem) are eiher null, or independenl obained, or irrelevan: z z z z z z 1 uz u,,,,,,,, z, u u 0 does no inervene in he problem REMARK This is an ideal elasic problem because i canno be eacl reproduced as a paricular case of he 3D elasic problem. There is no guaranee ha he soluion o u,, and u,, will allow obaining he soluion o u,, z, for he addiional geomeric eqns. z 14
Eamples of Plane Sress Analsis 3D problems which are picall assimilaed o a plane sress sae are characerized b: One of he bod s dimensions is significanl smaller han he oher wo. The acions are conained in he plane formed b he wo large dimensions. Slab loaded on he mean plane Deep beam 15
7.3 Plane Srain Ch.7. Plane Linear Elasici 16
Hpohesis on he Displacemen Field Simplifing hpohesis of a plane srain linear elasic problem: 1. The displacemen field is u u u 0. The displacemen variables associaed o he - plane are independen of he z direcion. u u,, u u,, 17
Geomer and Acions in Plane Srain These hpohesis are valid when: The bod being sudied is generaed b moving he plane of analsis along a generaional line. b u * * The acions,,, and, are conained in he plane of analsis and independen of he hird dimension, z. In he cenral secion, considered as he analsis secion he following holds (approimael) rue: uz 0 u 0 z u 0 z 18
Srain Field in Plane Srain uz 0 u 0 z u 0 z The srain field is obained from he geomeric equaions: u,, uz,, z 0 z u,, u,, uz,, z 0 z,, u,, u,, u uz,, z 0 z And he srain ensor for plane srain is: 1 0 1,, 0 0 0 0 REMARK The name plane srain arises from he fac ha all srain is conained in he - plane. 19
Sress Field in Plane Srain Inroducing he srain ensor ino Hooke s Law Tr 1 G and operaing on he resul ields: As G G G G z G z 0 G v( ) G 0 z z z,,,, z z,,,,,, And he sress ensor for plane srain is: 0,, 0 wih z 0 0 z 0
Consiuive equaion in Plane Srain Inroducing he srain ensor ino he consiuive equaion and operaing on he resul ields: E 1 G 11 1 Tr 1 E 1 G 11 1 E G 1 C plane srain 1 0 1 E 1 1 0 1 1 1 1 0 0 1 C plane srain Consiuive equaion in plane srain (Voig s noaion) 1
The Lineal Elasic Problem in Plane Srain (summar) The problem can be reduced o he wo dimensions of he plane of analsis. The unknowns are: u,,,,,, The addiional unknowns (wih respec o he general problem) are eiher null or obained from he unknowns of he problem: uz 0 z z z z z u u z 0
Eamples of Plane Srain Analsis 3D problems which are picall assimilaed o a plane srain sae are characerized b: The bod is generaed b ranslaing a generaional secion wih acions conained in is plane along a line perpendicular o his plane. The plane sress hpohesis z z z 0 mus be jusifiable. This picall occurs when: 1. One of he bod s dimensions is significanl larger han he oher wo. An secion no close o he eremes can be considered a smmer plane and saisfies: uz 0 u 0 z u 0 z u u u 0. The displacemen in z is blocked a he ereme secions. 3
Eamples of Plane Srain Analsis 3D problems which are picall assimilaed o a plane srain sae are: Pressure pipe Coninuous brake shoe Tunnel Solid wih blocked z displacemens a he ends 4
7.4 The Plane Linear Elasic Problem Ch.7. Plane Linear Elasici 5
Plane problem A lineal elasic solid is subjeced o bod forces and prescribed racion and displacemen Acions: On : On : u On : The Plane Linear Elasic problem is he se of equaions ha allow obaining he evoluion hrough ime of he corresponding displacemens u,,, srains,, and sresses,,. * * u * * u u * * b b b,,,,,,,,,,,, 6
Governing Equaions The Plane Linear Elasic Problem is governed b he equaions: 1. Cauch s Equaion of Moion. Linear Momenum Balance Equaion., b, 0 0 D u, u z z b z b z u z z z z z bz u 7
Governing Equaions The Plane Linear Elasic Problem is governed b he equaions:. Consiuive Equaion (Voig s noaion). Isoropic Linear Elasic Consiuive Equaion., C : D C Wih, and PLANE STRESS E E PLANE STRAIN E C 1 E E 1 1 1 0 1 0 0 0 1 8
Governing Equaions The Plane Linear Elasic Problem is governed b he equaions: 3. Geomerical Equaion. Kinemaic Compaibili. 1 S, u, u u D u u u u This is a PDE ssem of 8 eqns -8 unknowns: u,, unknowns 3 unknowns, 3 unknowns Which mus be solved in he space. 9
Boundar Condiions Boundar condiions in space Affec he spaial argumens of he unknowns Are applied on he conour of he solid, which is divided ino: u Prescribed displacemens on : u * * * u u,, * * u u,, Prescribed sresses on : * * *,, * *,, * n wih n n n 30
Boundar Condiions INTIAL CONDITIONS (boundar condiions in ime) Affec he ime argumen of he unknowns. Generall, he are he known values a 0 : Iniial displacemens: u u,,0 0 u Iniial veloci: u,, u v u u v 0,,0 v, 0 31
Unknowns The 8 unknowns o be solved in he problem are: u(,, ) u u,, 1 1,, Once hese are obained, he following are calculaed eplicil: PLANE STRESS PLANE STRAIN z 1 z 3
7.5 Represenaive Curves Ch.7. Plane Linear Elasici 33
Inroducion Tradiionall, plane sress saes where graphicall represened wih he aid of he following conour lines: Isosaics or sress rajecories Isoclines Isobars Maimum shear lines Ohers: isochromaics, isopachs, ec. 34
Isosaics or Sress Trajecories Ssem of curves which are angen o he principal aes of sress a each maerial poin. The are he envelopes of he principal sress vecor fields. There will eis wo families of curves a each poin: Isosaics 1, angens o he larges principal sress. Isosaics, angens o he smalles principal sress. REMARK The principal sresses are orhogonal o each oher, herefore, so will he wo families of isosaics orhogonal o each oher. 35
Singular and Neural Poins Singular poin: characerized b he sress sae 0 Neural poin: characerized b he sress sae 0 Mohr s Circle of a singular poin Mohr s Circle of a neural poin REMARK In a singular poin, all direcions are principal direcions. Thus, in singular poins isosaics end o loose heir regulari and can abrupl change direcion. 36
Differenial Equaion of he Isosaics Consider he general equaion of an isosaic curve: f 1 g g g 1g d d Solving he nd order eq.: Differenial equaion of he isosaics 10 ' 1, Known his funcion, he eq. can be inegraed o obain a famil of curves of he pe: f C 37
Isoclines Locus of he poins along which he principal sresses are in he same direcion. The principal sress vecors in all poins of an isocline are parallel o each oher, forming a consan angle wih he -ais. These curves can be direcl found using phooelasici mehods. 38
Equaion of he Isoclines To obain he general equaion of an isocline wih angle, he principal sress mus form an angle wih he -ais: Algebraic equaion of he isoclines 1 g, For each value of, he equaion of he famil of isoclines parameerized in funcion of is obained: f, REMARK Once he famil of isoclines is known, he principal sress direcions in an poin of he medium can be obained and, hus, he isosaics calculaed. 39
Isobars Locus of he poins along which he principal sress ( or 1 ) is consan. The isobars depend on he value of he principal sress bu no on heir direcion. There will eis wo families of isobars a each poin, and. 1 40
Equaion of he Isobars Take he equaion used o find he principal sresses and principal sress direcions given in a cerain se of aes: Algebraic equaion of he isobars, cnc 1 1 1, cnc This eq. implicil defines he famil of curves of he isobars: f, c 1 1 1 f, c 41
Maimum shear lines Envelopes of he maimum shear sress (in modulus) vecor fields. The are he curves on which he shear sress modulus is a maimum. Two planes of maimum shear sress correspond o each maerial poin, and. ma min These planes are easil deermined using Mohr s Circle. REMARK The wo planes form a 45º angle wih he principal sress direcions and, hus, are orhogonal o each oher. The form an angle of 45º wih he isosaics. 4
Equaion of he maimum shear lines Consider he general equaion of a slip line f, he relaion 1 an and an an 4 Then, 1 an an an 1 an d no an 1 d an 4 10 43
Equaion of he maimum shear lines Solving he nd order eq.: Differenial equaion of he slip lines ' 1, Known his funcion, he eq. can be inegraed o obain a famil of curves of he pe: f C 44
Summar Ch.7. Plane Linear Elasici 50
Summar For some problems he elasic problem can be solved independenl for one of he direcions. The analsis is reduced o a plane. Two problem pes: 1. Plane sress z 0 0 0 0 0 1 0 1,, 0 0 0 z,,,,,, wih z 1 The displacemen field is obained from, S u, PLANE ELASTICITY e L 51
Summar (con d). Plane srain u u u 0 1 0 1,, 0 0 0 0 u u,, u u,, In he cenral secion, uz 0 u 0 z u 0 z 0,, 0 wih z 0 0 z 5
Summar (con d) For boh problem pes he unknowns are: u u u,,,,,, The addiional unknowns (w.r.. he general problem) are eiher null, independenl obained or irrelevan: z z z z z z 1 uz PLANE STRESS,, z, 0 does no inervene in he problem PLANE STRAIN uz 0 z z z z z z 0 53
Summar (con d) The PLANE Linear Elasic Problem: Acions: On : On : u On : * * u b * * u u b b * *,,,,,,,,,,,, Responses: u,,, unknowns 3 unknowns 3 unknowns 54
Summar (con d) The PLANE Isoropic Linear Elasic Problem: u b u b PLANE STRAIN: This is a PDE ssem of 8 eqns -8 unknowns:, 3 unknowns Which mus be solved in he space. Cauch s Equaion of Moion 1 0 E C wih C 1 0 1 Consiuive Equaion 0 0 1 E ; PLANE E STRESS: E E; 1 1 u, R, R unknowns 3 unknowns u u u u ; ; Geomeric Equaion 55
Summar (con d) The PLANE Isoropic Linear Elasic Problem (con d): u : : u * * * * u u,, * * u u,, * *,, * *,, wih Boundar Condiions in Space * n u,,0 0 u,, 0 no u,,0 v, 0 Iniial Condiions 56
Summar (con d) Plane sress saes can be graphicall represened wih he curves: Isosaics or sress rajecories: curves which are a each maerial poin angen o he principal aes of sress. Isoclines: locus of he poins along which he principal sresses are in he same direcion. (Deermined using phooelasici mehods.) Isobars: locus of he poins along which he principal sress is consan. Maimum shear lines: envelopes of he maimum (in modulus) shear sress vecor fields. Isochromaics: curves along which he maimum shear sress is consan. (Deermined using phooelasici mehods.) Isopachs: curves along which he mean normal sress is consan. 57