Application of a Hierarchic Finite Element

8/13/2019 Application of a Hierarchic Finite Element

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APPLI CATI ON OF A HI ERARCHI C FI NI TE ELEMENt TO THE ANALYSI S OF SKELETAL STRUCTURES

Mans our Sedaghat Z. Ph. DDepar t ament o de I ngeni er ! aUni ver si dad de Concepci on

Wi l l i am Gi bson G.

Depar t ament o de I ngeni er ! aUni ver si dad de Concepci on

Ci vi lConcepci on - Chi l e

Ci vi lConcepci on - Chi l e

Par a e l anal i s i s de es t r uct ur as pl anas de secci ones e i ner c i as var i abl es , se pr opone un nuevo el ement o f i ni t o basado sobr e e l concept o j er ar ~qui co. El or den var i abl e de es t e e l ement o per mi t e un anal i s i s y model a-ci on r api do y ef i caz de est r uct ur as.

Se pr esent a e l desar r o l l o t eor i co del met odo j unt o con dos e j empl osde apl i caci on.

A new f i ni t e e l ement met hodol ogy based on t he hi er ar chi c concept i spr oposed f o r t he s t a t i c and dynami c anal ys i s o f t aper ed skel e t a l s t r uc -t ur es . The var i abl e o r de r o f t he hi e r ar chi c e l ement a l l ows f o r f a s t andef f i ci ent model i ng and anal ysi s.

Theor et i cal bas i s of t he met hod t oget her wi t h t wo i l l us t r at i ve exam-pl es ar e pr esent ed.


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To sat i s f y ar chi t ec t ur al and f unc t i onal r equi r ement s as wel l as abet t er di st r i but i on of wei ght and st r engt h, nonpr i smat i c beams ar e of t enused i n c i v i l engi neer i ng s t r uct ur es . Thi s paper deal s wi t h t he s t at i cand dynami c anal ysi s of l i near el ast i c st r uct ur es composed of beams wi t hvar i abl e cr oss sect i onal ar ea .

The i mpor t ance of t he anal ys i s of t aper ed beams was f i r s t s t r es s edby Ami r ki an (11 who used det ai l ed t abl es and by t he Por t l and CementAss oci at i on (21 wi t h t he i nt r oduct i on of a var i a t i on of t he momentdi s t r i but i on met hod. Newmar k (3) pr esent ed an appr oxi mat e numer i calmet hod t o det er mi ne s t at i c def l ect i on and moment s i n nonuni f or m beamsMore r ecent l y general pur pose f i ni t e el ement pr ogr ams, such as a SAP4 ( 4]have al l owed t he anal ysi s of t apered members by br eaki ng t hem i nt o anumber of uni f or m beam el ement s whi ch ar e t hen super i mposed t o pr oducet he desi r ed ef f ect of t aperi ng.

Gal l agher and Lee (51by comput i ng i t s st ef f nenpl acement f unct i ons. Thi sf ue vi br at i on anal ysi s.

i nt r oduced a gener al nonuni f or m beam el ement& consi s t ent man mat r i ces us i ng cubr i di s -el ement pr oved t o gi ve accur at e r esul t s i n

Resende & Doyl e ( 61t aper ed beams usi ng a 3

pr esent ed l at er an appr oxi mat e anal ysi sNode l i ne el ement .

Kar abal i s and Beskos (7) pr oposed t he st at i c and dynami c anal i si sof t aper ed pl anar beams whi ch exper i ence onl y axi al and f l exur al def or ma-t i ons t hei r met hod yi el ded t he exact st i f f enes mat r i ces f or r ect angul ar ,

box and I - sect i ons .I n t hi s paper a new f i ni t e el ement met hodol ogy based on t he hi er ar -

chi c concept , i s pr oposed f or t he st at i c and dynami c anal ysi s of t aperedskel et a l s t r uct ur es . Axi al , f l exur al and shear def or mat i ons ar e consi -der ed. The var i abl e or der of t he hi er ar chi c el ement al l ows f or f as tand ef f i c i ent r e- anal ys i s , t hi s , i n order t o i mpr ove accur acy and assur econver gence.

Most Ci vi l engi neer i ng skel et al st r uct ur es can be model ed wi t h t hepr esent el ement whi ch i s pr oven t o be more ef f i ci ent t han general pur posepr ogr ams of s t r uc t ur al anal ys i s s i nce some af f i c i ency i s l os t i n t hegener al i z at i on.

Theor et i cal bas i s f or t he anal ys i s i s pr es ent ed i n sec t i on 2,exampl er demost r at i ng t he capabi l i t y, accur acy and ef f i ci ency of met hod. ar e pr esent ed i n sect i on 3 and f i nal l y concl usi ons ar e dr awnsect i on 4.

t wot he

i n


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I t i s anumed t ha t t he t o t a l r o t a t i on o f a pl ane sec t i on,nor mal t o t he neut r a l axi s o f t he beam e l ement s , i " s due t o t heof t he t angent t o t he neut r a l axi s and t o t he shear def o r mat i on

or i gi nal l yr ot at i on

y

B = dv ydx -

1 L de 2 1 L d 2 1 L d 211 = - f EI ( - ) dx + f GAk( . . . . . Y- B ) dx + -2 f EA( . . . . . Y)dx

2 0 dx 2 0 dx 0 dx

SE Mi Bi -

i =1

P Lp v d x - f mBdx - f n u d x -

p

, < ffItn 1

'TIl.

Fi g. 1. - Appl i ed di s t r i but ed and Concent r aded l oads on t hebeam el ement .

A ba s es o f po l ynomi a l shape f unc t i ons i s f o r med such t ha t t he base sf or m a hi er ar c hy, i . e. The bas es of degr ee k c ont ai ns expl i c i t l ypol ynomi al bas es of degr ee 1, 2, , k - 1. Thi s pr oper t y i s l at erexpl oi t ed t o achi eve ef f i c i ency i n t he r e- anal ys i s of s t r uct ur a l pr obl emst o i mpr ove accur acy.

N (1) (E;) = 1 (1+1;)2 " 2


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~ N ( 1) +~ N ( 1) + a N ( 2)1 1 2 2 3 3

her e N3

( 2) i s a quadr at i c f unc t i on of t he f or m

N3

( 2) = Co + c l ~ + c 2 ~2

wi t h coef f i ci ent s ci chos en s o as t o gi ve N3 ( 2) = 0 at ~ = I It hi s i n or der t o pr eser ve t he r equi r ed CO- cont i nui t y of ~ bet weenel ement s . Thi s yi el ds

1...

d ~2 ~=o

Thus t he new unknown can be i nt er pr et ed as t he cur vat ur e at t he mi d-poi nt of t he el ement . The above pr ocedur e i s eas i l y gener al i zed t oobt ai n

~( ~) ~ N + ~ N + ~ ~ ( i ) N= 1 1 2 2 i =2 3 ( i +l )

=-1 0-~2)"2

= -1 (l - ~2H6 "

= -1 (l_~4)24

4( i )v m = vI N1 + v2 N2 + 1: v3 Ni +1 (0)i =2

4i

8 m = 81 N1 + 82 N2 + 1: 83 Ni +li =1wh e r e a l i near var i at i on f or t he axi al def or mat i on i s t hought t o besuf f i c i ent .


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To make f ul l us e of t he comput a t i onal ef f i c i ency of t heel ement t he gener al i zed despl acement vect or i s r eor der ed i nf ol l owi ng way

hi er ar chi ct he

(2) 0 (2) (3) 0(3) (4) 0(4)u 3 P3 v 3 P3 v 3 P3

and f or si mpl i ci t y t he condensed not at i on i s i nt r oduced f or t he aboveexpr ess i on

' "

{u( p) u( h) }T{u}~

and

u(~) N {u}u

v(~) N {V}v

8(0 N 8 {8}

The - s ame par t i t i on i s per f or med on t he shape f unc t i ons as soc i at ed wi t ht he t wo ext r eme nodal poi nt s and t hose associ at ed wi t h t he hi er ar chi cshape f unct i ons.

Nu [ N} NuhJ

Nv I

Nv ( p) : NvhJ

(14)

N8

[ N ( p) : N8hJ

8 I

The pot ent i al ener gy i s now mi ni mez ed as i n t he s t andar d f i ni t eel ement met hod, and t her ms ar e r ear r angea t o gi ve a st ef f ness and l oadmat r i x of t he f ol l owi ng condensed f or m.

[ : : : : : ] \ : : 1 { : : ~SUbmat r i x Kpp r el a t i ng t he t wo ext er nal nodes each wi t h t hr ee

degr ees of f r eedom wi l l a l ways be a 6 x 6 mat r i x . Submat r i ces Kph,Khp , and I t nn wi l l however have var i abl e di mensi ons dependi ng on t hedegr ee of appr oxi mat i on. Each added degr ee wi l l i ncr ease t he K mat r i xby t wo r ows and t wo col umns . Si nce onl y t he nodal unknowns v and 8ar e appr oxi mat ed by t he hi er ar chi c f unct i ons.

Ther ef or e on a f i xed mes h s ever a l anal ys i s wi t h pol ynomi al s of success i vel y hi gher degr ees ar e made by means of t he hi er ar chi c nest i ngof t he basi s f unct i ons . Comput a t i on of t he pr evi ous anal ys i s i s f ul l y


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ut i l i zed i n t he successi ve mor e accur at e anal ysi s .

Si nc e al l of t he {U( h) } unk nowns ar e as s oc i at ed wi t h a s i ngl ei nt e r na l node t hey can be condensed out , a t t he e l ement l eve l .

An i mpor t ant advant age of t he hi er ar chi c el ement i s t hat , moment sand shear var i a t i ons a t a l l po i nt s i n t he s t r uc t ur e can be accur a t l ycomput ed usi ng t he unknown associ at ed wi t h t he hi er ar chi c shape f unct i on.

2 .~=..1L+~

dx 2 EI GAK

dvdx - Y

~GAb

M = EI dedx

2Q = ( dI + dE I ) de + EI d e

dx dx dx d,1

2

P

(20)

and

Q E dI + dE . I ) . ~ ( _ . 2+~+e( 2) E; , ( 3) 1 E; , 2

dE; , dE; , L 2 2 3 +e3 ( - 6+2"

3( e( 2) e( 3) E; , e( 4) E; , 2 ( 21)+ e ( 4) f ) + EI + + 2) 43 6 3 3 3 L

We not i ce agai n t hat t he comput at i on of t he unknown associ at ed wi t ht he hi gher or der hi er ar c hi c s hape f unc t i on al l ows a s er i es t y pe of appr oxi mat i on f or bot h M and Q. I ner t i a I and modul us of e l as t i c i t yE, ar e assumed t o be known and var \ abl e al ong t he el ement .


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As mode of compar ai son i t was necessar y t o use 52 el ement s wi t h t her egul ar st i f f ness met hod t o obt ai n an accept abl e val ue f or t he def l ect i onat poi nt Cof 4. 181 r om.

The Fr ame of Fi gur e 3 i s next anal yz ed, not i ce t hat i n ever y cas et wo nodes def i ne each el ement i ndependent of t he degr ee of appr oxi mat i onsought .

Fi r s t , second and t hi r d f l oor have a di s t r i but ed l oad of 50 kg/ em,t he l ast f l oor i s l oaded wi t h 35 kg/ em. Shear and moment di agr ams ar eeasi l y f ound us i ng equat i ons ( 20) and ( 21) . The dynami c anal ys i s i scar r i ed out us i ng modal super posi t i on. Concent r at ed masses ar e used andei genval ues and ei genvect or are f ound usi ng t he subs pace i t erat i on al go-r i t hi m.

Fi gur e 4 shows moment & shear di agr am, & Fi gur e 5 t he f i r s t , secondand t hi r d mode of vi brat i on.

A new f i ni t e el ement speci al l y wel l sui t ed t o t he anal ys i s of non-pr i smat i c skel et al s t r uc t ur es i s i nt r oduced. The hi er ar chi c base of shape f unct i ons , al l ows f or r er et ed anal ys i s wi t h hi gher and hi gherdegr ees of appr oxi mat i ons wi t hout h modi f yi ng t he or i gi nal node and el ementnumber i ng. Al so each anal ysi s ut i l i zes f ul l y al l of t he comput at i ons of t he pr evi ous st eps. Very f ew el ement s ar e r equi ~ed t o model accur at el ycr oss sect i onal and i nert i a var i at i ons whi ch ar e appr oxi mat ed by pol yno-mi al s up t o degr ee f i ve.

A car ef ul l compar ai son wi t h t he st i f f ness met hod and t he st andar df i ni t e el ement met hod, showed t hi s el ement t o be mor e economj c comput at i onal y. Fi nal l y shear and moment val ues ar e accur at el y det er mi ned i n al lpoi nt s of t he s t r uct ur es us i ng t he pr i mar y var i abl es , and ar e not f oundas secondar y var i abl es as i n t he s t andar d f i ni t e el ement or s t i f f nessmet hod.


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' ? ' 7 e ;;

Element numberslie il~ ! 2 ' cI

I

+. . . . .-- ; ~" , , ,

l i E ,,-,-'" ,.1,!

+ e . . 4 1,

;: .~ ~t.:.t " . , )

i 2 " jI

i1 1 i'10

I

\ i

E J J :': "


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--~. . . . ~--~~

. . -. . . .


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;....._--- -_.~----- ----~. j,


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1.. Amirkian, A. ''Wedge- beam framing" Trans. ASCE117, P 596 - 652(1952)

2. P.C.A. Handbook of Frame Constants, Portland Cement Association,Old Orchard Rd. Skokie, Illinois, (1958)

3. Newmark, N.M. "Numerical Procedure for Computing deflections,moments and buckling loads". Trans. ASCE108, P 1161 -1188 (1943)

4. Bathe, K.J, Wilson, E.L., Peterson F.E. "SAPIV: A StructuralAnalysis Program for Static and Dynamic Response of Linear Systems",EERC 73-11, Univ. of California, Berkeley. June 1973.

Gallagher R.H., Lee, C.H.of non-uniform elements".( 1970)

"Matrix dynamic and Instability analysisInt. J. Num. Meth. Engng 2, p 265 -275

6. Resende, L. , Doyle, W.S. "An Effective Non-prismatic Three Dimen-sional beam Finite Element". Computers & Structures, Vol. 14 p. 71- 77 (1981)

7. Karabalis, D.L. and Beskos, D.E. "Static and Dynamic Analysis of Structures Composed of Tapered Beams". Computer & Structures Vol. 6

pp. 731 - 748 (1983)

8. Gibson, W.G. "Analisis de Estructuras Planas }'.ediante ElementosFinitos Jerarquicos". Tesis de Ingenieria Civil, Universidad deConcepcion, Depto. Ing. Civil, Concepcion - Chile (1984).

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Application of a Hierarchic Finite Element