A finite element for the elastic stability analysis of frameworks.

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A FINITE ELEMENT

for the

ELASTIC STABILITY ANALYSIS

OF FRAMEWORKS

A thesis suburitted in partial fulfil-nent of

tJle requirements of the degree of Doctor of Philosophy at the University of Auckland.

by B.J. Davidso:r,

Auckland UniwersLty, School of Engineering, De5nrtment of llheoretical anil Applietl lrtechanics,

April

^L976.

I

I

Futal

llke to

tltank

tlre

^{any people} whs hav.e hel;peii

,m6 iiurlng tlre p-eioil o€

reeerlqah that h.as lead

to

the Flesentatio.n

of thls €tes:i.s'

^an trlertlqrtrBt,

!

would ll,ke

to

tt8nti ag suFe:Ftrleor, D! r.C. uedla$d Hho inlti,ai,edl thte

rqrkr

alal has been an ever crgailabte strilr@ of, glrtdance andl ebdorrrdgeqent tq ^toe,

t

^would also

llke to

exte{d! Ey tjln$ks bo Profes.gor C.l{. $egedl*n ald the ,saaff of, the DepEltnrert

of

tbeo-:eticaL ,ard q!E{t!€d }tech4:[F!r rrlio have asgistedl ne

ta

rry eqlfu

nith thetr

b€lp andt f:ai,endshtp.

r 4

qtratefql foq ttle atd ^giveq .bt ^tAe ,leolrnLs.{anis ,of tbe &rgtneerdog ^Scbool FrhEheps trho be'lpait @ffl$aceu& tbe e$lilritFDt

ttlat

wae uset

to

^{tEst the} loaal-tlefleotlon

behavLo$r of, tbe alrved

r

^bea&| analysed ia, t$6s t$esis, ^andt Profreesor

$.1,

Uslebr&y &t'd

t[r

iI.A. Hoodl :o,f,

tjie

liqrartilent, 6f, Civ,i,l' Ebg.ibeerilr|g .tr.t-e rradb ,avallaUle :.abolratcEi' sFace

fot

^t":he tdsting

of

th!,E basn.

it

|ihe elastte and gemetrto sti.,ffness natrlces are (leveLoped fron

lst

prlne&rlea,

fu

^'a bemqqileli els{rfiEt rhlah

is

^.tp }te uEed

in

tlre

llnqr elaetic stabillty

^amfye,lg

of f;rilenorks..

^|!he el€ngrt fomrlatl,oa

ls

extensively testedl agalnst cLassical ritnil

etitr|grisehta1 re€nfltB

fer

b.eass, eo\wre, aadt

f-rmes.

A Fraetical agpl-lcatloa of the eleD€lrt

ts

denanstrated by us"ing

it to

investigate ttre latera.tr etabLl!.ty

of

^a. tfrrrbei

of

riE5id jotnteil trRrssssr. 8be braclngl leguiri4rrents

of

t&ese trusses 6r€ cqrpBred gt{:h _ttrB regulr,aerrts

of

the bEaci'rg to ^pi.nrred sollilmBr ntrl.gb are ^.fihg sEme alre ^gE the c.oogreeeio

@qpg

of tie

tqEgsee, andl have

ettlief

a constaot

or

steFr'ise parabolJ.c d[str-iboGlo.r of,

aril.al

load'. rf ia

fbundt .for nost braeilE ciases,

th.t

ttte crltl,c€rl load

rtrt

^{the braclng}

requfu,slenta

for

a tmssr €a$ be elosely estiinatoal

fM

^-a mralysls

of

a cohm

ulth r

rpa*abollci

axlal

loail

dls$bqtl.q.

Itt

NO:fAElON

q8er,troN'l

-

NEnourejlJor{

1.1

fritrneslrl<s

L.2

,t'Gthed,s

of

dletei*rdnlng tlie

elcstic

crtttcatr, :lqaCls

of

'

_ltanensrhe

I.3

^{?fie 6eope}

of

thi,s Iqdt

sqlr

.roH 2

- rt

DE{r,srcnmlT oF TrB- EE!{E!tt EI.!ASR, C D@ @OD-qUReg STIEFNESS UATRICES

Z,l

Introdluction

2.2

^ltttle va:rlat:lonal fo'in

of

ttre bneEllng

crlterlsn 2t3

Elrin-ualledt beans

2.4

llhe ftrf,tGe etriiirsent $oite!.

2.5 ln

atl5>ralsal .of the formulation SECIIO{ a

-

ONE DD'IENSICE{II, STII,CXIIRES

3.1

lhtrctaluctioD

3.2

^lEhe solutton ^Eeguene€

3..3

^lEhe 'dieptaceoent. analysia

3.t

^!.1le brrckllng an!-ysl,*

3.5

lronltnesrtties

in

tlre Hrist

of

a bean as ^a

rggttlt

df an applledl blnonent

3,6

^,llio de€ernlne the

er{tical

load.: brasc stlffneas

*lationships fbr

a collIi[t $rpPortea by dllsclete

lateral

braciag

3.? fo

darternlne the strenltf,h

of

di.seete ^{trate:aal and} toisloaal. braces attached

to

a esstlEesgedl nei$€8 gEglQlt 4-

-

^f'!0Q AIiP $iEEE DIJTEI{9IoNSL S!ffIg.rURBS

,0,.1

IntroilustLon

*.3

lhe eff,ects ^.aD .frimerroirh3 of tratp.i.ng restralats aPEil.teal

to

twigting netrbers

{.3

lhe out-of-plane buckling

of

a poatal ^faane

t.4

^ittre analysia

of

a etr,rved thin'ryalled ^bean uelng a sArlaight f,-,trn*te elenrent

S@-!ION: 5

-

T'tiE IAIERA;L A]lE .T0R,SrO!!A!, sRAq4{B RD$IIRE{E{TS

9r

P4,84[!4 cnoRp'l?ugsEs

5.1

Introaluotdo6

5.2

treselrtptlon

of

a tltEtca[

trust

5,3

Ffutite el,enrent, aaalysi.r

of

the ^t|ntsses

5n,d

B.tl.ff,ness of, trateiial arrCl tongicrral bracing

t6

a unlf,orn

coliuanr

5.5

fhe strengtlr

of

br.ictsgi

516.

lD&eeusslon

of

results

5.7

^Coneluslon

Prg€

,l

t,

5 6

.9t?

itl

25,

.t5

26 3(' XI

{3

45

{9

49 52

55

:63

65

58'

,81 BE 97

ldt

.ly

APEENDTCES E2.I

\2.2

A2;3 A2.4

lIhe iiefivat-loD of. th.e lbt-ential ^,€Ielrgiy terrg given

in

equatlors (2.i31 and t3.34).

ltte elgrent

elastic

and g€@etr{c sttfftlesa

nAtsice'E.

Ehe buclrltng nork dldn€'by the forces agttng oll a De.anr,

the natu:rial boudar| colrd:Ltion,S re,atrtrt{ng f_r@

€he f,Orfirq;

of

^the poten€ial. _€nerrgy assoc{ateil- nd.ch latotdl:tot.Eiona1 b-trcklilig userdl

iq

,thtre work andl

fn.tll.

Elbe sslutiani of,

t$s foijrtli

ot-der g,lenultaneous l!figgencouF gqrraiElqros xrith vaiiirble @efflcielrts url:ng the Runge-E.|rtla !Eth!d.,

El.fe Fertu&Etngp rettlg-al fo-r rjle _sqlut{on

of

the .nonllnegr _{S equat*on.}

te

girtel$Lae _ttle:

erltlcat

load: brace stl.f-fnesa

. t€lat{orurh.ip fOr a

celtrally

braeed untfo-rtil1r alontrlessed coLuuri.

Ahe st*ergth, reqpirenrents

qf

a lateratr

or

tor.atronal bracc..

Ea exllerittefitaXlt dete.cnlng gle _cen.tral

tieflestlo!

of

a r,lng quadrant Looded out-sf-p1ane.

She dteslgn Galculations f,or

tte

^trus€ea

ilesani,bedl

lrl

^n5.2

i

'

Convergenee

of

^rnaxlmun

erltiEal

load.s rtElr

ueah, re,f,futedeBt f,or

a

f,ou* ptrnel.

truss.!

|!be n€uiiprr[.ation

of

a ,stiff,Degs Datr.la to ., aclt:lerye a proportl,onallty betrveen sgreelfleil rdegrees

of freedq.

The ,adtlltiOnal telrns addedl to tbe stlf.;fnes-r

rEtrices

of

a trusE when tbe w€b

ls

attached

to

^the ccclpression chord

at

a position other

tian tts

shear cEnt.re.

:Ehe

lateral

buckling nodes

of

a Ftnried ^colutrn,

wlth a parabollo dlstrLbution

of axial

load.

'!the eritLoatr troad:

lateral

strengtb r-dlatiorr- shltrts

for

5 and. 7 braees

to

a cpfugtn.

Flogrlcan

Lletlng.

FaEe

lo9

u3 tt5

120

123

125 129'

139

l.lO

tt2

l4-tl

tl8

&3r1

a3.2 A9.3,

t\il.4

A4.1 e5.1

16.,2 tr5.;t

A5.4

133

llt

A5.5' ,\5.6 45.7 RmnENcEs

!r0E

ta9 r59

NOTATTON

4ll

v.artables an6 op€ratora afe deftned when t$oy

first

^aplrcqr

lr

tbg

t€ftt

Uri e-omolfy used orcs.ara sumnflsed bere.

I -

CroesrgectlgDa:. a'.$a

of

^a. Feq

%

Dlfferefrtlal elnro,ent

of

^area

of

a bodlg

a -

^gnc.ing betlireen braeea

l-r ! -

^{Positlon o.f}

lateral

^ltxrds

qr

^{a bean rElatlve}

to

^{1t9 ehear c€*rtrec}

.a-,

!ltE-tr

t, -

Fos.ttion of, attaclmieDt

of lateral

braseq

relatlva to

^bedit Abeat

t' Y

_centre

B -

^BfogES_t

D. b- -

Stlf&rese

of litatal

^aod tsorslc'lral lirraclng

tla.

o - - y'tolsrt

ct3ft -

^{f,at€Eial tes!,onEe f,uDctldr}

D _

Vo1Uire

of

a bo<ty

aD -

^Surface

of

^boity

a -

^Veetor

of

^nodal tltsglqcewst€ tn. local Eg*ordlil|ate3,

?

E -

^{Youngre reilulue}

e ,-

Non-dinensiopal fuEerfectlon sise

la

a gohun

.1' -

ttneS! straln

tglsor

^;

8--r

r-y G_ -

Poeltlm.of aSpllcatton

of

^{a polnt} load on an enil

of

a bean

F*r F"r

F"r -

^Fqrees

ln

the res[pct$e dLrectlogs

fDl fr -'

Non:€J,tne$s.tirtral, &atefal and Earsional brtcthg s.tlffnqeg

f-- -

Non${or€nsisna1, ^o:8n11 braei,ngF st{ffnese f,or'etttrer ,Iateral or

.cr

.toreioiral _b'iaces

G -

^{Shear rcthrlue}

9..

D'

-

Clsoss-s€ettor pfoE)eEtf!

of a

li*.rln-ralled beau

h - ebaracterlpli.c ili.nensl,gt-r of tlre ,crossiseetibn of a ddn-$Frlledl berr I -, I

r-y

-- &rcond Dirmentc of anea

t"" -

Polar rffirEnt

of

area about the sltear oedtsB o.f a be@

t -

^{Iorsional oohstant}

ESI

trgl - glaslig

rrl.i gecilnetrlc stXf,fnass ,lr€rtrloes

of

a

structqr eJ enl -.

^Elastic andl lFGmEtE.ic stilf,fness &alEtraes s.f, the ^e'lesrenL

t -

^I€ngth

of

^{a colurnn or bean}

-

,. -

!ength

of

an eldnent

vl

t|-

r.y't r tt-r ll_ -

I-'totBents about r€Steeti.9e axes

lt-, t' -p

u

- t&e{ts

ftoiln Erure torsl^on atld wErtr :estra,lned toqsion Eajspecti.irely.

n -

^$rilDer

of

bal,t ^$aves Ln an

inittally

irrtrlerf,ect

coh&

!l -

^{gSrread totque about.} t$e longLtudtl,nat aal6

of

an el€$ent

A -

^{Nuhb€r gf, brtsces atGached}

ts

a coltnn

r -

^Diatance

ln

dtirectlon

of

outmrd. nsnral

of

cunririnear co-ordLnatc Bystell

O -

^Odgtn

of

a eo-ordinate syeteu

ON, OY,

OA -

Direi:tlong

o!

tr:rttrgil,al ^{a!.es of,} a bed

P -, ITI*I

compresSlvg force

P -

^strenEth

of

btacing

F; r

ParabolLc loadl factoa

I -

^Any ltotat dr ^the sioss-.sedtio_n of, a beA

qL -

^{Effective length}

of

braee

l, L -

^Vector

of

no€al foteeq

io

global c9-ail.dinates, Gor

ihe

structrre and an element res;reetLvely.

l, i. - HT:i#ll.r*ff:tffiff" ln

elobal co-stdlaatea. f,o! tso stnetura

: t-

Veetor o-f nadal. fotccs

in

^roeal eo-oaeLaatgsl

a.c. -

^{|ltre Bb€cs aentre}

of

e beao

r -

^Dt'stanse ftao edge

of

bese toi..s@e point 0 ^i'n

its

uidld!.e srjrfac€

!f. -

^{SracLions acting oD a boalt}

t -

,|Ihlckness

of

bean

[tJr[t"l -

Transfornraltion

nattlx for

struoture ard elenrent respectlwly

U -

^{Straln eDnngy}

u,

@ne.lraltsed dislrlaeem€nt8

.

_{o3 vp} '.

,E r

Displaeeog!.t8 o-f

bes 4!

di.rections OX, Oy, OEl .regpectl,vrl. _r.

V , -

'Pstential assoc{ated wtth fgxceg

ffO

DLff,erential ielenent

of

v6lirsre,of E bolly

W

ltork doDe by f,qrcee

,r -

sPreaa loadl dn betqr

ln

ox

diFctlon

*r lt I -

Dlstarroee 'tn tbe dlreotioqs, O!t;, Oi., ^,q,9,

Ff 82 -

^tsearn c.ross-aectiohal p€olrertLes

f -

^{Wb'r.pinS consltent}

y -

^Order

of

par_ttal derivatllve

in

poteatl,al energly f.uneti.irnal

6 -

VarLatl:omal qperatpa

o'

*tJ - Ntlitinear

:straln teasor

G -

^srnatl Pataneter

vtt

tr Ir I - Dlsplaceltlents in the curvillnear co-ordl-nate systcm (n, s, z) 0- , 0-, 0_ - I\rist ^about respective axes

x-y-z

I - Itaal factor

\"*

\, II p

o

otJ

t

0

ol

"tJ

.+

- ^Value of load factor when bracing attached to structure is "full" or tiqid"

- Polsson's ratio - Potential ^energy

- Nondirnensionalised loail factor - Dlrect ^stress

- Stress ^tensor - ^{Shear stress}

- fi,rist ^about longitudinal axis of bearn

- Sectorial ^area - Rotatlon ^tensor - ^{A path}

F_,, x- !l_, B z- - ^Bar inilicati:rg applled force

viii