C PLANE ELASTICITY PROBLEM FORMULATIONS
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1 C M.. Tamn, CSMLab, UTM
2 Corse Content: A ITRODUCTIO AD OVERVIEW mercal method and Compter-Aded Engneerng; Phscal problems; Mathematcal models; Fnte element method. B REVIEW OF -D FORMULATIOS Elements and nodes, natral coordnates, nterpolaton fncton, bar elements, consttte eatons, stffness matr, bondar condtons, appled loads, theor of mnmm potental energ; Plane trss elements; Eamples. C Constant-stran tranglar (CST) elements; Plane stress, plane stran; Asmmetrc elements; Stress calclatons; Programmng strctre; mercal eamples. M.. Tamn, CSMLab, UTM
3 Fnte Element n Elastct Task: To dere the element stffness matr for a -node tranglar element nder plane stress / plane stran condton n elastct. Rerements: Consttte eatons (Generalzed stress-stran relatons) For aratonal approach - Eamne the fnctonal (based on potental energ) Mnmze the fnctonal - eld fnte element eatons: [k]{} {f} M.. Tamn, CSMLab, UTM
4 Stress tensor σ Hook's law for an sotropc materal ndergong nfntesmal deformaton. j σ τ τ z τ σ τ z τ τ σ z z zz where τ τ τ z z τ τ τ z z σ ε j ν ν σ j δ j σ Ε Ε kronecker delta : δ j kk f f j j τz τ z τ σ Stran tensor ε j ε γ γ z γ ε γ z γ z γ z ε zz where γ γ γ z z z γ γ γ z z M.. Tamn, CSMLab, UTM 4
5 Generalzed Internal States τ τz σ σ ε ε σ σ { } ( ) ( ) ( ) z w z z,,,,,, Dsplacement components M.. Tamn, CSMLab, UTM 5 τ z z { } z z z γ γ γ ε ε ε stran components { } z z z τ τ τ σ σ σ stress components w z w z z z z,, z w,, : deformato n For small γ γ γ ε ε ε
6 Generalzed Hooke s Law Ε z z ε ε ε ν ν ν ν ν ν ν ν ν ν σ σ σ M.. Tamn, CSMLab, UTM 6 ( )( ) Ε z z z z z z γ γ γ ε ν ν ν ν τ τ τ σ { } [ ]{ } [ ] matr or elastct materal s a where D D ε σ
7 -D Problems T z Plane Stress Plane Stran z ( τ ), τ ( ) σ z, τ z z τ z ε zz z ( γ ), γ ( ), γ z z γ z z ε ε ε γ zz on-zero stran components Ε Ε Ε ( σ ν ( σ σ ) ( σ ν ( σ σ )) ( σ ν ( σ σ ) zz ν τ Ε zz zz M.. Tamn, CSMLab, UTM 7
8 Stran Stress { σ} Plane Stress and Plane Stran ε γ { ε } ε σ σ τ [ ] Hooke s Law [ D] { σ} [ D]{ ε} Elastct matr E E D ( )( ) Plane stress Plane stran M.. Tamn, CSMLab, UTM 8
9 General Loadng Tpes Tracton force, T(/m) Dsplacement (,) Fed bondar (,) Concentrated force () (P, P ) Bod force (/m ) (f b, f b ) M.. Tamn, CSMLab, UTM 9
10 Fnte Element Dscretzaton Dsplacement components at node j Q j- s -dsplacement Q j s -dsplacement Use smaller-szed elements to mproe accrac M.. Tamn, CSMLab, UTM
11 Formlaton of Tranglar Element () Local node nmbers are assgned n conter-clockwse drecton. M.. Tamn, CSMLab, UTM { } 6 5 4
12 6 5 4 α α α α α α Assme a lnear araton of dsplacement feld wthn the element Formlaton of Tranglar Element () M.. Tamn, CSMLab, UTM : : node: α α α α α α α α α α α α α α α α α α
13 ( ) ( ) ( ) c c c b b b a a a where,, a a a ( ) ( ) ( ) c c c b b b a a a Formlaton of Tranglar Element () M.. Tamn, CSMLab, UTM c, c, b, b, c b area of element ( ),,, c b a s are nterpolaton fnctons.
14 Dsplacement Feld η ξ η ξ Alternate dsplacement fnctons M.. Tamn, CSMLab, UTM 4 { } [ ]{ } { } ξ η ( ) ( ) ( ) ( ) η ξ η ξ ( ),,, c b a
15 Shape Fnctons ξ η ξ η M.. Tamn, CSMLab, UTM 5
16 Area Coordnate Representaton The shape fnctons can be phscall represented b area coordnates, A ; A A ; A A A where A s the area of the tranglar element,.e. A A A A M.. Tamn, CSMLab, UTM
17 Isoparametrc Representaton When the same shape fnctons (, and ) are also sed to represent the coordnates (,) of an pont wthn the element n terms of the nodal coordnates M.. Tamn, CSMLab, UTM 7 ( ) ( ) ( ) ( ) η ξ η ξ η ξ η ξ or j j j - j
18 Eample Consder a tranglar element shown below. Ealate the shape fnctons,, and at an nteror pont P. The tranglar element for solton. M.. Tamn, CSMLab, UTM 8
19 Jacoban Matr [ J ] Area A det [ J ] ( ) M.. Tamn, CSMLab, UTM 9
20 { } γ ε ε ε Stran-Dsplacement Relatons () M.. Tamn, CSMLab, UTM Ν Ν Ν Ν Ν Ν Ν Ν Ν Ν Ν Ν γ
21 ε ε γ Stran-Dsplacement Relatons () det { ε} [ B]{ } [ J ] Stran-dsplacement matr Snce the strans for ths tranglar element depend onl on nodal coordnate ales that are constant, t s often termed constantstran-tranglar (CST) element. M.. Tamn, CSMLab, UTM
22 Dere the element eatons sng the mnmm potental energ prncple. General element eatons -dmensonal (plane) elastct problems Appl these eatons for case of plane stress, plane stran and asmmetrc solds. The fnctonal n the aratonal prncple s the potental energ of a -D elastc bod acted b srface and bod forces., T * C F * t(,) thckness, The area, A, s dded nto M dscrete elements. π M ( e (, ) π ) (, ) e M.. Tamn, CSMLab, UTM
23 To ensre conergence as element sze decreases, the chosen nterpolaton fnctons mst satsf compatblt and completeness rerements. Eamne the potental energ eaton: π ~ T ~ ~ T * (, ) [ δ [ B] [ C][ B]{ δ }] tda δ [ B] [ C]{ ε } - A ~ ~ * * F { δ } tda [ T ]{ δ }ds A c Compatblt Onl frst order derates of the dsplacement appear n the ntegrand of the fnctonal. Ths the nterpolaton fnctons mst ensre contnos dsplacement at element nterfaces..e. C contnt n dsplacement at nterfaces. Completeness The nterpolaton fnctons mst ensre that rgd bod dsplacement (nform dsplacement) and constant-stran states (nform frst derates of dsplacement) are represented n the lmt as element sze s redced..e. C contnt of dsplacement wthn elements. A o tda M.. Tamn, CSMLab, UTM
24 Polnomal contanng at least one constant and the lnear terms can sere as nterpolaton fnctons satsfng both compatblt and completeness rerements. For an element (e) wth r nmber of nodes: ( ) ( ) ( e ~, ) e { δ } (, ) r ~ { } ( e ) [ ]{ } ( e δ δ ) r (, ) (, ) nodal dsplacements { } { } ( e) M.. Tamn, CSMLab, UTM
25 Mnmm Potental Energ Prncple A aratonal prncple that hold throghot the sstem. Potental Energ stran energ work b eternal force Π (,, w) U (,, w) W (,, w) Stran energ of lnear elastc bod U Intal stran terms Ω Ω Ω ε { σ }d} ε [ C]{ ε} ~ d ~ T δ [ B] [ C][ B]{ δ }d U o C Ω ~ Ω ε [ ]{ ε } T δ [ B] [ C]{ d ε }d M.. Tamn, CSMLab, UTM
26 Work done b eternal force W Ω ~ ~ F { δ } dv T { δ } S ds Bod Force Tracton X Y Z F T T T Tz M.. Tamn, CSMLab, UTM
27 The potental energ fnctonal Π Ω ~ T ~ ~ T ~ ~ { δ [ Β] [ C][ Β]{ δ } δ [ Β] [ C]{ ε o } dv - F { δ } dv - T { δ } ds Ω s The elbrm dsplacement feld (,, w) mnmzes П and satsfes all the bondar condtons. M.. Tamn, CSMLab, UTM
28 Potental Energ The fnctonal n the aratonal prncple s the potental energ of a -D elastc bod acted b srface and bod forces. π π (, ) U (, ) W (, ) [ ] tda [ B] [ D]{ } T * ε T (, ) [ B] [ D][ B]{ } π M - A ( e (, ) π ) (, ) e A Fb { } tda T { }ds c A o tda M.. Tamn, CSMLab, UTM 8
29 Mnmm Potental Energ Theorem () The dsplacement feld (,) whch satsfes the elbrm, and the condtons at the bondar srface s the one that mnmzes the potental energ. δπ δπ ( Μ e (, ) δπ ) (, ) ( e ) (, ) e r π ( e) ( e) δ r π δ Snce δ, δ are ndependent aratons and not necessarl zero, ths π ( e) ( e) π ;,,...,r M.. Tamn, CSMLab, UTM 9
30 Mnmm Potental Energ Theorem () ( e) π π ( e A ) [ Β] ( e) T ( e) [ D][ Β]{ } tda ( e A ) Ν [ ] ( e k ) { } { F } ( e) { T} ( e) b ( e) ( e) { } { } ( e ) ( e) F da Ν T ds b ( e ) s Element stffness matr T [ k] [ B] [ D][ B] t da ( A e ) M.. Tamn, CSMLab, UTM
31 Eample: For a plane stress case, and a tpcal node, j [ k] [ B] ( e A ) T ( e ) [ ] ( e ) [ ] ( e ) ( e) ( e) C B t da, j,, b E b c c b ν 4 ν ν c ν c b ( e A ) t ( e) da ( e) If the thckness of the element s nform (t (e) constant) If the thckness of the element ares, estmate Usng table 5. (tet book) t ( e ) ( e ) ( e t da t ). ( e ) A ( e) ( e) ( e) ( e) t ( e ) t t ( ) ( ) ( ) ( ) ( e ) e e e t da t t t A ( e ) M.. Tamn, CSMLab, UTM
32 Force ector terms, {F} (e) Bod forces { F } B ( e A ) ( e) X Y t X Y * * * * t ( e) ( e) da Srface tracton * T * T { F } ds T c Ν ( s ) { F } t L s L s s - L L j j j T T s L j * * j T T L j * * T T * * ds M.. Tamn, CSMLab, UTM
33 Arrangement of terms dof at each node : o. of nodes for each element: r Element matr : [k] (e) Force-dsplacement eatons: [ ] ( e) { } ( e) { } ( e) k δ F [ k] [ k]... [ k] [ k] [ k]... [ k]... k [ ] ( e k ) [ ] [ k]... [ k] r r... r... [ ] [ ] [ ] r r rr k k... k dscrete nodal dsp. { δ } { } ( e ) { δ } δ... { δ} r... r r, { } ( e F ) { F} { F}... { F}... { } r F M.. Tamn, CSMLab, UTM
34 Ealent odal Force Vector STEPS: Calclate the total load actng on each element srface de to appled dstrbted loadng. Splt the total load proportonall to each of the node along the element edge. Sm the load ector at each node Resole the resltant nodal force nto the desred (-,-) component. M.. Tamn, CSMLab, UTM
35 M.. Tamn, CSMLab, UTM 5
36 Eample Determne the stffness matr for the straght-sded tranglar element of thckness t mm, as shown. Use E 7 GPa, ν. and assme a plane stress condton. Solton Element stffness matr s gen b where, e T [ k] t A [ B ] [ D ] [ B ] e t mm e e Ae det J.75 A.875 mm e [ ] () (Dmenson s n mm) M.. Tamn, CSMLab, UTM 6
37 The stran-dsplacement matr, [B] s gen b B det [ J ] [ ] [ B] M.. Tamn, CSMLab, UTM 7
38 The transpose of [B] matr s, [ B] T For a plane stress condton, the materal s matr [D] s gen b [ D] ν. E 7 ν. ν. ( ν ) (.) M.. Tamn, CSMLab, UTM 8
39 Sbstttng all the terms nto e.() we hae, [ k] e (. ) M.. Tamn, CSMLab, UTM 9
40 Mltplng and smplfng, we obtan smmetr 5.6 e 4 [ k] ote: Connectt wth the local DOFs s shown. M.. Tamn, CSMLab, UTM 4
41 Element Force Vector Bod Force Sppose bod force components, f and f, act at the centrod of a tranglar element. The work done b these forces s gen b, T { } { } e ( ) e f t da t f f da e M.. Tamn, CSMLab, UTM 4
42 Recall, Also, da e e A Sbstttng the aboe nto e.(), we get { } { f } t da { } { f } T T e e f t da f where {f} e s the element bod force ector, gen b t A f f f f f f f e e e { },,,,, ote: Phscal representaton of force ector {f} e s shown. T M.. Tamn, CSMLab, UTM 4
43 Tracton Force Sppose a lnearl arng tracton components act along edge - of a tranglar element. The potental energ de to the tracton force s, T { } { } ( ) e T tdl T T tdl l () Usng the relatons, 4 T T T T T T wth, ( ) ( ) l Also, dl l, dl l, dl l 6 l l l M.. Tamn, CSMLab, UTM 4
44 Sbsttton nto e.() elds, { } { } [,,, ] { } T T tdl T T e e 4 where {T} e s the ealent nodal force ector de to tracton force, gen b e tel { } [( T T ) ( T T ) ( T T ) ( T T )] T T 6 ote: The phscal representaton of the nodal force ector {T} e s shown. M.. Tamn, CSMLab, UTM 44
45 Specal Case: If the tracton forces are nform, then T T T ; T T T Ths, the nodal force ector n e.() becomes t l T T T T T e e { },,, T M.. Tamn, CSMLab, UTM 45
46 Concentrated Force The concentrated force term can be easl consdered b hang a node at the pont of applcaton of the force. If concentrated load components P and P are appled at a pont, then T { } { } P Q P Q P Ths, P and P,.e. the and components of {P} get added to the ( - )th component and ( )th components of the global force ector, {F}. ote: The contrbton of the bod, tracton and concentrated forces to the global force ector, {F} s represented b, ( e ) { } e { } { } { } F f T P e M.. Tamn, CSMLab, UTM 46
47 Eample Consder a porton of fnte element model of a plate as shown. A nform tracton force of k/m acts along the edges 7-8 and 8-9 of the model. Determne the ealent nodal forces at nodes 7, 8, and 9. M.. Tamn, CSMLab, UTM 47
48 Sggested solton We wll consder the two edges, 7-8 and 8-9 separatel, and then merge the fnal reslts. 4 4 cosθ T T cosθ.6 k/m.6 /mm snθ T T snθ. k/m. /mm For edge 7-8 (edge - local) l t e mm ( 85) ( 4) 5 mm Ealent nodal forces de to nform tracton force T k/m s, { } ( 5) [ ] T [ 5 5] T T M.. Tamn, CSMLab, UTM 48
49 For edge 8-9 (edge - local) l t e mm ( 85 7) ( 4 6) 5 mm Ealent nodal forces de to nform tracton force T k/m s, { } ( ) [ ] 5 T { T } [ ] T 5 5 These loads add to global forces F, F 4,,F 8 as shown. T M.. Tamn, CSMLab, UTM 49
50 Strans and Stress Calclatons a) Strans The strans n a tranglar element are, { ε} e d ε d d ε d ( d d γ ) d d e { ε} [ B]{ } ote: We obsered that {ε} e depends on the [B] matr, whch n trn depends onl on nodal coordnates (, ), whch are constant. Therefore, for a gen nodal dsplacements {}, the strans {ε} e wthn the element are constant. Hence the tranglar element s called a constant-stran trangle. M.. Tamn, CSMLab, UTM 5
51 b) Stresses The stresses n a tranglar element can be determned sng the stress-stran relaton, ote: σ τ e e e { σ} σ [ D]{ ε} [ D][ B]{ }. Snce the strans {ε} e are constant wthn the element, the stresses are also the same at an pont n the element.. Stresses for plane stress problem dffer from those for plane stran problem b the materal s matr [D].. For nterpolaton prposes, the calclated stresses ma be sed as the ales at the centrod of the element. 4. Prncpal stresses and ther drectons are calclated sng the Mohr crcle. M.. Tamn, CSMLab, UTM 5
52 Eample Consder a thn plate hang thckness t.5 n. beng modeled sng two CST elements, as shown. Assmng plane stress condton, (a) determne the dsplacements of nodes and, and (b) estmate the stresses n both elements. M.. Tamn, CSMLab, UTM
53 astran Solton Dstrbton of Total Translaton n the plate. CST elements sed. Magnfcaton 4X. M.. Tamn, CSMLab, UTM
54 astran Solton Dstrbton of Total Translaton n the plate. CST elements sed. Magnfcaton 4X. M.. Tamn, CSMLab, UTM
55 Sggested solton Element connectt Local odes Element o 4 4 For plane stress problem, the materals matr s gen b [ D] [ D] E ν ν ν ( ν ) M.. Tamn, CSMLab, UTM
56 Element Area of element, [ ] A The stran-dsplacement matr, det 6 n [ J ] ( ) B det [ J ] 6 Mltplng matrces [D][B] we get, [ ][ ] () 7 D B M.. Tamn, CSMLab, UTM
57 The stffness matr s gen b, () T [ k] t A [ B] [ D][ B] Sbsttte all parameters and mltplng the matrces, elds Q Q Q Q 4 Q 7 Q 8 [ ] () 7 k smmetrc.5. ote: Connectt wth global DOFs are shown. M.. Tamn, CSMLab, UTM
58 Element Area of element, A The stran-dsplacement matr s [ ] det 6 n [ J ] ( ) B det [ J ] 6 Mltplng matrces [D][B] we get, [ ][ ] () 7 D B M.. Tamn, CSMLab, UTM
59 The stffness matr s gen b, () T [ k] t A [ B] [ D][ B] Sbstttng all parameters and mltplng the matrces eld [ ] () 7 k Q 5 Q 6 Q 7 Q 8 Q Q smmetrc.5. ote: Connectt wth global DOFs are shown. M.. Tamn, CSMLab, UTM
60 Wrte the global sstem of lnear eatons, [K]{Q} {F}, and then appl the bondar condtons: Q, Q 5, Q 6, Q 7, and Q 8. The redced sstem of lnear eatons are, Q..4 Q Q 4 Solng the redced SLEs smltaneosl elds, Q.9 Q Q n. M.. Tamn, CSMLab, UTM
61 Stresses n element For element, the element nodal dsplacement ector s () 5 { } [ ].9,,.875, 7.46, T The element stresses, {σ} () are calclated from [D][B] () {} as () { σ } [ ] Stresses n element T 9., 8.7, 6. ps For element, the element nodal dsplacement ector s () 5 { } [ ],,,.875, 7.46 T The element stresses, {σ} () are calclated from [D][B] () {} as () { σ } [ ] T 9.4,.4, 97.4 ps M.. Tamn, CSMLab, UTM
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