ME 160 Introduction to Finite Element Method. Chapter 4 Finite Element Analysis in Stress Analysis of Elastic Solid Structures

Transcription

1 San Jose State Unerst Departent of Mechancal Engneerng ME Introducton to Fnte Eleent Method Chapter 4 Fnte Eleent Analss n Stress Analss of Elastc Sold Structures Instructor a-ran Hsu Professor

2 Part Introducton to Fundaentals of heor of near Elastct

3 What defnes elastc solds? A sold defors n response to eternal actons (e.g. forces heat etc.). he deforaton s copletel reersble eanng the sold returns to ts orgnal shape after the reoal of the eternal actons. wo pes of Elastc Solds pe : near elastct of aterals: hs tpe of elastct occurs to solds undergong sall deforatons such as sprngs that ehbt lnear relatonshps between the appled force (F) and the nduced elongaton () that can be represented b a atheatcal forula as: F k where k s a constant known as the rate or sprng constant. Man etallc aterals fall nto the categor of lnear elastc solds It can also be stated as a lnear relatonshp between stress (σ) and stran (ε) n stretchng or copressng a thn rod b he epresson: σ Eε where E Is known as the elastc odulus or Young's odulus pe : Nonlnear elastct of aterals: hs tpe of solds behaes as elastc aterals as descrbed aboe but can ehbt large deforatons such as rubbers and polers he FE forulaton presented n ths course wll be based on lnear elastct theor

4 near Elastc Behaor of Solds Fundaental assuptons () he ateral s treated as a contnuous edu (or a contnuu). In the words the ateral s hoogeneous wth no nternal defects or ods of sgnfcant szes () he ateral s sotropc eanng ts propertes are unfor n all drectons () he ateral has no eor (4) he ateral ehbts the sae propertes n tenson and copresson Defntons of stress and stran Unaal elongaton of a thn rod: A () Engneerng stran (e): It s defned wth reference to the orgnal shape of the sold. Matheatcall t s epressed as: Δ/ / - () rue stran (ε): It s easured on the bass of the edate proceedng length of the rod saple. Matheatcall t s epressed as: n ε n () Relatonshp between ε and e: ε ln(+e) d n F Δ (4) Engneerng stress (S): (5) rue stress (σ): σ P/A Insta tan eous load S Orgnal cross sectonal area P A () Relatonshp between S and σ: σ ( + e) S

5 Stress-Stran Cures of Materals Stretchng of thn rods of ost engneerng aterals wll ehbt the stress s. stran relatons llustrated n the fgure below: rue Stresses Strans Engneerng Desgnatons: A proportonal lt A elastc lt B eld pont neckng pont f rupture pont S Yeld strength of ateral S u ultate tensle strength of ateral pcal elastc deforaton of engneerng aterals: () Ver sall deforaton wth stran up to.% () Straght lnear relatonshp between the stress and stran resultng n a constant stffness of the. () he slope of lne OA s called the Young s odulus (E) representng the stffness of the ateral (4) Copletel recoerable stran (or deforaton) after the appled load s reoed (5) he eld stress (S o ) or σ s defned to be the ntercepton of ϵ offset.% of the σ-ϵ cure. It s a easure of aterals eceedng the elastc lt and undergong plastc deforaton (an rreersble deforaton)

6 Phscs of Deforable Solds subected to Eternal Forces Orgnal State: Phscal Consequences: RESPONDES o Alled Foreces: thngs wll happen: after appled forces:

7 he Man Coponents of Dsplaceents and Stresses: (Eleent) Dsplaceents: {U } : {U U U z } n a defored sold A sall eleent located at z: See detaled defntons on the net slde:

8 Induced Stress Coponents n Deforable Solds subected to Eternal Forces Sold subected to eternal forces: Result n: Because the forces appled to a general -D sold the nduced stresses are MUI-drectonal desgnated b σ αβ : σ agntude subscrpt α the as noral to the plane of acton subscrpt β the drecton the stress coponent ponts to So stress coponent σ stress coponent actng on the plane noral to the -as and pontng to the -drecton whereas stress coponent σ stress coponents actng on the face noral to the -as but ponts to the -drecton. In theor there are nne (9) stress coponents eerwhere nsde the sold: [ σ ] σ σ σ z σ σ σ z σ z σ z σ zz But n realt there are onl s () ndependent stress coponents wth: σ σ σ z σ z σ z σ z [ σ ] σ SYM σ σ σ z σ z σ zz σ σ σ σ 4 σ 5 σ For FE forulaton: σ σ σ σ σ { σ σ zz σ 4 σ } (4.) σ 5 σ z σ σ z

9 Induced Stresses n Deforable Solds subected to Eternal Forces Cont d A E -σ σ z D σ z σ σ zz σ z F σ z B σ σ z C σ G Stress coponents wth sae subscrpts such as: σ σ σ zz are called NORMA stress coponents wth unt: ps or Pascal (Pa) N/. Stress coponents wth dfferent subscrpts such as: σ σ z σ σ z σ z σ z are SHEARING stress coponents. Shear stress has the unt of change of angle fro the orgnal rght angle to the angle wth the stress.e (π/) θ. he unt s thus angle change n radans (rad) Effects of Noral and Shear Stresses to Sold Deforaton Noral stresses change sze: contracton σ Est NORMA to surfaces Shear stresses change shape: Est on the SURFACES -σ σ elongaton

10 Relatonshps between Prar unknown and Secondar Unknowns n FE analses Prar Unknown: Eleent dsplaceents: {U(z)} and Nodal dsplaceents: [N(z)]{u} n FE forulaton: {U} [N]{u} where {N(z)} Interpolaton functon Stran-dsplaceent relatons nkage to the st secondar unknown: Eleent strans: {ε(z)}[b(z)]{u(z)} correspondng to eleent stresses: {σ(z)} Generalzed Hook s aw Eleent strans and stresses and nodal dsplaceent are the solutons of FE analses. nd Secondar Unknown: Eleent stresses {σ(z)}[c]{ε}

11 Multaal Deforatons of Elastc Solds Unaal stress wth unaal deforaton: F ε Δ/ Stress: σ Stran: ε Δ Unaal stress wth baal deforaton wth Posson s rato: ateral contracton: - ε F ongtudnal stretchng: + ε F For solds wth sgnfcant cross-secton a unaal stress a produce baal deforaton such as llustrate n the fgure n the left. he lateral contracton deforaton s represented b: - ε Posson s rato defnes as: ateral contracton stran ν ongtudnal stretchng stran Posson rato ests n ost ult-aall loaded elastc solds and ths effect needs to be ncluded n all FE forulatons. hs effect also leads to the use of Generalze Hooke s aw for the forulatons ε ε

12 Generalzed Hooke s aw for Solds wth Mult-aal Stresses: Stress s. stran relatonshp for defored solds wth -D deforaton Elongaton n one-drecton causes contractons n other drectons and se ersa. otal strans n three drectons nduced b the three noral stresses n -drecton n -drecton n z-drecton n whch E Young s odulus and γ Posson s rato of the ateral he followng epresson epress the stresses n ters of strans the Hooke s law: σ σ σ zz E ( + ν )( ν ) ( ν ) ε + νε + νε z + ( ν ) ε + νε z + + ( ) νε ν ε z νε νε One a dere the unaal stress stuaton as a specal case fro the aboe epresson to obtan: σ Eε b substtutng: σ σ zz and ε -γε and ε z -γε n the generalzed Hooke s aw

13 Eleent Stran-Dsplaceent Relatons for FE Forulaton here are s () stran coponents correspondng to the stress coponents n nteror of the defored sold. hese stran coponents are related to the dsplaceents of the sold nduced b the eternal forces. Because deforaton of the sold CONINUOUSY arng throughout the sold the followng relatonshp ets: ( ) ( ) ( ) z U z U z U z z z z z z zz ε ε ε ε ε ε or: {ε(z)} [D]{U(z)} (4.) (4.) [ ] z z z D (4.4) Eleent Strans Correspondng to Eleent stresses Eleent Dsplaceents U(z) B theor of elastct: Eaple: For un-aal elongaton or contracton of a rod: Eleent dsplaceent {U} {U ()} he correspondng stran n eleent s: ( ) ( ) d du U ε

14 Eleent Stress-Stran Relatons the Generalzed Hooke s aw Accordng to generalzed Hooke s law for MUI-Aal stress state the followng relatonshp between the eleent stress and stran ests: ( )( ) + z z zz z z zz SYM E ε ε ε ε ε ε ν ν ν ν ν ν ν ν ν ν ν σ σ σ σ σ σ (4.5) where E Young s odulus and γ Posson s rato or: {σ} [C]{ε} (4.) where [C] s the elastct atr wth: [ ] ( )( ) + ν ν ν ν ν ν ν ν ν ν ν SYM E C (4.7) Eaple: For un-aall loaded rod: ( ) d du E E ε σ where U () s the dsplaceent n the rod

15 Phscs of Sold Deforaton b Eternal Forces Dsplaceents: {Φ(z)} : {Φ (zφ (zφ z (z} Sall eleent located at z: otal 9 stress coponents at (z) [ σ ] σ σ σ z σ σ σ z σ σ σ z z zz

16 Part FE Forulaton of Deforable Elastc Solds

17 Deraton of Eleent Equatons In Step 4 Chapter we dered the eleent equaton usng the Ralegh-Rtz ethod to take a for: We wll select tetrahedron eleents as the bass for FE forulaton for general -D sold structures [K e ]{q} {Q} (.7) n the tetbook where [K e ] Eleent atr {q} Vector of prar unknown quanttes at the nodes of the eleent {Q} Vector of eleent nodal actons (e.g. forces) Eleent equatons for each tetrahedron eleent n the FE odel for a structure are then assebled to establsh oerall stffness equaton for deternng nodal dsplaceents of all nodes n the structure. [K]{Φ} {R} FE Model for s Structure ade of etrahedron Eleents where [ K ] [ ] K e total nuber of eleents n the FE odel

18 Deraton of Eleent Equatons-Cont d Prncple of derng eleent equaton usng Ralegh-Rtz aratonal prncple Fro Chapter : et us deterne a sutable functonal to dere the eleent equaton. A general for of functonal: χ ( ) { } { φ }... d + g { } { φ φ φ φ } f... ds r s r and then appl the Varatonal prncple on: χ φ fro whch equatons of each eleent are dered: ( ) χ χ φ φ { χ} { χ} { χ} { φ}... φ φ φ he functonal for a defored sold subected to eternal forces s POENIA ENERGY (P) n the stuaton. he potental energ assocated wth a defored sold can be defned as: P U - W (4.8) olue s surface where U the stran energ n a defored sold and W the work done to the defored sold b eternal forces actng on the sold bod n the olue and surface of the sold

19 Deraton of Eleent Equatons-Cont d (a) (C) Stran energ n a defored sold: As we entoned earl n ths Chapter that a sold n (a) defors nto a new shape n (c) but not ndefntel. It stopped further deforaton after defored b certan aount. It reaches a new state of equlbru. WHY??? Iagne the followng phenoenon: Stretch a free-hung sprng b a weght W. he sprng wll elongate but onl b a fnte aount. (b) Ask ourself: WHY? Answer: the elongaton of the sprng deelops a resstance whch ncreases as the sprng elongates. he sprng ceases further elongaton when the resstance n the sprng balances the appled weght (force). We sa the sprng and the appled weght reaches a new state of equlbru whch stop the sprng (d) Fro further elongaton. Net: what wll happen to the sprng after the weght s reoed? You wll sa that the sprng returns to ts orgnal length but WHY?? Answer: because a for of ENERGY was stored n the stretched sprng when t s elongated. hs ENBERGY s released to restore Now ou know wh the sold n (a) ceases to defor further after the sprng to ts orgnal shape after the eternal force (the weght) the applcaton of the sste of eternal forces {p} has been appled was reoed. to the sold n (b). And ou would know that there s such ENERGY assocated wth the sold deforaton deeloped n the sold called SRAIN ENERGY. whch s responsble for restorng the sold to ts orgnal shape after the appled forces are reoed n EASIC solds. Matheatcal epresson of stran energ n State (c) s: U { ε} { σ }d (.9) tetbook

20 Deraton of Eleent Equatons-Cont d Potental energ n a defored sold subected to eternal forces: he potental energ n a defored sold s: P U - W (4.8) Stran energ: U σ σ zz d σ σ z σ σ z ( ε σ + ε σ + ε σ + ε σ + ε σ + ε σ )d { ε} { σ} d { ε ε ε zz ε ε z ε z} zz zz z z z z (4.9) Both the stran and stress coponents are functon of (z) and d (d)(d)(dz) the olue of gen ponts n the defored sold. Stran energ s a scalar quantt.

21 Work done to defor the sold: Defnton of work : Work (W) Force Dsplaceent (deforaton) wo knds of forces: () bod forces (unforl dstrbuted throughout the olue of the sold ()) e.g. the weght () surface tractons e.g. the pressure or concentrated forces actng on the boundar surface (s) Matheatcal epresson of work: { } { } ( ) { } { } ( ) ( ) ( ) { } ( ) ( ) ( ) { } ds t t t z z z d f f f z z z ds t z d f z W z s z z z s + + ) ( φ φ φ φ φ φ φ φ Deraton of Eleent Equatons cont d Potental energ n a defored sold subected to eternal forces: where {Φ(z)} the dsplaceent of the sold at (z) {f} bod forces and {t} the surface tractons and ds the part of the surface boundar on whch the surface tractons appl (4.)

22 Deraton of Eleent Equatons cont d Potental energ n a defored sold subected to eternal forces: So the potental energ stored n a defored sold s: P U W or: P U W ( ) ds { ε ( z) } { σ ( z) } d { φ( z) } { f } d { φ( z) } { t} + s (4.) Followng the Ralegh-Rtz Varatonal prncple the equlbru condton for the defored sold should satsf the followng condtons: ( ) { φ} P φ Fro whch equatons for each eleent a be dered fro: P φ ( φ) P( φ) P( φ) φ φ...

23 Deraton of Eleent Equatons cont d for FE esh of dscretzed solds What we had forulated was for contnuu solds. We wll now dere the EEMEN EQUAION for dscretzed solds n FE esh: We need to ake dstncton between the EEMEN quanttes and the NODA quanttes. he prar quantt n FE analss s DISPACEMENS. We need teo ake dstncton between the Eleent dsplaceents and the Nodal dsplaceents. he eleent dsplaceent s: {Φ(z)} wth three coponents: Φ (z) the eleent dsplaceent coponent along the -drecton Φ (z) the eleent dsplaceent coponent along the -drecton and Φ z (z) the eleent dsplaceent coponent along the z-drecton

24 Deraton of Eleent Equatons cont d for FE esh of dscretzed -D solds wth tetrahedron eleents We realze that ERAHEDRON and HEXAHEDRN eleents are used n FE odels for general -D sold structures. he tetrahedron eleents are the basc eleents for ths tpe of structures because heahedron eleents are ade up b 4 or ore tetrahedron eleents. Our FE forulaton for general -D sold structures wll thus be based on ERAHEDRON eleents We notce that tetrahedron eleents has four (4) assocate nodes: Φ Φ Φ and Φ 4 wth FIXED (specfed) COORDINAES. Each node has three () dsplaceent coponents too. hese nodal dsplaceent coponents are: { φ } { φ φ φ φ φ φ φ φ φ φ φ } z z z 4 4 φ4z where Φ Φ Φ z dsplaceents n Node n drectons; Φ Φ Φ z dsplaceents n Node n drectons; Φ Φ Φ z dsplaceents n Node n drectons; Φ 4 Φ 4 Φ 4z dsplaceents n Node 4 n drectons

25 Deraton of Eleent Equatons cont d for FE esh of dscretzed -D solds wth tetrahedron eleents We entoned preousl that the functonal that we wll use to dere the eleent equatons n FE forulaton of sold structures Is the potental functon n the sold as show below: P U W { ε ( z) } { σ ( z) } d ( { φ( z) } { f } d + { φ( z) } { t} ds) (4.) s Now because the EEMENS n dscretzed sold) Also these eleents are nterconnected b the NODES to sulate the orgnal sold structures. hs lnk requres the FE forulaton noles the functonal wth both the EEMEN and NODA quanttes n the forulaton. hs lnk s establshed usng the INERPOAION FUNCION that relates the eleent quanttes wth the correspondng nodal quanttes such s: ( z) ( z) ( z) he eleent dsplaceents: φ { φ( z) } φ φ z he nodal dsplaceents: [ N( z) ]{ φ} he INEERPOAION functon { φ } { φ φ φ φ φ φ φ φ φ φ φ } z z z 4 4 φ4z

26 Deraton of Eleent Equatons cont d Ke equatons to construct the functonal - the potental energ. he eleent dsplaceents s. nodal dsplaceents a Interpolaton functon: {Φ(z)} [N(z)] {Φ}. Eleent stran s. nodal dsplaceents: {ε(z)} [D]{Φ(z)} (4.) n whch [D] n Equaton (4.4) Hence {ε} [D][N(z)]{Φ} [B(z)]{Φ} (4.) wth [B(z)] [D][N(z)] (4.) pcal tetrahedron eleent for -D FE odels. Eleent stresses s. nodal dsplaceents: {σ} [C]{ε} (4.) n whch the elastct atr [C] n Equaton (4.7) Hence {σ} [C] [B(z)]{Φ} (4.4) 4. Stran energ wth nodal dsplaceents: U { ε} { σ }d Hence U ([ B( z) ]{ φ }) [ C] [ B( z){ φ} ] U (4.9) ( )d (4.5) { φ } [ B( z) ] [ C][ B( z) ]{ φ}d or (4.)

27 Deraton of Eleent Equatons cont d he functonal for Varatonal process ( ) { } ( ) { } ( ) { } { } ( ) { } { } ( ) + s ds t z d f z d z z W U P φ φ σ ε We entoned that the functonal for derng the eleent equatons for dscretzed sold structure s the POENIA ENERGY (P) as shown below: (4.) B substtutng the Stran energ epressed n Equaton (4.) nto the aboe equaton we get: { } [ ] [ ][ ]{ } { } [ ] { } { } [ ] { }ds t z N d f z N d z B C z B P s ) ( ) ( ) ( ) ( ) ( φ φ φ φ φ (4.7) { } { } [ ] [ ][ ] { } { } [ ] { } ( ) { } [ ] { } ds t z N d f z N d z B C z B P s ) ( ) ( ) ( ) ( ) ( φ φ φ φ φ φ Due to the fact that nodal dsplaceent {Φ} hae fed alue but not a functon og (z) so the can be factore Out of the ntegraton wth respect to (z). We thus hae the followng: (4.8)

28 Deraton of Eleent Equatons cont d Eleent equaton b Varatonal process All eleents n dscretzed solds subected to eternal force requre to satsf the followng condton that: ( ) { } φ φ φ φ P P P (4.9) B substtutng the potental energ P n Equaton (4.8) nto the aboe equaton: ( ) { } { } [ ] [ ][ ] { } { } [ ] { } ( ) { } [ ] { } ) ( ) ( ) ( ) ( ds t z N d f z N d z B C z B P s φ φ φ φ φ φ φ φ

29 Deraton of Eleent Equatons cont d Eleent equaton b Varatonal process he aboe araton results n: ( [ B( z) ] [ C][ B( z) ] d){ φ} ( [ ( )] { } ) N z f d [ N( z) ] { t} ds Upon ong the last two tes to the rght-hand sde: ( [ B( z) ] [ C][ B( z) ] d){ φ} ( [ N( z) ] { f } d) + [ N( z) ] { t} We a represent Equaton b the followng eleent equaton: where [ K ] [ B( z) ] [ C] B( z) [K e ] {Φ} {q} s ( ds) s (4.) (4.) [ ]d Eleent stffness atr (4.) e { φ} Nodal dsplaceent copponents { q} [ N( z) ] { f } d + [ N( z) ] { t} ds Nodal force atr {f} Bod forces {t} Surface tractons [N(z)] n Step Chapter [B(z)] n Equaton (4.) [C] n Equaton (4.7) s (4.)

30 Eaples of FE Stress Analss of Sold Structures NOE: In FE stress analss of sold structures t s custoer to represent the eleent dsplaceents b: and nodal dsplaceents b: {u}. { U ( z) } U U U z ( z) ( z) ( z) he relatonshp between the eleent dsplaceents and the nodal dsplaceents are: {U} [N(z)] {u} where [N(z)] the Interpolaton functon. It s a row atr for -D bar eleents rectangular atrces for - or -D eleents. Interpolaton functon enables the deternaton of the prar quanttes n the eleent wth specfed coordnate (z) wth the sae prar quanttes of the assocate nodes

31 Part Fnte Eleent Forulaton for One-densonal Bar eleents. Bar eleents subected to undrectonal load. russ bar eleents. Bea bendng eleents

32 One-Densonal Stress Analss of Bar Eleents Dere the nterpolaton functon usng general forulaton: Φ We wll need frst to dere the [h] atr n Equaton (.) b the followng coputatons: ( ) α α φ + Φ Fgure. near Interpolaton Functon of a Bar Eleent φ α + α We hae: φ α + α and Fro whch we wll hae φ φ { φ} [ A]{ a} α α wth the atr [A] to be: [ A] and A B followng Equaton (.5) we hae the nterpolaton functon of a bar eleent to be: he nerse atr of [A] s: [ A ] [ h] N( ) { R} [ h] { } {( ) ( + ) } For the present case wth and we hae the nterpolaton functon to be: N( ) (.7)

33 Dere the eleent equaton for a tpcal -D bar eleent: a bar eleent ade of one ateral wth Young s odulus E. he bar s subected two forces F and F as shown n the fgure below. Establsh the eleent equaton Node Node u u F F Because the bar s ade of one ateral and the appled forces are along the length of the bar. So the bar wll onl defor along the length of the bar e.g. the eleent dsplaceent U s a functon of coordnate onl: We thus hae the eleent dsplaceent to be U U(). he correspondng dsplaceents at the two nodes are: {u} {u u }. he relatonshp between the eleent dsplaceent and the correspondng nodal dsplaceents are: {U()} {N()} {u} (4.4) n whch {N()} s the nterpolaton functon for one-d bar eleent. It s a row atr wth two coluns We had dered ths nterpolaton functon before wth assupton that the eleent dsplaceent {U()} follows a lnear polnoal functon: U() α + α and found the nterpolaton functon usng ths lner functon to be In the sae for as n Equaton (.7) we dered usng general forulaton ethod: { N( ) } wth (4.5)

34 Because the bar eleent s subected to force along the -coordnate onl and both the nduced stress and stran are n the drecton of -coordnate onl. We thus obtan fro Equaton (4.4) and obtan: [ ] d d D Fro Equaton (4.) we thus hae the [B] atr to be: [ ] [ ]{ } { } ) ( ) ( d d N D B For the sae reason of beng unaal stress dstrbuton.e. σ Eε b unaal Hooke s law we hae the elastct atr [C] E the Young s odulus fro Equaton (4.7) wth Posson s rato γ. We thus has the eleent equaton of the defored bar epressed n the for: [K e ] {Φ} {q} (4.) wth [ ] [ ] [ ] ( ) [ ] ( ) [ ] ( ) [ ] { } ( ) { } { } ( ) ( ) ( ) ( ) ) ( ) ( EA EA d EA d EA Ad E Ad B E B d z B C z B K e (4.7)

35 Eaple 4- Show the eleent equaton for the bar eleent n the fgure and deterne dsplaceent at Node u u F Node Node he nodal forces {q} for the bar eleent are epressed as: {q} {-F F } We a thus epress the eleent equaton for the bar eleent to be: EA u u F F Because there s onl one bar eleent n the structure the oerall stffness equaton n Equaton (.8) s dentcal to that of eleent equaton n Equaton (4.4). Howeer adustents of the now oerall stffness equaton wth the three atrces n Equaton (4.4) would be requred as epressed: K K aa ba F K K ab bb q q a b R R a b where {q a } specfed (known) nodal quanttes; {R b } specfed (known) appled resultng actons fro whch we a obtan: {q b } [K bb ] - ({R b } [K ba ]{q a }) (.) (PRINCIPA REF) (a) he specfed nodal unknown {q a } n ths case s u and F the requred unknown {q b } s u We thus need no adustent of the Matrces n the oerall stffness equaton n (4.4). he requred unknown quantt u s obtaned fro Equaton (. REF) to be: EA EA u ( F ( )( ) ) F ( )

36 Eaple 4- Deforaton and stress n a copound bar ade of two dfferent aterals Use the FEM to deterne the dsplaceents at the ont of a copound road ade of copper and alunu nduced b a unaal force P N at of the end of the rod as shown n the Fgure A below: Fgure A Copound Rod subected to a Unaal Force he copound rod has a cross-sectonal area A 5 and the Young s odul E cu MPa and E al 9 MPa. Soluton: he stuaton shown n Fgure A ndcates that the rod s epected to elongate along the sae drecton n the -as as shown n Fgure B. Fgure B FE odel for a Copound Bar

37 Eaple 4- Cont d he FE odel n Fgure B ndcates the followng: () here are total nodes n the structure () Nodal coordnates: Node at ; Node at 95 and Node at () he length of Eleent 95 ; the length of Eleent 5 (4) Both eleents hae a cross-sectonal area: A A A 5 (5) Dsplaceents at the nodes are: {u} {u u u } wth u Our soluton begns wth deelopng the eleent equatons for both eleents n the FE odel: Eleent ade of copper: Coeffcent atr for Eleent : E A [ ] N / K e 95 Eleent equaton for Eleent : u u p p (a) where p and p are forces art Node and respectel

38 Eaple 4- Cont d Eleent equaton for Eleent : u u p p Eleent ade of alunu: Coeffcent atr for eleent : E A [ ] N / K e 5 Eleent equaton for Eleent : u u p p (b) Due to the fact that the present case noles WO eleents wth Node beng coon to both these eleent we need to asseble the coeffcent atrces b followng the establshed rule b sung up the alues of Node fro both eleent coeffcent atrces.

39 Eaple 4- Cont d We thus asseble the oerall stffness atr b addng [ Ke ] In Equaton (a) and [ K ] e n Equaton (b) n the followng wa: [ K ] ( ) (c) he nubers n boldface n Equaton (c) are those assocated wth Node. he oerall stffness equaton for the bar structure wth specfed loadng/boundar condtons s thus epressed n the followng parttoned atrces as: u 47.5 u 47.5 u p p p he two unknown nodal dsplaceents u and u a be obtaned b the followng equatons usng the parttons n Equaton (d): u 47.5 u resultng n the dsplaceent of the copound bar at the ont (Node ) to be u.94 and the dsplaceent at the free end u.4. he total elongaton of the rod s.98. (d)

40 Eaple 4- Cont d Stran n Eleents Now that we hae soled the dsplaceents at the nodes we a use the tran-dsplaceent relatons to deterne the nduced strans n both these eleents: { ε } { ε ε } where ε and ε are the strans n Eleent and respectel he stran-dsplaceent relatonshp s aalable fro the epresson: {ε} [B]{u} n Equaton (4.) n whch the atr {B} s: d d [ B( ) ] { N( ) } d d We hae: Node at ; Node at 95 and Node at leadng to: the length of Eleent 95 ; the length of Eleent 5. We a thus epress the [B()] for both eleents to be: [ B ] and [ ] B

41 Eaple 4- Cont d Fro the [B] atrces for both eleents we a copute the strans n Eleent and as follows: ε u u u % for eleent and ε u u u + 5 5u % for eleent Stresses n eleents We a use the Hooke s aw to deterne the stresses n each of these two eleents fro ther correspondng strans For eleent wth E cu MPa: σ [ C ] ε E.. MPa ε For eleent wth E al 9 MPa: σ cu [ C ] ε E 9.. MPa ε al 77

42 Fnte Eleent Forulaton of Elastc Sold Structures One-Densonal Bar Eleents for russ Structures

43 FE Forulaton of russ Eleents Usng -D Bar Eleents Coon russes

44 FE Forulaton of russ Eleents Usng -D Bar Eleents - Cont d he characterstcs of a truss eleent can be suarzed as follows (a quote fro Dr. Agarwal s lecture notes):. russ s a slender eber (length s uch larger than the cross-secton).. It s a two-force eber.e. t can onl support an aal load and cannot support a bendng load. Mebers are oned b pns (no translaton at the constraned node but free to rotate n an drecton). Beng a two-force eber the force actng n the eber s n the length-drecton onl.. he cross-sectonal densons and elastc propertes of each eber are constant along ts length. 4. he eleent a nterconnect n a -D or -D confguraton n space. We wll forulate -D confguraton onl. Meanng the truss eber defors n a plane defned b - coordnates. 5. he bar eleents for truss structures s echancall equalent to a sprng snce t has no stffness aganst appled loads ecept those actng along the as (the length drecton) of the eber.. Howeer unlke a sprng eleent dscussed n preous chapters a truss eleent can be orented n an drecton n a plane and the sae eleent s capable of tenson as well as copresson along ts longtudnal drectons.

45 FE Forulaton of russ Eleents Usng -D Bar Eleents Cont d Sple Brdge Structure: FE odel wth -D bar eleents wth desgnated eleent and node nubers. All eleents are nterconnected b frctonless hnges : Eleent () Node ο () ο ο () (4) (5) () (7) ο (8) () ο ο ο 4 5 (9) () 7 ο ο ο ο P P P

46 FE Forulaton of russ Eleents Usng -D Bar Eleents Cont d A fundaental phenoenon n Statcs s that hnged ont of an eber n A frae or truss cannot resst oent. Consequentl an eber n s truss As shown n the fgure at the left wth hnge onts (as llustrated n the lower Fgure wll hae forces actng along the length of the eber. Consequentl the deraton of the nterpolaton functon n the FE forulaton of these ebers a begn wth the bar eleent n Eaple 4- russ ebers along -drecton of -drecton onl: he or these eleent equatons for these eleents wll be n slar fors as for the -D bar eleents n Eaples 4- and 4-. he eleent equaton for those truss ebers that are nclned wth the -coordnate wth angle θ Howeer wll be n dfferent fors as wll be dered n the followng procedures.

47 FE Forulaton of Inclned russ Eleents Usng -D Bar Eleents Inclned truss bar eleent n - plane: Eleent () (5) (7) and (): F ( ) u he nclned truss bar eleents such as shown n the fgure at rght a hae WO dsplaceent coponents but these eleents can onl elongate or contract n the ONGIUDINA drecton onl. he sae apples to the nduced stran and stress. So hese are regarded as a specal bar eleents. F ( ) hese bar eleents rean hang two nodes: Node and Node ocated at specfed coordnates ( ) and ( ) respectel. Each node has two dsplaceent coponents: u and dsplaceent of Node n respecte - and - drectons u and dsplaceent of Node n respecte - and -drectons F and F are the forces actng at Node I and Node respectel u

48 FE Forulaton of russ Eleents Usng -D Bar Eleents Cont d Deraton of eleent equaton he stuaton n the bar eleent n truss structures s unque n the wa that these bar eleents are sold eleent of two-force ebers eanng that the force s actng along the length of the bar onl. Consequentl the nduced dsplaceents that ae effecte to the elongaton or contracton of the bar (truss) eber eleents are the ones n the length-drecton onl. he stran and stress n the truss ebers are along the length of the bar too. he dsplaceent coponents n an nclned bar eleent n the plane defned b the () coordnate sste n the fgure below do not contrbute n producng the dsplaceent stran and stress n the bar eleent for ths reason. Howeer ther equalence to the ones along the length drecton of the bar eleent do. Consequentl we need to conert the current stuaton n the nclnes bar eleent n the left to the equalent stuaton n the rght of the fgure through a coordnate transforaton process of the nodal dsplaceent coponents and the appled nodal forces Equalent pcal Bar Eleent

49 FE Forulaton of russ Eleents Usng -D Bar Eleents Cont d Deraton of eleent equaton Cont d ransforaton of Nodal dsplaceent coponents: russ bar eleent: Fro the global coordnates to ocal coordnates - he latter s used for the FE forulaton as wth -D bar eleents pcal bar eleent: Conertng truss bar eleent to equalent tpcal bar eleent B referrng to the aboe fgure we hae the followng relatonshp n transforng the nodal dsplaceent coponents fro he global coordnates () n the left fgure to the local coordnate () n the rght of the fgure: at Node : δ u cosθ + snθ cu + s (4.8a) at Node : δ u cos θ + snθ cu + s (4.8b) where c cosθ and s snθ Equatons (4.8ab) a be epressed n atr for: { δ } or n a shorthand erson of: {δ} []{u} wth the atr [] beng the transforaton atr δ c δ s c u s u (4.9a) (4.9b)

50 FE Forulaton of russ Eleents Usng -D Bar Eleents Cont d Deraton of eleent equaton Cont d ransforaton of Nodal force coponents: Fro global coordnates to ocal coordnates ransforaton of nodal forces can be done n a slar wa as we dd wth nodal dsplaceents. Howeer t would be easer to do t usng the work done to the eleent b these forces as work done s a scalar quantt whch s easer to transfor n space than the ector quanttes such as dsplaceents. Snce the work done to the eleent b nodal forces a be epressed as: W δ coordnate sstes or n a long-hand for: F F f W δ δ u u or δ f f F F { } { f } { u} { F} { } { } { } { } { u} { F} for the sae eleent n both local and global Substtutng {δ} []{u} n Equaton (b) nto the aboe epresson we get: ([]{u}]) {f} {u} {F} leadng to:{u} [] {f} {u}{f} We wll thus hae the nodal force transforaton b the followng relatonshp: [] {f} {F} (4.) where {f} nodal forces n tpcal bar eleent {F} nodal forces n nclned truss eleent

51 FE Forulaton of russ Eleents Usng -D Bar Eleents Cont d Deraton of eleent equaton Cont d Eleent equaton n ocal Coordnate Sste Now that we hae ade the real stuaton n the Global coordnates () to be equal to the stuaton n -D ocal coordnate () stuaton through a transforaton process: Inclned truss bar eleent: ransforaton Matr: Equalent bar eleent: c [ ] s c s Real stuaton n Global coordnates (4.a) Coordnate ransfor: Preousl dered nterpolaton functon: { N( ) } Preousl dered eleent equaton wth odfed notaton of nodal quanttes: or EA δ δ f f [ K e ]{ δ } { f } wth [ K e ] AE AE (4.4) AE AE (4.7)

52 FE Forulaton of russ Eleents Usng -D Bar Eleents Cont d Deraton of eleent equaton Cont d Eleent equaton n ocal Coordnate Sste EA [ K e ]{ δ } { f } wth [ K e ] δ δ f AE AE f AE AE (4.4) (4.7) Relatonshp of nodal forces between the two coordnate sstes: {F} n global coordnate sste [] {f} n ocal coordnate sste. he eleent equaton n global coordnate sste thus hae the for: [ K e ]{ δ} { f } he nodal forces n the ocal coordnate sste {f} can be ewed as a transfored forces {F} ro the global coordnate sstes wth the relatonshp: [] {f} {F} as shown n Equaton (4.7) or {f} ([] ) - leadng to: [K e ]{δ} ([] ) - {F} But we also hae alread dered the followng relatonshps: {δ} []{u} n Equaton (4/b) and [] {F} n Equaton (4.7). B substtutng these relatonshps nto the aboe epresson we wll get the followng reersed transforaton of eleent equaton fro the ocal coordnate sste to Global coordnate sstes: [K e ][]{u} ([] ) - {F} (4.) (4.)

53 FE Forulaton of russ Eleents Usng -D Bar Eleents Cont d Deraton of eleent equaton Cont d Eleent equaton n ocal Coordnate Sste he Eleent Equaton of nclned truss eleents s: [K e ][]{u} ([] ) - {F} (4.) We realze the fact that the transforaton atr: [] [] - (Reference No. ). hus Equaton (4.) has a new for of: [K e ][]{u} ([] - ) - {F} leads to: [K e ][]{u} []{F} wth ([] - ) - [] n the last epresson. Now f we ultpl both sdes b [] - and use the relatonshp of [] - [] we hae the eleent equaton of the nclned truss eleent (.e. the bar eleent n the global coordnate sste) to be: [] [K e ][]{u} {F} (4.4) AE where [ ] and [ ] K e AE AE AE c s c s wth A cross-sectonal area and the length of the truss eleent respectel ccosθ and ssnθ

54 FE Forulaton of russ Eleents Usng -D Bar Eleents Cont d Eleent equaton for Inclned russ Mebers he stffness atr: where [ ] K e { } u u [ K e ]{ u} { F} u and { F} AE c cs c cs cs s cs s c cs c cs F F F F cs s cs s (4.5) (4.a) (4.b) (4.7) ) hs stffness atr represents the stffness of a sngle eleent wth nclne angle θ ) It s setrc about the dagonal lne of the square atr ) Snce there 4 unknown nodal dsplaceents - eanng 4 degree-of-freedo (dof) the atr s of the sze of (44) 4) he ters c and s represent the cosne and sne alues of the orentaton of the eleent wth the horzontal plane rotated n a counter clockwse drecton (+e drecton)

55 Eaple 4- FE Stress Analss of a russ Structure: A truss wth ebers oned b frctonless hnges as shown n the fgure. Eleent and are ade of alunu and eleent s ade of steel. Materals Cross-sectonal Area (A) ( - ) Young s Modulus (E) ( N/ ) Yeld Strength (σ ) ( N/ ) Meber () Alunu 7 7 Meber () Alunu 7 7 Meber () Steel Soluton: We realze the fact that there are ebers each has ts own denson and ateral wth propertes lsted n the aboe table. So we a conenentl construct the FE odel of the truss structure b usng eleents wth node par - eleent wth node par - and eleent wth node par - as shown n the fgure. We wll dere the eleent equatons for all these eleents b usng the forulatons for truss eleents wth: Equaton (4-4) for the horzontal Eleent and Equatons (4.5) and (4.7) for the nclned eleents and.

56 Eaple 4- FE Stress Analss of a russ Structure - cont d Deraton of eleent equatons Eleent :. E 7 MPa A - : u () Node Node. u Beng horzontal we hae the nclne angle θ whch leads to: c cosθ c ssnθ s and cs We also calculate the coeffcent of the stffness atr of eleent as follows: ( 7 )( ) 5.84 N E A. Usng Equaton (4.7) the stffness atr for Eleent s thus: [ ] K e E A c cs c cs cs s cs s c cs c cs cs s cs s 5.84 (a)

57 Eaple 4- FE Stress Analss of a russ Structure - cont d Deraton of eleent equatons Eleent :.5 E 7 MPa A - : Node u In ths Eleent we hae the nclned angle θ9 whch lead to: 5 () c cos 9 o c ; s sn 9 o cos o s and cs and ( 7 )( ). N E A.5 Node u Hence the stffness atr for Eleent s: [ K ] e. (b)

58 Eaple 4- FE Stress Analss of a russ Structure - cont d Deraton of eleent equatons Eleent :. E MPa A - : Eleent has an nclned angle θ o leadng to: Node u c cos o.8 c.75 s sn o.5 s.5 and cs.4; Also the coeffcent of the stffness atr: Node u X ( )( ).7 N E A. he stffness atr for eleent s: [ ] K e EA c cs c cs cs s cs s c cs c cs cs s cs s (c)

59 Deraton of eleent equatons Eaple 4- FE Stress Analss of a russ Structure - cont d he Eleent equatons for the truss structure: [K e ] {Φ} {q} Fro Equaton (4.): wth a general epresson for eleent equatons we are now n the poston to epress the sae equatons for the eleents n the current truss structure as follows. Eleent () wth Node and f f f f u u Eleent () wth Node and f f f f u u.... Eleent () wth Node and f f f f u u n whch u u and u dsplaceent coponents n -drecton of Node and respectel and dsplaceent coponents n -drecton of Node and respectel f f and f Nodal force at Node and respectel f f and f Nodal force at Node and respectel

60 Eaple 4- FE Stress Analss of a russ Structure - cont d Assebl of eleent equatons for Oerall Stffness Equaton B followng the descrpton on assebl of eleent stffness atrces n Step 5 wth dagra of Chapter Steps n Fnte Eleent Analss we a asseble the three () truss eleent stffness atrces shown aboe n the followng for: or n a neat for as shown n the net slde:

61 Eaple 4- FE Stress Analss of a russ Structure - cont d Eleent stffness atr and Oerall Stffness Equaton of the russ Structure he oerall stffness atr of the truss structure: (d) Wth ths oerall stffness atr we a establsh the oerall stffness equaton for the truss structure as shown below: (e)

62 Boundar and oadng condtons of the russ Structure Eaple 4- FE Stress Analss of a russ Structure - cont d We recognze the followng specfed condtons for the dscretzed truss structures: he boundar condtons (dsplaceent at nodes): ) Wth Node beng copletel fed: u ) Node s allowed to oe n ertcal () drecton: u he loadng condtons: All ecept Node has one force actng n the -drecton: f f f f f and f.4 kn 4 N he oerall stffness equaton n Equaton (e) after the substtuton of the aboe specfed boundar and loadng condtons has the for: f f f f f f u u u

63 Eaple 4- FE Stress Analss of a russ Structure - cont d Oerall Structure Stffness Equaton wth gen Boundar and oadng Condtons for the russ Structure u Now f we follow Step n Chapter on Steps n FE Analss we a partton Equaton (f) n the followng wa: (f) where {q a } specfed (known) nodal quanttes; {R b } specfed (known) appled resultng actons fro whch we a obtan: {q b } [K bb ] - ({R b } [K ba ]{q a }) { } { } q a { q } b u { } { } R a { } R b [ K ] [ ] ba K bb

64 Eaple 4- FE Stress Analss of a russ Structure - cont d Soluton of unknown nodal dsplaceents n truss structure We wll get the unknown nodal dsplaceents fro the portoned oer stffness equaton b the epresson: {q b } [K bb ] - {R b }: u Sole for the unknown dsplaceent coponents at Node and : u u.8 - and wth negate sgns eanng downward drecton

65 Eaple 4- FE Stress Analss of a russ Structure - cont d Soluton of secondar unknowns of nduced strans and stresses n all three eleents n the truss structure u u ο ο X X We wll deterne the stran coponents and then stress coponents n each of the eleents n the truss structure. We should bear n nd that truss ebers are two force ebers n whch onl the n-lne dsplaceent coponents wll produce strans and stresses. Eleent : wth Node and Node We hae the relatonshp between the eleent dsplaceent U() and the correspondng nodal dsplaceent coponents {u u } along the lne of the bar eleent n the fgure I the left: u U ( ) [ N( ) ]{ u} u + u u where the nterpolaton functon of ths sple bar eleent s gen n Equaton (.5) tetbook We wll use Equaton (4.): the eleent stran {ε} [B()]{u} to copute the onl stran ε n the eleent b usng Equaton (4.): { ( ) } ε [ B( ) ]{ u} ε n whch the atr [B()] s gen n Equaton (4.) to be: [B()] [D][N()] wth [ D ] for the present one-densonal case. We thus hae: [ B( ) ] d d We thus hae the stran ε n eleent to be: d d

66 Eaple 4- FE Stress Analss of a russ Structure - cont d Soluton of secondar unknowns of nduced strans and stresses n all three eleents n the truss structure We wll deterne the stran coponents and then stress coponents n each of the eleents n the truss structure. We should bear n nd that truss ebers are two force ebers n whch onl the n-lne dsplaceent coponents wll produce strans and stresses. u Eleent : wth Node and Node n global coordnates. u ο ο X X [] wth θ o Eleent n local coordnate he transforaton atr: [ ] he relatonshp of nodal dsplaceents n ocal coordnate sste and the correspondng global coordnate sstes s: {δ} []{u} or: { δ} δ δ But we obtaned the nodal dsplaceents fro the preous calculatons to be: u u u u u u

67 Eaple 4- FE Stress Analss of a russ Structure - cont d Soluton of secondar unknowns of nduced strans and stresses n all three eleents n the truss structure δ { δ} δ We hae the stran n the eleent s obtaned b usng Equaton (4.): {ϵ} [B()]{Φ} and [B()] [D]{N()] (.) We thus hae: d [ B] [ D] { N( ) } d he stran n Eleent s thus: δ { ε } 49.4 / δ...8 he correspondng stress n Eleent can be obtaned fro Equaton (4.): {σ}[c]{ϵ}(7 )( ).4 MPa

68 Soluton of secondar unknowns of nduced strans and stresses n all three eleents n the truss structure - Cont d Eaple 4- FE Stress Analss of a russ Structure - cont d Eleent : wth Node and Node Node Node u u We hae coputed that u u and u et us transfor the coordnate sste n the lower left fgure to one-densonal bar eleent usng Equaton (4.9) to the followng stuaton: Node Node Node Node δ δ { } u u s c s c δ δ δ We hae : c cos 9 o and s sn 9 o. leadng to: { }.5 u u u u u δ δ δ he aboe epresson leads to: δ - - and δ or: { } δ δ δ We hae the stran n the eleent s obtaned b usng Equaton (4.): {ϵ} [B()]{Φ} and [B()] [D]{N()] (.) We thus hae: [ ] [ ] ( ) { } d d N D B he stran n Eleent s thus: { } / δ δ ε he correspondng stress n Eleent can be obtaned fro Equaton (4.): {σ}[c]{ϵ}(7 )(8 - )5. MPa

69 Eaple 4- FE Stress Analss of a russ Structure - cont d Soluton of secondar unknowns of nduced strans and stresses n all three eleents n the truss structure - Cont d Eleent : wth Node and Node We obtaned the nodal Dsplaceents fro he preous calculaton to be as shown Node u Node u.8 - ransforaton atr [] δ u δ u ο ο. he transforaton atr [] has the followng for wth θ o : cos sn.8 o [ ] cos o sn o.5.8.5

70 Eaple 4- FE Stress Analss of a russ Structure - cont d Soluton of secondar unknowns of nduced strans and stresses n all three eleents n the truss structure - Cont d Eleent : wth Node and Node he nodal dsplaceents n the equalent Eleent n the ocal coordnates can be related to the real Eleent n the global coordnates b the transforaton atr to be: u u.5 δ { δ} ( ) 8. δ u u But snce the nterpolaton functon of the equalent Eleent n the local coordnate s: { ( )} N (.7) And the atr [B()] [D][N(z)] fro Equaton (4.) n the present case to be: [ B( ) ] d d

71 Eaple 4- FE Stress Analss of a russ Structure - cont d Soluton of secondar unknowns of nduced strans and stresses n all three eleents n the truss structure - Cont d Eleent : wth Node and Node he stran {ε} n ters of nodal dsplaceents n Eleent a be obtaned b usng Equaton (4.): {ε}[d]{n()}: { ε} δ ε δ. (.8.7) he nduced stress n Eleent s {σ} can be coputed b usng Equaton (4.) or {σ} [C]{ε} wth [C] E Pa (N/ ) We thus hae the stress n eleent to be: {σ} σ E ε ( )( ) N/ or -.44 MPa (a copresse stress)

72 Fnte Eleent Forulaton of Elastc Sold Structures One-Densonal Bar Eleents for Bendng of Beas

73 Quck Reew of Sple Bea heor ) ke truss ebers beas are slender ebers (length s uch larger than the cross-secton) n structures. ) Howeer unlke the truss ebers bea ebers can resst forces appled laterall or transersel to ther aes. Orgnal straght bea: Induced deflecton b appled forces ) Beas can also resst the appled oents that cause the beas to bend.e. beas can hae deflecton nduced b appled oents (e.g. torque n the - plane) 4) Beas a be straght or cured. We wll focus on straght beas onl. 5) he applcaton of unlateral forces or oents wll not onl cause the bea to bend n shapes but also hang deflectons () perpendcular to the bea aes as shown n the aboe fgure. ) So the actons to bea can be: concentrated forces dstrbuted load and oents he nduced reactons are: Deflecton of the bea () bendng oents M() and shear forces V() and the bendng stresses (noral stress σ n () noral to the cross- secton of the bea and the shear stress σ s () - actng on the cross-secton of the bea) P concentrated force W() dstrbuted load per unt length Deflected state Induced deflecton: ()

74 Quck Reew of Sple Bea heor Cont d Constructon of nduced bendng oent and shear force dstrbutons (dagras) P Unfor dstrbuton load: w N/ A B A B R a Pb a b R b Pa R a w R b w Bendng Moent Dagra: M() V() pb Pb M ( ) Pab M Pa ( ) ( ) pa Shear Force Dagra: M() V() w w 8 w

75 Quck Reew of Sple Bea heor Cont d Induced Shear Stresses n Bent Beas b Appled Force here are two tpes of stresses nduced to the bea subected to eternal forces: () he noral stress along the -coordnate. It ests n perpendcular to the cross-secton of the bea (σ ) and () he shear stress (σ z ) that ests on the surfaces of the bea cross-secton. It also ests on the face along the -coordnate (σ ) wth the sae agntude Epresson for noral stress (σ ): M ( ) ( ) σ (4.8) I where M() bendng oent at locaton dstance fro center of the bea cross-secton to the depth of the cross-secton n drecton and I secton oent of nerta b For beas wth rectangular cross-secton: bh I z H For beas wth crcular cross-secton: I πd 4 4 Secton oent of nerta for beas of other cross-sectons ncludng I-cross-sectons are aalable fro echancal engneerng handbooks z d

76 Quck Reew of Sple Bea heor Cont d Induced Shear Stresses n Bent Beas b Appled Force Epresson for shearng (σ z ) or (σ ): hese stresses are nduced b the shear force V() at the arous cross-sectons along the bea da z c H ( ) c V ( ) da Q V σ Ib Ib (4.9) b where Qshear oent We hae ( ) ( ) V I For beas wth rectangular cross-sectons da bd resultng n: Q h 4 h b bd h σ wth ( ) b h 4 σ V at the center lne of the bea where a bh

77 Quck Reew of Sple Bea heor Cont d Euler-Bernoull heor of Bea Bendng hs theor relates the nduced deflecton (deforaton of beas) and the appled forces d d Orgnal straght bea: Deflecton and Bendng Moent M() and Shear force V() relatons: ( ) M ( ) EI Induced deflecton b appled forces P concentrated force w() dstrbuted load per unt length Deflected state Induced deflecton: () Induced Deflecton () a e obtaned b solng the 4 th order dfferental equaton: ( ) 4 d EI d ( ) 4 ( ) where E Young s odulus of the bea ateral I Secton oent of nerta Appled Forces d d M ( ) EI and V ( ) EI (4.4ab) d d Induced Deflecton () & M() and V()

78 A FE Forulaton of Bea Eleents Prncpal reference: A Frst Course n the FEM th Edton Cengage earnng b Darl ogan 7 Unfor dstrbuton load: w N/ Prar Quanttes In the eleent: he deflecton () B At the nodes: he deflectons and slope θ or as epressed as: { d} A () θ θ B at Node and Node Bea Eleent Node ρ ρ Radus of curature deflecton cure at Θ slope of deflecton cure at V() Deflecton at () Node Datu lne

79 FE Forulaton of Bea Eleents Prncpal reference: A Frst Course n the FEM th Edton Cengage earnng b Darl ogan 7 Bea eleents Contrar to the truss eleents bea eleents defors fro ts orgnal straght shape nto cured bent shape due to lateral (or transerse) forces or appled couples (oents). Actons and Induced Reactons n Bea Eleents Node M V Defored shape Orgnal shape θ() Node X X V M X Actons ateral forces V and V Induced Reactons n the eleent at Node at Node near dsplaceent near dsplaceent near dsplaceent () Moents M and M Rotaton Rotaton Rotaton θ() θ θ

80 FE Forulaton of Bea Eleents Cont d Prncpal reference: A Frst Course n the FEM th Edton Cengage earnng b Darl ogan 7 Dere Interpolaton Functon We assue the traerse dsplaceent of the bea eleent follows a lnear polnoal functon o the for: ( ) a + a + a a4 + (4.4) n whch a a a and a 4 are constant coeffcents B substtutng the coordnates of Node I and Node nto the assued eleent deflecton n Equaton (4.8) we get: At Node : ( ) 4 a θ d ( ) d a + a + a + a At Node : ( ) 4 θ d ( ) a + a + d a a

81 FE Forulaton of Bea Eleents Cont d Prncpal reference: A Frst Course n the FEM th Edton Cengage earnng b Darl ogan 7 Dere Interpolaton Functon We thus hae: () [N()]{d} We a thus epress the eleent deflecton () n ters of the nodal deflectons b substtutng the 4 constant coeffcents nto Equaton (.8) and after re-arranged ters to eld: ( ) ( ) + ( θ + θ ) + ( ) ( θ + θ ) + θ + he aboe epresson can also be epressed as: ( ) { N N N N } θ θ θ (4.4) θ N θ + (4.4) where N ( + ) ( ) ( + ) N θ ( ) N wth Interpolaton functon: ( ) [ ] { N N N N } N (4.44) θ θ he eleent deflectons he nodal deflectons

82 FE Forulaton of Bea Eleents Cont d Prncpal reference: A Frst Course n the FEM th Edton Cengage earnng b Darl ogan 7 Dere Eleent Stran ε Dsplaceent () Relaton We realze the fact that the rotaton (or the slope) of the deflected bea s: d( ) An epanded ew of a defored bea n -drecton: Contracted top edge θ for d -u Undefored neutral as Rotaton of the bea secton at Stretched botto edge X d d( ) We a fnd fro the aboe dagra of epanded bea that the stretch u can be obtaned b: u( ) θ d But fro theor of elastct: or ε ( ) ( ) d ( ) du d d ε (4.45) d d d d

83 FE Forulaton of Bea Eleents Cont d Prncpal reference: A Frst Course n the FEM th Edton Cengage earnng b Darl ogan 7 Dere Eleent Stress σ stran ε Relaton M ( ) We wll hae the noral stress σ Eε (4.8) I and the shear stress: ( ) σ V ( ) Q Ib (4.9) wth Q to be the shear oent he relatonshps between the bendng oent M() and Shear force V() are epressed n Equaton (4.4a) and (4.4b) respectel: ( ) d M ( ) EI and V ( ) d EI d d ( )

84 FE Forulaton of Bea Eleents Cont d Prncpal reference: A Frst Course n the FEM th Edton Cengage earnng b Darl ogan 7 Dere the Eleent Stffness Equatons We a epress the appled actons to the bea eleent n appled forces f f and oents n ters of the eleent deflectons V() represented b the assued polnoal functon n Equaton (4. 4) to be: Nodal forces: d f V EI d f he bea eleent: d V EI d Node f Defored shape Orgnal shape θ() Node X X ( ) EI ( + θ + θ ) ( ) EI ( θ + θ ) Nodal oents: M M d EI d ( ) -f ( ) EI ( + 4 θ + θ ) d EI d ( ) EI ( + θ + 4 θ ) X

85 he aboe epressons a e epress n the followng atr for for the eleent stffness equatons: f f EI 4 4 θ θ Dere the Eleent Stffness Equatons FE Forulaton of Bea Eleents Cont d Prncpal reference: A Frst Course n the FEM th Edton Cengage earnng b Darl ogan 7 (4.4) We thus hae the stffness equaton of a bea eleent to be: [ ] 4 4 EI k e (4.47) Nodal unknown quanttes(transerse dsplaceents & rotatons Stffness atr [Ke] Appled nodal transerse forces & oents

86 FE Forulaton of Bea Eleents Cont d Prncpal reference: A Frst Course n the FEM th Edton Cengage earnng b Darl ogan 7 Eaple 4 Deterne the deflecton of a cantleer bea at the half span and at the pont under the load. Densons and appled lateral force are shown n the fgure. he bea s ade of a ateral wth Young s odulus E MPa l/.5 Node Node l Bea Cross-secton: b c h c P N Node Soluton: We realze the fact that we are seekng solutons at the d-span and at the pont under the appled force. It d reasonable to dscretze the bea nto to two () eleents as shown below: Eleent : Node Node θ.5 θ Eleent : Node Node θ.5 θ I Secton oent of nerta (I) s: bh ( )( ) EI ( )(.7-8 ).7 We wll frst dere the eleent equatons gen n Equaton (4.4) wth eleent stffness atrces shown n Equaton (4.47) for both Eleent and.

87 FE Forulaton of Bea Eleents Cont d Eaple 4 [ ] EI k e For Eleent wth.5 : Eleent equaton for Eleent s: f f θ θ n whch θ θ are the respecte deflectons and rotatons n Node and Node respectel whereas f and f are the appled lateral forces and oents at Node and Node respectel (a) (b)

88 FE Forulaton of Bea Eleents Cont d Eaple 4 Due to the fact that Eleent has the sae length and s ade of the sae ateral as Eleent wth the onl dfference of the assocated nodes we can epress the sae eleent equaton for Eleent for eleent as shown below: f f θ θ (c) Assebl of eleent equatons n (b) and (c) for Oerall stffness equatons: Because Node happens to be coon node shared b Eleent and we need to asseble the eleent equaton followng the MAP stpulated n Step 5 n Chapter : [ ] [ ] [ ] + K e K e K Su Node Appled loads at Node n Eleent and should be sued up too n the load atr

89 FE Forulaton of Bea Eleents Cont d Eaple 4 We thus hae the assebled oerall stffness equatons as: ( ) ( ) ( ) ( ) f f f θ θ θ f f f θ θ θ he assebled oerall stffness equaton for the bea structure s: (d) We a sole the s unknown responses at the three nodes n the bea structure fro Equaton (d)

90 FE Forulaton of Bea Eleents Cont d Eaple 4 We realze that the followng boundar and appled loadng condtons appl: and θ for hang Node beng fed at the bult-n end and f f and. the onl non-zero load s the appled force f - N at Node We thus hae the oerall stffness equaton of the bea structure wth s[ecfed boundar and loadng condtons take the for: θ θ (e) he four unknown at Node and can be soled fro Equaton (e) b parttonng the aboe atr equatons followng Step n Chapter.

91 FE Forulaton of Bea Eleents Cont d Eaple θ θ b a b a bb ba ab aa R R q q K K K K In the for of: where {q a } specfed (known) nodal quanttes; {R b } specfed (known) appled resultng actons fro whch we a obtan: {q b } [K bb ] - ({R b } [K ba ]{q a }) (. REF) In the present case we hae {q a } { } We thus hae the 4 unknowns obtaned b: {q b } [K bb ] - {R b }

92 FE Forulaton of Bea Eleents Cont d Eaple θ θ {q b } [K bb ] - {R b }: { } θ θ q b [ ].5 4 bb K { } b R One wll fnd that: [ ] K bb

93 FE Forulaton of Bea Eleents Cont d Eaple 4 Gen condtons: θ We a thus sole for the 4 prar unknown quanttes n {q b } fro the followng equatons: θ θ Fro whch we obtan the followng solutons: ( ). ( ).4 ( ) ( ) Deflectons at Node c at Node c Slope at Node θ -.4 rad at Node θ rad. Check wth solutons fro classcal bea theor: P l he nduced deflecton n the cantleer bea b the appled force P N s A X P ( ) ( + ) l/ EI One a fnd the deflecton at Pont A at (equalent to Node ) () X and the deflecton at Pont B at l/ (equalent to Node ) -.. Both these alues full agree wth the solutons we obtaned fro the FE analss.

94 Part 4 Fnte eleent Forulaton for wo-densonal Stress Analss of Solds wth Plate Eleents

95 he Need for Stress Analss of Solds of Plane Geoetr here are an achne coponents that nole the geoetr that preent engneers usng tradtonal ethods to conduct stress Analss n ther desgn process. For eaple perforated tapered plates are coonl use wth the holes for bolted onts wth Other parts of the achne. he fgure on the left below shows a tapered perforated plate and the perforated plate n the rght also nole the plate wth two cure notches. Conentonal wsdo shows that serous stress concentraton can occur n the cnt of a structure wth drastc geoetr changes. he parts near the holes and the tapered areas and the notches n both plates are ulnerable for stress concentratons. We hae entoned that the proble of plate wth a sall hole a be soled b the theor of adanced strength of aterals. Howeer plates wth tapered and notched edges as shown aboe can not be handled b an estng ethod fro classcal theores of elastct and the FEM appears the onl aalable ethod of the soluton.

96 Eaples of FE Stress Analss of Sold Structures wo-densonal Stress Analss of Solds wth Plate Eleents Cont d hs tpe of structures are tpcall thn n ther thckness n coparson to the bulk olue of the oerall structure. As such onl three () out of total s () ndependent stress coponents need to be consdered n the analss as llustrated below: Plane structure subect to n-plane forces: Induced stresses: ο Note: Shear stresses est on the thckness or edges. he n-plane dsplaceent coponents n the sold: { U ( ) } u( ) ( ) u() the coponent along the -coordnate V() the coponent along the -coordnate he stress coponents n the sold: {σ()} {σ ( σ () σ ()} he stran coponents n the sold: {ε()} {ε () ε () ε ()}

97 Eaples of FE Stress Analss of Sold Structures he stran dsplaceent relaton n Equaton (4.) s odfed to the new for: wo-densonal Stress Analss of Solds wth Plate Eleents Cont d ( ) { } ( ) ( ) ( ) ( ) ( ) + u u u ) ( ε (4.48) and the stress-stran relaton n Equaton (4.5) becoes: { } E ε ε ε ν ν ν ν σ ) ( (4.49)

98 Forulaton FE analss for wo-densonal Stress Analss of Solds wth Plate Eleents et us now forulatng the FE for plate structures such as wth the dscretzaton n a tapered plate as llustrated In the fgure below: We begn our FE forulaton of the plate structure wth the epresson of the eleent dsplaceent coponents {U()} n ters of the correspondng nodal dsplaceents usng an nterpolaton functon {N()} as follows: ( ) ( ) u { U ( ) } { N( ) }{ u} (4.) u Eleent dsplaceents Interpolaton Nodal dsplaceents u functon u u (4.) where the nodal dsplaceent coponents { }

99 Forulaton FE analss for wo-densonal Stress Analss of Solds wth Plate Eleents Cont d Deraton of nterpolaton functon {N()} We assue the eleents used n ths FE analss are the sple eleents eanng that the Eleent dsplaceents follow lnear polnoal functons n relatng ther nodal dsplaceents: and We wll thus hae: { U ( ) } u ( ) α + α + α ( ) α + α + α u along the -coordnate 4 5 ( ) ( ) or n an alternate atr for: { U ( ) } [ R( ) ]{ α} along the -coordnate { α α α α α α } 4 5 (4.8) where α α α α 4 α 5 α n the atr {α} are constants to be deterned wth specfed nodal coordnates later. he atr [R()] n Equaton (4.8) has the for: [ R( ) ] (4.9)

100 Forulaton FE analss for wo-densonal Stress Analss of Solds wth Plate Eleents Cont d Wth the specfed nodal coordnates: u α + α + α u α + α + α u α + α + α and α 4 + α 5 + α α 4 + α 5 + α α 4 + α 5 + α We are able to epand Equaton (4.8) nto the followng for for the nodal dsplaceents: { } 5 4 α α α α α α u u u u (4.) Dsplaceent coponents of nodes n the eleent Specfed coordnates of the nodes n the eleent Constant coeffcents Deraton of nterpolaton functon {N()} - cont d

101 Forulaton FE analss for wo- Densonal Stress Analss of Solds wth Plate Eleents Cont d Equaton (4.) a be epressed n a dfferent for of: {u} [A]{α} (4.) n whch the atr [A] has the for: [ ] A (4.) We a obtan the soluton of the unknown coeffcent atr n Equaton (4.) as: { } [ ] { } [ ]{ } u h u A α (4.) Deraton of nterpolaton functon {N()} - cont d wth [A] - the nerse of the nodal coordnate atr [A] n Equaton (4.) where the atr: [h] [A] - (4.)

102 Forulaton FE analss for wo- Densonal Stress Analss of Solds wth Plate Eleents Cont d Deraton of nterpolaton functon {N()} - cont d We recall Equaton (4.8) wth: { U ( ) } u ( ) ( ) { α α α α α α } 4 5 (4.8) Also wth Equaton (4.9) wth: [ R( ) ] (4.9) We wll obtan the followng epresson after substtutng the atrc {α} n Equaton (4.) nto Equaton (4.8) eldng: { U ( ) } [ R( ) ][ h]{ u} (4.5) Dsplaceent coponents n Eleent Dsplaceent coponents of nodes n the eleent B coparng Equaton (4.5) wth Equaton (4.) we hae the nterpolaton functon of ths sple eleent to be: wth Matrces [R()] n Equaton (4.9) and [h] n Equaton (4.4) [N()] [R()][h] (4.)

103 Forulaton FE analss for wo- Densonal Stress Analss of Solds wth Plate Eleents Cont d he nterpolaton functon {N()} - cont d ( ) ( ) ( ) ( ) [ ] A N + + ( ) ( ) ( ) ( ) [ ] A N + + ( ) ( ) ( ) ( ) [ ] A N + + ( ) ( ) ( ) ) ( trangle ade of the eleent the area of A + + where (.) We a refer to the nterpolaton functon {N()} that we dered for the -node plate eleents n Chapter for a coparable case wth the current analss wth the for of: ( ) { } ( ) ( ) ( ) { }{ } ( ) ( ) ( ) { } u u u N N N N N N u u u N N N u N u U (4. a) (.a) ( ) { } ( ) ( ) ( ) { }{ } u u u N N N N N N u N u U An alternate epresson: (4. b)

104 Forulaton FE analss for wo- Densonal Stress Analss of Solds wth Plate Eleents Cont d Deraton of Eleent equaton Because the eleent equaton s dered b nzng the Potental energ n gthe defored sold we need to dere the epresson of stran energ n ters of nodal dsplaceents (the prar quanttes n the analss). We wll frst epress the eleent stran s. nodal dsplaceents b Equaton (4.) as: {ϵ(.)} [B]{u} (4.) [B(z)] [D][N(z)] (4.) where B usng Equaton (4.4) for [D] and Equaton (4.b) for [N()} we get the atr [B()] n the followng: [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) A B ) ( (4.) he potental energ n the defored eleent accordng to Equaton (4.8) s: { } ( ) { } [ ] [ ][ ]{ } { } ( ) [ ] { } { } ( ) [ ] { }ds t N u d f N u d u B C B u u P s (4.)

105 Forulaton FE analss for wo- Densonal Stress Analss of Solds wth Plate Eleents Cont d Deraton of Eleent equaton where he eleent equaton s obtan b nzng the potental energ n Equaton (4.8) as: P ({ u} ) { u} eadng to the followng eleent equaton: [ ]{ u} { p} [ K ] Eleent stffness atr [ B] [ C][ B]d e and the nodal force atr K e (4.) { p} Nodal forcwe atr [ N( ) ] { f } d [ N( ) ] { t}ds s (4.) + (4.) he ntegraton n Equaton (4.4) wth respect to the olue of the eleent a turn out to be tedous. Howeer f the sze of the eleent s not too large ths ntegraton a be approated b the followng epresson wthout sgnfcant error: [K e ] [B] [C][B] (wa) (4.) n whch w s the thckness of the plane eleent and A s the plane area that can be coputed b Equaton (.a). A ( ) + ( ) + ( ) the area of the eleent ade of trangle ()

106 Nuercal eaple on FE analss for wo-densonal Stress Analss of Solds wth a Plate Eleent Eaple (Eaple 7. fro the cted reference): A structure ade of a trangular plate defned b three corners at AB and C. A force P s appled at corner C as shownn the fgure. Fnd the followng: (a) he dsplaceent of the plate at corner C (b) he dsplaceent n the plate (c) he stresses and strans n the plate and (d) he reactons at the two fed corners Soluton: We assue that onl ONE eleent s used for the analss. hs trangular plate eleent has three nodes I and located as shown at the rght: he coordnates of the nodes are: { ) ( ) and ( 4)

107 Nuercal eaple on FE analss for wo-densonal Stress Analss of Solds wth a Plate Eleent Cont d (a) o deterne the nodal dsplaceents: We wll frst obtan the [A] atr usng Equaton (4.) wth the specfed nodal coordnates: and wth Equaton (4.4) to obtan the [h] atr: And then usng Equaton (4.) to obtan the nterpolaton functon n the for of Equaton (4.b) as: [ N( ) ] (.7.5) (.7.5).5 (.7.5) (.7.5).5

108 Nuercal eaple on FE analss for wo-densonal Stress Analss of Solds wth a Plate Eleent Cont d We are now read to dere the eleent atr for the structure wth the newl dered nterpolaton functon. We wll obtan frst the [B] atr fro usng Equaton (4.7) as: [ B] and the [C] atr FROM Equaton (4.7) and the coeffcent atr of Equaton (4.5): [ C].99. We are now read to deterne the eleent stffness atr n Equaton (4.9) fro Equatons (4.4) and (4.4) as: [ ] K e SYM

109 Nuercal eaple on FE analss for wo-densonal Stress Analss of Solds wth a Plate Eleent Cont d Constructng the eleent equaton: We realze the followng boundar and appled force condtons: u (fed ends) and the appled nodal forces: p p p p and p p cos o 8 lb and p p sn o -5 lb he eleent equaton accordng to Equaton (4.9) becoes: SYM u p 7. u p u p 8 8 p 8 p p 5 Because there s onl one eleent n the structure the aboe eleent equaton s also the oerall stffness equaton of the structure fro whch we a sole the dsplaceent coponents of A nodes after Makng necessar nterchange of rows and coluns n the aboe equaton (accordng to the rule stpulatedn Step 5 n Chapter.

110 Nuercal eaple on FE analss for wo-densonal Stress Analss of Solds wth a Plate Eleent Cont d hus after the necessar nterchanges of rows and coluns we reached the partton of the eleent equaton In the for of Equaton (. Ref) as: u u 8 5 he nonzero nodal dsplaceents can be sole fro the aboe portoned oerall stffness equaton b the followng sultaneous equatons: u u 8 5 he aboe sultaneous equatons a be soled b ether atr nerson ethod or Gaussan elnaton ethod wth: u. - nch u.8 - nch and nch

111 Nuercal eaple on FE analss for wo-densonal Stress Analss of Solds wth a Plate Eleent Cont d (b) he dsplaceents n the eleent: We a use the nterpolaton functon {N()} to deterne the dsplaceent coponents eerwhere n the eleent: For the dsplaceent n the -drecton: u() ( -.7.5)u + (.7 -.5)u +.5 u +(.7-.5) V() ( -.7.5) + (.7 -.5) P For nstance the dsplaceents at Pont P() hae the alues of: u() +(.7-.5) nch () nch (c) he stran coponents n the eleent b usng Equaton (4.) wth {ε} [B]{u}: ( ) { } ( ) ( ) ( ) u u u ε ε ε ε

112 Nuercal eaple on FE analss for wo-densonal Stress Analss of Solds wth a Plate Eleent Cont d he stresses n the eleent a be obtaned b usng the generalzed Hooke s aw n Equaton (4.5): { } ps E ) ( ε ε ε ν ν ν ν σ (d) he reactons at all nodes: One a dere the followng epresson for the nodal forces: { } [ ] { } [ ] { }( ) area plane A and plte the thckness of wthw WA B d B R σ σ { } R R R R R R R he reactons at Node therefore hae nuercal alues at: R 8 lb towards left and R 7. lb n the downward drecton

113 Nuercal eaple on FE analss for wo-densonal Stress Analss of Solds wth a Plate Eleent Cont d Iportant lesson learned fro ths nuercal eaple: We notced that the stresses and strans n the eleent (and thus the trangular plate structure re CONSAN: ε ( ).5 σ.5 ( ) ε ( ). and σ ( ) σ.7 ( ) 75 ε σ 88. { } ε { } ps eanng there s no araton of stresses and strans throughout the entre structure. hs s obousl not realstc!! he reason for what has happened n ths (and the other)nuercal eaple s because we used lnear polnoal n derng the nterpolaton functon resultng n usng SIMPEX eleent n the FE analss. Sple eleents offers sple atheatcal forulaton n FEA but results n constant stresses and strans n eleents. hat was the reason wh engneers need to place an ore (saller) eleents n the area wth conceable hgh gradents of prar unknown quanttes such as n the followng cases: hs s good lesson for an ntellgent FE user to learn and eercse