(5.1.1) v Here, u and v are the displacements parallel to x and y directions respectively.

Transcription

1 Lecture : Constant Stran rangle he trangular elements wth dfferent numbers of nodes are used for solvng two dmensonal sold members. he lnear trangular element was the frst tpe of element developed for the fnte element analss of D solds. However, t s observed that the lnear trangular element s less accurate compared to lnear quadrlateral elements. But the trangular element s stll a ver useful element for ts adaptvt to complex geometr. hese are used f the geometr of the D model s complex n nature. Constant stran trangle (CS) s the smplest element to develop mathematcall. In CS, stran nsde the element has no varaton (Ref. module, lecture ) and hence element sze should be small enough to obtan accurate results. As ndcated earler, the dsplacement s expressed n two orthogonal drectons n case of D sold elements. hus the dsplacement feld can be wrtten as u d (5..) v Here, u and v are the dsplacements parallel to x and drectons respectvel. 5.. Element Stffness Matrx for CS A tpcal trangular element assumed to represent a subdoman of a plane bod under plane stress/stran condton s represented n Fg he dsplacement (u, v) of an pont P s represented n terms of nodal dsplacements u u + u + u (5..) v v + v + v Where,,, are the shape functons as descrbed n module, lecture. Fg. 5.. Lnear trangular element for plane stress/stran

2 he stran-dsplacement relatonshp for two dmensonal plane stress/stran problem can be smplfed n the followng form from three dmensonal cases (eq...9 to..). u u v x x x x v u v v u u u v v x x x x (5..) In case of small ampltude of dsplacement, one can gnore the nonlnear term of the above equaton and wll reach the followng expresson. u ε x x v ε (5..) v u γ x + x Hence the element stran components can be represented as, u ε x u+ u + u x x x x v ε ε v+ v + v u v γ x + u+ u + u+ v+ v + v x x x x Or, u ε x x x x u ε v γ x v ε Or, [ B]{ d} x x x v u (5..5) ε (5..6)

3 In the above equaton [B] s called as stran dsplacement relatonshp matrx. he shape functons for the node trangular element n Cartesan coordnate s represented as, x x x x x A x x xxx A x x xxx A Or, x A x (5..7) A x A Where, x x, x x,,,, x x, x x, x x, Hence the requred partal dervatves of shape functons are, β β, β, x A x A x A γ γ, γ, A x A x A Hence the value of [B] becomes: [ B] Or, [ B] x x x x x x β β β γ γ γ A γ γ γ β β β,, x x, (5..8) (5..9)

4 Accordng to Varatonal prncple descrbed n module, lecture, the stffness matrx s represented as, [ k] [ B] [ D][ Bd ] Ω (5..) Ω Snce, [B] and [D] are constant matrces; the above expresson can be expressed as [ ] [ ] [ ][ ] [ ] [ ][ ] k B D B d V B D B V (5..) V For a constant thckness (t), the volume of the element wll become A.t. Hence the above equaton becomes, [ ] [ ] [ ][ ] k B D B At (5..) For plane stress condton, [D] matrx wll become: µ E [ D] µ µ µ herefore, for a plane stress problem, the element stffness matrx becomes, β γ β γ µ β β β Et β γ [ k] µ γ γ γ A( ) γ β µ µ γ γ γ β β β γ β γ β Or, ( + µ ) [ k] Et A ( µ ) β+ Cγ ββ + Cγγ ββ + Cγγ βγ µβγ + Cβγ µβγ + Cβγ ( + µ ) β + Cγ ββ + Cγγ µβγ + Cβγ βγ µβγ + Cβγ ( + µ ) β + Cγ µβγ + Cβγ µβγ + Cβγ βγ γ+ Cβ γγ + Cββ γγ + Cββ Sm. γ + Cβ γγ + Cββ γ + Cβ ( µ ) Where, C Smlarl for plane stran condton, [D] matrx s equal to, (5..) (5..) (5..5)

5 ( µ ) µ E [ D] µ ( µ ) ( µ )( µ (5..6) + ) µ Hence the element stffness matrx wll become: [ k] Mβ+ γ Mββ + γγ Mββ + γγ ( µ + ) βγ µβγ + βγ µβγ + βγ Mβ + γ Mββ + γγ µβγ + βγ ( µ + ) βγ µβγ + βγ Et Mβ + γ µβγ + βγ µβγ + Cβγ ( µ + ) βγ A( + µ ) Mγ+ β Mγγ + ββ Mγγ + ββ Sm. Mγ + β Mγγ + ββ Mγ + β Where M ( µ ) (5..7) 5.. odal Load Vector for CS From the prncple of vrtual work, d u F d u F d (5..8) Where, F Γ, and F Ω are the surface and bod forces respectvel. Usng the relatonshp between stress-stan and stran dsplacement, one can derve the followng expressons: σ D B d, δ ε B δ d and δ u δ d { } [ ][ ]{ } { } [ ] { } { } [ ] { } (5..9) Hence eq. (5..8) can be rewrtten as, Or, S d B DBdd d F d d F d S B DBdd F d F d (5..) (5..) Here, [ s ] s the shape functon along the boundar where forces are prescrbed. Eq.(5..) s k d F, and thus, the nodal load vector becomes equvalent to S F F d F d For a constant thckness of the trangular element eq.(5..) can be rewrtten as (5..) S F t F ds t F da (5..) S A For the a three node trangular two dmensonal element, one can represent F and F as,

6 F F F x and F F F For example, n case of gravt load on CS element, For ths case, the shape functons n terms of area coordnates are: x L L L L L L F F x F g L F t da t da da A A A L L g L (5..) As a result, the force vector on the element consderng onl gravt load, wll become, L L L g L L gt L g L g L (5..5) he ntegraton n terms of area coordnate s gven b, p q r p!q!r! LL LdA A (5..6) ( p+ q+ r + )! A hus, the nodal load vector wll fnall become F!!! A gat gt ( )! gat!!! gat A ( )!!!! gat A ( )! (5..7)

7 Lecture : Lnear Stran rangle 5.. Element Stffness Matrx for LS In case of CS, t s observed that the stran wthn the element remans constant. hough, these elements are able to provde enough nformaton about dsplacement pattern of the element, but t s unable to provde adequate nformaton about stress nsde an element. hs lmtaton wll be sgnfcant enough n regons of hgh stran gradents. he use of a hgher order trangular element called Lnear Stran rangle (LS) sgnfcantl mproves the results at these areas as the strn nsde the element s varng. he LS element has sx nodes (Fg. 5..) and hence, twelve degrees of freedom. hus the dsplacement functon can be chosen as follows. u α + αx + α + αx + αx + α5 (5..) v a + a x+ a + a x + a x+ a Fg. 5.. Lnear stran trangle element herefore, the element stran matrx s obtaned as u x x x v 8 x v u x ( 7) ( 9)x ( 5 ) x (5..)

8 In the area coordnate sstem as dscussed n module, lecture we can wrte the shape functon for the sx node trangular element as LL LL LL (5..) L L L L L L 5 6 he dsplacement (u,v) of an pont wthn the element can be represented n terms of ther nodal dsplacements wth the use of nterpolaton functon. u v 6 6 u v Usng eq.(5..) we can rewrte eq.(5..) as, u u u 5 u 6 u 5 x x x x x x u ε v v x x x x x x v v v5 v 6 Or, Where, [ B]{ d} (5..) ε (5..5) [ B] 5 6 x x x x x x x x x x x x 5 6 Usng Chan rule, L L L x L x L x L x As dscussed n module, lecture, we can wrte the above expresson as, (5..6)

9 b b b x A L A L A L b. ( L ) x A Smlarl we can evaluate expressons for other terms and can be wrtten as, b b b... x A x A x A x x x And, a a a. ( L ). ( L ). ( L) A A A 5 6 La + La La + La La + La Where, ( L ) ( L ) ( L ) 5 6 ( Lb Lb ) ( Lb Lb ) ( Lb Lb ) ( ) ( ) ( ) a x x a x x a x x b b b he stffness matrx of the element s represented b, [ k] [ B] [ D][ Bd ] Ω (5..7) Ω he, [D] matrx s the consttutve matrx whch wll be taken accordng to plane stress or plane stran condton. he nodal stran and stress vectors are gven b, { εn} { εx εx εx ε ε ε γ x γ x γ x} (5..9a) { σn} { σx σx σx σ σ σ τx τx τx} (5..9b) [ Bn ] [ ] [ ] [ Bn ] { d} [ B ] [ B ] { ε n} (5..) n n Referrng to secton.., usng proper values of area coordnates n [B] matrx, one can fnd b b b b b [ Bn ] b b b b b A b b b b b (5..a)

10 And, a a a a a Bn a a a a a A a a a a a [ ] (5..b) hus, the element stffness can be evaluated b puttng the values from eq. (5..) n eq. (5..7). 5.. odal Load Vector for LS Smlar to -node trangular element, the load wll be lumped at each node whch can be computed usng the earler expresson, S F Fd Fd (5..) And for element wth constant thckness, S F t F ds t F da (5..) S A 5.. umercal Example usng CS Determne the dsplacements at the nodes for the followng D sold contnuum consderng a constant thckness of 5 mm, Posson s rato, µ as.5 and modulus of elastct E as x 5 /mm. he contnuum s dscrtzed wth two CS plane stress elements. Fg. 5.. Geometr and dscretzaton of the contnuum he element s connected wth node, and and let assume ts Cartesan coordnates are (x, ), (x, ) and (x, ) respectvel. If we consder nodes, and are smlar to node, and n eq.(5..9) then the [B] can be wrtten as

11 [ B] β β β γ γ γ A γ γ γ β β β B ntroducng values of β & γ dscussed n prevous lecture note, we can get value of [B] as [ B] 5 For plan stress problem, puttng the values of E and µ one can fnd the followng values. µ 6 [ D] µ µ 6 herefore the stffness matrx for the element wll be E µ 6 [ k] ta[ B] [ D][ B] Puttng values of t, A, [B] & [D]we wll get, [ k ] Smlarl element s connected wth nodes, and and global coordnates of these nodes are (x, ), (x, ) and (x, ) respectvel. For ths element, b proceedng n a smlar manner to element we can calculate [B] matrx as, [ B] 5 Hence, the elemental stffness matrx becomes,

12 k [ ] B assemblng the stffness matrces nto global stffness matrx [K], [ K ] ow, applng equaton[ F] [ K]{ d}, the followng expresson can be wrtten. Fu F u F u Fu Fv F v Fv F v u u u u v v v v Puttng boundar condtons u v u u v and adoptng elmnaton technque for applng boundar condton we get expresson, v u 8.v Solvng the above expresson, the unknown nodal dsplacements ma be obtaned as follows. v mm, u. 5 mm and 5 mm. v 96.9

13 Lecture : Rectangular Elements he rectangular elements are wdel used for solvng two dmensonal contnuums. he man advantage of ths tpe of element s the eas formulaton and eas development of computer code. he element stffness of such elements s derved here usng the concept of soparametrc formulaton. 5.. Computaton of Element Stffness In case of a four node rectangular element, the geometr and dsplacement fled can be expressed n terms of ther nodal values wth the help of nterpolaton functon. As the formulaton wll be soparametrc, the nterpolaton functon wll become same for expressng both the varables. hus, coordnates and dsplacements at an pont nsde the element (Fg. 5..) can be expressed as x x x x x (5..) And u u u u u (5..) v vv v v he above equatons can be wrtten n matrx form as x x x x x x And (5..)

14 u v u u v (5..) v u v u v he shape functon four node rectangular element s derved and shown n module, lecture. However the shape functons are reproduced here for eas reference for the dervaton of the stffness matrx. x h x h (5..5) xh x h Fg. 5.. Four node rectangular element

15 he stran-dsplacement relatonshp for two dmensonal plane stress/stran problem can be smplfed n the followng form from three dmensonal cases (eq...9 to..). u u v x x x x v u v (5..6) v u u u v v x x x x In case of small ampltude of dsplacement, one can gnore the nonlnear term of the above equaton and wll reach the followng expresson. u ε x x v ε (5..7) v u γ x + x Usng the shape functon the above expresson can be wrtten as u v x x x x u ε x v ε u γ x v x x x x u v [ B]{ d} (5..8) Here, [B] s known as stran dsplacement relatonshp matrx. he dervatves of the shape functons are calculated usng the chan rule. x. +. ξ x ξ ξ (5..9) x. +. x Here, s referred to number of nodes n an element and wll be n ths case. Convertng above expresson n matrx form

16 x x x x x x (5..) x h h h he matrx [] s referred to acoban matrx whch s dscussed n Lecture 7, module. Usng eq. (5..) one can wrte x x x x x (5..) x x x x x Puttng the values of the nodal coordnates and shape functons of the four node element n the above equaton the followng relatons wll be obtaned. h h h h x a a a x (5..a) Smlarl, h h h h b b x (5..b) x x x x x a a h x x x x b b b h (5..d) Substtutng above values n acoban matrx the followng relatons wll be obtaned. (5..c) a a and (5..) b b hus, eq.(5..) can be wrtten as. x a a x x x (5..). b b h h h After dervaton of the shape functons expressed n eq.(5..5), the followng values wll be obtaned.

17 h h h h. ; ; ; x a x a x a x a x a x x x x. ; ; ; b h b b b b So, the stran dsplacement relatonshp matrx, [B] wll become as follows. [ B] ( ) ( ) ( + ) ( + ) a a a a b b b b b a b a b a b a ( ξ) ( + ξ) ( + ξ) ( ξ) ( ξ) ( ) ( + ξ) ( ) ( + ξ) ( + ) ( ξ) ( + ) he element stffness matrx wll become [ k] t [ B] [ D][ B] dx d t [ B] [ D][ B] dξ d (5..7) (5..5) (5..6) It s seen that the above s expressed n terms of ξ and and hence can be numercall ntegrated b the Gauss Quadrature rule. he stffness matrx for each element can be found whch needs to be globall assembled for gettng the global stffness matrx to obtan the soluton. he stffness matrx of hgher order rectangular element can be derved n a smlar fashon. For example, n case of eght node rectangle element, the sze of [B] matrx wll become 6 whch was 8 for four node element. hus the sze of element stffness for eght node element wll become Computaton of odal Loads If a dstrbuted load acts on a sde of a four node rectangular element, the nodal load vector can be calculated the smlar procedure as dscussed n case of trangular element. If an element as shown below s subjected to a lnearl varng ntenstes of load at ts one sde, then the magntude of ths at an pont on the sde can be expressed b ts nterpolaton functon as follows. + qx qx (5..8) qx Here, q x and q x are the force ntenstes per unt length at nodes and respectvel. he load at nodes can be calculated from the followng expresson. S { Fx} { } qd x Γ (5..9) As ξ along the sde -, the nterpolaton functon wll become

18 S { } ( ξ)( ) ( + ξ)( ) ( ) ( + ξ)( + ) ( + ) ( ξ)( + ) (5..) If the element thckness s t, then dγ t.dl. hus the eq.(5..9) can be replaced as ( ) + + qx { Fx } t dl ( ) (5..) q + x Fg. 5.. Varng load on a four node element After ntegratng the above expresson, the nodal load vector along x drecton wll become as follows. qx q t + x { Fx } (5..) qx + qx

19

20 Lecture : umercal Evaluaton of Element Stffness Dervaton of element stffness for a four node rectangle element has been demonstrated n last lecture. he stffness matrx of each element can be calculated easl b developng a sutable computer algorthm. o help students for developng ther own computer code, a numercal example has been solved and demonstrated here. 5.. umercal Example Calculate the stffness matrx for the gven four node rectangular element b the Gauss Quadrature ntegraton rule usng one pont and two pont formula assumng plane stress formulaton. Consder, the thckness of element cm, E k/cm and µ. Fg. 5.. Element Dmenson 5.. Evaluaton of Stffness usng One Pont Gauss Quadrature For the calculaton of stffness matrx, frst, Gauss Quadrature ntegraton procedure has been carred out. hus, the natural coordnate of the samplng pont wll become, and weght wll become. whch s shown n the fgure below. Fg. 5.. atural coordnates for one pont Gauss Quadrature

21 For a four node quadrlateral element, the shape functons and ther dervatves are as follows. ( ξ )( ), ( ξ ) ; ξ ( ) ; ( +ξ )( ) and ( ξ ) ; ξ ( + ) ; ( +ξ )( + ) ( ξ )( + ) and ( + ξ ) ( + ξ ) ; ξ ξ ( + ) ; ( ) he acoban matrx can be found from the followng relatons. x ( ξ) + ( ξ) + ( + ξ) ( + ξ) x ξ ξ ξ ξ x x ( ) ( + ) + ( + ) + ( ) x x x x Consderng the samplng pont, (ξ and ), the value of the acoban, [] s + 7 [ ] hus, [ ] 5 and [ ] ; and ow, the stran vector for the element wll become [ ε ] u ξ u v ξ v

22 ] [ v u v u v u v u ξ ξ ξ ξ ξ ξ ξ ξ ε [ ] } ]{ [ } { ' ] [ d B d B ε ] [ / B and [ ] ' ] [ B B [B] For plane stress condton / ] [ µ µ µ µ E C

23 [ C][ B] / Assume the values of gauss weght, w, the stffness matrx [k] at ths samplng pont s [ k] tw [ B] [ C] [ B], Where t s thckness of the element. hus, / [ k] Evaluaton of Stffness usng wo Pont Gauss Quadrature

24 In ths case, Gauss Quadrature ntegraton procedure has been carred out to the calculate the stffness matrx of the same element for a comparson. he natural coordnate of the samplng pont s shown n the fgure below. Fg. 5.. atural Coordnates for wo Ponts Gauss Quadrature he natural co-ordnates of the samplng ponts for Gauss Quadrature ntegraton are For a four node quadrlateral element, the shape functons and ther dervatves are as follows. ( ξ )( ), ( ξ ) ; ξ ( ) ; he acoban matrx wll be ( +ξ )( ), ( ξ ) ; ξ ( + x ) ; ( +ξ )( + ) and ( + ξ ) ; ξ ( + ) ; ( ξ )( + ) ( + ξ ) ξ ( ) ( ξ) + ( ξ) + ( + ξ) ( + ξ) x ξ ξ ξ ξ x x ( ) ( + ) + ( + ) + ( ) x x x x

25 (a) At samplng pont, (ξ.5775,.5775) he value of the acoban, [] at samplng pont wll become ) (.5775) (.5775) (.5775) (.5775) (.5775) (.5775) (.5775) ( ] [ hus, 5 5 ] [ and 5 [ ] 5 ; hus 875 ξ ξ ε v v u u ] [ ] [ v u v u v u v u ξ ξ ξ ξ ξ ξ ξ ξ ε [ ] } ]{ [ } { ' ] [ d B d B ε

26 [ B ] [ B '] hus, [ B] [B] For plane stress condton E [ C ] µ µ µ µ /

27 [C][B] / he values of gauss weghts are w w j.. herefore, the stffness matrx [k] at ths samplng pont s [ k] tw w [ B] [ C] [ B], where t s thckness of the element. hus at samplng pont, j j j j / [ k ]

28 [ k ] sm (b) At samplng pont, (ξ-.5775,.5775) he value of the acoban, [] at samplng pont can be calculated n a smlar wa and fnall the stran-dsplacement relatonshp matrx and then the stffness matrx [k ] can be evaluated and s shown below [ k ] 875 [ k ] sm

29 (c) At samplng pont, (ξ-.5775, ) he value of the stran-dsplacement relatonshp matrx and then the stffness matrx [k ] can be evaluated and s shown below [ k ] 875 [ k ] sm (d) At samplng pont, (ξ.5775, ) he value of the stran-dsplacement relatonshp matrx and then the stffness matrx [k ] can be evaluated and s shown below.

30 [ k ] 875 [ k ] sm he stffness matrx of the element can be computed as the sum of the values at the four samplng ponts: [ k] [ k] + [ k] + [ k] + [ k]. hus, the fnal value of the stffness matrx wll become sm [ k]

31 Lecture 5: Computaton of Stresses, Geometrc onlneart and Statc Condensaton 5.5. Computaton of Stresses After solvng the statc equaton of {F} [K]{d}, the nodal dsplacement {d} can be obtaned n global coordnate sstem. he element nodal dsplacementd can then be calculated from the nodal connectvt of the element. Usng stran-dsplacement relaton and then stress-stran relaton, the stress at the element level are derved. DDB d (5.5.) εhere, s the stress at the Gauss pont of the element as the samplng ponts for the ntegraton has been consdered as Gauss ponts. Here, [D] s the consttutve matrx, [B] s the stran dsplacement matrx of the element. As a result these stresses at Gauss ponts need to extrapolate to the correspondng nodes of the element. It s well establshed that Gauss ntegraton ponts are the optmal samplng ponts for two dmensonal soparametrc elements. he local stress smoothng s a technque that can be used to extrapolate stresses computed at Gauss ponts to nodal ponts. he stresses are computed at four Gauss ponts (I, II, III and IV) of an 8 node element as shown n Fg For example, at pont III, r s and ξ. herefore the factor of proportonalt s σ ;.e., r ξ and s (5.5.) Stresses at an pont P n the element are found b the usual shape functon as σ P d σ for,,, (.5.) In the above equaton, σ P s σ x, σ and τ x at pont P. d are the blnear shape functons wrtten n terms of r and s rather than ξ and as d ( ± r)( ± s) (5.5.) d are evaluated at r and s coordnates of pont P. Let the pont P concdes wth the corner. o calculate stress σ x at corner from σ x values at the four Gauss ponts, substtuton of r and s nto the shape functons wll gve σ.8666σ.5σ +.σ. 5σ (5.5.5) xi xi xii xiii xiv

32 Fg atural coordnate sstems used n extrapolaton of stresses from Gauss ponts he resultant extrapolaton matrx thus obtaned ma be wrtten as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) IV III II I σ σ σ σ σ σ σ σ σ σ σ σ (5.5.6) Here, σ, σ.. 8 σ are the smoothened nodal values and I σ IV σ are the stresses at the Gauss ponts. Smoothened nodal stress values for four node rectangular element can be also be evaluated n a smlar fashon. he relaton between the stresses at Gauss ponts and nodal pont for four nodel element wll be

33 I II III IV (5.5.7) he stress at partcular node jonng wth more than one element wll have dfferent magntude as calculated from adjacent elements (Fg. 5.5.(a)). he stress resultants are then modfed b fndng the average of resultants of all elements meetng at a common node. A tpcal stress dstrbuton for adjacent elements s shown n Fg. 5.5.(b) after stress smoothenng.

34 Fg Stress smoothenng at common node 5.5. Geometrc onlneart As dscussed earler, nonlnear analss s manl of two tpes: () Geometrc nonlneart and () Materal nonlneart. For geometrc nonlneart consderaton, the relaton between stran and dsplacement s of utmost mportance n the fnte element formulaton for stress analss problems. In case of plane stress/stran problem, the nonlnear term of the stran expresson are dropped for the sake of smplct n the analss. However, for large dsplacement problems, the nonlnear stran term plas a vtal role to obtan accurate response. he generalzed strandsplacement relatons for the two-dmensonal plane stress/stran problems are rewrtten here to derve the nonlnear soluton. u u v ε x + + x x x v u v ε + + γ x v u u u v v x x x (5.5.8) he dsplacements at an pont nsde the node are expressed n terms of ther nodla dsplacements. hus, herefore, n [ ]{ } and v v [ ]{ v} u u u n (5.5.9)

35 [ ] { u } [ B ]{ u } u x x v [ B ]{ v} x u [ B ]{ u} v [ B ]{ v} (5.5.) Here, [B ] and [B ] are the dervatve of the shape functon [ ] wth respect to x and respectvel. he vectors {u } and {v } represent the nodal dsplacements vectors n x and drectons respectvel. he vector of strans at an pont nsde an element,{ ε } ma be expressed n terms of nodal dsplacement as { ε } [ B] { d} (5.5.) where [B] s the stran dsplacement matrx. {d} s the nodal dsplacement vector and ma be expressed as { d} { u} { v } he matrx [B] ma be expressed wth two components as [ B] [ B ] + [ B ] l nl (5.5.) (5.5.) where, [B l ] and [B nl ] are the lnear and nonlnear part of the stran-dsplacement matrx respectvel and are expressed as follows: and [ ] B l [ B ] [ ] [ ] [ B ] [ ] [ ] B B (5.5.)

36 { u} [ B ] [ B ] { v} [ B ] [ B ] [ Bnl ] { u} [ B ] [ B ] { v} [ B ] [ B ] { u} [ B ] [ ] { } [ ] [ ] B v B B (5.5.5) Steps to nclude effect of geometrcal nonlneart he nonlnear geometrc effect of the structure at a partcular nstant of tme can be obtaned b performng the followng steps.. Calculaton of dsplacement {d} consderng lnear part of stran matrx [B l ].. Evaluaton of nonlnear part of the stran matrx [B nl ] (eq5.5.5) adoptng {d} from prevous step.. Evaluaton of total stran matrx [B] [B l ] + [B nl ].. Calculaton of dsplacement {d} consderng both lnear and nonlnear part of stran matrx [B]. 5. Repetton of steps to wth {d}, from whch modfed dsplacement, {d} are obtaned. 6. Step 5 s carred out untl the dsplacements for two consecutve teraton converge.e., { d} { d} j { d} j j < ε Where ε s an pre-assgned small value and j s the number of teratons Statc Condensaton he hgher order Lagrangan elements (.e., nne node, sxteen node rectangular element) contan number of nternal nodes. hs s necessar sometmes for the completeness of the desred polnomal used n dsplacement functon for dervaton of nterpolaton functon. hese nternal nodes are not connected to the adjonng elements n the assemblage (Fg. 5.5.).

37 Fg Internal nodes of ne node elements hus, the dsplacements of these nodes are not requred to formulate overall equlbrum equatons of the structure. hs lmts the usefulness of these elements. A technque known as statc condensaton can be used to suppress the degrees of freedom assocated wth the nternal nodes n the fnal computaton. he technque of statc condensaton s explaned below. he equlbrum equaton for a sstem are expressed n the fnte element form as F K d (5.5.6) Where, {F}, [K] and {d} are the load vector, stffness matrx and dsplacement vector for the entre structure. he above equaton can be rearranged b separatng the relevant terms correspondng to nternal and external nodes of the elements. F K K d e F e Ke K ee d e (5.5.7) Here, {d } and {d e } are the dsplacement vectors correspondng to nternal and external nodes respectvel. Smlarl, {F } and {F e } are force vectors correspondng to nternal and external nodes. ow, the above expresson can be wrtten n the followng form separatel. F K d K d (5.5.8) e e F K d K d (5.5.9) e e ee e he stffness matrx and nodal load vector corresponds to the nternal nodes can be separated out. For ths, eq.(5.5.8) can be rewrtten as d K F K K d (5.5.) e e Substtutng the value of {d } obtaned from the above equaton n eq.(5.5.9), the followng expresson wll be obtaned.

38 F K K F K K d K d (5.5.) e e e e ee e Here, the equatons are reduced to a form nvolvng onl the external nodes of the elements. he above reduced substructure equatons are assembled to acheve the overall equatons nvolvng onl the boundar unknowns. hus the above equaton can be rewrtten as F e Ke K F Kee Ke K Ke de de (5.5.) Or F ck cd e (5.5.) Where, - and - F F -K K F K K -K K K d. Here, [K c ] c e e c ee e e e s called condensed or reduced stffness matrx and {F c } s the condensed or effectve nodal load vector correspondng to external nodes of the elements. In ths process, the sze of the matrx for nverson wll be comparatvel small. he unknown dsplacements of the exteror nodes, {d e } can be obtaned b nvertng the matrx [K c ] n eq.(5.5.). Once, the values of {d e } are obtaned, the dsplacements of nternal nodes {d } can be found from eq.(5.5.).

39 Lecture 6: Axsmmetrc Element 5.6. Introducton Man three-dmensonal problems show smmetr about an axs of rotaton. If the problem geometr s smmetrc about an axs and the loadng and boundar condtons are smmetrc about the same axs, the problem s sad to be axsmmetrc. Such three-dmensonal problems can be solved usng two-dmensonal fnte elements. he axsmmetrc problem are most convenentl defned b polar coordnate sstem wth coordnates (r, θ, z) as shown n Fg hus, for axsmmetrc analss, followng condtons are to be satsfed.. he doman should have an axs of smmetr and s consdered as z axs.. he loadngs on the doman has to be smmetrc about the axs of revoluton, thus the are ndependent of crcumferental coordnate θ.. he boundar condton and materal propertes are smmetrc about the same axs and wll be ndependent of crcumferental coordnate. Fg Clndrcal coordnates Axsmmetrc solds are of total smmetr about the axs of revoluton (.e., z-axs), the feld varables, such as the stress and deformaton s ndependent of rotatonal angle θ. herefore, the feld varables can be defned as a functon of (r, z) and hence the problem becomes a two dmensonal problem smlar to those of plane stress/stran problems. Axsmmetrc problems ncludes, crcular clnder loaded wth unform external or nternal pressure, crcular water tank, pressure vessels, chmne, boler, crcular footng restng P( r,z, on sol θ ) mass, etc Relaton between Stran and Dsplacement An axsmmetrc problem s readl descrbed n clndrcal polar coordnate sstem: r, z and θ. Here, θ measures the angle between the plane contanng the pont and the axs of the coordnate

40 sstem. At θ, the radal and axal coordnates concde wth the global Cartesan X and Y coordnates. Fg shows a clndrcal coordnate sstem and the defnton of the poston vectors. Let r, ˆ zˆ and ˆ be unt vectors n the radal, axal, and crcumferental drectons at a pont n the clndrcal coordnate sstem. Fg Clndrcal Coordnate Sstem If the loadng conssts of radal and axal components that are ndependent of θ and the materal s ether sotropc or orthotropc and the materal propertes are ndependent of θ, the dsplacement at an pont wll onl have radal (u r ) and axal (u z ) components. he onl stress components that wll be nonzero are σ rr, σ zz, σ θθ and τ rz. (a) Element n r-z plane (b) Element n r-θ plane

41 Fg Deformaton of the axsmmetrc element A dfferental element of the bod n the r-z plane s shown n Fg. 5.6.(a). he element undergoes deformaton n the radal drecton. herefore, t ntates ncrease n crcumference and assocated crcumferental stran. Let denote the radal dsplacement as u, the crcumferental dsplacement as v, and the axal dsplacement as w. Dashed lne represents the deformed postons of the bod n Fg. 5.6.(b). he radal stran can be calculated from the above dagram as u u ε r dr u+ - u dr r (5.6.) r Snce the rz plane s effectvel the same as a rectangular coordnate sstem, the axal stran wll become w w ε z dz w+ - w dz z (5.6.) z Consderng the orgnal arc length versus the deformed arc length, the dfferental element undergoes an expanson n the crcumferental drecton. Before deformaton, let the arc length s assumed as ds rdθ. After deformaton, the arc length wll become ds (r+u) dθ. hus, the tangental stran wll be r +ud- rd u ε (5.6.) rd r Smlarl, the shear stran wll be u w rz z r and r z (5.6.) hus, there are four stran components present n ths case and s gven b { ε} u r r ε r w ε z z z u ε θ u w γ r r rz u w + z r z r (5.6.5)

42 5.6. Relaton between Stress and Stran he stress stran relaton for axsmmetrc case can be derved from the three dmensonal consttutve relatons. We know the stress-stran relaton for a three-dmensonal sold s σ x σ σ z τ x τ z τ zx E (+μ)(μ) μ μ μ μ μ μ μ μ μ μ μ μ ϵ x ϵ ϵ z ν x ν z ν zx (5.6.6) he stresses actng on a dfferental volume of an axsmmetrc sold under axsmmetrc loadng s shown n Fg Fg Stresses actng on a dfferental volume ow, comparng the stress-stran components present n the axsmmetrc case, the stress-stran relaton can be expressed from the above expresson as follows

43 σ r σ z σ θ τ rz E (+μ)(μ) μ μ μ μ μ μ μ μ μ μ ε r ε z ε θ ν rz hus, the consttutve matrx [D] for the axsmmetrc elastc sold wll be μ μ μ μ μ μ E [D] (+μ)(μ) μ μ μ μ (5.6.7) (5.6.8) 5.6. Axsmmetrc Shell Element A clndrcal lqud storage contaner lke structures (Fg ) ma be dealzed usng axsmmetrc shell element for the fnte element analss. It ma be noted that the lqud n the contaner ma be dealzed wth two dmensonal axsmmetrc elements. Let us consder the radus, heght and, thckness of the crcular tank are R, H and h respectvel. Fg hn wall clndrcal contaner

44 he stran energ of the axsmmetrc shell element (Fg ) ncludng the effect of both stretchng and bendng are expressed as H U ε + ( ε θ θ+ M χ πrd) (5.6.9) Here, and θ are the membrane force resultants and M s the bendng moment resultant. he shell s assumed to be lnearl elastc, homogeneous and sotropc. hus the force and moment resultants can be expressed n terms of the md-surface change n curvature χ as follows. Fg Axsmmetrc plate element Here, the stran-dsplacement relaton s gven b { σ } [ D]{ ε} (5.6.) In whch, ε µ Eh { σ} θ, { ε} εθ and [ D] µ (5.6.) M χ µ h he generalzed stran vector can be expressed n terms of the dsplacement vectors as follows. Where, { } [ B]{ d} ε (5.6.)

45 { d} u v and [ B] R (5.6.) Here, u and v are the dsplacement components n two perpendcular drectons. Wth the use of stress and stran vectors, the potental energ expresson are wrtten n terms of dsplacement vectors as H U πr ({ d} [ B] [ D][ B]{ dd} ) (5.6.) hus, the element stffness are derved as H [ ] π [ ] [ ][ ] k R B D B d (5.6.5) Smlarl, neglectng the rotar nerta, the knetc energ can be expressed as H πr ({ d} [ ] m[ ]{ dd } ) (5.6.6) Where, m denotes the mass of the shell element per unt area and { d } represents the veloct vector. hus, the element mass matrx s gven b [ ] π [ ] [ ] L e M Rm d (5.6.7) Lecture 7: Fnte Element Formulaton of Axsmmetrc Element Fnte element formulaton for the axsmmetrc problem wll be smlar to that of the two dmensonal sold elements. As the feld varables, such as the stress and stran s ndependent of rotatonal angle θ, crcumferental dsplacement wll not appear. hus, the dsplacement feld varables are expressed as n u r,z n w r,z r,z u r,z w (5.7.) Here, u and w represent radal and axal dsplacements respectvel at nodes. (r, z) are the shape functons. As the geometr and feld varables are ndependent of rotatonal angle θ, the nterpolaton functon (r, z) can be expressed smlar to -dmensonal problems b replacng the x and terms wth r and z terms respectvel.

46 5.7. Stffness Matrx of a rangular Element Fg shows the clndrcal coordnates of a three node trangular element. Hence the analss of the axsmmetrc element can be approached n a smlar wa as the CS element. hus the feld varables of such an element can be expressed as u α + αr+ αz (5.7.) w α + α r+ α z Or, 5 { d} [ φ]{ α} (5.7.) Where, u r z { d}, [ φ] w r z Usng end condtons, u r z α u rj z j α u rk zj α w r zα w rj z j α w rk zk α 5 Or, d A α { } [ ]{ } { α} [ A] { d} Here { d} are the nodal dsplacement vectors. and { α} { α α α α α α } 5 (5.7.) (5.7.5)

47 Fg Axsmmetrc three node trangle n clndrcal coordnates Puttng above values n eq.(5.7.), the followng relatons wll be obtaned. Or, { d} [ φ][ A] { d} [ ] { d} { d} (5.7.6) r r u j k r w j kz z z (5.7.7) Usng a smlar approach as n case of CS elements, the three shape functons [,, ] be assumed as, ( r, z) ( rz rz ) ( z z) r ( r r) z A + + ( r, z) ( rz rz ) ( z z) r ( r r) z A + + ( r, z) ( rz rz ) + ( z z) r+ ( rr) z A can

48 Or, rz A r z rz A r z rz A r z (, ) ( α + β + γ ) (, ) ( α + β + γ ) j j j j (, ) ( α + β + γ ) k k k k (5.7.8) Where, α r z r z α r z rz α rz r z j k k j j k k k j j β z z β z z β z z j k j k k j γ r r γ r r γ r r k j k j A ( rz j + rjzk + rkz rz k rjz rkzj) Puttng the value of {u,w} n eq. (5.7.7) from eq. (5.6.5), j k r r r r r j k r j z k z z z z r r r { ε} [ B]{ d} j k j k z z z z r r r hus, the stran dsplacement matrx can be expressed as, β β j βk j k B r r r A γ γ j γ k γ γ j γk β β j βk [ ] (5.7.9) (5.7.) (5.7.) r + rj + rk Where, r. hus the stffness matrx wll become [ ] [ ] [ ][ ] k B D BdΩ π Or, [ k] [ B] [ D][ B] r dθ da π [ B] [ D][ B] r dr dz (5.7.)

49 Snce, the term [B] s dependent of r terms; the term [ B] [ D][ B] cannot be taken out of ntegraton. Yet, a reasonabl accurate soluton can be obtaned b evaluatng the [B] (denoted as [B]) matrx at the centrod. Hence, [ ] π [ ] [ ][ ] Or, k r B D B dr dz [ ] [ ] [ ][ ] k B D B π ra (5.7.) 5.7. Stffness Matrx of a Quadrlateral Element

50 5 he stran-dsplacement relaton for axsmmetrc problem derved earler (eq.(5.6.5)) can be rewrtten as u u r r ε u r w z ε z z { ε} w (5.7.) εθ u r r γ r rz w u w + z z r u Applng chan rule of dfferentaton equaton we get, u u ξ r u u z w w rξ w w z u u Hence, the stran components are calculated as u ξ u εr ε z w εθ r ξ γ rz w u Or, (5.7.5)

51 5 u u r z w (5.7.6) r rz w u u u w w Wth the use of nterpolaton functon and nodal dsplacements,,,, x h x h can be expressed for a four node quadrlateral element as u u x x x x x u u u h h h h h u w w x x x x x w w w h h h h h w (5.7.7) Puttng eq. (5.7.7) n eq. (5.7.6) we get, u u r u z u w r rz w w w (5.7.8) hus, the stran dsplacement relatonshp matrx [B] becomes

52 5 B r For a four node quadrlateral element, ( ξ ) ( ) ( ) ( ξ ) and ξ ( ξ+ ) ( ) ( ) ( ξ+ ) and ξ ( ξ+ ) + ( ) ( + ) ( ξ+ ) and ξ ( ξ ) + ( ) ( + ) ( ξ ) and ξ hus, the [B] matrx wll become (5.7.9) (5.7.) B r (5.7.) he stffness matrx for the axsmmetrc element fnall can be found from the followng expresson after numercal ntegraton. ++ [ k] [ B] [ D][ Bd ] Ω [ B] [ D][ πr. B ]..dξd (5.7.) Ω

53 5 Lecture 8: Fnte Element Formulaton for Dmensonal Elements 5.8. Introducton Sold elements can easl be formulated b the extenson of the procedure followed for two dmensonal sold elements. A doman n D can be dscrtzed usng tetrahedral or hexahedral elements. For example, the eght node sold brck element s analogous to the four node rectangular element. Regardless of the possble curvature of edges or number of nodes, the sold element can be mapped nto the space of natural co-ordnates,.e theξ ±, ±, ζ ± just lke a plane element. Fg. 5.8.Eght node brck element For three dmensonal cases, each node has three degrees of freedom havng u, v, and w as dsplacement feld n three perpendcular drectons (X, Y and Z). In ths case, one addtonal dmenson ncreases the computatonal expense manfolds Stran Dsplacement Relaton he stran vector for three dmensonal cases can be wrtten n the followng form

54 5 x z x z zx u x u u z v x v v z w x w w z ε ε ε γ γ γ (5.8.) he followng relaton exsts between the dervatve operators n the global co-ordnates and the natural co-ordnate sstem b the use of chan rule of partal dfferentaton. z x z x z x z x ζ ζ ζ ξ ξ ξ ζ ξ (5.8.) Where the acoban Matrx wll be [ ] ζ ζ ζ ξ ξ ξ z x z x z x (5.8.) For an soparamatrc element the coordnates at a pont nsde the element can be expressed b ts nodal coordnate. and n n n x x; z z (5.8.)

55 55 Substtutng the above equatons nto the acoban matrx for an eght node brck element, we get [ ] 8 x ξ x x ζ ξ ζ z ξ z z ζ he stran dsplacement relaton s gven b { ε } [ B]{ d}, where, { } u d v. w (5.8.5) he dsplacements n the x, and z drecton are u, v, and w respectvel. Let consder the nverse of acoban matrx as [ ] (5.8.6) hus, the relaton between two coordnate sstems can be rewrtten as x z x ξ ξ ξ ξ ξ x z x z z ζ ζ ζ ζ ζ (5.8.7) hus, one can wrte the followng relatons Smlarl, u u u u u au x ξ ζ ξ ζ 8 8 u u u u u bu ξ ζ ξ ζ 8 8 u u u u u cu z ξ ζ ξ ζ v v v w w w av; bv; cv; aw; bw and cw x z x z Usng above relatons, the stran vector can be wrtten as

56 56 { ε} u x v ε x a ε w b u (5.8.8) 8 ε z z c v u v γ x b a + w γ x z c b γ v w zx + c a z u w + z x ow, the stran dsplacement relatonshp matrx [B] can be dentfed from the above equaton b comparng t to { } [ B]{ d} ε Element Stffness Matrx he element stffness matrx can be generated smlar to two dmensonal case usng the followng relatons k [B] [D][B]dξddζ [ ] (5.8.9) he sze of the consttutve matrx [D] for sold element wll be 6 6 and s alread dscussed n module, lectures. For eght node brck element, the sze of stffness matrx wll become as number of nodes n one element s 8 and the degrees of freedom at each node s. It s well establshed that Gauss ntegraton ponts are the optmal samplng ponts for eght node soparametrc brck elements Element Load Vector he forces on an element can be generated due to ts self weght or externall appled force whch ma be concentrated or dstrbuted n nature. he dstrbuted load ma be unform or non-unform. All these tpes of loads are to redstrbuted to the nodes usng fnte element formulaton Gravt load he load vector due to bod forces n general s gven b { Q } [ ] { X }dω (5.8.) Ω

57 57 where {X} s the bod forces per unt volume. he nodal load vector at an node ma be expressed as { Q } [ ] { X }dω (5.8.) Ω In case of gravt load, the force wll act n the global negatve Z drecton. herefore, [ ] and { X} (5.8.) ρg Here, the mass denst of the materal s ρ and the acceleraton due to gravt s g. hus, eq.(5.8.) wll become { Q } dω (5.8.) Ω ρg For soparametrc element the, the above expresson wll become Q { } dξddζ (5.8.) ρ Usng Gauss Quadrature ntegraton rule, the above expresson ma be evaluated as n n n {Q } ww jwk ( ξ, j, ζk ) j k ρg ξ ζ (5.8.5) (,, ) Where, n s the number of nodes n an element. For eght node lnear brck element the value of n wll be 8 and the ntegraton order suggested s. Smlarl, for twent node quadratc brck element, the value of n wll be and the ntegraton order suggested s. j k Surface pressure Let assume a unform surface pressure of ntenst q s actng normal to the element face. he load vector due to surface pressure s gven b s Q [ ] { p} da (5.8.6) { } he nodal load at an node ma be expressed as s Q [ ] { p da (5.8.7) { } } s In case of surface load, the value of [ ] n the above equaton wll become

58 58 Here s s s [ ] (5.8.8) s s s the nterpolaton functon for the node. For example, the value of b substtutng ξ n s for face. hus, the surface pressure s expressed as, s can be obtaned ql { p } qm (5.8.9) qn Where, l, m, n are the drecton cosnes. hus, eq.(5.8.7) can be expressed usng eq.(5.8.9) n the followng form. s ql + + s { Q } qm da (5.8.) s qn he value of da s can be evaluated consderng the cross product of vectors along the natural coordnates parallel to the loaded faces of the element. hus, da e e ddζ (5.8.) Stress Computaton Usng the relaton of {F} [K]{d}, the unknown nodal dsplacement vector {d} are calculated n global coordnate sstem. Once the nodal dsplacements are obtaned, the stran components as each node can be computed usng stran-dsplacement relatons for each element. Smlarl element stress can be calculated usng stress-stran relaton. hese stresses at Gauss ponts are extrapolated to the correspondng nodes of the element to fnd the nodal stresses. In general, for three dmensonal state of stress there are at least three planes, called prncpal planes. he correspondng stress vector s perpendcular to the plane and where there are no normal shear stresses. hese three stresses whch are normal to these prncpal planes are called prncpal stresses. he prncpal stresses σ, σ, and σ are computed from the roots of the cubc equaton represented b the determnant of the flowng. σx σ τx τxz τx σ σ τ z τ τ σ σ xz z z (5.8.) he characterstc equaton has three real roots σ, due to the smmetr of the stress tensor. he prncpal stresses are arranged so that σ > σ > σ. he maxmum shear stress can be computed from the followng relatons.

59 59 max largest of σ σ, σ τ σ and σ σ (5.8.) hese three shear stress components wll occur on planes orented at 5 from the prncpal planes. he dstorton energ theor suggests that the total stran energ can be dvded nto two components. he are () volumetrc stran energ and () dstorton or shear stran energ. It s antcpated that eld develops f the dstorton component exceeds that at the eld pont for a smple tensle test. From the concept of dstorton energ theor, the equvalent stress whch s hstorcall known as Von Mses stress are defned as σ e ( σ σ ) + ( σ σ ) + ( σ σ ) (5.8.) he Von Mses stresses offer a measure of the shear or dstortonal stress n the materal. In general, ths tpe of stress tends to cause eldng n metals.

60 6 Worked out Examples Example 5. Calculaton of nodal loads on a trangular element A CS element as shown n Fg. 5.I gets axal loadng of (F x ) k/m n X drecton and (F ) k/m n Y drecton. Compute the nodal loads n the element. Fg. 5.I Dstrbuted loadng on a trangular element From the above fgure, the length of sdes, and are calculated and wll be, 8 and 6 cm respectvel. Frst, let consder sde : Or, Puttng, l { } [ ] { } F F ds Γ { } L L L l l l L Fx L L F ds ds ds F L L L L p q pq!! L L ds l, we wll get, p+ q+! ( )

61 6.5 l.5. { F} Smlarl for sde, L L l l L Fx L l F ds ds k L F L.8 L L.8 { } Snce no force s actng on sde, { F} Hence, the nodal load vector n all the nodes n x and drectons wll become,.5.5 { F} k..8.8