Nonlinear FE Analysis 2016

8/19/2019 Nonlinear FE Analysis 2016

1/41

INTRODUCTION TO NONLINEAR FINITE

ELEMENT ANALYSIS

Doc. Ing. Vladimír Ivančo !"D.

Fac#l$% o& M'c"anical Engin''ringT'c"nical Univ'r(i$% o& )o*ic' Slova+ia

,S -i(mar #n' /001


2/41

NONLINEAR FINITE ELEMENT ANALYSIS 2


3/41


CONTENTS

2. S$r#c$#ral nonlin'ari$i'(...............................................................................................3

2.2 In$rod#c$ion...................................................................................................3

2./ T%4'( o& ($r#c$#ral nonlin'ari$i'(...................................................................5

2.6 Conc'4$ o& $im' c#rv'(..................................................................................5

/. 7'om'$ricall% nonlin'ar &ini$' 'l'm'n$ anal%(i(..........................................................8

/.2 Larg' di(4lac'm'n$ and (mall ($rain 9'"avior...............................................8

/./ Incr'm'n$al : i$'ra$iv' (ol#$ion(...................................................................2/

/./.2 Incr'm'n$al m'$"od.........................................................................23/././ N';$on:Ra4"(on m'$"od................................................................25

/./.6 Modi&i'd N';$on:Ra4"(on m'$"od.................................................28

/./.3 ............................................................................................................................60E>am4l'( &or COSMOS?M 4rogram..........................................................................62


4/41


1. STRUCTURAL NONLINEARITIES

1.1 Introduction

Man% 'ngin''ring 4ro9l'm( can 9' (olv'd #(ing a lin'ar a44ro>ima$ion. In $"' Fini$'El'm'n$ Anal%(i( @FEA $"' ('$ o& '=#a$ion( d'(cri9ing $"' ($r#c$#ral 9'"avio#r i( $"'nlin'ar

F d K [email protected]

In $"i( ma$ri> '=#a$ion K i( $"' ($i&&n'(( ma$ri> o& $"' ($r#c$#r' d i( $"' nodaldi(4lac'm'n$( v'c$or and F i( $"' '>$'rnal nodal &orc' v'c$or.

C"arac$'ri($ic( o& lin'ar 4ro9l'm( i( $"a$

• $"' di(4lac'm'n$( ar' 4ro4or$ional $o $"' load(

• $"' ($i&&n'(( o& $"' ($r#c$#r' i( ind'4'nd'n$ on $"' val#' o& $"' loadl'v'l.

T"o#g" 9'"avio#r o& r'al ($r#c$#r'( i( nonlin'ar '.g. di(4lac'm'n$( ar' no$ 4ro4or$ional $o$"' load(B nonlin'ari$i'( ar' #(#all% #nim4or$an$ and ma% 9' n'gl'c$'d in mo($ 4rac$ical

4ro9l'm(.

T"' ('$ o& lin'ar alg'9raic '=#a$ion( @2.2 i( r'c'iv'd i& a((#ming $"a$

• di(4lac'm'n$( ar' (mall and can 9' n'gl'c$'d in '=#ili9ri#m '=#a$ion(

• $"' ($rain i( 4ro4or$ional $o $"' ($r'(( @lin'ar ,oo+'an ma$'rial mod'l

• load( ar' con('rva$iv' ind'4'nd'n$ on di(4lac'm'n$( and (#44or$( o&$"' ($r#c$#r' r'main #nc"ang'd.

Sol#$ion o& man% 4ro9l'm( n''d( a9andonm'n$ o& $"'(' a44ro>ima$ion(. T"'di(4lac'm'n$( ma% 9' (o larg' c"ang'( o& $"' ($r#c$#r' ("a4' @or con&ig#ra$ion c"ang'(canno$ 9' n'gl'c$'d. Man% ma$'rial( 9'"av' nonlin'arl% or lin'ar ma$'rial mod'l canno$ 9'#('d i& ($r'(( '>c''d( (om' val#'. Mor'ov'r load( ma% c"ang' $"'ir ori'n$a$ion(according $o di(4lac'm'n$( and r'($rain$( ma% c"ang' d#ring loading. Con('=#'n$l%($r#c$#r' 9'"av'( nonlin'arl%. I& $"'(' 4"'nom'na ar' incl#d'd in a FEA $"' ('$ o& '=#ili9ri#m '=#a$ion( 9'com'( nonlin'ar.


5/41


1.2 Types of structural nonlinearities

S$r#c$#ral nonlin'ari$i'( can 9' (4'ci&i'd a(

1. Geoetrical nonlinearities T"' '&&'c$ o& larg' di(4lac'm'n$( on $"'ov'rall g'om'$ric con&ig#ra$ion o& $"' ($r#c$#r'.

2. !aterial nonlinearities Ma$'rial 9'"avio#r i( nonlin'ar. !o((i9l'ma$'rial mod'l( ar'

a nonlin'ar 'la($ic

9 'la($o4la($ic

c vi(co'la($ic

d vi(co4la($ic.". #oundary nonlinearities i.'. di(4lac'm'n$ d'4'nd'n$ 9o#ndar%condi$ion(. T"' mo($ &r'=#'n$ 9o#ndar% nonlin'ari$i'( ar' 'nco#n$'r'd in con$ac$

4ro9l'm(.

Con('=#'nc'( o& nonlin'ar ($r#c$#ral 9'"avio#r $"a$ "av' $o 9' r'cogni'd ar'

a T"' 4rinci4l' o& (#4'r4o(i$ion canno$ 9' a44li'd. T"#( &or '>am4l' $"'r'(#l$( o& ('v'ral load ca('( canno$ 9' com9in'd. R'(#l$( o& $"'nonlin'ar anal%(i( canno$ 9' (cal'd.

9 Onl% on' load ca(' can 9' "andl'd a$ a $im'.

c T"' ('=#'nc' o& a44lica$ion o& load( @loading "i($or% ma% 9'im4or$an$. E(4'ciall% 4la($ic d'&orma$ion( d'4'nd on a mann'r o& loading. T"i( i( a r'a(on &or dividing load( in$o (mall incr'm'n$( innonlin'ar FE anal%(i(.

d T"' ($r#c$#ral 9'"avio#r can 9' mar+'dl% non:4ro4or$ional $o $"'a44li'd load.

' T"' ini$ial ($a$' o& ($r'(( @'.g. r'(id#al ($r'(('( &rom "'a$ $r'a$m'n$;'lding '$c. ma% 9' im4or$an$.

1." Concept of tie cur$es

For nonlin'ar ($a$ic anal%(i( $"' load( ar' a44li'd in incr'm'n$al ($'4( #(ing time curves.T"' $im' varia9l' r'4r'('n$( a 4('#do $im' ;"ic" d'no$'( $"' in$'n(i$% o& $"' a44li'dload( a$ c'r$ain ($'4.For nonlin'ar d%namic anal%(i( and nonlin'ar ($a$ic anal%(i( ;i$" $im':d'4'nd'n$ ma$'rial

4ro4'r$i'(2 $im' r'4r'('n$( $"' r'al $im' a((ocia$'d ;i$" $"' load(G a44lica$ion.

2 i.'. anal%(i( o& cr''4 and r'la>a$ion 4ro9l'm( 9% #(' o& vi(co'la($ic or vi(co4la($ic ma$'rialmod'l(.


6/41


A( an '>am4l' $im' c#rv'( o& &orc'( F 2 and F / loading (im4l' 9'am ar' di(4la%'d inFig#r' 2.2. Val#'( o& &orc'( a$ an% $im' ar' d'&in'd a(

( ) 222 f t F λ = and ( ) /// f t F λ =

;"'r' f 2 and f / ar' nominal @in4#$ val#'( o& &orc'( and λ 2 and λ 2 ar' load 4aram'$'r( $"a$ar' nc$ion( o& $im' t .

Fig#r' 2.2 E>am4l' o& $im' c#rv'(

T"' c"oic' o& $im' ($'4 (i' d'4'nd( on ('v'ral &ac$or( (#c" a( $"' l'v'l o& nonlin'ari$i'( /

o& $"' 4ro9l'm and $"' (ol#$ion 4roc'd#r'. 7'n'rall% (#&&ici'n$l% (mall ($'4( ar' n'c'((ar%

$o (im#la$' nonlin'ar r'(4on(' o& a ($r#c$#r' ;i$" (a$i(&ac$or% acc#rac%. On $"' o$"'r "andlarg' n#m9'r o& $oo (mall $im' ($'4( #('l'((l% incr'a('( con(#m4$ion o& C!U $im'.Com4#$'r 4rogram( ar' #(#all% '=#i44'd ;i$" an ada4$iv' a#$oma$ic ($'44ing algori$"m $o&acili$a$' $"' anal%(i( and $o r'd#c' $"' (ol#$ion $im' d'mand(.

/ ,ig"l% nonlin'ar 4ro9l'm( n''d (mall'r load incr'm'n$(.


7/41


2. GEO!ETRICALL% NONLINEAR

&INITE ELE!ENT ANAL%SIS

2.1 Lar'e displaceent and sall strain (e)a$ior

To '>amin' g'om'$ricall% nonlin'ar 9'"avio#r ;' ;ill ($ar$ ;i$" an '>am4l'. -' a((#m'larg' di(4lac'm'n$ 9#$ (mall ro$a$ion and ;"a$ i( $"' mo($ im4or$an$ (mall ($rain. T"'($r#c$#r' i( v'r% (im4l' H onl% on' 9ar $r#(( a( i( ("o;n in Fig#r' /.2. A$ $"' 9'ginning;"'n $"' &orc' P i( 'ro $"' a>ial &orc' N in $"' 9ar i( 'ro $oo and 9ar "a( i$( ini$iall'ng$" L0.

Fig#r' /./ E>am4l' o& nonlin'ar ($r#c$#r' H (ingl' 9ar $r#((

U(ing $"' &r'' 9od% diagram ("o;n in Fig#r' /.2 $"' '=#ili9ri#m '=#a$ion i(

0(in =− P N α or

.0=−+

P L

u N @/.2

A((#m' $"a$ ma$'rial i( lin'arl% 'la($ic ;i$" Yo#ngG( mod#l#( E! T"' a((#m4$ion o& (mall($rain( m'an( "'r' $"a$ c"ang'( o& $"' 9ar cro(( ('c$ional ar'a A can 9' n'gl'c$'d. T"'na>ial &orc' in $"' 9ar i(


8/41

NONLINEAR FINITE ELEMENT ANALYSIS "

ε 0 A E N = @/./

;"'r' A0 i( $"' ini$ial cro(( ('c$ional ar'a and ε i( $"' e#$i#eeri#$ str%i# d'&in'd a(

0

0

L

L L−=ε . @/.6

A( l'ng$"( ar' giv'n a(

//

0 % L += and // A@ u% L ++= @/.3

$"' '>4r'((ion &or ($rain i( g'$$ing ra$"'r com4lica$'d. -' can ov'rcam' $"i( 4ro9l'm 9%in$rod#cing &ree#'s str%i# d'&in'd a(

/0

/0

/

/ L

L L&

−

=ε @/.5

;"ic" &or o#r 4ro9l'm 9'com'(

/

000 /

2

+=

L

u

L

u

L

&ε . @/.

U(' o& $"i( n'; m'a(#r' o& ($rain i( 4o((i9l' 9'ca#(' ;' can d'&in' ($rain ar9i$raril%. T"'onl% condi$ion i( $"a$ $"' ($rain m'a(#r' m#($ 9' o9J'c$iv' ;"ic" m'an( $"a$ i( "av' $o 9'ind'4'nd'n$ on c"oic' o& coordina$' (%($'m and in('n(i$iv' $o a rigid 9od% mov'm'n$.

From '=#a$ion( @/.6 and @/.5 i$ &ollo;( $"a$

+

−=

−+

+=

+=

+−=

−=

0

0

0

0

0

0

0

0

0

0

0

0

/0

/0

/ /

/

222

/

2

/

2

// L

L

L

L L

L

L L

L

L L

L

L L

L

L L

L

L L& ε ε ε ε

or

/

/

2ε ε ε +=& . @/.8

No$ing $"a$ $"' con($i$#$ion '=#a$ion ;a( m'a(#r'd a(

ε σ E A

N ==0

@/.K

$"' (am' con($i$#$iv' '=#a$ion ;"'n #(ing 7r''nG( ($rain ("o#ld 9'

&&&

&

E E E ε

ε

ε

ε ε

ε ε

ε

ε σ

/

22

/

2 / +=

+==

. @/.1


9/41

NONLINEAR FINITE ELEMENT ANALYSIS (

T"i( m'an( $"a$ ;' ("o#ld #(' val#' LA/?2@2?M ε +=∗ E E in($'ad o& E in $"' con($i$#$ion'=#a$ion. For$#na$'l% ;' can ignor' $"i( com4lica$ion no; 9'ca#(' &or (mall 'ngin''ring($rain i( $"' di&&'r'nc' 9'$;''n 'ngin''ring and 7r''nG( ($rain n'gligi9l'.

For '>am4l' con(id'r $"a$ ε 000/ @'.g. mild ($''l %i'ld( a$ a9o#$ $"i( val#' $"'nε & 000/ 05⋅000// 000/00/. T"i( m'an( $"a$ di&&'r'nc' i( onl% 02P i.'. a val#'

$"a$ can 9' #(#all% n'gl'c$'d. A((#ming $"a$ ($rain i( (mall ;' can ;ri$' & E ε σ ≈ andaccording $o '=#a$ion @/.

+= /

/

0

0

/

2uu

L

A E N . @/.20

S#9($i$#$ing @/.20 $o '=#ili9ri#m '=#a$ion @/.2 and a((#ming $"a$ &or (mall ($rain i(

0 L L ≈ giv'( $"' '=#ili9ri#m '=#a$ion

( ) ./6/

//6

6

0

0 P uuu L

A E =++

@/.22

O9vio#(l% $"' '=#a$ion i( nonlin'ar ;i$" r'(4'c$ $o di(4lac'm'n$ u. T"a$ m'an( $"a$r'la$ion 9'$;''n load P and di(4lac'm'n$ u i( r'4r'('n$'d no$ 9% a ($raig"$ lin' a( i$ i(;"'n c"ang'( o& con&ig#ra$ion ar' n'gl'c$'d 9#$ 9% a c#rv'. T"i( nonlin'ar c"arac$'ri($ic&or E /2⋅205 M!a A0 200 mm/ % /00 mm and /0 mm i( ("o;n in Fig#r' /./.

-1000

-500

0

500

1000

1500

2000

2500

-10 -8 -6 -4 -2 0 2 4 6 8 10

u [mm]

P [N]

Fig#r' /.6 7'om'$ricall% nonlin'ar 9'"avio#r o& a (ingl' 9ar $r#((

T"'r' i( ano$"'r 4o((i9ili$% $o o9$ain '=#a$ion o& '=#ili9ri#m @/.2 or @/.22. From 4rinci4l' o& vir$#al di(4lac'm'n$( i$ &ollo;( $"a$ )e# te structure is i# e*ui+i,rium-


10/41

NONLINEAR FINITE ELEMENT ANALYSIS .0

virtu%+ )/rs /f i#ter#%+ %#1 eter#%+ f/rces %re e*u%+ f/r ever i#em%tic%++ %1missi,+e

set /f virtu%+ 1is+%ceme#ts. For o#r ($r#c$#r' ;i$" on' d'gr'' o& &r''dom onl% on' vir$#aldi(4lac'm'n$ uδ i( 4o((i9l' and 4rinci4l' o& vir$#al di(4lac'm'n$( "a( a &orm

u P 5

5

& δ δε σ =∫ d

@/.2/

;"'r' &δε i( vir$#al ($rain corr'(4onding $o vir$#al di(4lac'm'n$ uδ . T"' vir$#al ($raincan 9' '>4r'(('d &rom '=#a$ion @/. a(

./

2

d

d

d

d/0

/

000

u L

uu

L

u

L

u

L

uu

u

&& δ δ δ

ε δε

+=

+== @/.26

I$ i( a((#m'd in 4rinci4l' o& vir$#al di(4lac'm'n$( $"a$ vir$#al di(4lac'm'n$ i( in&ini$'(imaland "'nc' $"' ($r'(( A?@ A N =σ r'main( #nc"ang'd. No$ing $"a$ σ and &δε ar'con($an$ ov'r $"' ;"ol' vol#m' in $"i( ca(' and a((#ming $"a$ c"ang'( o& $"' vol#m'can 9' n'gl'c$'d d#' $o (mall ($rain i.'. 000 L A5 5 =≈ '=#a$ion @/.2/ 9'com'(

u P L Au L

u

A

N δ δ =

+0/

00

and &rom $"i( '=#a$ion i$ &ollo;( $"a$

P L

u N =+

0

.

T"i( i( $"' (am' '=#a$ion a( $"' '=#a$ion o& '=#ili9ri#m @/.2. A&$'r (#9($i$#$ing &or N &rom@/.20 $"' '=#a$ion @/.22 ;ill 9' r'c'iv'd again.

U$ilia$ion o& 4rinci4l' o& vir$#al di(4lac'm'n$( @!VD i( a conv'ni'n$ ;a% $o o9$aincondi$ion( o& '=#ili9ri#m &or com4l'> ($r#c$#r'(. For g'n'ral $"r'':dim'n(ional ca(' ;'"av' $"r'' com4on'n$( o& di(4lac'm'n$ u v ) and (i> com4on'n$( o& 7r''nG( ($rain

∂

∂

+

∂

∂

+

∂

∂

+∂

∂

=

///

/

2

2

)

2

v

2

u

2

u

2ε

∂∂

+

∂∂

+

∂∂

+∂∂

=///

/

2

3

)

3

v

3

u

3

v 3ε

∂∂

+

∂∂

+

∂∂

+∂∂

=///

/

2

6

)

6

v

6

u

6

) 2ε

@/.23


11/41

NONLINEAR FINITE ELEMENT ANALYSIS ..

3

)

2

)

3

v

2

v

3

u

2

u

2

v

3

u 23

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂+

∂

∂=γ

6

)

2

)

6

v

2

v

6

u

2

u

2

)

6

u 26 ∂

∂∂∂+

∂∂

∂∂+

∂∂

∂∂+

∂∂+

∂∂=γ

6

)

3

)

6

v

3

v

6

u

3

u

3

)

6

v 36

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂+

∂

∂=γ .

In &ini$' 'l'm'n$ m'$"od ar' di(4lac'm'n$ in$'r4ola$'d ;i$"in $"' &ini$' 'l'm'n$( a(

i

i

i u N u ∑= ii

i v N v ∑= ii

i ) N ) ∑= @/.25

;"'r' ui vi )i ar' nodal di(4lac'm'n$( and N i ar' ("a4' nc$ion(. S#9($i$#$ing $"'(''=#a$ion( in$o '>4r'((ion( o& 7r''nG( ($rain com4on'n$( ;' o9$ain

d BBε A/

2@ N L += .

@/.2

In ma$ri> '=#a$ion @/.2

=

36

26

23

6

3

2

γ

γ

γ

ε

ε

ε

ε @/.28

and d i( ma$ri> o& nodal di(4lac'm'n$(. Ma$ri> LB i( $"' #(#al (mall di(4lac'm'n$ ma$ri>

and ma$ri> N B r'&l'c$( $"' &ac$ $"a$ 7r''nG( ($rain i( a nonlin'ar nc$ion o& di(4lac'm'n$(. El'm'n$( o& $"i( ma$ri> ar' lin'ar nc$ion( o& nodal di(4lac'm'n$( d . I$mig"$ 9' ("o;n $"a$ vir$#al ($rain corr'(4onding $o $"' vir$#al nodal di(4lac'm'n$( d δ i(

.A@ d Bd BBε δ δ δ =+= N L @/.2K

According $o $"' 4rinci4l' o& vir$#al di(4lac'm'n$( vir$#al ;or+ o& in$'rnal &orc'( m#($ 9''=#al $o vir$#al ;or+ o& '>$'rnal &orc'( i& $"' ($r#c$#r' i( in '=#ili9ri#m. T"i( i(r'4r'('n$'d 9% $"' '=#a$ion

F d σ ε T

5

T 5 δ δ ∫ =d @/.21


12/41


;"'r' F i( ma$ri> o& nodal &orc'(.-' (#44o(' lin'ar r'la$ion 9'$;''n ($r'(( and ($rain com4on'n$( "'nc'

Dε σ =

;"'r' D i( ma$ri> o& ma$'rial 'la($ic con($an$(.S#9($i$#$ing @/.2K in$o @/.21 giv'(

F d σ Bd T T

5

T 5 δ δ ∫ =d @/./0

&or an% +in'ma$icall% admi((i9l' ('$ o& vir$#al di(4lac'm'n$( d δ . T"'n

F σ BT ∫ =

5

5 d . @/./2

T"' la($ '=#a$ion i( a ma$ri> r'4r'('n$a$ion o& a ('$ o& nonlin'ar alg'9raic '=#a$ion( &or #n+no;n nodal di(4lac'm'n$( d .

F d R =A@ . @/.//

2.2 Increental * iterati$e solutions

-' "av' (''n $"a$ a((#m4$ion o& larg' di(4lac'm'n$( l'ad( $o nonlin'ar '=#a$ion o& '=#ili9ri#m @/.2 or @/.22 &or a (im4l' 9ar $r#(( '>am4l'. 7'n'rall% in &ini$' 'l'm'n$anal%(i( ;' "av' a ('$ o& nonlin'ar '=#a$ion( @/.//.

L'$ #( ($ar$ ;i$" $"' 9ar:$r#(( '>am4l'. T"' '=#a$ion o& '=#ili9ri#m @/.2 or @/.22 can 9' ;ri$$'n in a &orm

P u R =A@ @/./6

;"'r'

( )./6/A@//6

600

uuu L

A E

L

u N u R ++=

+= @/./3

r'4r'('n$( a com4on'n$ o& in$'rnal &orc'.T"' 9a(ic ($'4 $o (olv' $"' nonlin'ar '=#a$ion @/./3 i( a lin'ar a44ro>ima$ion &or (mall

incr'm'n$ o& &orc' and corr'(4onding incr'm'n$ o& di(4lac'm'n$.A((#m' $"a$ &or a 4r'(cri9'd val#' o& &orc' P ;' manag'd $o &ind @'.g. 9% 'rror and $rialm'$"od a di(4lac'm'n$ u (a$i(&%ing $"' '=#a$ion @/./6. In$'rnal &orc' Ru 8 du9 &or n';'>$'rnal &orc' P 2 P d P and r'la$'d di(4lac'm'n$ u2 u du can 9' a44ro>ima$'d 9%$"' lin'ar nc$ion


13/41


uu

Ru Ruu R

u

dd

dA@Ad@

+=+

and a44ro>ima$' condi$ion o& '=#ili9ri#m i(

P P uu

Ru R

u

ddd

dA@ +=

+ .

A((#ming '=#a$ion @/./6 giv'(

P uu

R

u

ddd

d=

@/./5

or

P uu : T ddA@ = @/./;"'r'

uT

u

R :

=d

d@/./8

i( call'd $"' t%#$e#t stiff#ess. For $"' 4ar$ic#lar ca(' o& $"' 9ar:$r#(( $ang'n$ ($i&&n'(( can 9' 'a(il% &o#nd a(

u

N

L

u N

L

u

u : T

d

d

d

d

00

++

+= .

U(ing $"' '=#a$ion @/.20 giv'(

00d

d

L

u

L

EA

u

N +=

&rom ;"ic"

σ : : : :

uT ++=0 @/./K

;"'r'/

00

0

=

L

L

A E : i( $"' +i#e%r stiff#ess

/

0

/

0/

+= L

u

u

L

A E : u i( $"' i#iti%+ 1is+%ceme#t stiff#ess

0 L

N : =

σ i( $"' i#iti%+ stress stiff#ess!

T"' lin'ar ($i&&n'(( ;"ic" i( ind'4'nd'n$ on di(4lac'm'n$ i( &amiliar &rom (malldi(4lac'm'n$ ($r#c$#ral anal%(i(. T"' ini$ial di(4lac'm'n$ ($i&&n'(( r'&l'c$( $"' '&&'c$ o&


14/41


di(4lac'm'n$ on ($i&&n'((6. T"' ini$ial ($r'(( ($i&&n'(( r'&l'c$( $"' &ac$ $"a$ $"'r' i( an a>ial&orc' in $"' 9ar 4rior $o load incr'm'n$.

In li+' mann'r ;' can 4r'c'd' in a g'n'ral ca(' d'(cri9'd 9% $"' '=#a$ion @/./2 or

@/.// and d'riv'

d K d R d d R dA@Ad@ T +=+

andF d K dd =T @/./1

;"'r'

d

R K

∂∂

=T

i( $"' t%#$e#t stiff#ess m%tri. -' can al(o &ind o#$ $"a$

σ K K K K ++= uT 0 . @/.60

;"'r' T K i( +i#e%r stiff#ess m%tri uK i( i#iti%+ 1is+%ceme#t stiff#ess m%tri and σ K

i( i#iti%+ stress stiff#ess ma$ri>.In$rod#c$ion o& $ang'n$ ($i&&n'(( ma$ri> i( cr#cial &or (ol#$ion o& nonlin'ar '=#a$ion(

@/.//. T"' mo($ ;id'l% #('d m'$"od( ar' 9ri'&l% in$rod#c'd in $"' &ollo;ing $'>$

2.2.1 Increental et)od

T"' load i( divid'd in$o a ('$ o& (mall incr'm'n$( iF ∆ . Incr'm'n$( o& di(4lac'm'n$( id ∆ar' calc#la$'d &rom $"' ('$ o& lin'ar (im#l$an'o#( '=#a$ion(

iiiT F d K ∆=∆− A2@ .62

and an #4da$'d (ol#$ion i( o9$ain'd a(

.2 iii d d d ∆+= − 6/

T"' 4roc'd#r' i( ("o;n in Fig#r' /.6. I$ i( o9vio#( $"a$ (ol#$ion 'rror H di&&'r'nc' &rom'>ac$ (ol#$ion grad#all% c#m#la$'(. To r'd#c' 'rror larg' n#m9'r o& (mall incr'm'n$al($'4( "a( $o 9' don' $"a$ i( in'&&ici'n$. On $"' o$"'r "and divi(ion o& loading 4roc'(( in$o(#&&ici'n$l% (mall incr'm'n$( i( n'c'((ar% $o mod'l load 4a$" d'4'nd'n$ 9'"avio#r o& a($r#c$#r'. D'4'nd'nc' o& r'(4on(' on a mann'r o& loading no$ onl% o& &inal val#'( o& load(i( $%4ical &or 4ro9l'm( ;i$" 4la($ic d'&orma$ion and ;i$" &ric$ion. In $"'(' 4ro9l'm(incr'm'n$al m'$"od i( #(#all% com9in'd ;i$" on' o& &ollo;ing m'$"od(.

6 '.g. i$ can 9' (''n &rom $"' diagram in Fig#r' /./ $"a$ &or com4r'((iv' load P ($i&&n'(( d'cr'a('(and &or $'n(ional &orc' P incr'a('(.


15/41


Fig#r' /.3 Incr'm'n$al m'$"od

2.2.2 Ne+ton*Rap)son et)od

S#44o(' $"a$ ini$ial di(4lac'm'n$( 0d ar' +no;n. T"' &ir($ g#'(( o& nodal di(4lac'm'n$(

&or load F i( calc#la$'d 9% (olving ('$ o& lin'ar alg'9raic '=#a$ion(

F d K =2A0@T @/.66

;"'r' A@ 0A0@ d K K T T = i( $ang'n$ ($i&&n'(( ma$ri> calc#la$'d &or ini$ial di(4lac'm'n$(.

A( $"' di(4lac'm'n$( 2d ar' @mo($ 4ro9a9l% no$ acc#ra$' $"' '=#ili9ri#m '=#a$ion @/.//i( no$ (a$i(&i'd and

F d R ≠A@ 2

$"a$ m'an( $"'r' ar' #n9alanc'd @or r'(id#al nodal &orc'(

F d R r −= A@ 22 . @/.63

Q% com4#$ing n'; $ang'n$ial ($i&&n'(( ma$ri> A@ 2A2@ d K K T T = and (olving n'; ('$ o&alg'9raic lin'ar '=#a$ion(

22A2@ r d K =∆T @/.65

;' ;ill o9$ain an im4rov'd (ol#$ion


16/41


22/ d d d ∆+= . @/.6

I& 0 F d R r ≠−= A@ // $"' 4roc'd#r' i( r'4'a$'d #n$il $"' (#&&ici'n$l% acc#ra$' (ol#$ion i(o9$ain'd. T"' i$'ra$ion( ar' (c"'ma$icall% ("o;n in Fig#r' /.3.T"i( m'$"od +no;n a( N';$on:Ra4"(on3 m'$"od @NR i( o&$'n com9in'd ;i$"incr'm'n$al m'$"od a( di(4la%'d in Fig#r' /.5.

Fig#r' /.3 S$andard N';$on:Ra4"(on @NR m'$"od.

3 o('4" Ra4"(on @23K:2825 ;a( an Engli(" ma$"'ma$ician a F'llo; o& $"' Ro%al Soci'$% o& London and &ri'nd o& N';$on.


17/41


Fig#r' /.5 Com9ina$ion o& N';$on:Ra4"(on and incr'm'n$al m'$"od(.

2.2." !odified Ne+ton*Rap)son et)od

T"' ($andard N';$on:Ra4"(on m'$"od al$"o#g" '&&'c$iv' in man% ca('( n''d( $"'(ol#$ion o& $"' ('$ o& lin'ar '=#a$ion( @/.65 ;"ic" i( $im' d'manding &or larg' (%($'m(.Modi&i'd N';$on:Ra4"(on m'$"od @MNR di&&'r( &rom ($andard NR algori$"m in $"a$ $"'($i&&n'(( ma$ri> i( onl% #4da$'d occa(ionall%. In $"' '>am4l' ("o;n in Fig#r' /. $"'

$ang'n$ial ($i&&n'(( ma$ri> i( &orm'd and d'com4o('d a$ $"' 9'ginning and #('d $"ro#g"o#$$"' i$'ra$ion(. Advan$ag' o& $"' m'$"od i( in (aving com4#$'r $im' 9'ca#(' &ac$ori(a$iono& $"' $ang'n$ ($i&&n'(( ma$ri> i( 4'r&orm'd onl% onc' &or $"' load incr'm'n$. On $"' o$"'r "and n#m9'r o& i$'ra$ion( n''d'd i( #(#all% larg'r.


18/41

NONLINEAR FINITE ELEMENT ANALYSIS ."

Fig#r' /. Modi&i'd N';$on:Ra4"(on@MNR m'$"od.

2.2., -uasi*Ne+ton et)ods

T"'r' '>i($ man% o$"'r m'$"od( &or (ol#$ion o& $"' ('$ o& nonlin'ar alg'9raic '=#a$ion( (ocall'd =#a(i:N';$on m'$"od(. T"' mo($ 4o4#lar among $"'m i( Qro%d'n H Fl'$c"'r H 7old&ar9 H S"anno @QF7S m'$"od.

2." Linear sta(ility analysis

T"'or'$icall% 9'lo; a c'r$ain cri$ical load a ($r#c$#r' i( in 4o(i$ion o& ($a9l' '=#ili9ri#m;"il($ a9ov' $"a$ load $"' '=#ili9ri#m ma% 9' #n($a9l'. Un($a9l' '=#ili9ri#m m'an( $"a$$"o#g" $"' ($r#c$#r' i( in '=#ili9ri#m an% ar9i$rar% (mall di($#r9anc' ;ill ca#(' lo(( o& $"i( '=#ili9ri#m. In man% 4rac$ical 4ro9l'm( $"' di(4lac'm'n$( ar' (mall &or load l'(( $"ancri$ical and 9'"avio#r o& $"' ($r#c$#r' can 9' con(id'r'd a( a lin'ar nc$ion o& a44li'd

load. T"' $%4ical '>am4l' i( E#l'r ($r#$ 9#c+ling Fig#r' /.8.For a>ial &orc' N $"a$ i( l'(( $"an cri$ical $"' ($r#$ i( in ($a9il' '=#ili9ri#m. T"i('=#ili9ri#m i( 4o((i9l' i& a la$'ral load ! $"'n d'&l'c$( $"' ($r#$ a( ;'ll. I& $"' la$'ral load i(r'mov'd $"' ($r#$ ;ill r'$#rn $o i$( ($raig"$ ("a4'.

I& $"' &orc' N i( gr'a$'r $"an cri$ical $"' ($r#$ can r'main @$"'or'$icall% ($raig"$ 9#$ i$('=#ili9ri#m i( #n($a9l' an% (mall la$'ral load ;ill ca#(' d'&l'c$ion incr'a(ing #n$il $"'colla4('5.

5 In r'ali$% #n($a9l' '=#ili9ri#m i( d#' $o ini$ial im4'r&'c$ion( @'.g. 'cc'n$rici$% o& &orc' N ini$ialc#rva$#r' o& $"' ($r#$ '$c. im4o((i9l' 9#$ '($ima$ion o& cri$ical load ma% 9' #('l in man% ca('(.


19/41

NONLINEAR FINITE ELEMENT ANALYSIS .(

Fig#r' /.8 Q#c+ling o& a ($r#$

For load l'(( $"an cri$ical (mall longi$#dinal @in 4lan' and la$'ral di(4lac'm'n$( allo;$"' ini$ial di(4lac'm'n$ ($i&&n'(( ma$ri> K u $o 9' ignor'd. T"' '=#ili9ri#m '=#a$ion can 9';ri$$'n a(

P d N K K =+ AL@M 0 σ λ @/.68

T"' 'la($ic cri$ical @9#c+ling load i( giv'n 9% $"' lo;'($ val#' o& load 4aram'$'r λ &or ;"ic" d ≠ 0 ;"'n $"' la$'ral load P 0 . !"%(icall% $"i( m'an( $"a$ '=#ili9ri#m i( 4o((i9l';i$" v'r% (mall la$'ral di(4lac'm'n$( in $"' a9('nc' o& an% la$'ral load. In ma$"'ma$ical('n(' ;' "av' $o (olv' $"' 'ig'nval#' 4ro9l'm

0 d N K K =+ AL@M 0 σ λ @/.6K

;"'r' λ i( $"' 'ig'nval#' and d i( $"' corr'(4onding 'ig'nv'c$or.I$ ("o#ld 9' no$'d $"a$ d#' $o a((#m4$ion( acc'4$'d $"' (ol#$ion r'4r'('n$( i$('l& onl% an'($ima$ion o& $"' #44'r 9o#nd o& $"' ($r#c$#r' load ca4aci$%.


20/41


2., Lar'e displaceent and lar'e strain (e)a$iour

-"'n ($rain i( larg' i$ i( inadmi((i9l' $o n'gl'c$ ("a4' and vol#m' c"ang'( o& a ($r#c$#r'.For '>am4l' in $"' (im4l' 9ar '>am4l' ;' "av' $o in$rod#c' c#rr'n$ cro(( ('c$ional Ain($'ad o& ini$ial A0 and c#rr'n$ l'ng$" L in($'ad o& ini$ial l'ng$" L0 in $"' '=#a$ion( @/.20and @/.22.

Accordingl% in$'gra$ion in $"' '=#a$ion @/.21 '>4r'((ing $"' 4rinci4l' o& vir$#aldi(4lac'm'n$( "a( $o 9' $a+'n ov'r $"' c#rr'n$ vol#m'. T"i( 9ring( 4ro9l'm( a( $"' c#rr'n$vol#m' i( #n+no;n 9'ca#(' i$ d'4'nd( on di(4lac'm'n$( $"a$ ar' #n+no;n $oo and m#($

9' calc#la$'d &ir($. To (olv' $"i( 4ro9l'm i$ i( n'c'((ar% $o in$rod#c' a $ran(&orma$ion (o$"a$ in$'gral( ar' $a+'n ov'r +no;n vol#m'. T;o 4o((i9l' ;a%( ar' 9ri'&l% d'(cri9'd

9'llo;

2.,.1 Total La'ran'ian forulation

In a To$al Lagrangian @TL &orm#la$ion all in$'gral( ar' calc#la$'d ;i$" r'(4'c$ $o $"'ini$ial #nd'&orm'd con&ig#ra$ion o& $"' ($r#c$#r'

F d σ ε T

5

P T & 5 δ δ ∫ =

0

d @/.61

;"'r' 0 i( $"' ini$ial vol#m'. D#' $o $ran(&orma$ion n'; m'a(#r' &or ($r'(( (o call'd('cond !iola:)irc""o&& ($r'(( $'n(or P "a( $o 9' in$rod#c'd ;i$" 7r''nG( ($rain $'n(or &.

2.,.2 Updated La'ran'ian forulation

In an U4da$'d Lagrangian @UL &orm#la$ion a +no;n d'&orm'd con&ig#ra$ion i i( $a+'n a(an ini$ial ($a$' &or (#9('=#'n$ con&ig#ra$ion @i82 and $"i( i( con$in#all% #4da$'d a( $"'calc#la$ion 4roc''d(

( ) F d σ ε T 5

; iT

Ai

5 δ δ ∫ =++

0

dA2@A2@

@/.30

In $"' l'&$ (id' o& $"' '=#a$ion @/.30 ; i( Ca#c"% ($r'(( $'n(or and A i( Alman(i ($rain

$'n(or r'(4'c$iv'l%. No$a$ion @i2 A and @i2 ; m'an( $"a$ $"' ($rain and ($r'(( ar' incon&ig#ra$ion @i82. In$'gra$ion i( don' ov'r vol#m' i $"a$ i( in c#rr'n$ con&ig#ra$ion i.U(' o& di&&'r'n$ m'a(#r'( &or ($r'(( and ($rain in TL and UL &orm#la$ion &ollo;( condi$ion$"a$ vir$#al ;or+ o& in$'rnal &orc'( m#($ 9' $"' (am' irr'(4'c$iv' o& $"' vol#m' ov'r ;"ic"i( in$'gra$ion $a+'n.

T"a$ m'an( $"a$ ($r'(( and ($rain m'a(#r'( m#($ 9' ;or+ conJ#ga$'.


21/41

NONLINEAR FINITE ELEMENT ANALYSIS 2.

". !ATERIAL NONLINEARITIES

".1 Introduction

Lin'ar 'la($ic FE anal%(i( i( 9a('d on lin'ar con($i$#$iv' ($r'((:($rain '=#a$ion(

ε Dσ = @6.2

in ;"ic" $"' $'rm( o& ma$'rial ma$ri> D ar' '>4r'(('d a( nc$ion( o& con($an$ val#'( o& mod#l#( o& 'la($ici$% and !oi((onG( ra$io. T"' con($an$ D ma$ri> l'ad( $o a con($an$($i&&n'(( ma$ri> K ;"ic" i( &or ($rain:di(4lac'm'n$ r'la$ion("i4

d Bε = @6./

giv'n 9%

5

T

5

d BD BK ∫ = @6.6

D'4ar$#r' &rom lin'ar 'la($ici$% im4li'( $"a$ $"' lin'ar 'la($ic con($i$#$iv' '=#a$ion( ar'no long'r valid a( $"' ma$'rial ma$ri> i( no long'r con($an$. T"' non:con($an$ ma$'rial

ma$ri>D

r'4r'('n$( nonlin'ar con($i$#$iv' '=#a$ion( corr'(4onding $o $"' ado4$'dnonlin'ar ma$'rial mod'l. Con('=#'n$l% $"' condi$ion( o& '=#ili9ri#m d'riv'd in FEM&rom 4rinci4l' o& vir$#al di(4lac'm'n$( ar' nonlin'ar li+' '=#a$ion( @/./2 and @/.//.Sol#$ion o& $"'(' '=#a$ion( i( 9a('d on $"' (am' m'$"od( a( in g'om'$ricall% nonlin'ar ca('. U(#all% i$ i( n'c'((ar% $o divid' load in$o incr'm'n$( and 4'r&orm '=#ili9ri#mi$'ra$ion( @'.g. 9% MNR or NR m'$"od &or 'ac" incr'm'n$. Mor'ov'r &or 'ac" loadincr'm'n$ $"'r' m#($ 9' 4'r&orm'd ($r'(( i$'ra$ion( a( $"' ma$'rial ma$ri> i( nc$ion o& ($rain. T"' ($rain i( #n+no;n % ri/ri and ;ill 9' com4#$'d onl%.

Ma$'rial nonlin'ari$i'( ar' o&$'n com9in'd ;i$" g'om'$rical and?or 9o#ndar%nonlin'ari$i'(.

".2 Nonlinear elasticity odels

Nonlin'ar 'la($ic 9'"avior o& ma$'rial( can 9' &orm#la$'d in ('v'ral ;a%(. T"' (im4l'($ i(t/t%+ f/rm%t ;"'r' $"' ($r'(( and ($rain( ar' d'&in'd in $'rm( o& $"' ('can$ mod#l#( o& 'la($ici$% E ( ('' Fig#r' 6.2

ε ε σ A@ s E = @6.3


22/41


In /am4l' i( $"' Moon'%:Rivlin ma$'rial mod'l #('d &or mod'lling r#99'r:li+' ma$'rial(.


23/41


Fig#r' 6./ S$rain 'n'rg% d'n(i$% nc$ion( = and = ∗

For an% nonlin'ar 'la($ic ma$'rial mod'l i$ i( 4o((i9l' $o d'&in' r'la$ion 9'$;''n ($r'(( and($rain incr'm'n$( a(

ε Dσ dd T = @6.K

Ma$ri> DT i( nc$ion o& ($rain( . Con('=#'n$l% a ('$ o& '=#ili9ri#m '=#a$ion( ;' r'c'iv'

in FEM i( nonlin'ar and m#($ 9' (olv'd 9% #(' o& an% m'$"od @'.g. NR d'(cri9'd a9ov'.

"." Elastoplastic aterial odel

".".1 %ieldin' criterion

E>4'rim'n$( indica$' $"a$ lin'ar 'la($ic mod'l i( acc'4$a9l' onl% ;i$"in a limi$'d rang' o& ($r'((. A( an '>am4l' $"' ($r'((:($rain c#rv' &rom $'n(ion $'($ o& ($''l (4'cim'n i( ("o;n

in Fig#r' 6.6. Un$il $"' %i'ld ($r'(( r'4r'('n$'d 9% 4oin$ A @in $"' giv'n ca('σ /K0 M!a $"' d'&orma$ion( ar' 'la($ic and ($r'((:($rain r'la$ion ma% 9' d'(cri9'd a(σ E ε . -"'n $"' ($r'(( l'v'l '>c''d( $"' %i'ld ($r'(( an 'la($o4la($ic con($i$#$iv' la;gov'rn( $"' r'la$ion("i4 9'$;''n incr'm'n$( o& ($r'(( and ($rain.

D#' $o lac+ o& in&orma$ionK a44ro>ima$' ($r'((:($rain c#rv'( ar' #(#all% #('d inanal%(i(. Qilin'ar a44ro>ima$ion d'&in'd 9% %i'ld ($r'(( mod#l#( o& 'la($ici$% E and$ang'n$ial mod#l#( E T i( ("o;n in Fig#r' 6.3. I& E T 0 ma$'rial mod'l i( 'la($ic:4'r&'c$l%

4la($ic. I& E T ≠ 0 ma$'rial mod'l a((#m'( ($rain "ard'ning.

K In a d'(ign 4roc'(( $"' r'al ma$'rial c#rv' i( #(#all% #n+no;n onl% 9a(ic val#'( li+' %i'ld ($r'(('$c. ar' availa9l'. Mor'ov'r $"' ma$'rial 4ro4'r$i'( (lig"$l% di&&'r 9% di&&'r'n$ (#44li'(.


24/41


0

50

100

150

200

250

300

350

400

450

0.00 0.02 0.04 0.06 0.08 0.10 0.12

[ M P a ] A

Fig#r' 6.6 T%4ical ($r'((:($rain c#rv' &or mild ($''l

I$ ("o#ld 9' no$'d $"a$ c#rv'( in Fig#r'( 6.6 and 6.3 ar' &or $'n(il' 9'"avio#r. I$ i( #(#all%a((#m'd $"a$ (imilar c#rv'( &or com4r'((iv' 9'"avio#r ar' a44lica9l' i& $"'r' "a( 9''n no"i($or% o& 4la($ic d'&orma$ion.

Fig#r' 6.3 Ela($o4la($ic mod'l ;i$" lin'ar ($rain "ard'ning

T"' indica$ion o& %i'lding #nd'r m#l$ia>ial condi$ion( in m'$al( i( o9$ain'd &rom'>4'rim'n$( #(#all% cond#c$'d on c%lindrical (am4l'( (#9J'c$'d $o com9in'd a>ial loadand $or=#'. E>4'rim'n$( (#gg'($ $"a$ $"'r' i( no (igni&ican$ di&&'r'nc' in 9'"avio#r o& m'$al( in $'n(ion or com4r'((ion and no vol#m' c"ang' a((ocia$'d ;i$" %i'lding and no'&&'c$ o& m'an ($r'(( l'v'l on %i'lding can 9' a((#m'd.In a ma$"'ma$ical d'(cri4$ion on('$ o& %i'lding ma% 9' r'4r'('n$'d 9% a (calar nc$ion$'rm'd $"' ie+1 fu#cti/# F! T"' %i'ld nc$ion i( ;ri$$'n in a &orm ;"ic" l'ad( $o $"'condi$ion(


25/41


0


26/41


".".2 /ost yieldin' (e)a$iour

T"' ndam'n$al a((#m4$ion in d'(cri9ing 4o($:%i'lding 9'"avior i( $"' d'com4o(i$ion o&

$"' $o$al ($rain incr'm'n$ in$o an 'la($ic @r'cov'ra9l' 4ar$ and a 4la($ic @irr'v'r(i9l' 4ar$.For #nia>ial ($r'(( ($a$' i( according $o Fig#r' 6.

T E =ε

σ

d

d E

e =

ε

σ

d

[email protected]

and 4la($ic ($rain incr'm'n$ i( $"'n

σ ε ε ε ddddT

T e 4

E E

E E −=−= . @6.25

Fig#r' 6./ D'com4o(i$ion o& $"' $o$al ($rain incr'm'n$

Q% analog% in m#l$ia>ial ($r'(( ($a$' $"' $o$al ($rain( ar' d'com4o('d in$o 'la($ic and 4la($ic 4ar$( $oo

4eε ε ε += @6.2

In m#l$ia>ial ca('( (#9('=#'n$ loading a&$'r &ir($ %i'ld 4rod#c'( 4la($ic d'&orma$ion $"a$can 9' accom4ani'd 9% a modi&ica$ion o& $"' ("a4' and?or 4o(i$ion o& $"' %i'ld (#r&ac'.

For a 4'r&'c$l% 4la($ic ma$'rial $"'r' i( no c"ang' in $"' %i'ld (#r&ac' d#ring 4la($icd'&orma$ion. For a ($rain "ard'ning ma$'rial 4la($ic d'&orma$ion 4rod#c'( a c"ang' in $"'%i'ld (#r&ac'. T"' ini$ial %i'ld (#r&ac' i( r'4lac'd 9% $"' (#9('=#'n$ %i'ld (#r&ac'. Amodi&i'd %i'ld nc$ion i( ado4$'d ;"ic" "a( a &orm (#c" a(

0A@ = : F 4ε σ . @6.28

T"i( %i'ld nc$ion d'4'nd( on $"' ($r'(('( 9#$ al(o $"' 4la($ic ($rain( and a "ard'ning 4aram'$'r :! T"' ;a% in ;"ic" $"' 4la($ic ($rain( modi&% $"' %i'ld nc$ion i( d'&in'd 9%"ard'ning r#l'(


27/41


Fig#r' 6.8 I(o$ro4ic "ard'ning.

An is/tr/ic %r1e#i#$ la; im4li'( $"a$ $"' %i'ld (#r&ac' incr'a('( in (i' 9#$ main$ain( i$(original ("a4' #nd'r loading condi$ion(. Sc"'ma$ic r'4r'('n$a$ion o& i(o$ro4ic "ard'ning&or #nia>ial and 9ia>ial ($r'(( ($a$' i( ("o;n in Fig#r' 6.8.

In i#em%tic %r1e#i#$ $"' original %i'ld (#r&ac' i( $ran(la$'d $o a n'; 4o(i$ion in ($r'(((4ac' ;i$" no c"ang' o& i$( ("a4' and (i' a( ("o;n in Fig#r' 6.K. )in'ma$ic "ard'ning"a( 4aramo#n$ im4or$anc' in mod'lling c%clic 9'"avior.

T"' com9ina$ion o& $"' $;o 4rinci4al "ard'ning la;( l'ad( $o a mie1 %r1e#i#$ la;;"'r' $"' ini$ial %i'ld (#r&ac' 9o$" '>4and( and $ran(la$'( a( a con('=#'nc' o& 4la($ic &lo;.

Fig#r' 6.K )in'ma$ic "ard'ning.


28/41

NONLINEAR FINITE ELEMENT ANALYSIS 2"

"."." Constitutional e0uations of elastoplastic aterial

T"' %i'ld cri$'rion (a%( ;"'$"'r 4la($ic d'&orma$ion ;ill occ#r 9#$ (a%( no$"ing a9o#$ $"'

4la($ic 9'"avio#r o& a ma$'rial a&$'r on('$ o& 4la($ic d'&orma$ion(. T"i( i( d'&in'd 9% (o:call'd f+/) ru+e in ;"ic" i( $"' ra$' and $"' dir'c$ion o& 4la($ic ($rain( i( r'la$'d $o $"'($r'(( ($a$' and $"' ($r'(( ra$'. T"i( r'la$ion can 9' '>4r'(('d a(

i>

4i>

?

σ λ ε ∂∂

= dd @6.28

or in ma$ri> &orm a(

σ ε

∂∂

=? 4

λ dd @6.2K

;"'r' dλ i( a (calar val#' @$o 9' d'$'rmin'd and ? i( a (calar val#'d nc$ion o& ($r'((com4on'n$( call'd +%stic /te#ti%+ .

"."., Nuerical procedures

In a #nia>ial ($r'(( ($a$' $"' 4la($ic 9'"avio#r o& ma$'rial can 9' d'(cri9'd a(

ε σ dd T E = @6.2K

For a 9ilin'ar ma$'rial i( E T con($an$ a( o9vio#( &rom $"' '=#a$ion( @6.23 and @6.25.

In a m#l$ia>ial ($r'(( ($a$' a (imilar con($i$#$iv' '=#a$ion can 9' &orm#la$'d a(

ε Dσ dd T = @6.21

T"' $ang'n$ial ma$'rial ma$ri> DT i( #('d $o &orm a $ang'n$ial ($i&&n'(( ma$ri> K T . -"'n$"' $ang'n$ial ($i&&n'(( ma$ri> i( d'&in'd $"' di(4lac'm'n$ incr'm'n$ i( o9$ain'd &or a+no;n load incr'm'n$

F d K ∆=∆T @6./0

A( load and di(4lac'm'n$ incr'm'n$( ar' &inal no$ in&ini$'(imal di(4lac'm'n$( o9$ain'd

9% (ol#$ion o& $"i( ('$ o& lin'ar alg'9raic '=#a$ion ;ill 9' a44ro>ima$' onl%. T"a$ m'an(condi$ion( o& '=#ili9ri#m o& in$'rnal and '>$'rnal nodal &orc'( ;ill no$ 9' (a$i(&i'd andi$'ra$iv' 4roc'(( i( n'c'((ar%. An% o& m'$"od( m'n$ion'd a9ov' ma% 9' #('d.T"' 4ro9l'm $"a$ ari('( no; i( $"' ndam'n$al 4ro9l'm in com4#$a$ional 'la($o4la($ici$% :no$ onl% '=#ili9ri#m '=#a$ion( 9#$ al(o con($i$#$iv' '=#a$ion( o& ma$'rial m#($ 9' (a$i(&i'd.T"a$ m'an( $"a$ ;i$"in $"' 'ac" '=#ili9ri#m i$'ra$ion ($'4 c"'c+ o& ($r'(( ($a$' andi$'ra$ion( $o &ind 'la($ic and 4la($ic 4ar$ o& ($rain( a$ 'v'r% in$'gra$ion 4oin$ m#($ 9'incl#d'd. T"' i$'ra$ion 4roc'(( con$in#'( #n$il 9o$" '=#ili9ri#m condi$ion( andcon($i$#$iv' '=#a$ion( ar' (a$i(&i'd (im#l$an'o#(l%. T"' conv'rg'd (ol#$ion a$ $"' 'nd o& load incr'm'n$ i( $"'n #('d a$ $"' ($ar$ o& n'; load incr'm'n$.


29/41

NONLINEAR FINITE ELEMENT ANALYSIS 2(


30/41


RE&ERENCES

2 ,in$on E In$rod#c$ion $o Nonlin'ar Fini$' El'm'n$ Anal%(i(. NAFEMS 7la(go;211/

/ Cri(&i'ld M. A. Non:lin'ar Fini$' El'm'n$ Anal%(i( o& Solid( and S$r#c$#r'(. o"n-il'% Son( 2112

6 Qi$$nar . 'Jno"a . N#m'ric+ m'$od% m'c"ani+% @N#m'rical m'$"od( o& m'c"anic(. VUT !ra"a 211/

3 O+ro#"lí+ M. ,W("l C. !l'*'+ . !$X+ S. Nadrc"al . M'c"ani+a 4oddaJnc"$Zl'( n#m'ric+X ma$'ma$i+a a (#4'r4očí$ač' @M'c"anic o& (olid( n#m'ricalma$"'ma$ic( and (#4'rcom4#$'r(. C'c" Acad'm% o& Sci'nc' !rag#' 2118.

5 O+ro#"lí+ M. Im4l'm'n$a$ion o& Nonlin'ar Con$in##m M'c"anic( in Fini$' El'm'n$Cod'(. In($i$#$' o& T"'rmod%namic( !rag#' 2115.

,in$on E. Ea$$ M. ,. F#ndam'n$al T'($( &or T;o and T"r'' Dim'n(ional SmallS$rain Ela($o4la($ic Fini$' El'm'n$ Anal%(i(. NAFEMS 7la(go; 21K8.

8 El'c$ronic doc#m'n$a$ion o& 4rogram COSMOS?M v'r(ion /.15. SRAC Lo( Ang'l'(/005 ;;;co(mo(m.com.

K Falon Q. 7. ,i$c"ing( D. An In$rod#c$ion $o Mod'ling Q#c+ling and Colla4('. NAFEMS 7la(go; /00

1 E#rocod' 6 D'(ign o& ($''l ($r#c$#r'( : !ar$ 2:2 7'n'ral r#l'( and r#l'( &or 9#ilding(.CEN Qr#(('l( /005.


31/41


32/41


EA!/LES &OR COS!OS3! /ROGRA!

1.1 Copressed slender (ea

!ro9l'm d'(cri4$ion Sl'nd'r 'la($ic 9'am i( com4r'(('d 9% &orc' F and la$'rall% load'd 9%a (mall &orc' P . T"' &orc'( incr'a(' according $o $im' c#rv'( 1 TC 2 and TC /. Forc'

P /00 N i( a((ocia$'d ;i$" $"' c#rv' TC 2 and &orc' F 200000 N ;i$" $"' c#rv'TC /20. Dim'n(ion( o& $"' 9'am ar' L 200 mm % K0 mm and t 6 mm. Mod#l#( o& 'la($ici$% i( E /2⋅205 M!a.

Fig. 2 Com4r'(('d 9'am

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

TC 1

TC 2

Fig. / Tim' c#rv'(

1 T"' $im' i( no; onl% a 4aram'$'r con$rolling $"' loading 9'ca#(' $"' 4ro9l'm i( ($a$ic.20 A((ocia$ion ;i$" a $im' c#rv' m'an( $"a$ a$ 'v'r% $im' $"' val#' o& &orc' i( m#l$i4li'd 9%in($an$an'o#( val#' o& $im' c#rv' ordina$'.


33/41


Mod'lling "in$(

2. T"' QEAM/D 'l'm'n$ gro#4 i( #('d $o mod'l 9'am. A( $"' 4ro9l'm i(g'om'$ricall% non:lin'ar &in'r &ini$' 'l'm'n$ m'(" m#($ 9' #('d $"an &or a lin'ar 4ro9l'm. 7'om'$ric nonlin'ari$i'( ar' $a+'n in$o acco#n$ i& o4$ion &or $"''l'm'n$ gro#4 QEAM/D i( ('$ $o 2 @larg' di(4lac'm'n$ &orm#la$ion.

/. I$ i( #('l $o 4'r&orm lin'ar and 9#c+ling anal%('( 9'&or' $"' nonlin'ar anal%(i(.

For com4ari(on $"' E#l'r cri$ical &orc' i( giv'n 9%/

/

A/@ L

I E F 6 cr

π = . For $"'

dim'n(ion( and mod#l#( o& 'la($ici$% giv'n a9ov' i( F cr 2K50KK8 N.

6. Load ("o#ld 9' divid'd in$o (mall incr'm'n$( and NR or MNR i$'ra$ion( o& '=#ili9ri#m ("o#ld 9' #('d &or 'ac" incr'm'n$al ($'4.

3. Fin'r load incr'm'n$( ar' n'c'((ar% ;"'n F → F cr .5. R'($ar$ &rom $"' la($ (#cc'((ll% accom4li("'d load incr'm'n$ can 9' 4'r&orm'd.

Command(

C* 1. Geometry creatio!"#$%0%0%1%0PT%1%0%0%0C'T&%1%1%1%(%1600C'T&%2%2%1%(%1600C* 2. )eii+ materia, roertie

MP&/P%1%#'%2.1#5C* 3. )eii+ e,emt +rou a eam ectio#G&/P%1%#AM2)%0%0%0%0%0%1%0%0M#C)#%1%1%4%1%%80%80%3%3%0%0%0%0%0C* 4. Creati+ # me7MC&%1%2%1%2%5%1NM#&G#%1%12%1%1%0%0%0NC/MP%1%12C* 5. )eii+ ouary coitioC* 5.1 recrie i,acemet)PT%1%'%0%3%2%)PT%1%(%0%1%1%C* 5.2 eii+ orce P

C&)#%T"M#%1%1%0%0%.1%1%10%1PT%2%'%200%2%1C* 5.3 orce C&)#%T"M#%2%1%0%0%10%10%10PT%3%(%-1#5%3%1C* 6. A9uti+ o,iear aa,yi AN/N:"N#A&%%1%1%20%0.001%0%N%0%0%1#;010%0.001%0.01%0%1%0%0N:P:/T%1%100%1%0C*


34/41


TA&T%1T"M#%1.


35/41


0

50

100

150

200

250

300

0 0.005 0.01 0.015 0.02 0.025 0.03

trai

. t r e . .

Fig. 5 Qilin'ar ma$'rial mod'

Tim' c#rv' r'&l'c$( load "i($or%. D#ring $'($ i( in$'rnal 4r'((#r' incr'a('d $o 2/ × nominalval#' ;"ic" i( 2/0 M!a.

Fig. Load "i($or% H $im' c#rv'

Mod'lling "in$(

2. D#' $o a>ial (%mm'$r% onl% a 4ar$ o& $"' c%lind'r longi$#dinal ('c$ion ma% 9'mod'll'd.

/. K nod' !LANE/D 'l'm'n$( ar' #('d ;i$" o4$ion 5 ('$ $o Von Mi('( +in'ma$ic"ard'ning.

6. Load incr'm'n$a$ion ;i$" NR i$'ra$ion( ("o#ld 9' #('d.3. R'($ar$ i( #('d $o c"ang' load @$im' incr'm'n$.


36/41


Fig. 8 FE m'(" and 9o#ndar% condi$ion(

Command(

C* G#/M#T&( CAT"/N!"#$%0%0%1%0PT%1%45%0%0C'T&%1%1%1%'%35#'T&%1%1%1%(%10C* )#"N"NG #:#M#NT G&/P#G&/P%1%P:AN#2)%0%2%1%0%2%0%0%0C* )#"N"NG MAT#&"A: P&/P#&T"#MP&/P%1%#'%2.1#5MP&/P%1%N'(%.3MP&/P%1%#TAN%100

MP&/P%1%"G(:)%260C* M#="NGM%1%1%1%8%8%2%1%1C* )#"N"NG /N)A&( C/N)"T"/N)C&%2%(%0%1%1%C&)#%T"M#%1%1%0%0%1%1.2%1.1%1.2%2%0%3%1PC&%3%120%3%1%120%4C* A)>T"NG N/N:"N#A& ANA:("N:C/NT&/:%0%1N:P&"NT%0%0%0%0%0%0%0N:P:/T%1%50%1%0C* T#T "M:AT/NT"M#%0%1%.2

&N/N:"NC*TA&T%1T"M#%1%1.1%.1&N/N:"NC* N:/A)"NGT"M#%1.1%2%0.0&N/N:"NC*C* N/M"NA: P :/A)"NGT"M#%2%3%0.2&N/N:"NC*


37/41


R'(#l$( No$' $"a$

2. Lin'ar anal%(i( giv'( non:r'ali($ic val#'( o& ($r'(( @ma>imal von Mi('( ($r'(( i( a9o#$603 M!a.

/. !la($ic d'&orma$ion( occ#r d#ring $'($ loading.

6. A&$'r $'($ r'(id#al ($r'(( i( 4r'('n$ in $"' c%lind'r. T"i( r'di($ri9#$ion o& ($r'(( i(#('l and d'(ir'd.

3. D#' $o ($r'(( r'di($ri9#$ion onl% ($r'(( ;i$"in 'la($ic rang' i( 4r'('n$ d#ring nominalloading a&$'r $"' $'($.

1." eforation of a t)in*+alled tan4

!ro9l'm d'(cri4$ion T"in:;all'd 4r'((#r' v'(('l H $an+ i( (#9J'c$'d $o in$'rnal 4r'((#r'.Com4#$' ($r'(( di($ri9#$ion according $o lin'ar and nonlin'ar ($a$ic anal%(i(. In $"'nonlin'ar anal%(i( 9o$" g'om'$rical and ma$'rial nonlin'ari$i'( ("o#ld 9' $a+'n in$oacco#n$. Ma$'rial i( 'la($o4la($ic $"' (am' a( in 4r'vio#( '>am4l' @Fig#r' 5. Nominalval#' o& in$'rnal 4r'((#r' i( 0 M!a. S%mm'$ric "al& o& $"' $an+ i( ("o;n in Fig#r' K.Diam'$'r o& $"' man"ol' cov'r @$"ic+n'(( o& /5 mm i( 00 mm.

Fig. K T"in:;all'd $an+


38/41

NONLINEAR FINITE ELEMENT ANALYSIS 3"

Mod'lling "in$(

2. S,ELL3T 'l'm'n$( ;i$" o4$ion( &or larg' di(4lac'm'n$ (mall 4la($ic ($rain

von Mi('( %i'ld cri$'rion and +in'ma$ic "ard'ning ar' #('d./. D#' $o (%mm'$r% onl% on' 'ig"$" o& $"' $an+ ma% 9' mod'll'd.6. S$r'(('( ar' calc#la$'d in 'l'm'n$ co:ordina$' (%($'m(. To r'c'iv' com4a$i9l' co:

ordina$' (%($'m( c#rv'( ar' r'ori'n$'d.3. El'm'n$ co:ordina$' (%($'m o& S,ELL3T 'l'm'n$ i( d'&in'd according $o Fig#r' 1.

T"' 'l'm'n$ :a>i( go'( &rom $"' &ir($ nod' $o $"' ('cond. T"' :a>i( li'( in $"' 4lan'd'&in'd 9% $"' &ir($ $"r'' nod'( 4'r4'ndic#lar $o $"' :a>i( $o;ard $"' &o#r$" nod'. T"'

:a>i( com4l'$'( a rig"$:"and Car$'(ian (%($'m. -"'n m'("ing (#r&ac'( 'l'm'n$ :a>'( ar' 4arall'l $o $"' &ir($ (#r&ac' c#rv' mar+'d 9% a($'ri(+( in Fig#r' 20. FE m'(" i(("o;n in Fig#r' 22.

5. T"' a#$oma$ic $im' ($'44ing and r'($ar$ o4$ion( ar' #('d &or nonlin'ar FE anal%(i(.

1

2

3 x

y

z

T/P AC#

/TT/M AC#

Fig. 1 El'm'n$ co:ordina$' (%($'m @ECS

Fig. 20 Ori'n$a$ion o& 'l'm'n$(


39/41

NONLINEAR FINITE ELEMENT ANALYSIS 3(

Fig. 22 Fini$' 'l'm'n$ m'("

Command(

C*

C* 1. DEFININ7 7EOMETRYC*

C* 1.1. creating points and curvesC*!"#$%0%0%1%0PT%1%0%0%0PT%2%200%0%0P:AN#%%0%1C&PC"&C:#%1%2%1%200%-80%1PT%4%0%1


40/41


!"#$%-1%2%3%0C&C/MP%1%<C*

C* 1.2 creating surfacesC*$##P%1%4%1%'%0%2C*

C* 2. FE MES, DEFINITIONC* 2.1 defining element type and real constant sets#G&/P%1%=#::4T%1%0%0%1%2%1%0%0&C/NT%1%1%1%6%6%0%0%0%0%0&C/NT%1%2%1%6%25%0%0%0%0%0C* 2.2 defining material propertiesMP&/P%1%#'%2.1#5%N'(%.3%"G(:)%260MP&/P%1%#TAN%100C* 2.3 meshing

ACT#T%&C%1M%5%6%1%4%10%6%1.8%1!"#$%-1.5%2%2%0 ACT#T%&C%2MAN%


41/41

NONLINEAR FINITE ELEMENT ANALYSIS 4.

N:AT/T#P%1%.001%.1%5N:P:/T%1%20%1%0N:P&"NT%0%0%0%0%0%0%0

AN/N:"N#A&%%1%1%20%0.001%0%N%0%1%1#;010%0.001%0.01%0%1%0%0C* 4.3 running nonlinear analysisT"M#%0%1%.1&N/N:"NC*C* 4.4 continuing in "# analysisTA&T%1T"M#%1%1.5%.1&N/N:"NC*C* 5. P/TP&/C#"NGC* plotting stress$ displacements and xy graphs.C*

R'(#l$(

2. T"'r' ar' larg' di&&'r'nc'( 9'$;''n r'(#l$( o& lin'ar and nonlin'ar anal%(i(. In lin'ar anal%(i( $"' von Mi('( ($r'(( i( a9ov' $"' %i'ld ($r'(( /0 M!a. Di&&'r'nc' 9'$;''nlin'ar and nonlin'ar anal%(i( r'(#l$( i( a con('=#'nc' o& di&&'r'n$ (ol#$ion m'$"od(.-"il' in $"' lin'ar anal%(i( i( ($r'(( con(id'r'd a( 'la($ic and g'om'$r% o& $"' $an+ i(con(id'r'd #nc"ang'd in $"' nonlin'ar anal%(i( c"ang'( o& $"' $an+ ("a4' ar' $a+'nin$o acco#n$ $og'$"'r ;i$" ma$'rial nonlin'ari$i'(.

/. On('$ o& 4la($ic d'&orma$ion i( o9vio#( &rom $"' gra4" in Fig. 2/.

0

40

80

120

160

200

240

280

0 0.3 0.6 0. 1.2 1.5

time

σ [

M P a ]

Fig. 2/ Co#r(' o& '=#ival'n$ von Mi('( ($r'(( in nod' 288

Documents

Nonlinear FE Analysis 2016