Analysis of Structural Vibration using the Finite Element Method

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1 Sminar: Vibrations and Structur-Born Sound in Civil Enginring Thor and Applications Analsis of Structural Vibration using th Finit Elmnt thod John.A. Shiwua 5 th April, 006 Abstract Structural vibration tsting and analsis contributs to progrss in man industris, including arospac, auto-making, manufacturing, wood and papr production, powr gnration, dfns, consumr lctronics, tlcommunications and transportation. Th most common application is idntification and supprssion of unwantd vibration to improv product qualit..

2 Part. Introduction and Basic concpts. Basic Trminolog of Structural Vibration Th trm vibration dscribs rptitiv motion that can b masurd and obsrvd in a structur. Unwantd vibration can caus fatigu or dgrad th prformanc of th structur. Thrfor it is dsirabl to liminat or rduc th ffcts of vibration. In othr cass, vibration is unavoidabl or vn dsirabl. In this cas, th goal ma b to undrstand th ffct on th structur, or to control or modif th vibration, or to isolat it from th structur and minimiz structural rspons.... Common Vibration Sourcs It can b criticall important to nsur that th natural frquncis of th structural sstm do not match th oprating frquncis of th quipmnt. Th dnamic amplification which

3 can occur if th frquncis do coincid can lad to hazardous structural fatigu situations, or shortn th lif of th quipmnt... Forms of Vibration Fr vibration is th natural rspons of a structur to som impact or displacmnt. Th rspons is compltl dtrmind b th proprtis of th structur, and its vibration can b undrstood b amining th structur's mchanical proprtis. Forcd vibration is th rspons of a structur to a rptitiv forcing function that causs th structur to vibrat at th frqunc of th citation. Sinusoidal vibration is a spcial class of vibration. Th structur is citd b a forcing function that is a pur ton with a singl frqunc.. Th motion of an point on th structur can b dscribd as a sinusoidal function of tim 0. Random vibration is vr common in natur. Th vibration ou fl whn driving a car rsult from a compl combination of th rough road surfac, ngin vibration, wind buffting th car's trior, tc. Rotating imbalanc is anothr common sourc of vibration. Th rotation of an unbalancd machin part can caus th ntir structur to vibrat. Eampls includ a washing machin, an automobil ngin, shafts, stam or gas turbins, and computr disk drivs..3 Structural Vibration Th simplst vibration modl is th singl-dgr-of-frdom, or mass-spring-dampr modl. It consists of a simpl mass () that is suspndd b an idal spring with a known stiffnss (K) and a dashpot dampr from a fid support. A dashpot dampr is lik a shock absorbr in a car. K C Fig.. 3. Simpl ass-spring-dampr vibration odl

4 If ou displac th mass b pulling it down and rlasing it, th mass will rspond with motion similar to Figur. 4. Th mass will oscillat about th quilibrium point and aftr vr oscillation, th maimum displacmnt will dcras du to th dampr, until th motion bcoms so small that it is undtctabl. Evntuall th mass stops moving. Figur.4. Fr Vibration of ass-spring-dampr Figur.4 shows that th tim btwn vr oscillation is th sam. rlatd to th frqunc of oscillation Assuming th damping is small, thn th mathmatical rlationship is givn b K ω n = ()

5 A largr stiffnss will rsult in a highr n, and a largr mass will rsult in a lowr n. Figur (.4). also rvals somthing about damping. Thor tlls us that th amplitud of ach oscillation will diminish at a prdictabl rat. Th rat is rlatd to th damping factor C. Usuall damping is dscribd in trms of th damping ratio--. That ratio is rlatd to C b C ζ = () ωn Whn a structur is citd, it will dform, vibrat and tak on diffrnt shaps dpnding on th frqunc of th citation and th mounting of th nds. For a continuous structur such as a bam fid at both nd and whr th mass is distributd ovr volum. Th bam will hav a first rsonant frqunc at which all its points will mov in unison; at th first rsonant frqunc, th bam will tak th shap shown to th right in Figur (.5) labld First od Shap. Bam st od shap nd od shap Pinnd Ends 3rd od shap Fig.. 5 A bam fid at both nds. At a highr frqunc, th bam will hav a scond rsonant frqunc and mod shap, and a third, and fourth, tc. Thorticall thr ar an infinit numbr of rsonant frquncis and mod shaps. Howvr at highr frquncis, th structur acts lik a low-pass filtr and th vibration lvls gt smallr and smallr. Th highr mods ar hardr to dtct and hav lss ffct on th ovrall vibration of th structur.

6 Part. Vibration Plat.. Gnral Thor In thr-dimnsional lasticit thor th strss at a point is spcifid b th si quantitis: σ, σ, σz th componnts of dirct strss; σ, σ z, σz th componnts of shar strss. Th componnts of strss on th fac ABCD of an lmnt ar shown in Fig. (.), from which th sign convntions ar sn to b: dirct Z J d H z D C Y F B 0 d A X Fig... Strss componnts on th fac ABCD of an lmnt Strsss ar positiv whn tnsil; th shar strss σ z acts on a fac prpndicular to th X- ais in a dirction paralll to th Z-ais and is positiv if it acts in th positiv dirction of th Z- ais on a fac for which th positiv dirct strss is in th dirction If th lngths of th sids of th lmnt ar d d and dz, th shar strsss σ z on th facs ABCD and OFHJ form a

7 coupl about th Y-ais of magnitud ( σ z dz d ) d Considring th othr componnts of shar strss, onl th componnts σ z acting on th facs DCHJ and OABF form a coupl about th Y-ais. Thus taking momnts about th Y-ais for th quilibrium of th lmnt, σz = σz. Similarl, σ = σ and σ z = σz Th strain at a point is dfind similarl b: ε, ε, εz, -th componnts of dirct strain; ε, ε z, εz, -th componnts of shar strain. Th componnts of displacmnt at an point (,, z) ar u, v and w, positiv in th dirctions OA, OY and OZ, rspctivl. Th componnts of dirct strss and strain ar rlatd b Hook's law, tndd to includ Poisson's ratio ff ( ) ε = σ v( σ + σz) E (3) with analogous prssions for ε, andεz, Th componnts of shar strss and strain ar rlatd b ε ( ) σ, = tc (4) G In ths quations E, G and v ar th lastic constants, Young's modulus, shar modulus (or modulus of rigidit) and Poisson's ratio, rspctivl. In this papr onl homognous isotropic lastic solids will b considrd; for ths solids thr ar onl two indpndnt lastic constants and E = G( + v) (5) Lt us stablish th strain-displacmnt rlations. If th displacmnt in th X-dirction of th point ( z,, ) is u, thn th displacmnt in th sam dirction of th adjacnt point ( + dz,, ) is[ u+ ( u/ d ) ]. Thus th dirct strain in th X-dirction ε = Incras in lngth of lmnt Initial lngth of lmnt

8 [ u+ ( u/ d ) ] d = d =u (6) Similarl, ε == v and ε z == w z Considring an lmnt ABCD, initiall rctangular and having sids of lngth d and d paralll to th X- and F-as, th cornr A, initiall at th point (,, ) is displacd to A in th plan OXY, with componnts of displacmnt u and in th X- and F-dirctions, rspctivl (Fig..). Th point is displacd to B [ v+ ( v/ d ) ] th point D is displacd to with a displacmnt in th -dirction of Y u D C D A B v A B X Fig... Displacmnt in th plan OXY D, with a displacmnt in th X-dirction of[ u+ ( u/ d ) ]. Th dformd shap of th lmnt is th paralllogram A B C D and th shar strain is (a + ). Thus for small angls u u ε = α + β = + (7) Similarl,

9 ε z u u = + z and ε z u u = + z For an lmnt of sids d, d and dz (Fig. 5.) th forc du to th strss σ is σ ddz ; if th corrsponding strain is ε, th tnsion in th X-dirction isε d and th work don on th lmnt is ( σddz)( εd) = σε dv (8) whr dv = d, ddz, is th volum of th lmnt. Considring th lmnt ABCD of Fig.., of thicknss dz in th Z-dirction, th forc corrsponding to th shar strss σ on th fac BC is σ ddz ; thr is an qual and opposit forc on th fac AD, th two forcs forming a coupl of magnitud σ dddz ; th rotation du to this coupl is. Du to th complmntar shar strss on th facs AB and CD, thr is a coupl of work don on th lmnt is qual magnitud with corrsponding rotation a. Using, ε = α + β, th dv σ ε Gnralizing for a thr-dimnsional stat of strss, th strain nrg in an lastic bod δ = ( σε + σ ε + σzεz + σε + σ zεz + σzε z ) dv (9) V Using quations (3) and (4) th strain nrg ma b prssd in trms of strain onl or of strss onl.. Transvrs Vibrations of Rctangular Plat Considr th plat lmnt shown blow Fig.(.5 ). Th undformd middl plan of th plat is dfind as OX Y, with th X- and Y-as paralll to th dgs of th plat; th Z-ais will b takn as positiv upwards. Th following assumptions ar mad:

10 . Th plat is thin and of uniform thicknss h; thus th fr surfacs of th plat ar th plans z =± h. Th dirct strss in th transvrs dirctionσ z is zro. This strss must b zro at th fr surfacs, and providd that th plat is thin, it is rasonabl to assum that it is zro at an sction z. 3. Strsss in th middl plan of th plat (mmbran strsss) ar nglctd, i.. transvrs forcs ar supportd b bnding strsss, as in flur of a bam. For mmbran action not to occur, th displacmnts must b small compard with th thicknss of th plat. 4. Plan sctions that ar initiall normal to th middl plan rmain plan and normal to it. A similar assumption was mad in th lmntar thor for bams and implis that dformation du to transvrs shar is nglctd. Thus with this assumption th shar strains σ and σ z ar zro. z 5. Onl th transvrs displacmnt w (in th Z-dirction) has to b considrd. Figur (.3a) shows an lmnt of th plat of lngth d in th unstraind stat and Fig..3b th corrsponding lmnt in th straind stat. If OA rprsnts th middl surfac of th plat, thn OA = from th third assumption; also = Rdθ, whr R is th radius of curvatur of th dformd middl surfac. Thus th strain in BC, at a distanc z from th middl surfac, ε ( ) BC BC R + z R z BC R R = = =

11 Z B C Z B C O A z O A R (a) (b) Fig..3. Elmnt of plat. (a) Unstraind. (b) Straind Th rlation btwn th curvatur and th displacmnt of th middl surfac, w, is: R w = Thus ε w = z (0) Similarl ε w = z From quation (7) th shar strain ε, at a distanc z from th middl surfac is u v +, whr u and v ar th displacmnts at dpth z in th X- and Y-dirctions, rspctivl. Using th assumption hat sctions normal to th middl plan rmain normal to it, w u =. Similarl w v =

12 Thus ε w = z () (In quation () th trm w is th twist of th surfac.) Z u= z w A Z w X Fig..4 Rlation btwn u and w From th scond assumption and quations (3) and (4), th strss-strain rlations for a thin plat ar σ E = v ε + ε ( v ) ; σ = ( ε + v ε ) E v () Also σ E = ( v) ε Substituting from quations (0), () and () in th strain nrg prssion (9), ab h / E δ = ε + ε + vεε + ( v ) ε ε dzdd 00 h/ ( v ) a b D w w w w w = v ( v) dd (3) aftr intgrating with rspct to z, whr

13 Eh D = 3 ( v ) Th limits in th abov intgrals impl that th plat is boundd b th lins = 0, = a, = 0 and = b. If is th dnsit of th plat, th kintic nrg ab h/ w I= ρ t 0 0 h/ dzdd ab 00 w = ρ dd t (4) aftr intgrating with rspct to z. Eprssions (3) and (4) will b usd with th finit lmnt mthods. Th lmnt of th plat with sids d and d and thicknss h, shown in Fig..5, is subjctd to a bnding momnt a twisting momnt and a transvrs shar forc S pr unit lngth on th fac OB; on th fac A hr ar, pr unit lngth, a bnding momnt and a shar forc S., a twisting momnt Y S + S d + d + d B d p (, ) f ( t) S + S d C + d Z S d h + d X O A S Fig..5 Forcs and momnts on an lmnt of a plat

14 Th bnding momnts and ar th rsultant momnts du to th dirct strsss σ and σ rspctivl, aftr intgrating through th thicknss of th plat. Similarl, th twisting momnts ar th rsultants du to th shar strssσ. aintaining a consistnt sign convntion btwn th dfinitions for strsss in Fig.. and th momnts shown in Fig..5, w hav h/ = σ zdz ; h / h/ = h/ σ zdz and h/ = = σ zdz (5) h / Figur.5 shows also th incrmntal quantitis acting on th facs AC and BC and th applid forc pr unit ara, p(, ) f (t), in th Z-dirction. In addition thr is an inrtia forc w pr unit ara, ρh, in th Z-dirction. Th quilibrium quations, obtaind b rsolving in t th Z-dirction and taking momnts about th Y- and X-as, ar, aftr dividing b d d: S S w + + p (, ) f( t) = ρh t + S = 0 and + + S = 0 Eliminating S and S w p (, ) f( t) = ρh t (6) Substituting from quations (0) to () in quations (5) and intgrating with rspct to z. w w = D + v ; w w = D + v, and

15 w = D( v) (7) Substituting from quations (7) in quation (6) givs th quilibrium quation for an lmnt of th plat in trms of w and its drivativs, w D + + ρh pf (, ) ( t) = t (8) For a dnamic problm wt (,,) must satisf quation (8) togthr with th boundar conditions. Th standard simpl boundar conditions ar simpl supportd, clampd and fr. Part 3. Application of Finit Elmnt thod. 3. Finit Elmnt thod. Th basic concpt of th finit lmnt approach is to subdivid a larg compl structur into a finit numbr of simpl lmnts, such as bam lmnts, quadrilatrals, or triangls and th compl diffrntial quations ar thn solvd for th simpl lmnts. Assmblag of th lmnts into a global matri transforms from a diffrntial quations formulation ovr a continuum to a linar algbra problm that is radil solvabl b using computrs.. In static analsis. Th finit lmnt mthod solvs quations of th form [ K]{ U} = { P} (9) whr [K] is a global stiffnss matri, {U} is a vctor of nodal displacmnts, and {P} is a vctor of nodal forcs. Th dnamic analsis usuall starts with an quation of th form ([ K] ω [ ] ){ U} { P} = (0) whr in this quation dnots th frquncis of vibration and [] is th mass matri. For a fr vibration, w hav {P} = 0 and Eq. 0. rducs to th following prssion: ([ K] ω [ ] ){ U} { 0} = () For a nontrivial solution, w hav

16 [ K] ω [ ] = () which rprsnts th gnralizd Eignvalu problms. 3. Elmnt Stiffnss and ass atrics 3.. Aial Elmnts Considr a truss lmnt shown in Fig. (.6), which has th dgr of frdom u and u at points and, rspctivl. Sinc th mmbrs of th truss ar loadd aiall, th lmnt would b subjct to aial forcs f and f as shown in th figur, and consquntl, u and u would rprsnt aial displacmnts. W can assum hr that th displacmnt function u() ma b rprsntd b th following linar quation: u u f f Figur (.6). Truss lmnt. u( ) = b+ b (3) whr b and b ar constants. In matri form Equation (3) ma b writtn as follows: u ( ) { } b = b B using Eq. (3) and appling th boundar conditions u(0) = u and u(l) = u, whr L is th lngth of th lmnts, w find ( 0 ) (0) (4) u = b + b (5) u( L) = b + Lb (6) Equations (5) and (6) ar writtn in matri form as follows u 0 b = u L b Th sam two quations ma also b writtn in a matri form that rlats b and b and u and u in th following wa (7)

17 b L 0 u = b u B substituting Eq. (8) into Eq. (4), w find u u ( ) = L L u (8) (9) or, in gnral { } u ( ) H H u = u whr H, in Eq. (30) ar rfrrd to as shap functions or intrpolation functions (30) Th aial strain ε of th lmnt in Fig. (.7) is th rat of chang of u ( ) with rspct to. Thrfor, b diffrntiating Eq. (9) with rspct to, w find ε L u L u = B appling Hook's law, th normal strss at cross sctions along th lngth of th lmnt is givn b th prssion σ = Eε (3) whr is Young's modulus of lasticit and is givn b Eq. (3). Th total nrg stord in th lmnt is dfind b th prssion = L 0 T du EA d du d d f u f u whr A is th cross-sctional ara of th mmbr. B using Eq. (30), th prssion givn b Eq. (33) ma b writtn as follows: (3) (33) = L 0 EA u L L Whn A is constant, w hav u L u { u } L d { f f } u u (34) L d = L 0 (35) and quation 34. ilds

18 EAL = L L L L u u u u { u u } { f f } AE u u = { u u} { f f} L u u (36) For a stationar condition, w hav [ K ] AE = L In a similar mannr, various othr lmnt stiffnss matrics ma b obtaind. (39) 3.3 Finit Elmnt thod for In-plan Vibration of Plat In th convntional thor of plat vibration (Sction.) th strsss in th middl plan of th plat ar nglctd. This is a valid assumption for small vibrations of uniform plats. Howvr, for a plat rinforcd b ccntric stiffnrs or for a sstm of plats, built up to rprsnt approimatl a curvd or shll structur, bnding and mmbran, or in-plan, dformations ar coupld. In sction finit lmnts for in-plan and vibrations will b dvlopd. Th lmnts for inplan vibrations ar chosn, as th ar simplr in concpt.

19 Y Y V 3 V 4 U 3 U (5) (6) (9) V l l V U U Fig..7 Rctangular plat for inplan dformation X 3 (3) (4) (8) () () (7) 3 Fig..8 Plat consisting of four inplan dformation X Figur.7 shows a rctangular lmnt of sids l andl. Th middl surfac of th lmnt lis in th plan OXY. W considr a stat of plan strss; i.. th non-zro componnts of strss ar σ, σ andσ. As th strsss ar assumd to b uniform through th thicknss h, th strain nrg of th lmnt l l δ = h ( σ ε + σ ε + σ ε ) dd 00 (40) from quation (9). If u and ar th displacmnts in th X- and Y-dirctions, rspctivl, th strain nrg can b prssd in trms of displacmnts, using th strss-strain rlations () and th strain-displacmnt rlations (7) and (8), δ ( v) l l Åh u v u v u v = v v 0 0 dd

20 u l l u v u v v =,, D dd u v + (4) Th lmnt of Fig..7s has nods at th four cornrs; thus nodal variabls j =,, 3 and 4. Th assumd displacmnt functions ar u j ar v j and with u = a+ a + a 3 + a 4 v= a5 + a 6 + a 7 + a 8 (4) or u g 0 a v = 0 g whr g [ ] = and a is a vctor containing th cofficints a, a,... a 8. Substituting th nodal valus u u a u l a u 0 l a 3 u l l ll a v l 0 0 a6 v l 0 a 4 4 = = Na = v a5 3 7 v l 4 l ll a 8 (43) Thus a= Bu whr B= N As u = a + a4, v = a7 + a8

21 u v and + = a a a a u v Ga = u v + whr G = (44) Substituting from quation (44) in quation (4) δ = l l 0 0 T T agdgadd l l T T T = u B GDGdd Bu 0 0 = u K u T whr th lmnt stiffnss matri (45) l l T T K = B G DGddB 0 0 Th kintic nrg of th lmnt l l u v I = ρh + t t 0 0 dd u l l u v t = ρh dd t t v 0 0 t T = uu & & (46)

22 whr th lmnt mass matri l l T T g g 0 = B ρh ddb T 00 0 g g (47) Bfor considring th assmbl of lmnts to rprsnt th structur th conditions to b satisfid b th assumd displacmnt functions (4) will b discussd. Ths will b prsntd in gnral trms, so that th ar applicabl to othr tps of lmnt.. Displacmnts and thir drivativs up to th ordr on lss than that occurring in th strain nrg prssion should b continuous across lmnt boundaris.. Th displacmnt functions should b abl to rprsnt appropriat rigid-bod motions. 3. Th displacmnt functions should b abl to rprsnt stats of constant strain. If ths conditions ar satisfid, w hav conforming or compatibl lmnts; for ignvalu calculations with conforming lmnts rprsnting a structur th ignvalus convrg monotonicall from abov to th corrct valus with progrssiv subdivision of th lmnt msh. For non-conforming lmnts with condition (3) satisfid ignvalus convrg to th corrct valus vntuall, as th msh is rfind, but convrgnc is not monotonic. If condition (3) is not satisfid, th lmnts ar too stiff and ignvalus convrg on valus highr than th corrct ons. 3.4 Finit Elmnt thod for Transvrs Vibrations of Plats Figur (.9) shows a rctangular lmnt with sids of lngth l and l and having nods at th four cornrs. Th nodal variabls ar th transvrs displacmnt φ j ( wj ) and j( wj ) w j and th rotations ψ with j ; =,, 3 and 4. (For a right-hand st of as th Z-ais in Fig. (.9) is outwards from th plan of th diagram. Th displacmnt w is in th Z-dirction; φ and ψ ar positiv (i.. clockwis) rotations about th Y- and X-as, rspctivl. Ths dfinitions caus th ngativ sign in th rlation btwn φ and w

23 Y ( w3, φψ 3, 3) 3 4 ( w3, φψ 3, 3) (, w φψ, ) (, w φψ, ) X Fig..9 Rctangular plat for flur of a plat Th assumd displacmnt function is (48) w= a + a+ a+ a + a+ a + a + a + a + a + a + a = ga Whr g is a row matri of polnomial trms and th vctor a contains th twlv cofficints a i. Th trm in a nsurs rigid-bod translation; thos in a and a3 nsur rigidbod rotations; thos in a4 and a6 nsur stats of uniform curvatur; and that in a5 nsurs a stat of uniform twist. Along OX, th lin joining nods and in Fig..9, putting = 0 in quation (48), w= a + a+ a + a φ = a + a4+ 3a7 Thus th cofficints a, a, a4 and a7 ar uniqul dfind in trms of th four nodal valus w, φ, w and φ. As th lattr ar common to th two lmnts, for which OX is a common boundar, thr is continuit of w and across th intr-lmnt boundar. Also along OX, ψ = a + a + a + a

24 Substituting nodal valus of and in quation (48), w w w Na φ φ ψ ψ = = and a N w Bw = = (49) From quation (3) th strain nrg prssion for th lmnt can b writtn 0 0,, l l w w w w w C dd w δ = (50) whr ( ) v C D v v = From Equation (48) w w w a Ga = = (5)

25 Using Eqs. (49) and (5) in quation (50) δ = w K w T from Eqs. (48 whr th stiffnss matri (5) l l T T K = B GCGddB 00 Th kintic nrg of th lmnt l l w I = ρh dd t 00 w ) and (49) = gbw & t Thus T I = ww & & whr th lmnt mass matri (53) l l T T = B ρh g ddb 0 0 Th structur matrics ar assmbld in th mannr dscribd in Sction 5.4, noting that lmnt matrics ar now of ordr and thr ar thr dgrs of frdom pr nod. Eplicit prssions for matrics in books on finit lmnt mthod. For forcd vibration th matri quation K and can b found w&& + Kw = p (54) This is obtaind from th Lagrang quation. In quation (54) K and ar th stiffnss and mass matrics of th structur, obtaind b assmbling lmnt

26 matrics aftr convrting from local to global coordinats if ncssar and aftr liminating rows and columns associatd with zro nodal valus at boundaris. Th vctor w compriss all th dgrs of frdom of th constraind structur; th vctor p consists of th gnralizd forcs associatd with ach nodal variabl. Considring th contribution to p from a particular lmnt, p, and supposing that this lmnt is subjctd to a transvrs applid forc pr unit ara of p (, ) f( t ),application of th principl of virtual work givs. l l T p δw pft (, ) ( ) δw( dd, ) = 0 0 Whr δ w list th virtual incrmnts in th lmnt nodal valus and δ w (, ) is th virtual displacmnt a point (,). Using th Eqs. (48) and (49), l l T δ = () (, ) δ 0 0 p w f t pgb wdd and l l T p = f() t p( gdd, ) B 0 0 (55) With th aid of quation (55) th gnralizd forc vctor for th structur, p can b assmbld. Aftr solving quation (54), th strss rsultants (i.. th bnding momnts and th twisting momnt and, and hnc th strsss, for a particular lmnt can b dtrmind, as th strss rsultants can b prssd in trms of th vctor of nodal displacmnts w th rlation [from quations (7), (49) and (5)] = DGB (56)

27 Rfrnc matrials:. Dmtr G. Frtis chanical and Structural Vibration.. J. F. Hall - Finit Elmnt Analsis in Earthquak Enginring to th intrnational handbook of Earthquak Enginring and Enginring Sismolog part B, G. D.anolis and D.E. Bskos - Boundar Elmnt thod in Elastodnamics, 988.

AS 5850 Finite Element Analysis

AS 5850 Finite Element Analysis AS 5850 Finit Elmnt Analysis Two-Dimnsional Linar Elasticity Instructor Prof. IIT Madras Equations of Plan Elasticity - 1 displacmnt fild strain- displacmnt rlations (infinitsimal strain) in matrix form