A Quadratic Serendipity Plane Stress Rectangular Element

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1 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt In Chaptr 2, w larnd two diffrnt nrgy-bad mthod of: 1. Turning diffrntial quation into intgral (or nrgy) quation 2. Uing thi form of th quation to gnrat dicrt approximation uing hap function In Chaptr 3, w larnd how crtain hap function may b drivd In Chaptr 4, w larnd om baic rult from laticity thory. Namly, th form of th tr quilibrium quation and how tr rlat to train via om form of Hook Law In Thi chaptr, w d lik to put all th ida togthr to how th finit lmnt mthod i ud a gnral 2011 Alx Grihin MAE 323 Chaptr 5 1

2 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt Th firt thing w d lik to do i crat a plan tr lmnt uing th quadratic Srndipity hap function drivd in Chaptr 3 ovr a rctangular domain. What w man by thi i w d lik to b abl to fully dfin th dicrt algbraic (wak) form of it govrning latotaticquation r ( r 1)( 1)( r + + 1) 4 1 ( r + 1)( 1)( r + 1) 4 1 ( r + 1)( + 1)( + r 1) 4 1 ( r 1)( + 1)( r 1) 4 ( r, ) = 1 ( 2 r 1)( 1) 2 1 ( 1)( 2 r + 1) 2 1 ( 2 r 1)(1 + ) 2 1 ( 1)( 2 r 1) 2 N (1) 2011 Alx Grihin MAE 323 Chaptr 5 2

3 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt Formally, th way w d do thi i to tart with th diffrntial quation (from Chaptr 4 rmmbring that th indic rang ovr patial coordinat): σ + = ij, j bi 0 Thn, uing th Galrkin formulation*, w would multiply thi with a trial function. In thi contxt, it would b a vctor-valud trial function, w i ( ) ij, j bi wi 0 σ + = Thn intgrat ovr an lmnt volum ( ) σ + b w d = ij, j i i 0 *Altrnativly, w could intgrat th train nrgy dnity and quat thi to th work don by xtrnal nodal forc (i.. th Rayligh-Ritz Mthod) 2011 Alx Grihin MAE 323 Chaptr 5 3

4 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt Now, w won t go through th complt drivation bcau it involv om mathmatic mot tudnt havn t n yt (motly concpt from Advancd Calculu). Thi i only bcau w ar now working in two patial dimnion. W will jut giv th rulting wak form: t σ δε d = t b w + t F w ds ij ij i i i i S whr: δε ij = ( wi, j + wj, i ) 1 2 ( chaptr 4 for th dfinition of train) And ti th thru-thickn (normal to th plan) of th domain. Now rplac th tr and train tnor with thir vctor countrpart (a wll a th forc), a w aw in Chaptr 4, and lt aum a unit thickn for t: T σ δ εd = bw + F wds (2) S Str Strain Body Load vctor vctor vctor Extrnal urfac load vctor 2011 Alx Grihin MAE 323 Chaptr 5 4

5 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt Now, rcall that th Srndipity lmnt ar ioparamtric, which man that if w ar going to prform th intgral in (2), w nd an xplicit mapping btwn th ioparamtric coordinat and th global coordinat y For th train matrix, thi mapping i upplid x 7 by th Jacobian of (x,y) with rpct to (r,): 4 x y r r 8 J = 5 x y 1 r uch that: Ni x y Ni Ni r r r x x = N N = J i x y i N (3) i y y 2011 Alx Grihin MAE 323 Chaptr 5 5

6 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt If w wantd to convrt th quation to th global coordinat ytm, w would nd th invr of (3): Ni y y Ni Ni x 1 r r 1 r N = = J i dt J x x Ni Ni y r whr dt Ji th dtrminant of J givn by : x y y x dt J = r r (4) Although w could intgrat (4) dirctly, it a littl inconvnint bcau it rprnt a full coordinat tranformation at vry point in th intgral w r going to prform. Fortunatly, w can u a hortcut 2011 Alx Grihin MAE 323 Chaptr 5 6

7 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt Th hortcut w u will allow u to intgrat ovr th paramtric coordinat (which rmmbr, alway rang from/to ±1) intad of tranforming into global coordinat. Th hortcut i mad poibl by th concpt of ubtitution of variabl*. Sinc w ar intgrating hap function (or drivativ of hap function), and th function ar ioparamtric, w know that: x y J x, y ± 1, ± 1 f ( x, y) dxdy = f ( N ( r, ), N ( r, ))dt drd Whr N x (r,), N y (r,) ar our hap function for th xand y-dirction, rpctivly. Thi i bcau dt Jactually rprnt a diffrntial volum ditortion (a mapping of th diffrntial volum in on coordinat ytm to othr): dt J= ( x, y) ( r, ) *W r howing th multivariat vrion, which i byond th cop of lmntary calculu. S: and: Alx Grihin MAE 323 Chaptr 5 7

8 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt Now, rturning to th govrning quation: T ( σ ) δ ε d = b w + FwdS W ar going to dicrtiz thi quation with our hap function. W hav now attachd a uprcript to all trm which will b valuatd on an lmnt bai. Bfor doing o, w mak u of Hook Law for an iotropic matrial to covrt th tr in th LHS to train (w want th quation in trm of a ingl primary unknown variabl. In our ca, thi will b diplacmnt): T ( ε ) Cδ ε d = b wd + FwdS So, w nd to writ th train vctor in trm of hap function. You alrady got a hint of how w will do in thi in Chaptr 2. W r going to writ th train vctor in trm of a train hap function matrix tim diplacmnt: ε = B u = d (6) S S (5) 2011 Alx Grihin MAE 323 Chaptr 5 8

9 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt Bi th train hap function matrix, and it i dfind by: B = N (7) Whr, i a train oprator. In th thr dimnion, it i givn a: 0 0 x 0 0 y 0 0 z = 0 y x 0 z y 0 z x N 0 0 = 0 0 N N 0 N 0 d u = v w 2011 Alx Grihin MAE 323 Chaptr 5 9

10 Putting It All Togthr MAE 323: Chaptr Alx Grihin MAE 323 Chaptr 5 10 In two dimnion: A Quadratic Srndipity Plan Str Rctangular Elmnt u v = d 0 0 x y x y = Subtituting in our hap function and convrting to paramtric coordinat: 0 0 r r = 1 1 (, ) (, ) n i i i n i i i N r u N r v = = = d = = N u N 0 u d N v 0 N v Or

11 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt Nxt, r-arrang dfor mor convnint torag and manipulation. Firt, xpand it in matrix form: d u1 u 2 u 3 u4 u 5 u6 u 7 N N N N N N N N u = N1 N2 N3 N4 N5 N6 N7 N 8 v1 v 2 v3 v 4 v5 v 6 v7 v Alx Grihin MAE 323 Chaptr 5 11

12 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt Now, rarrang: d u1 v 1 u 2 v2 u 3 v3 u 4 N 0 N 0 N 0 N 0 N 0 N 0 N 0 N 0 v = 0 N1 0 N2 0 N3 0 N4 0 N5 0 N6 0 N7 0 N 8 u5 v 5 u6 v 6 u7 v 7 u8 v Alx Grihin MAE 323 Chaptr 5 12

13 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt Now, calculat B according to (7): B N1 N2 N3 N4 N5 N6 N7 N r r r r r r r r N N N N N N N N r N1 N1 N2 N2 N3 N3 N4 N4 N5 N5 N6 N6 N7 N7 N8 N8 r r r r r r r r r = Bfor continuing, lt pau and rviw what w v don. W v calculatd an lmnt train hap function matrix, B according to: B = N 0 r 0 N 0 = 0 N r 2011 Alx Grihin MAE 323 Chaptr 5 13

14 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt So, w now hav an xprion of train in trm of local paramtric hap function: 0 r ε = B u = N u 0 N 0 u = (8) 0 N v r So, lt go back to quation (5) and plug in what w v got o far: T ( ε ) Cδ ε d = b wd + FwdS ( ) ( ) ( ) T ( B u ) C B u dt Jd = b N u dt Jd + F N u dt J ds S S 2011 Alx Grihin MAE 323 Chaptr 5 14

15 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt Aftr implification: ( ) T ( B ) C B u dt Jd = b N dt Jd + F N dt J ds S (9) Thi i th final t of quation which rult in th algbraic ytm: k u = b + F Elmnt tiffn Elmnt diplacmnt Elmnt body forc Global xtrnal load vctor Compar thi to th gnral (but non-paramtric) quation offrd in Chaptr 2 (quation (21)) for th ca of no body forc: T = = u C d F 2011 Alx Grihin MAE 323 Chaptr 5 15

16 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt Lt focu on th LHS of quation (9). W hould rmark that all finit lmnt quation involv a trm with thi form. In mathmatic, it i rfrrd to a a bilinar form. It alway involv an outr product of hap function and uually rprnt th intrnal nrgy of th ytm. In tructural mchanic, it provid u with th tiffn matrix ( ) T ( B ) C B u dt J d W hav almot all th ingrdint w nd now to calculat th tiffn matrix of a quadratic rctangular Srndipity lmnt for plan tr problm. Equation (1) provid u with th hap function, quation (7) provid u with B, and C for plan tr i providd from Chaptr 4: E C = 2 1 ν 1 ν 0 ν (1 ν ) / Alx Grihin MAE 323 Chaptr 5 16

17 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt Thr till on thing miing! How do w calculat th intgral ovr th domain hown blow? ( ) T ( B ) C B u dt J d r Thi actually can b don analytically. Eithr manually or with a Computr Algbra Sytm (CAS). Howvr, both tchniqu ar too low in gnral. What i ndd i a vry accurat and robut (aily programmd and widly applicabl) mthod of doing thi vn if it till only approximat. Hitorically, th mthod almot univrally adoptd i calld Gauian Quadratur, which tnd to giv vry good rult for th intgral of mooth (or picwi mooth) function 2011 Alx Grihin MAE 323 Chaptr 5 17

18 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt Gauian Quadratur Gauian Quadratur (omtim calld Lgndr-Gau Quadratur*) work by ampling th intgrand at point prcribd ovr th domain by th quadraturrul. Th point ar thn wightd and ummd, producing th an approximation of th intgral. n n ( ) ( ) T T ( B ) C B u dt J d B ( r, ) C B ( r, ) dt J ( r, ) w w i= 1 j= 1 i j i j i j i j Th two-dimnional quadraturrul i gnratd by taking th outr product of on-dimnional rul. Thu, if a thr-point rul i ud, th on dimnion location, r i and corrponding wight, w i ar found (lookd up from a tabl or calculatd). Th two dimnional point and wight ar thn found by th taking th outr product of ach (thu a thr point rul rult in nin point in two dimnion, and 27 point in thr dimnion). * Alx Grihin MAE 323 Chaptr 5 18

19 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt Gauian Quadratur Quadraturpoint ar uually givn on th intrval -1<r i <1, and o thi i anothr convninc providd by th ioparamtric coordinat Blow th coordinat for a two-point rul ar hown = r 1/ 3 r = 1/ 3 r = 1/ 3 r = 1/ Alx Grihin MAE 323 Chaptr 5 19

20 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt Gauian Quadratur Blow ar th point for a thr-point quadratur = 3 / 5 r r = 3 / 5 r = 3 / 5 r=0.0 r = 3 / Alx Grihin MAE 323 Chaptr 5 20

21 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt Gauian Quadratur Blow i a tabl of quadratur point and corrponding wight 2-point and 3- point quadratur Point Location Wight 2 1/ / / 5 5 / / / / So, how do w know how many point to u whn w intgrat uing Gauian Quadratur? Th rul ar drivd (in on dimnion) o a to intgrat all polynomial up to dgr 2m-1 xactly, whr m i th numbr of point ud in th quadratur! So, in principl, a 2-point rul would intgrat 2 nd and 3 rd dgr function xactly Alx Grihin MAE 323 Chaptr 5 21

22 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt Gauian Quadratur Blow, two curv: on 2 nd dgr and on 3 rd dgr ar intgratd xactly with a two-point Gauian quadratur p 1 p = 2 5x + 3x 1 p = x + 2x 4x p p1 ( x) dx (1) p1 + (1) p1 = = Exact! p2( x) dx (1) p2 + (1) p2 = = Exact! So, how do w know how many point to u whn intgrating lmnt? 2011 Alx Grihin MAE 323 Chaptr 5 22

23 MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt Gauian Quadratur Th guidlin for xact intgration i uually not followd in finit lmnt. On raon i that for 2 nd and 3 rd dgr hap function, th prcding formula would only b rliabl if th id of th rctangl wr traight (if th mid-id nod lay on a traight lin conncting cornr nod). Whn thi i not th ca, w hav a Jacobian with diffrnt valu at all point within th domain thi introduc rror into th intgral. Othr raon hav to do with mh intabiliti (which w ll dicu latr) and matrix ambly fficincy. In practic, a two-point quadraturrul i uually ud for linar lmnt, whra a thr-point rul i frquntly ud for quadratic lmnt Alx Grihin MAE 323 Chaptr 5 23

(1) Then we could wave our hands over this and it would become:

(1) Then we could wave our hands over this and it would become: MAT* K285 Spring 28 Anthony Bnoit 4/17/28 Wk 12: Laplac Tranform Rading: Kohlr & Johnon, Chaptr 5 to p. 35 HW: 5.1: 3, 7, 1*, 19 5.2: 1, 5*, 13*, 19, 45* 5.3: 1, 11*, 19 * Pla writ-up th problm natly and