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; 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. -node tranglar element 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 () To dere nterpolaton fnctons (sng drect approach) M.. Tamn, CSMLab, UTM : : node: α α α α α α α α α α α α α α α α α α Sole for α s

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 M.. Tamn, CSMLab, UTM 4 { } [ ]{ } { } ( ),,, c b a { } 6 5 4

15 η ξ η ξ Interpolaton fnctons n terms of natral coordnates (ξ, η) M.. Tamn, CSMLab, UTM 5 η ξ

16 η ξ η ξ Interpolaton fnctons n terms of natral coordnates (ξ, η) η ξ M.. Tamn, CSMLab, UTM ( ) ( ) ( ) ( ) η ξ η ξ

17 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

18 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 8 ( ) ( ) ( ) ( ) η ξ η ξ η ξ η ξ or j j j - j

19 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 9

20 Jacoban Matr [ J ] Area A det [ J ] ( ) M.. Tamn, CSMLab, UTM

21 { } γ ε ε ε Stran-Dsplacement Relatons () M.. Tamn, CSMLab, UTM γ

22 Stran-Dsplacement Relatons () ε ε γ det [ J ] { ε} [ B]{ } 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

23 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 A Fb { } ( e (, ) π ) (, ) e tda c T { }ds A o tda M.. Tamn, CSMLab, UTM

24 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 4

25 Mnmm Potental Energ Theorem () π π ( e A ) ( e) [ 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 5

26 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

27 The stran-dsplacement matr, [B] s gen b B det [ J ] [ ] [ B] M.. Tamn, CSMLab, UTM 7

28 The transpose of [B] matr s, [ B] T For a plane stress condton, the materal matr [D] s gen b [ D] ν. E 7 ν. ν. ( ν ) (.) M.. Tamn, CSMLab, UTM 8

29 Sbstttng all the terms nto e.() we hae, [ k] e (. ) M.. Tamn, CSMLab, UTM 9

30 Mltplng and smplfng, we obtan smmetr 5.6 e 4 [ k] ote: Connectt wth the local DOFs s shown. M.. Tamn, CSMLab, UTM

31 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

32 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

33 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

34 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 4

35 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 5

36 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 6

37 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 7

38 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 8

39 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 9

40 Strans and Stress Calclatons a) Strans The strans n a tranglar element are, { ε} e ε ε γ d d d d ( d d ) 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 4

41 b) Stresses The stresses n a tranglar element can be determned sng the stress-stran relaton, ote: σ e e e { σ} σ [ D]{ ε} [ D][ B]{ } τ. 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 4

42 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

43 astran Solton Dstrbton of Total Translaton n the plate. CST elements sed. Magnfcaton 4X. M.. Tamn, CSMLab, UTM

44 astran Solton Dstrbton of Total Translaton n the plate. CST elements sed. Magnfcaton 4X. M.. Tamn, CSMLab, UTM

45 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

46 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

47 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

48 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

49 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

50 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

51 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