Module 2: Analysis of Stress

Transcription

1 Module/Leon Module : Anali of Sre.. INTRODUCTION A bod under he acion of eernal force, undergoe diorion and he effec due o hi em of force i ranmied hroughou he bod developing inernal force in i. To eamine hee inernal force a a poin O in Figure. (a), inide he bod, conider a plane MN paing hrough he poin O. If he plane i divided ino a number of mall area, a in he Figure. (b), and he force acing on each of hee are meaured, i will be oberved ha hee force var from one mall area o he ne. On he mall area D A a poin O, a force DF will be acing a hown in he Figure. (b). From hi he concep of re a he inernal force per uni area can be underood. Auming ha he maerial i coninuou, he erm "re" a an poin acro a mall area D A can be defined b he limiing equaion a below. (a) Figure. Force acing on a bod (b) Sre = DF lim DA 0DA (.0) where DF i he inernal force on he area DA urrounding he given poin. Sre i omeime referred o a force ineni. Applied Elaici for Engineer T.G.Siharam & L.GovindaRaju

2 .. NOTATION OF STRESS Module/Leon Here, a ingle uffi for noaion, like,,, i ued for he direc ree and double uffi for noaion i ued for hear ree like, z, ec. mean a re, produced b an inernal force in he direcion of Y, acing on a urface, having a normal in he direcion of X...3 CONCEPT OF DIRECT STRESS AND SHEAR STRESS z Figure. Force componen of DF acing on mall area cenered on poin O Figure. how he recangular componen of he force vecor DF referred o DF DF DFz correponding ae. Taking he raio,,, we have hree quaniie ha DA DA DA eablih he average ineni of he force on he area DA. In he limi a DA 0, he above raio define he force ineni acing on he X-face a poin O. Thee value of he hree ineniie are defined a he "Sre componen" aociaed wih he X-face a poin O. The re componen parallel o he urface are called "Shear re componen" denoed b. The hear re componen acing on he X-face in he -direcion i idenified a. The re componen perpendicular o he face i called "Normal Sre" or "Direc re" componen and i denoed b. Thi i idenified a along X-direcion. Applied Elaici for Engineer T.G.Siharam & L.GovindaRaju

3 Module/Leon From he above dicuion, he re componen on he X -face a poin O are defined a follow in erm of force ineni raio = DF lim DA DA 0 DF = lim (.) DA 0DA z = DA 0 DFz lim DA The above re componen are illuraed in he Figure.3 below. Figure.3 Sre componen a poin O Applied Elaici for Engineer 3 T.G.Siharam & L.GovindaRaju

4 Module/Leon..4 STRESS TENSOR Le O be he poin in a bod hown in Figure. (a). Paing hrough ha poin, infiniel man plane ma be drawn. A he reulan force acing on hee plane i he ame, he ree on hee plane are differen becaue he area and he inclinaion of hee plane are differen. Therefore, for a complee decripion of re, we have o pecif no onl i magniude, direcion and ene bu alo he urface on which i ac. For hi reaon, he re i called a "Tenor". Figure.4 Sre componen acing on parallelopiped Figure.4 depic hree-orhogonal co-ordinae plane repreening a parallelopiped on which are nine componen of re. Of hee hree are direc ree and i are hear ree. In enor noaion, hee can be epreed b he enor ij, where i =,, z and j =,, z. In mari noaion, i i ofen wrien a Applied Elaici for Engineer 4 T.G.Siharam & L.GovindaRaju

5 Module/Leon ij = é ê ê ê ë z z I i alo wrien a zù ú zú ú zzû (.) S = é ê ê ê ë z z zù ú zú ú zû (.3)..5 SPHERICAL AND DEVIATORIAL STRESS TENSORS A general re-enor can be convenienl divided ino wo par a hown above. Le u now define a new re erm ( m ) a he mean re, o ha + + z m = (.4) 3 Imagine a hdroaic pe of re having all he normal ree equal o m, and all he hear ree are zero. We can divide he re enor ino wo par, one having onl he "hdroaic re" and he oher, "deviaorial re". The hdroaic pe of re i given b é ê ê 0 êë 0 m 0 0 m 0 ù 0 ú ú ú mû The deviaorial pe of re i given b (.5) é - ê ê ê ë z m - z m z z z - m ù ú ú ú û (.6) Here he hdroaic pe of re i known a "pherical re enor" and he oher i known a he "deviaorial re enor". I will be een laer ha he deviaorial par produce change in hape of he bod and finall caue failure. The pherical par i raher harmle, produce onl uniform volume change wihou an change of hape, and doe no necearil caue failure. Applied Elaici for Engineer 5 T.G.Siharam & L.GovindaRaju

6 ..6 INDICIAL NOTATION Module/Leon An alernae noaion called inde or indicial noaion for re i more convenien for general dicuion in elaici. In indicial noaion, he co-ordinae ae, and z are replaced b numbered ae, and 3 repecivel. The componen of he force DF of Figure. (a) i wrien a DF, DF and DF 3, where he numerical ubcrip indicae he componen wih repec o he numbered coordinae ae. The definiion of he componen of re acing on he face can be wrien in indicial form a follow: DF = lim DA 0D A DF A = lim (.7) DA 0D 3 = lim DA 0 DF D A 3 Here, he mbol i ued for boh normal and hear ree. In general, all componen of re can now be defined b a ingle equaion a below. DFj ij = lim (.8) DAi 0DAi Here i and j ake on he value, or TYPES OF STRESS Sree ma be claified in wo wa, i.e., according o he pe of bod on which he ac, or he naure of he re ielf. Thu ree could be one-dimenional, wo-dimenional or hree-dimenional a hown in he Figure.5. (a) One-dimenional Sre Applied Elaici for Engineer 6 T.G.Siharam & L.GovindaRaju

7 Module/Leon (b) Two-dimenional Sre Figure.5 Tpe of Sre (c) Three-dimenional Sre..8 TWO-DIMENSIONAL STRESS AT A POINT A wo-dimenional ae-of-re ei when he ree and bod force are independen of one of he co-ordinae. Such a ae i decribed b ree, and and he X and Y bod force (Here z i aken a he independen co-ordinae ai). We hall now deermine he equaion for ranformaion of he re componen, and a an poin of a bod repreened b infinieimal elemen a hown in he Figure.6. Figure.6 Thin bod ubjeced o ree in plane Applied Elaici for Engineer 7 T.G.Siharam & L.GovindaRaju

8 Module/Leon Figure.7 Sre componen acing on face of a mall wedge cu from bod of Figure.6 Conider an infinieimal wedge a hown in Fig..7 cu from he loaded bod in Figure.6. I i required o deermine he ree and, ha refer o ae, making an angle q wih ae X, Y a hown in he Figure. Le ide MN be normal o he ai. Conidering and area coq and inq, repecivel. a poiive and area of ide MN a uni, he ide MP and PN have Equilibrium of he force in he and direcion require ha T = coq + inq T = coq + inq (.9) Applied Elaici for Engineer 8 T.G.Siharam & L.GovindaRaju

9 Module/Leon where T and T are he componen of re reulan acing on MN in he and direcion repecivel. The normal and hear ree on he ' plane (MN plane) are obained b projecing T and T in he and direcion. = T coq + T inq (.0) = T coq - T inq Upon ubiuion of re reulan from Equaion (.9), he Equaion (.0) become = co q + in q + inq coq = (co q - in q )+( - ) inq coq (.) The re i obained b ubiuing B mean of rigonomeric ideniie æ p ö çq + for q in he epreion for. è ø co q = (+coq), inq coq = inq, (.) in q = (-coq) The ranformaion equaion for ree are now wrien in he following form: ( + ) + ( - ) co q in q = + (.a) = ( + ) - ( - ) co q - in q (.b) =- ( - ) in q + co q (.c)..9 PRINCIPAL STRESSES IN TWO DIMENSIONS To acerain he orienaion of correponding o maimum or minimum, he d necear condiion = 0, i applied o Equaion (.a), ielding dq -( - ) inq + coq = 0 (.3) Therefore, an q = - (.4) A q = an (p +q), wo direcion, muuall perpendicular, are found o aif equaion (.4). Thee are he principal direcion, along which he principal or maimum and minimum normal ree ac. Applied Elaici for Engineer 9 T.G.Siharam & L.GovindaRaju

10 Module/Leon When Equaion (.c) i compared wih Equaion (.3), i become clear ha = 0 on a principal plane. A principal plane i hu a plane of zero hear. The principal ree are deermined b ubiuing Equaion (.4) ino Equaion (.a), = + ± æ - ö ç è ø + (.5) Algebraicall, larger re given above i he maimum principal re, denoed b. The minimum principal re i repreened b. Similarl, b uing he above approach and emploing Equaion (.c), an epreion for he maimum hear re ma alo be derived...0 CAUCHY S STRESS PRINCIPLE According o he general heor of re b Cauch (83), he re principle can be aed a follow: Conider an cloed urface S wihin a coninuum of region B ha eparae he region B ino ubregion B and B. The ineracion beween hee ubregion can be repreened b a T nˆ defined on S. B combining hi principle wih Euler field of re vecor ( ) equaion ha epree balance of linear momenum and momen of momenum in an kind of bod, Cauch derived he following relaionhip. T ( nˆ ) = -T (- nˆ ) T ( nˆ ) = T ( ) nˆ (.6) nˆ i he uni normal o S and i he re mari. Furhermore, in region where he field variable have ufficienl mooh variaion o allow paial derivaive upo an order, we have where ( ) ra = div + f where r = maerial ma deni A = acceleraion field f = Bod force per uni volume. (.7) Thi reul epree a necear and ufficien condiion for he balance of linear momenum. When epreion (.7) i aified, = T (.8) which i equivalen o he balance of momen of momenum wih repec o an arbirar poin. In deriving (.8), i i implied ha here are no bod couple. If bod couple and/or couple ree are preen, Equaion (.8) i modified bu Equaion (.7) remain unchanged. Cauch Sre principle ha four eenial ingradien Applied Elaici for Engineer 0 T.G.Siharam & L.GovindaRaju

11 Module/Leon (i) The phical dimenion of re are (force)/(area). (ii) Sre i defined on an imaginar urface ha eparae he region under conideraion ino wo par. (iii) Sre i a vecor or vecor field equipollen o he acion of one par of he maerial on he oher. (iv) The direcion of he re vecor i no rericed... DIRECTION COSINES Conider a plane ABC having an ouward normal n. The direcion of hi normal can be defined in erm of direcion coine. Le he angle of inclinaion of he normal wih, and z ae be b P,, z be a poin on he normal a a radial a, and g repecivel. Le ( ) diance r from he origin O. Figure.8 Terahedron wih arbirar plane Applied Elaici for Engineer T.G.Siharam & L.GovindaRaju

12 From figure, or Le Therefore, co a =, cob = and r r z co g = r = r coa, = r cob and z= r cog co a = l, cob = m and co g = n z = l, = m and = n r r r Module/Leon Here, l, m and n are known a direcion coine of he line OP. Alo, i can be wrien a = + + z r (ince r i he polar co-ordinae of P) z or + + = r r r l + m + n =.. STRESS COMPONENTS ON AN ARBITRARY PLANE Conider a mall erahedron iolaed from a coninuou medium (Figure.9) ubjeced o a general ae of re. The bod force are aken o be negligible. Le he arbirar plane ABC be idenified b i ouward normal n whoe direcion coine are l, m and n. In he Figure.9, T, T, T are he Careian componen of re reulan T, acing on z oblique plane ABC. I i required o relae he ree on he perpendicular plane inerecing a he origin o he normal and hear ree acing on ABC. The orienaion of he plane ABC ma be defined in erm of he angle beween a uni normal n o he plane and he,, z direcion. The direcion coine aociaed wih hee angle are co (n, ) = l co (n, ) = m and (.9) co (n, z) = n The hree direcion coine for he n direcion are relaed b l + m + n = (.0) Applied Elaici for Engineer T.G.Siharam & L.GovindaRaju

13 Module/Leon Figure.9 Sree acing on face of he erahedron The area of he perpendicular plane PAB, PAC, PBC ma now be epreed in erm of A, he area of ABC, and he direcion coine. Therefore, Area of PAB = A PAB = A = A.i = A (li + mj + nk) i Hence, A PAB = Al The oher wo area are imilarl obained. In doing o, we have alogeher A PAB = Al, A PAC = Am, A PBC = An (.) Here i, j and k are uni vecor in, and z direcion, repecivel. Now, for equilibrium of he erahedron, he um of force in, and z direcion mu be zero. Therefore, T A = Al + Am + z An (.) Applied Elaici for Engineer 3 T.G.Siharam & L.GovindaRaju

14 Dividing hroughou b A, we ge T = l + m + z n Similarl, for equilibrium in and z direcion, T = l + m + z n T z = z l + z m + z n Module/Leon (.a) (.b) (.c) The re reulan on A i hu deermined on he bai of known ree,,,, and a knowledge of he orienaion of A. z z, z The Equaion (.a), (.b) and (.c) are known a Cauch re formula. Thee equaion how ha he nine recangular re componen a P will enable one o deermine he re componen on an arbirar plane paing hrough poin P...3 STRESS TRANSFORMATION When he ae or re a a poin i pecified in erm of he i componen wih reference o a given co-ordinae em, hen for he ame poin, he re componen wih reference o anoher co-ordinae em obained b roaing he original ae can be deermined uing he direcion coine. Conider a careian co-ordinae em X, Y and Z a hown in he Figure.0. Le hi given co-ordinae em be roaed o a new co-ordinae em,, z where in lie on an oblique plane. and X, Y, Z em are relaed b he direcion coine.,, z l = co (, X) m = co (, Y) (.3) n = co (, Z) (The noaion correponding o a complee e of direcion coine i hown in Table.0). Table.0 Direcion coine relaing differen ae X Y Z ' l m n l m n z l 3 m 3 n 3 Applied Elaici for Engineer 4 T.G.Siharam & L.GovindaRaju

15 Module/Leon Figure.0 Tranformaion of co-ordinae The normal re i found b projecing T, T and T z in he direcion and adding: = T l + T m + T z n (.4) Equaion (.a), (.b), (.c) and (.4) are combined o ield = l + m + z n + ( l m + z m n + z l n ) (.5) Similarl b projecing T, T, T in he and z direcion, we obain, repecivel z = l l + m m + z n n + (l m + m l )+ z (m n + n m ) + z (n l + l n ) (.5a) z = l l 3 + m m 3 + z n n 3 + (l m 3 + m l 3 )+ z (m n 3 + n m 3 )+ z (n l 3 + l n 3 ) (.5b) Recalling ha he ree on hree muuall perpendicular plane are required o pecif he re a a poin (one of hee plane being he oblique plane in queion), he remaining componen are found b conidering hoe plane perpendicular o he oblique plane. For one uch plane n would now coincide wih direcion, and epreion for he ree Applied Elaici for Engineer 5 T.G.Siharam & L.GovindaRaju

16 Module/Leon,, z would be derived. In a imilar manner he ree z, z, z are deermined when n coincide wih he z direcion. Owing o he mmer of re enor, onl i of he nine re componen hu developed are unique. The remaining re componen are a follow: = l + m + z n + ( l m + z m n + z l n ) (.5c) z = l + 3 m + 3 z n + ( 3 l 3 m 3 + z m 3 n 3 + z l 3 n 3 ) (.5d) = l l 3 + m m 3 + z n n 3 + (m l 3 + l m 3 )+ z (n m 3 + m n 3 )+ z (l n 3 + n l 3 ) z (.5e) The Equaion (.5 o.5e) repreen epreion ranforming he quaniie,,,, o compleel define he ae of re. z z I i o be noed ha, becaue, and z are orhogonal, he nine direcion coine mu aif rigonomeric relaion of he following form. l i + m i + n i = (i =,,3) and l l + m m + n n = 0 l l 3 + m m 3 + n n 3 = 0 (.6) l l 3 + m m 3 + n n 3 = 0 Applied Elaici for Engineer 6 T.G.Siharam & L.GovindaRaju